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Potential Games Are Necessary to Ensure Pure NashEquilibria in Cost Sharing GamesRagavendran Gopalakrishnan, Jason R. Marden, Adam Wierman

To cite this article:Ragavendran Gopalakrishnan, Jason R. Marden, Adam Wierman (2014) Potential Games Are Necessary to Ensure PureNash Equilibria in Cost Sharing Games. Mathematics of Operations Research 39(4):1252-1296. http://dx.doi.org/10.1287/moor.2014.0651

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MATHEMATICS OF OPERATIONS RESEARCH

Vol. 39, No. 4, November 2014, pp. 1252–1296ISSN 0364-765X (print) � ISSN 1526-5471 (online)

http://dx.doi.org/10.1287/moor.2014.0651© 2014 INFORMS

Potential Games Are Necessary to Ensure Pure Nash Equilibriain Cost Sharing Games

Ragavendran Gopalakrishnan, Jason R. MardenDepartment of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Colorado 80309

{[email protected], [email protected]}

Adam WiermanDepartment of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California 91125,

[email protected]

We consider the problem of designing distribution rules to share “welfare” (cost or revenue) among individually strategic agents.There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., theShapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantees theexistence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class ofscalable and separable games that includes a variety of applications such as facility location, routing, network formation, andcoverage games. Given arbitrary local welfare functions �, we prove that a distribution rule guarantees equilibrium existencefor all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weightedShapley value on some “ground” welfare functions �′ , which can be distinct from �. However, if budget-balance is required inaddition to the existence of a Nash equilibrium, then �′ must be the same as �. We also provide an alternate characterizationof this space in terms of “generalized” marginal contributions, which is more appealing from the point of view of computationaltractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games withany fixed local welfare functions, it is necessary to work within the class of potential games.

Keywords : cost sharing; game theory; marginal contribution; Nash equilibrium; Shapley valueMSC2000 subject classification : Primary: 91A10; secondary: 91A40, 91A80OR/MS subject classification : Primary: games/group decisions: noncooperative; secondary: networks/graphs: applicationsHistory : Received April 12, 2013; revised September 3, 2013. Published online in Articles in Advance May 27, 2014.

1. Introduction. Fair division is an issue that is at the heart of social science—how should the cost incurred(revenue generated) by a group of self-interested agents be shared among them? This central question has led to alarge literature in economics over the last decades (Young [50, 51]; Moulin [33]) and more recently in computerscience (Anshelevich et al. [4], Jain and Mahdian [21], Moulin [35]). A standard framework within which to studythis question is that of cost sharing games, in which there is a set of agents making strategic choices of whichresources to utilize. Each resource generates a welfare (cost or revenue) depending on the set of agents thatchoose the resource. The focus is on finding distribution rules that lead to stable and/or fair allocations, which istraditionally formalized by the concept of the core in the cooperative theory and, more recently, by the Nashequilibrium in the noncooperative theory.

Cost sharing has traditionally been studied in the cooperative framework. Here, the problems studied typicallyinvolve a cost value v4S5 for each subset of players S, which usually stems from the optimal solution to anunderlying combinatorial optimization problem.1 A canonical example is the multicast network formation game(Granot and Huberman [17]), where a set of agents (consumers) N wishes to connect to a common source(a broadcaster) s by utilizing links of an underlying graph. Each link (resource) has a cost associated with itsusage, and the total cost of all the links used needs to be split among the agents. In such a situation, any subset Sof agents, if they cooperate, can form a coalition, and the best they can do is to choose the links of the minimumcost spanning tree for the set of vertices S ∪ 8s9, and incur its cost—denote it by v4S5. Here, the core consists ofall possible ways of distributing v4N5 to the players in N in such a way that it is in their best interest to fullycooperate to form the grand coalition. That is, a distribution rule f v2 N →� is in the core, if

∑

i∈N f v4i5= v4N 5,and for every subset S ⊆N ,

∑

i∈S fv4i5≤ v4S5. In general, the core can be empty, though for multicast games

it is not.A cooperative framework, in effect, models a “binary choice” for the agents—opt out or opt in and cooperate.

In large distributed (and often unregulated) systems such as the Internet, agents’ options are more complex becausethey have the opportunity to strategically choose the best action from multiple available options. Accordingly, thereis an emerging focus within cost sharing games on weaker notions of stability such as Nash equilibria. This focus

1 Note that our focus is on cost sharing games and not cost sharing mechanisms (Feigenbaum et al. [14]), which additionally involve solicitingagents’ exogenous private valuations of attaining the end goal. We briefly discuss the applicability of our results to cost sharing mechanismsin §4.3.

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Gopalakrishnan, Marden, and Wierman: Potential Games Are Necessary to Ensure PNE in Cost Sharing GamesMathematics of Operations Research 39(4), pp. 1252–1296, © 2014 INFORMS 1253

is driven by applications such as network-cost sharing (Anshelevich et al. [4], Chen et al. [7]) where individuallystrategic behavior is commonly assumed.

Our previous example of multicast games also provides a useful illustration of the noncooperative cost sharingframework. Multicast games were first modeled as noncooperative games in Chekuri et al. [6], whose model alsogeneralized facility location games, an important class of problems in operations research. The principal differencefrom the cooperative model is that here, the global cost share of an agent stems from local distribution rules thatspecify how the local cost (cost of each link) is split between the agents using that link. Accordingly, an agent’stotal cost share is simply the sum of its cost shares across all the links it uses. In addition, each agent can choosebetween potentially several link combinations that connect to the source. A pure Nash equilibrium corresponds to achoice of links by each agent such that each agent incurs the least possible cost given the links chosen by the otheragents. Similar to the core perhaps being empty in the cooperative model, a pure Nash equilibrium may not existin general, but for multicast games it does (Chekuri et al. [6]).

Existing literature on noncooperative cost sharing games focuses on designing distribution rules that guaranteeequilibrium existence and studying the “efficiency” of the resulting equilibria. Perhaps, the most famous suchdistribution rule is the Shapley value (Shapley [43]), which is budget-balanced; guarantees the existence of aNash equilibrium in any game; and for some classes of games, such as convex games, is always in the core.Generalizations of the Shapley value, e.g., weighted and generalized weighted Shapley values (Shapley [42]),exhibit many of the same properties.

In addition to guaranteeing equilibrium existence, it is also of paramount importance that these equilibria be“efficient.” That is, they should result in a system cost (usually, the total cost incurred by all the agents) that iswithin a small factor of the optimum. For example, in the noncooperative multicast game, which (effectively) usesthe Shapley value distribution rule, a Nash equilibrium choice of links by the agents may not collectively result inthe minimum spanning tree for N ∪ 8s9.

With these goals in mind, researchers have recently sought to provide characterizations of the class of (local)distribution rules that guarantee equilibrium existence. The first step toward this goal was made in Chen et al. [8],which proves that the only budget-balanced distribution rules that guarantee equilibrium existence in all costsharing games are generalized weighted Shapley value distribution rules. Following on Chen et al. [8], Marden andWierman [29] provides the parallel characterization in the context of revenue sharing games. Though thesecharacterizations seem general, they are actually just worst-case characterizations. In particular, the proofs in Chenet al. [8] and Marden and Wierman [29] consist of exhibiting a specific “worst-case” welfare function that requiresthat generalized weighted Shapley value distribution rules be used. Thus, characterizing the space of distributionrules (not necessarily budget-balanced) for specific local welfare functions remains an important open problem.In practice, it is exactly this issue that is important: when designing a distribution rule, one knows the specific localwelfare functions for the situation, wherein there may be distribution rules other than generalized weighted Shapleyvalues that also guarantee the existence of an equilibrium.

Our contribution. In this article, we provide a complete characterization of the space of distribution rules (notnecessarily budget-balanced) that guarantee the existence of a pure Nash equilibrium (which we will henceforthrefer to as just an equilibrium) for any specific local welfare functions. The principal contributions of this articleare as follows.

1. Our main result (Theorem 1) states that all games conditioned on any fixed local welfare functions possessan equilibrium if and only if the distribution rules are equivalent to generalized weighted Shapley value distributionrules on some “ground” welfare functions. This shows, perhaps surprisingly, that the results in Chen et al. [8] andMarden and Wierman [29] hold much more generally. In particular, it is neither the existence of some worst-casewelfare function nor the restriction of budget-balance that limits the design of distribution rules to generalizedweighted Shapley values.

2. Our second result (Theorem 2) provides an alternative characterization of the set of distribution rules thatguarantee equilibrium existence. In particular, it states that all games conditioned on any fixed local welfarefunctions possess an equilibrium if and only if the distribution rules are equivalent to generalized weightedmarginal contribution distribution rules on some “ground” welfare functions. This result is actually a consequenceof a connection between Shapley values and marginal contributions, namely that they can be viewed as equivalentgiven a transformation connecting their ground welfare functions (Proposition 2).These characterizations provide two alternative approaches for the problem of designing distribution rules, withdifferent design trade-offs, e.g., between budget-balance and tractability. More specifically, a design throughgeneralized weighted Shapley values provides direct control over how close to budget-balanced the distribution rulewill be; however, computing these distribution rules often requires computing exponentially many marginal

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Gopalakrishnan, Marden, and Wierman: Potential Games Are Necessary to Ensure PNE in Cost Sharing Games1254 Mathematics of Operations Research 39(4), pp. 1252–1296, © 2014 INFORMS

contributions (Matsui and Matsui [31], Conitzer and Sandholm [10]). On the other hand, a design throughgeneralized weighted marginal contributions requires computing only one marginal contribution; however, it ismore difficult to provide bounds on the degree of budget-balance.

Another important consequence of our characterizations is that potential games are necessary to guarantee theexistence of an equilibrium in all games with fixed local welfare functions since generalized weighted Shapleyvalue and generalized weighted marginal contribution distribution rules result in (“weighted”) potential games (Hartand Mas-Colell [18], Ui [46]). This is particularly surprising since the class of potential games is a relatively smallsubset of the class of games that possess an equilibrium (Sandholm [41]), and our characterizations imply thatsuch a relaxation in game structure would offer no advantage in guaranteeing equilibria.

In addition to the implications of the characterizations themselves, their proofs develop tools for analyzing costsharing games that could be useful for related models, such as cost sharing mechanisms. The proofs consist of asequence of counterexamples that establish novel necessary conditions for distribution rules to guarantee theexistence of an equilibrium. Within this analysis, new tools for studying distribution rules using their basisrepresentation (see §3) are developed, including an inclusion-exclusion framework that is crucial for our proof.Additionally, the proofs expose a relationship between Shapley value and marginal contribution distribution rules,leading to a novel closed form expression for the potential function of the resulting games.

2. Model. In this work we consider a simple, but general, model of a welfare (cost or revenue) sharing game,where there is a set of self-interested agents/players N = 811 : : : 1 n9 (n> 1) that each select a collection ofresources from a set R= 8r11 : : : 1 rm9 (m> 1). That is, each agent i ∈N is capable of selecting potentiallymultiple resources in R; therefore, we say that agent i has an action set ¡i ⊆ 2R. The resulting action profile,or (joint) allocation, is a tuple a= 4a11 : : : 1 an5 ∈¡ where the set of all possible allocations is denoted by¡=¡1 ×· · ·×¡n. We occasionally denote an action profile a by 4ai1 a−i5 where a−i ∈¡−i denotes the actions ofall agents except agent i.

Each allocation generates a welfare, ·4a5, that needs to be shared among the agents. In this work, we assume·4a5 is 4linearly5 separable across resources, i.e.,

·4a5=∑

r∈RWr48a9r51

where 8a9r = 8i ∈N 2 r ∈ ai9 is the set of agents that are allocated to resource r in a, and Wr 2 2N →� is the localwelfare function at resource r . This is a standard assumption (Anshelevich et al. [4], Chekuri et al. [6], Chenet al. [8], Marden and Wierman [28]) and is quite general. Note that we incorporate both revenue and cost sharinggames since we allow for the local welfare functions Wr to be either positive or negative.

The manner in which the welfare is shared among the agents determines the utility function Ui2 ¡→� thatagent i seeks to maximize. Because the welfare is assumed to be separable, it is natural that the utility functionsshould follow suit. Separability corresponds to welfare garnered from each resource being distributed among onlythe agents allocated to that resource, which is most often appropriate, e.g., in revenue and cost sharing. Thisresults in

Ui4a5= ∑

r∈aif r4i1 8a9r51

where f r 2 N × 2N →� is the local distribution rule at resource r ; i.e., f r4i1 S5 is the portion of the local welfareWr that is allocated to agent i ∈ S when sharing with S. In addition, we assume that resources with identical localwelfare functions have identical distribution rules; i.e., for any two resources r1 r ′ ∈R,

Wr =Wr ′ =⇒ f r = f r ′0

In light of this assumption, for the rest of this article, we write f Wr instead of f r . For completeness, we definef Wr 4i1 S5 2= 0 whenever i y S. A distribution rule f Wr is said to be budget-balanced if, for any player set S ⊆N ,∑

i∈S fWr 4i1 S5=Wr4S5.

We represent a welfare sharing game as G= 4N 1R1 8¡i9i∈N 1 8fWr 9r∈R1 8Wr9r∈R5, and the design of f Wr is the

focus of this article. When there is only one local welfare function, i.e., when Wr =W for all r ∈R, we drop thesubscripts and denote the local welfare function and its corresponding distribution rule by W and f W , respectively.

The primary goals when designing the distribution rules f Wr are to guarantee (i) equilibrium existence and(ii) equilibrium efficiency. Our focus in this work is entirely on (i), and we consider pure Nash equilibria;however it should be noted that other equilibrium concepts are also of interest (Adlakha et al. [2], Su and van derSchaar [44], Marden [24]). Recall that a 4pure Nash5 equilibrium is an action profile a∗ ∈¡ such that

4∀ i ∈N5 Ui4a∗i 1 a

∗−i5= max

ai∈¡i

Ui4ai1 a∗−i50

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Table 1. Example distribution rules.

Name Parameter Formula

Equal share None f WEQ4i1 S5= W4S5

�S�

Proportional share×= 4�11 : : : 1�n5

where �i > 0for all 1 ≤ i ≤ n

f WPR6×74i1 S5= �i

∑

j∈S �j

W4S5

Shapley value

None

f WSV4i1 S5= ∑

T⊆S\8i9

4�T �5!4�S� − �T � − 15!�S�! 4W4T ∪ 8i95−W4T 55

Marginal contribution f WMC4i1 S5=W4S5−W4S − 8i95

Weighted Shapley value ×= 4�11 : : : 1�n5


f WWSV6×74i1 S5= ∑

T⊆S2 i∈T

�i∑

j∈T �j

(

∑

R⊆T

4−15�T �−�R�W4R5

)

Weighted marginal contribution f WWMC6×74i1 S5=�i4W4S5−W4S − 8i955

Generalized weightedShapley value

�= 4Ë1è5

Ë= 4�11 : : : 1 �n5

è= 4S11 : : : 1 Sm5


and Si ∩ Sj = �for i 6= j

and⋃

è=N

f WGWSV6�74i1 S5= ∑

T⊆S2 i∈±T

�i∑

j∈±T �j

(

∑

R⊆T

4−15�T �−�R�W4R5

)

where ±T = T = T ∩ Sk and k = min8j � Sj ∩ T 6= �9

Generalized weightedmarginal contribution

f WGWMC6�74i1 S5= �i4W4±Sk5−W4±Sk − 8i955

where ±Sk = S −⋃k−1l=1 Sl and i ∈ Sk

2.1. Examples of distribution rules. Existing literature on cost sharing games predominantly focuses on thedesign and analysis of specific distribution rules. As such, there are a wide variety of distribution rules that areknown to guarantee the existence of an equilibrium. Table 1 summarizes several well-known distribution rules(both budget-balanced and non-budget-balanced) from existing literature on cost sharing, and we discuss theirsalient features in the following.

2.1.1. Equal/proportional share distribution rules. Most prior work in network cost sharing (Anshelevichet al. [4], Corbo and Parkes [11], Fiat et al. [15], Chekuri et al. [6], Christodoulou et al. [9]) deals with theequal share distribution rule, f W

EQ, defined in Table 1. Here, the welfare is divided equally among the players.The proportional share distribution rule, f W

PR6×7, is a generalization, parameterized (exogenously) by × ∈��N �++, a

vector of strictly positive player-specific weights, and the welfare is divided among the players in proportion totheir weights.

Both f WEQ and f W

PR are budget-balanced distribution rules. However, for general welfare functions, they do notguarantee an equilibrium for all games.2

2.1.2. The Shapley value family of distribution rules. One of the oldest and most commonly studieddistribution rules in the cost sharing literature is the Shapley value (Shapley [43]). Its extensions include theweighted Shapley value and the generalized weighted Shapley value, as defined in Table 1.

The Shapley value family of distribution rules can be interpreted as follows. For any given subset of players S,imagine the players of S arriving one at a time to the resource, according to some order � . Each player i can bethought of as contributing W4P�

i ∪ 8i95−W4P�i 5 to the welfare W4S5, where P�

i denotes the set of players in S thatarrived before i in �. This is the “marginal contribution” of player i to the welfare, according to the order �.The Shapley value, f W

SV4i1 S5, is simply the average marginal contribution of player i to W4S5, under the assumptionthat all �S�! orders are equally likely. The weighted Shapley value, f W

WSV6×74i1 S5, is then a weighted average of the

2 When the local welfare functions 8Wr9 are “anonymous,” i.e., when Wr4S5 is purely a function of �S� for all S ⊆N and r ∈R, 8f WrEQ 9

guarantees an equilibrium for all games. This is a consequence of it being identical to the Shapley value distribution rule (§2.1.2) in this case.However, the analogous property for f W

PR6×7 does not hold.

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marginal contributions, according to a distribution with full support on all the �S�! orders, determined by theparameter × ∈��N �

++, a strictly positive vector of player weights. The (symmetric) Shapley value is recovered whenall weights are equal.

The generalized weighted Shapley value, f WGWSV6�7, generalizes the weighted Shapley value to allow for the

possibility of player weights being zero. It is parameterized by a weight system given by �= 4Ë1è5, whereË ∈��N �

++ is a vector of strictly positive player weights and è= 4S11 S21 : : : 1 Sm5 is an ordered partition of the setof players N . Once again, players get a weighted average of their marginal contributions, but according to adistribution determined by Ë, with support only on orders that conform to è; i.e., for 1 ≤ k < l ≤m, players in Slarrive before players in Sk. Note that the weighted Shapley value is recovered when �è� = 1, i.e., when è is thetrivial partition, 4N 5.

The importance of the Shapley value family of distribution rules is that all distribution rules are budget-balanced,guarantee equilibrium existence in any game and also guarantee that the resulting games are so-called “potentialgames” (Hart and Mas-Colell [18], Ui [46]).3 However, they have one key drawback—computing them is often4

intractable (Matsui and Matsui [31], Conitzer and Sandholm [10]) since it requires computing the sum ofexponentially many marginal contributions.5

2.1.3. The marginal contribution family of distribution rules. Another classic and commonly studieddistribution rule is f W

MC, the marginal contribution distribution rule (Wolpert and Tumer [48]), where each player’sshare is simply his marginal contribution to the welfare, see Table 1. Clearly, f W

MC is not always budget-balanced.However, an equilibrium is always guaranteed to exist, and the resulting game is an exact potential game, wherethe potential function is the same as the welfare function. Accordingly, the marginal contribution distribution rulealways guarantees that the welfare maximizing allocation is an equilibrium; i.e., the “price of stability” is one.Finally, unlike the Shapley value family of distribution rules, note that it is easy to compute, as only two calls tothe welfare function are required.

Note that it is natural to consider weighted and generalized weighted marginal contribution distribution ruleswhich parallel those for the Shapley value described above. These are defined formally in Table 1, and they inheritthe equilibrium existence and potential game properties of f W

MC, in an analogous manner to their Shapley valuecounterparts. These rules have, to the best of our knowledge, not been considered previously in the literature;however, they are crucial to the characterizations provided in this article.

2.2. Important families of cost/revenue sharing games. Our model for welfare sharing games generalizesseveral existing families of games that have received significant attention in the literature. We illustrate a fewexamples below, in all of which the typical distribution rule studied is the equal share or Shapley value distributionrule:

(i) Multicast and facility location games (Chekuri et al. [6]) are a special case where N is the set of users;R is the set of links of the underlying graph; ¡i consists of all feasible paths from user i to the source; and for allr ∈R, Wr = crW is the local welfare function, where cr is the cost of the link r and W is given by

4∀S ⊆N5 W4S5={

−11 S 6= �01 S = � (1)

(ii) Congestion games (Rosenthal [38]) are a special case where, for each r ∈R, the local welfare function Wr

is “anonymous”; i.e., Wr4S5 is purely a function of �S� and is given by �S� times the negative of the delay functionat r , for all S ⊆N .

(iii) Atomic routing games with unsplittable flows (Roughgarden and Tardos [40]) are a special case where N isthe set of source-destination pairs 4si1 ti5, each of which is associated with �i units of flow, R is the set of edges ofthe underlying graph, and ¡i consists of all feasible si − ti paths. If cr4x5 denotes the latency function on edge r ,then Wr is the negative of the cost of the total flow due to the players in S; i.e., Wr4S5= −�S�cr4

∑

i∈S �i5, for allS ⊆N .

3 Shapley value distribution rules result in exact potential games, weighted Shapley value distribution rules result in weighted potential games,and generalized weighted Shapley value distribution rules result in a slight variation of weighted potential games (see Appendix C for details).4 The Shapley value has been shown to be efficiently computable in several applications (Deng and Papadimitriou [12], Mishra andRangarajan [32], Aadithya et al. [1]), where specific welfare functions and special structures on the action sets enable simplifications of thegeneral Shapley value formula.5 Technically, if the entire welfare function is taken as an input, then the input size is already O42n5 and Shapley values can be computed“efficiently.” However, if access to the welfare function is by means of an oracle (Liben-Nowell et al. [23]), then the actual input size is stillO4n5, and the hardness is exposed.

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(iv) Network formation games (Anshelevich et al. [4]) are a special subcase of the previous case, with asuitable encoding of the players. Suppose the set of players is N = 80111 : : : 1 n− 19, and the cost of constructingeach edge r is Cr4S5 when S ⊆N is the set of players who choose that edge. Then one possibility is to set�i = 10i so that

∑

i∈S �i can be decoded to obtain the set of players S. Therefore, cr can be defined such that forall S ⊆N , cr4

∑

i∈S �i5=Cr4S5/�S�.Other notable specializations of our model that focus on the design of distribution rules are network coding

(Marden and Effros [25]), graph coloring (Panagopoulou and Spirakis [37]), and coverage problems (Marden andWierman [27, 28]). Designing distribution rules in our cost sharing model also has applications in distributedcontrol (Gopalakrishnan et al. [16]).

3. Basis representations. To gain a deeper understanding of the structural form of some of the distributionrules discussed in §2.1, it is useful to consider their “basis” representations. Not only do these representationsprovide insight, they are crucial to the proofs in this paper. The basis framework we adopt was first introduced inShapley [42] in the context of the Shapley value and corresponds to the set of “inclusion functions.” We start bydefining a basis for the local welfare functions below and then move to introducing the basis representation of thedistribution rules we introduced in §2.1.

3.1. A basis for welfare functions. Instead of working with W directly, it is often easier to represent W as alinear combination of simple basis welfare functions. A natural basis, first defined in Shapley [42], is the set ofinclusion functions. The inclusion function of a player subset T ⊆N , denoted by W T , is defined as

W T 4S5 2={

11 T ⊆ S1

01 otherwise0(2)

In the context of cooperative game theory, inclusion functions are identified with unanimity games. It is wellknown (Shapley [42]) that the set of all inclusion functions, 8W T 2 T ⊆N9, constitutes a basis for the space of allwelfare functions, i.e., given any welfare function W , there exists a unique support set ´W ⊆ 2N and a uniquesequence QW = 8qW

T 9T∈´W of nonzero weights indexed by ´W such that

W = ∑

T∈´W

qWT W T 0 (3)

We sometimes denote the welfare function W by the tuple 4´W 1QW 5.

3.2. A basis for distribution rules. The basis representation for welfare functions introduced above naturallyyields a basis representation for distribution rules. To simplify notation in the following, we denote f W T

by f T , foreach T ∈´W . That is, f T 2 N × 2N →� is a basis distribution rule corresponding to the unanimity game W T ,where f T 4i1 S5 is the portion of W T 4S5 allocated to agent i ∈ S when sharing with S.

Given a set of basis distribution rules 8f T 2 T ⊆N9, by linearity, the function f W ,

f W 2= ∑

T∈´W

qWT f T 1 (4)

defines a distribution rule corresponding to the welfare function W . Note that if each f T is budget-balanced,meaning that for any player set S ⊆N ,

∑

i∈S fT 4i1 S5=W T 4S5, then f W is also budget-balanced. However, unlike

the basis for welfare functions, some distribution rules do not have a basis representation of the form (4), e.g.,equal and proportional share distribution rules (see §2.1.1). But well-known distribution rules of interest to us, likethe Shapley value family of distribution rules, were originally defined in this manner. Further, our characterizationshighlight that any distribution rule that guarantees equilibrium existence must have a basis representation.

Table 2 restates the distribution rules shown in Table 1 in terms of their basis representations, which, as can beseen, tend to be simpler and provide more intuition.

For example, the Shapley value distribution rule on a welfare function W is quite naturally defined throughits basis—for each unanimity game W T , the welfare is shared equally among the players; see (5). In otherwords, whenever there is welfare generated (when all the players in T are present), the resulting welfare is splitequally among the contributing players (players in T ). Similarly, the weighted Shapley value, for each unanimitygame W T , distributes the welfare among the players in proportion to their weights; see (7). Finally, the basisrepresentation highlights that the generalized weighted Shapley value can be interpreted with è as representing a

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Table 2. Definition of basis distribution rules.

Name Parameter Definition

Shapley value f TSV4i1 S5= f WT

EQ 4i1 S5=

1�T � 1 i ∈ T and T ⊆ S

01 otherwise(5)

Marginal contribution f TMC4i1 S5= �T �f T

SV4i1 S5={

11 i ∈ T and T ⊆ S

01 otherwise(6)

None

Weighted Shapley ×= 4�11 : : : 1�n5 f TWSV6×74i1 S5= f WT

PR 6×74i1 S5=

�i∑

j∈T �j

1 i ∈ T and T ⊆ S

01 otherwise(7)

value where �i > 0 for all 1 ≤ i ≤ n

Weighted marginal f TWMC6×74i1 S5=

(

∑

j∈T�j

)

f TWSV6×74i1 S5=

{

�i1 i ∈ T and T ⊆ S

01 otherwisecontribution

Generalized weightedShapley value

f TGWSV6�74i1 S5=

�i∑

j∈±T �j

1 i ∈ ±T and T ⊆ S

01 otherwise

where ±T = T ∩ Sk and k = min8j � Sj ∩ T 6= �9 (8)

�= 4Ë1è5 Ë= 4�11 : : : 1 �n5

è= 4S11 : : : 1 Sm5

where �i > 0 for all 1 ≤ i ≤ n

and Si ∩ Sj = � for i 6= j

and⋃

è=N

Generalized weighted TGWMC6�74i1 S5=(

∑

j∈±T�j

)

f TGWSV6�74i1 S5=

{

�i1 i ∈ ±T and T ⊆ S

01 otherwise

where ±T = T ∩ Sk and k = min8j � Sj ∩ T 6= �9 (9)marginal contribution

grouping of players into priority classes, and the welfare being distributed only among the contributing players ofthe highest priority, in proportion to their weights; see (8).

Interestingly, the marginal contribution distribution rule, though it was not originally defined this way, has abasis representation that highlights a core similarity to the Shapley value. In particular, though the definitions inTable 1 make f W

MC and f WSV seem radically different, from Table 2, their basis distribution rules, f T

MC and f TSV, are, in

fact, quite intimately related; see (5) and (6). We formalize this connection between the Shapley value family ofdistribution rules and the marginal contribution family of distribution rules in §4.2.

4. Results and discussion. Our goal is to characterize the space of distribution rules that guarantee theexistence of an equilibrium in welfare sharing games. Toward this end, this paper builds on the recent works ofChen et al. [8] and Marden and Wierman [29] that take the first steps toward providing such a characterization.Proposition 1 combines the main contributions of these two papers into one statement. Let � denote a nonemptyset of welfare functions. Let f� = 8f W 9W∈� denote the set of corresponding distribution rules. Let §4N 1 f�1�5denote the class of all welfare sharing games with player set N , local welfare functions Wr ∈�, and correspondingdistribution rules f Wr ∈ f�. We refer to � as the set of local welfare functions of the class §4N 1 f�1�5. Notethat this class is quite general; in particular, it includes games with arbitrary resources and action sets. When thereis only one local welfare function, i.e., when �= 8W9, we denote this class simply by §4N 1 f W 1W5. Note that§4N 1 f W 1W5⊆§4N 1 f�1�5 for all W ∈�.

Proposition 1 (Chen et al. [8], Marden and Wierman [29]). There exists a local welfare function Wfor which all games in §4N 1 f W 1W5 possess a pure Nash equilibrium for a budget-balanced f W if and only ifthere exists a weight system � for which f W is the generalized weighted Shapley value distribution rule, f W

GWSV6�7.

Less formally, Proposition 1 states that if one wants to use a distribution rule that is budget-balanced andguarantees equilibrium existence for all possible welfare functions and action sets, then one is limited to the classof generalized weighted Shapley value distribution rules.6 This result is shown by exhibiting a specific “worst-case”local welfare function W ∗ (the one in (1)) for which this limitation holds. In reality, when designing a distributionrule, one knows the specific set of local welfare functions � for the situation, and Proposition 1 claims nothing in

6 The authors of Chen et al. [8] and Marden and Wierman [29] use the term ordered protocols to refer to generalized weighted Shapley valuedistribution rules with �è� = �N �, i.e., where è defines a total order on the set of players N . They state their characterizations in terms ofconcatenations of positive ordered protocols, which are generalized weighted Shapley value distribution rules with an arbitrary è.

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the case where it does not include W ∗, where, in particular, there may be other budget-balanced distribution rulesthat guarantee equilibrium existence for all games. Recent work has shown that there are settings where this is thecase (Marden and Wierman [28]), at least when the agents are not allowed to choose more than one resource. Inaddition, the marginal contribution family of distribution rules is a non-budget-balanced class of distribution rulesthat guarantee equilibrium existence in all games (no matter what the local welfare functions �), and there couldpotentially be others as well.

In the rest of this section, we provide two equivalent characterizations of the space of distribution rules thatguarantee equilibrium existence for all games with a fixed set of local welfare functions—one in terms ofgeneralized weighted Shapley values and the other in terms of generalized weighted marginal contributions.We defer complete proofs to the appendices. However, we sketch an outline in §6, highlighting the proof techniqueand the key steps involved.

4.1. Characterization in terms of generalized weighted Shapley values. Our first characterization states thatfor any fixed set of local welfare functions, even if the distribution rules are not budget-balanced, the conclusion ofProposition 1 is still valid. That is, every distribution rule that guarantees the existence of an equilibrium in allgames is equivalent to a generalized weighted Shapley value distribution rule:

Theorem 1. Given any set of local welfare functions �, all games in §4N 1 f�1�5 possess a pure Nashequilibrium if and only if there exists a weight system �, and a mapping gSV that maps each local welfare functionW ∈� to a corresponding ground welfare function gSV4W5 such that its distribution rule f W ∈ f� is equivalent tothe generalized weighted Shapley value distribution rule, f W ′

GWSV6�7, where W ′ = gSV4W5 is the actual welfarethat is distributed7 by f W , defined as,

4∀S ⊆N5 W ′4S5=∑

i∈Sf W 4i1 S50 (10)

Refer to Appendix A for the complete proof and §6 for an outline. Although Proposition 1 states that thereexists a local welfare function for which any budget-balanced distribution rule is required to be equivalent toa generalized weighted Shapley value (on that welfare function) in order to guarantee equilibrium existence,Theorem 1 states a much stronger result that, for any set of local welfare functions, the corresponding distributionrules must be equivalent to generalized weighted Shapley values on some ground welfare functions to guaranteeequilibrium existence. This holds true even when the distribution rules are not budget-balanced. Proving Theorem 1requires working with arbitrary local welfare functions, which is a clear distinction from the proof of Proposition 1,which exhibits a specific local welfare function, showing the result for that case.

From Theorem 1, it follows that designing distribution rules to ensure the existence of an equilibrium merelyrequires selecting a weight system �= 4Ë1è5 and a ground welfare function W ′ for each local welfare functionW ∈� (this defines the mapping gSV) and then applying the distribution rules 8f W ′

GWSV6�79W∈�. Budget-balance, ifrequired, can be directly controlled through proper choice of W ′ since 8W ′9 are the actual welfares distributed. Forexample, if exact budget-balance is desired, then W ′ =W for all W ∈�. Notions of approximate budget-balance(Roughgarden and Sundararajan [39]) can be similarly accommodated by keeping W ′ “close” to W .

An important implication of Theorem 1 is that if one hopes to use a distribution rule that always guaranteesequilibrium existence in games with any fixed set of local welfare functions, then one is limited to working withinthe class of “potential games.” This is perhaps surprising since a priori, potential games are often thought to be asmall, special class of games (Sandholm [41]).8 More specifically, generalized weighted Shapley value distributionrules result in a slight variation of weighted potential games (Hart and Mas-Colell [18], Ui [46]),9 whose potentialfunction can be computed recursively as

ê6�74a5=∑

r∈RÔr 6�748a9r51

where Ôr 6�72 2N →�m is the local potential function at resource r (we denote the kth element of this vector by4�r 6�75k), and for any 1 ≤ k ≤m and any subset S ⊆N ,

4�r 6�75k4S5= 1∑

i∈S �i

(

W ′r 4±Sm−k+15+∑

i∈S�i4�r 6�75k4S − 8i95

)

1 (11)

where W ′r = gSV4Wr5 and ±Sk = S −⋃k−1

l=1 Sl. Refer to Appendix C for the proof.

7 Note that W ′ =W if and only if f W is budget-balanced.8 In spite of this limitation, it is useful to point out that there are many well understood learning dynamics that guarantee equilibriumconvergence in potential games (Blume [5], Marden et al. [30], Marden and Shamma [26]).9 See Definition 1 in Appendix C.

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Theorem 1 also has some negative implications. First, the limitation to generalized weighted Shapley valuedistribution rules means that one is forced to use distribution rules that may require computing exponentially manymarginal contributions, as discussed in §2.1. Second, if one desires budget-balance, then there are efficiency limitsfor games in §4N 1 f�

WSV1�5. In particular, there exists a submodular welfare function W such that for any weightvector ×, there exists a game in §4N 1 f W

WSV6×71W5 where the best equilibrium has welfare that is a multiplicativefactor of two worse than the optimal welfare (Marden and Wierman [29]).

4.2. Characterization in terms of generalized weighted marginal contributions. Our second characterizationis in terms of the marginal contribution family of distribution rules. The key to obtaining this contribution isthe connection between the marginal contribution and Shapley value distribution rules that we observed in §3.We formalize this in the following proposition. Refer to Appendix B for the proof.

Proposition 2. For any two welfare functions W ′ = 4´′1Q′5 and W ′′ = 4´′′1Q′′5 and any weight system�= 4Ë1è5,

f W ′GWSV6�7= f W ′′

GWMC6�7 ⇐⇒ ´′ =´′′ and 4∀T ∈´′5 q′T =

(

∑

j∈±T�j

)

q′′T 0 (12)

Informally, Proposition 2 says that generalized weighted Shapley values and generalized weighted marginalcontributions are equivalent, except with respect to different ground welfare functions whose relationship is throughtheir basis coefficients, as indicated in (12). This proposition immediately leads to the following equivalentstatement of Theorem 1.

Theorem 2. Given any set of local welfare functions �, all games in §4N 1 f�1�5 possess a pure Nashequilibrium if and only if there exists a weight system �, and a mapping gMC that maps each local welfare functionW ∈� to a corresponding ground welfare function gMC4W5 such that its distribution rule f W ∈ f� is equivalentto the generalized weighted marginal contribution distribution rule, f W ′′

GWMC6�7, where W ′′ = gMC4W5 is defined as

W ′′ = h4gSV4W551 (13)

where h denotes the mapping that maps W ′ to W ′′ according to (12).

Importantly, Theorem 2 provides an alternative way of designing distribution rules that guarantee equilibriumexistence. The advantage of this alternative design is that marginal contributions are much easier to compute thanthe Shapley value is, which requires computing exponentially many marginal contributions. However, it is muchmore difficult to control the budget-balance of marginal contribution distribution rules. Specifically, 8W ′′9 are notthe actual welfares distributed so there is no direct control over budget-balance as was the case for generalizedweighted Shapley value distribution rules. Instead, it is necessary to start with desired welfares 8W ′9 to bedistributed (equivalently, the desired mapping gSV) and then perform a “preprocessing” step of transforming it intothe ground welfare functions 8W ′′9 using (13), which requires exponentially many calls to each W ′. However, thisis truly a preprocessing step and thus only needs to be performed once.

Another simplification that Theorem 2 provides when compared to Theorem 1 is in terms of the potentialfunction. In particular, in light of Proposition 2, the distribution rules f W ′

GWSV6�7 and f W ′′GWMC6�7, where W ′′ = h4W ′5,

result in the same “weighted” potential game with the same potential function ê6�7. However, in terms of W ′′,there is a clear closed-form expression for the local potential function at resource r , Ôr 6�72 2N →�m. For any1 ≤ k ≤m and any subset S ⊆N ,

4�r 6�75k4S5=W ′′r 4±Sm−k+151

where W ′′r = gMC4Wr5 and ±Sk = S −⋃k−1

l=1 Sl. In other words, we have

4∀S ⊆N5 Ôr 6�74S5= 4W ′′r 4±Sm51W ′′

r 4±Sm−151 : : : 1W′′r 4±S1550

Refer to Appendix C for the proof. Having a simple closed form potential function is helpful for many reasons.For example, it aids in the analysis of learning dynamics and in characterizing efficiency bounds through thewell-known potential function method (Tardos and Wexler [45]).

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4.3. Limitations and extensions. It is important to highlight that our characterizations in Theorems 1 and 2crucially depend on the fact that an equilibrium must be guaranteed in all games, i.e., for all possibilities ofresources, action sets, and choice of local welfare functions from �. (This is the same for the characterizationsgiven in the previous work in Chen et al. [8] and Marden and Wierman [29].) If this requirement is relaxed, it maybe possible to find situations where distribution rules that are not equivalent to generalized weighted Shapleyvalues can guarantee equilibrium existence. For example, Marden and Wierman [28] gives such a rule for acoverage game where players can select only one resource at a time. A challenging open problem is to determinethe structure of the action sets that is necessary for the characterizations in Theorems 1 and 2 to hold.

A second remark is that our entire focus has been on characterizing distribution rules that guaranteeequilibrium existence. However, guaranteeing efficient equilibria is also an important goal for distribution rules.The characterizations in Theorems 1 and 2 provide important new tools to optimize the efficiency, e.g., the price ofanarchy, of distribution rules for general cost sharing and revenue sharing games through proper choice of theweight system and ground welfare functions. An important open problem in this direction is to understand theresulting trade-offs between budget-balance and efficiency.

Finally, it is important to remember that our focus has been on cost sharing games; however, it is natural to askif similar characterizations can be obtained for cost sharing mechanisms (Moulin and Shenker [36], Dobzinskiet al. [13], Immorlica and Pountourakis [20], Johari and Tsitsiklis [22], Yang and Hajek [49], Moulin [34]). Morespecifically, the model considered in this paper extends immediately to situations where players have independentheterogeneous valuations over actions by adding more welfare functions to �.10 However, in cost sharingmechanisms, player valuations are private, which adds a challenging wrinkle to this translation. Thus, extendingour characterizations to the setting of cost sharing mechanisms is a difficult, but important, open problem.

5. Prior work in noncooperative cost sharing games. As noted previously, the first steps toward characterizingthe space of distribution rules that guarantee equilibrium existence were provided in Chen et al. [8] and Mardenand Wierman [29]. Prior to that, almost all the literature in cost sharing games (Anshelevich et al. [4], Corbo andParkes [11], Fiat et al. [15], Chekuri et al. [6], Christodoulou et al. [9]) considered a fixed distribution rule thatguarantees equilibrium existence, namely equal share (dubbed the “fair cost allocation rule,” equivalent to theShapley value in these settings), and the focus was directed toward characterizing the efficiency of equilibria.

A recent example of work in this direction is von Falkenhausen and Harks [47], which considers games wherethe action sets (strategy spaces) of the agents are either singletons or bases of a matroid defined on the ground setof resources. For such games, the authors focus on designing (possibly nonseparable, nonscalable) distribution rulesthat result in efficient equilibria. They tackle the question of equilibrium existence with a novel characterization ofthe set of possible equilibria independent of the distribution rule and then exhibit a family of distribution rules thatresult in any given equilibrium in this set. Our goal is fundamentally different from theirs, in that we seek tocharacterize distribution rules that guarantee equilibrium existence for a class of games, whereas they directlycharacterize the best and worst achievable equilibria of a given game.

An alternative approach for distribution rule design is studied in Anshelevich et al. [3] and Hoefer andKrysta [19]. The authors consider a fundamentally different model of a cost sharing game where agents not onlychoose resources but also indicate their demands for the shares of the resulting welfare at these resources. Theirmodel essentially defers the choice of the distribution rule to the agents. In such settings, they prove that anequilibrium may not exist in general.

6. Proof sketch of Theorem 1. We now sketch an outline of the proof of Theorem 1 for the special casewhere there is just one local welfare function W , i.e., �= 8W9, highlighting the key stages. For an independent,self-contained account of the complete proof, refer to Appendix A.

10 To see this, consider a welfare sharing game G ∈§4N 1 f�1�5. Let the action set of player i be ¡i = 8a11 : : : 1 al9, and suppose he valuesaction aj at uj , for 1 ≤ j ≤ l. Then we modify G to G′ by adding l more resources r11 : : : 1 rl to R, setting

Wrj4S5=

{

uj 1 i ∈ S

01 i y S

and fWrj = f

WrjSV for 1 ≤ j ≤ l, and augmenting each action in ¡i with its corresponding resource so that aj → aj ∪8rj9. Then G′ ∈§4N 1 f�′

1�′5,where �′ =�∪ 4

⋃lj=18Wrj

95 and f�′ = f� ∪ 4⋃l

j=18fWrj 95. Notice that all games in §4N 1 f�1�5 have an equilibrium if and only if all

games in §4N 1 f�′1�′5 have an equilibrium.

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First, note that we only need to prove one direction since it is known that for any weight system � and any twowelfare functions W1W ′, all games in §4N 1 f W ′

GWSV6�71W5 have an equilibrium (Hart and Mas-Colell [18]).11 Thus,our focus is solely on proving that for distribution rules f W that are not generalized weighted Shapley values onsome ground welfare function, there exists G ∈§4N 1 f W 1W5 with no equilibrium.

The general proof technique is as follows. First, we present a quick reduction to characterizing only budget-balanced distribution rules f W that guarantees the existence of an equilibrium for all games in §4N 1 f W 1W5. Thenwe establish several necessary conditions for a budget-balanced distribution rule f W that guarantees the existenceof an equilibrium for all games in §4N 1 f W 1W5, which effectively eliminates all but generalized weighted Shapleyvalues on W , giving us our desired result. We establish these conditions by a series of counterexamples thatamount to choosing a resource set R and the action sets 8¡i9i∈N , for which failure to satisfy a necessary conditionwould lead to nonexistence of an equilibrium.

Throughout, we work with the basis representation of the welfare function W that was introduced in §3.1. Sincewe are dealing with only one welfare function W , we drop the superscripts from ´W , QW , and qW

T in order tosimplify notation. It is useful to think of the sets in ´ as being “coalitions” of players that contribute to thewelfare function W (also referred to as contributing coalitions), and the corresponding coefficients in Q as beingtheir respective contributions. Also, for simplicity, we normalize W by setting W4�5= 0 and, therefore, � y´.Before proceeding, we introduce some notation below:

1. For any subset S ⊆N , ´4S5 denotes the set of contributing coalitions in S:

´4S5= 8T ∈´ � T ⊆ S90

2. For any subset S ⊆N , N4S5 denotes the set of contributing players in S:

N4S5=⋃

´4S50

3. For any two players i1 j ∈N , ´ij denotes the set of all coalitions containing i and j:

´ij = 8T ∈´ � 8i1 j9⊆ T 90

4. Let ¢⊆ 2N denote any collection of subsets of a set N . Then the relation ⊆ induces a partial order on ¢.¢min denotes the set of minimal elements of the poset 4¢1⊆5:

¢min = 8B ∈¢ � 4@B′ ∈¢5 s.t. B′ ( B90

Example 1. Let N = 8i1 j1 k1 l9 be the set of players. Table 3(a) defines a W2 2N →�, as well as five differentdistribution rules for W . Table 3(b) shows the basis representation of W , and Table 3(c) illustrates the notationdefined above for W . Throughout the proof sketch, we periodically revisit these distribution rules to illustrate thekey ideas.

The proof is divided into five sections—each section incrementally builds on the structure imposed on thedistribution rule f by previous sections.

6.1. Reduction to budget-balanced distribution rules. First, we reduce the problem of characterizingall distribution rules f W that guarantee equilibrium existence for all G∈§4N 1 f W 1W5 to characterizing onlybudget-balanced distribution rules f W that guarantee equilibrium existence for all G ∈§4N 1 f W 1W5:

Proposition 3. For all welfare functions W , a distribution rule f W guarantees the existence of an equilibriumfor all games in §4N 1 f W 1W5 if and only if it guarantees the existence of an equilibrium for all games in§4N 1 f W 1W ′5, where, for all subsets S ⊆N , W ′4S5 2=∑

i∈S fW 4i1 S5.

Proof. This proposition is actually a subtlety of our notation. For games G∈§4N 1 f W 1W5, the welfarefunction W does not directly affect strategic behavior (it only does so through the distribution rule f W ). Therefore,in terms of strategic behavior and equilibrium existence, the classes §4N 1 f W 1W5 and §4N 1 f W 1W ′5 are identical,for any two welfare functions W1W ′. Therefore, a distribution rule f W guarantees equilibrium existence forall games in §4N 1 f W 1W5 if and only if it guarantees equilibrium existence for all games in §4N 1 f W 1W ′5.To complete the proof, simply pick W ′ to be the actual welfare distributed by f W , as defined in (10). �

11 Notice that W has no role to play as far as equilibrium existence of games G ∈§4N 1 f W ′GWSV6�71W5 is concerned since it does not affect player

utilities. This observation will prove crucial later.

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Table 3. Tables for Examples 6.x.

(a) Definitions of welfare functions and distribution rules

S ⊆N � 8i9 8j9 8k9 8l9 8i1 j9 8i1 k9 8i1 l9 8j1 k9 8j1 l9 8k1 l9 8i1 j1 k9 8i1 j1 l9 8i1 k1 l9 8j1 k1 l9 8i1 j1 k1 l9

W4S5 0 5 3 0 3 6 2 8 0 6 3 0 7 5 3 1

f W1 4·1 S5 − 455 435 405 435 43135 41115 44145 40105 43135

(

321 3

2

)

4010105(

731 7

31 7

3

) (

531 5

31 5

3

)

4111115(

141 1

41 1

41 1

4

)

f W2 4·1 S5 − 455 435 405 435 44125 441−25 45135 421−25 43135 40135 43111−45 4311135 441−2135 421−2135 42101−4135

f W3 4·1 S5 − 455 435 405 435 44125 441−25 45135 421−25 43135 40135 43111−45

(

1031 2

313)

441−2135 421−2135(

731− 1

31−413

)

f W4 4·1 S5 − 455 435 405 435 44125 441−25 45135 411−15 43135 40135 43101−35 4311135 441−2135 411−1135 421−11−3135

f W5 4·1 S5 − 455 435 405 435 43115 421−35 45135 401−35 43135 40135 401−21−65 411−1115 421−3135 401−3135 4−21−41−6115

W ′4S5 0 5 3 0 3 4 −1 8 −3 6 3 −8 1 2 0 −11

(b) Basis representation of W (c) Notation (d) Basis distribution rules computed by recursion (A14)

Coalition Contribution CoalitionT ∈´ qT ∈Q Symbol Value T ∈´ f T

1 4·1 T 5 f T2 4·1 T 5 f T

3 4·1 T 5 f T4 4·1 T 5 f T

5 4·1 T 5

8i9 5 ´48i1 l95 88i91 8l99 8i9 415 415 415 415 415

8j9 3 ´48j1 k95 88j91 8j1 k99 8j9 415 415 415 415 415

8l9 3 N48i1 k95 8i1 k9 8l9 415 415 415 415 415

8i1 j9 −2 ´ij 88i1 j91 8i1 j1 l99 8i1 j9 41105(

121 1

2

) (

121 1

2

) (

121 1

2

)

41115

8i1 k9 −3 ´minij 88i1 j99 8i1 k9

(

431− 1

3

) (

131 2

3

) (

131 2

3

) (

131 2

3

)

41115

8j1 k9 −3 8j1 k9 41105(

131 2

3

) (

131 2

3

) (

231 1

3

)

41115

8i1 j1 l9 −2 8i1 j1 l9(

131 1

31 1

3

) (

121 1

210) (

131 2

310) (

121 1

210)

4111115

Notice that f W is a budget-balanced distribution rule for the actual welfare it distributes, namely, W ′ as definedin (10). Hence, it is sufficient to prove that for budget-balanced distribution rules f W that are not generalizedweighted Shapley values, there exists a game in §4N 1 f W 1W5 for which no equilibrium exists.

Example 2. Note that f W1 through f W

4 are budget-balanced, whereas f W5 , which is the marginal contribution

distribution rule f WMC, is not. Let W ′, shown in Table 3(a), be the actual welfare distributed by f W

5 , as definedin (10). Then f W

5 is a budget-balanced distribution rule for W ′. In fact, it is the Shapley value distribution rule f W ′SV .

6.2. Three necessary conditions. The second step of the proof is to establish that for every subset S ⊆N ofplayers, any budget-balanced distribution rule f W must distribute the welfare W4S5 only among contributingplayers and do so as if the noncontributing players were absent:

Proposition 4. If f W is a budget-balanced distribution rule that guarantees the existence of an equilibrium inall games G ∈§4N 1 f W 1W5, then

4∀S ⊆N5 4∀ i ∈ S5 f W 4i1 S5= f W 4i1N 4S550

Section A.2.1 is devoted to the proof, which consists of incrementally establishing the following necessaryconditions, for any subset S ⊆N :

(a) If no contributing coalition is formed in S, then the distribution rule f W does not allocate any utility to theplayers in S (Lemma 1).

(b) The distribution rule f W distributes the welfare only among the contributing players in S (Lemma 2).(c) The distribution rule f W distributes the welfare among the contributing players in S as if all other players

were absent (Lemma 3).

Example 3. For the welfare function W , S =N4S5 for all subsets S, making Proposition 4 trivial exceptfor the two subsets 8k9 and 8k1 l9, for which k is not a contributing player. Note that f W

2 through f W4 allocate

no welfare to k in these subsets, and l gets the same whether k is present or not. But f W1 4k1 8k1 l95 6= 0 and

f W1 4l1 8k1 l95 6= f W

1 4l1 8l95. Therefore, f W1 , which is the equal share distribution rule f W

EQ, violates conditions (b)–(c),and hence Proposition 4, so it does not guarantee equilibrium existence in all games; see Counterexample 2 in theproof of Lemma 3.

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6.3. Decomposition of the distribution rule. The third step of the proof establishes that f W must have abasis representation of the form (4), where the basis distribution rules are generalized weighted Shapley values:

Proposition 5. If f is a budget-balanced distribution rule that guarantees the existence of an equilibrium inall games G ∈§4N 1 f W 1W5, then there exists a sequence of weight systems ì= 8�T 9T∈´ such that

f W = ∑

T∈´qT f

TGWSV6�

T 70

Note that for now, the weight systems �T could be arbitrary and need not be related in any way. We deal withhow they should be “consistent” in the next section.

In §A.2.2, we prove Proposition 5 by describing a procedure to compute the basis distribution rules, f T ,assuming they exist, and then showing the following properties of f T :

(a) Each f T is a budget-balanced distribution rule for W T (Lemmas 4–5).(b) The distribution rule f W and the basis distribution rules 8f T 9T∈´, satisfy (4) (Lemma 7).(c) Each f T is nonnegative, so f T = f T

GWSV6�T 7 for some �T (Lemma 8).

Example 4. Table 3(d) shows the basis distribution rules computed by our recursive procedure in (A14). Notethat f T

j 11 ≤ j ≤ 4, are budget-balanced distribution rules for W T (for each T , the shares sum up to 1). It can beverified that f T

2 1 f T3 1 f T

4 are nonnegative and satisfy (4). Next, observe that from Table 3(a), f W1 4i1 8i1 l95= 4, but

from Table 3(d),∑

T∈´ qT fT

1 4i1 8i1 l95= q8i9f8i9

1 4i1 8i1 l95= 5, so f W1 violates condition (b). Also, f 8i1 k9

1 4k1 8i1 k95 < 0,violating condition (c). Thus, f W

1 , the equal share distribution rule, does not have a basis representation and hencedoes not guarantee equilibrium existence in all games; see Counterexamples 3–4 in the proofs of Lemmas 7 and 8.

6.4. Consistency of basis distribution rules. The fourth part of the proof establishes two important consistencyproperties that the basis distribution rules f T must satisfy:

(a) Global consistency: If there is a pair of players i1 j common to two coalitions T 1 T ′, then their local sharesfrom these two coalitions must satisfy (Lemma 9):

f T 4i1 T 5f T ′4j1 T ′5= f T ′

4i1 T ′5f T 4j1 T 50

(b) Cyclic consistency: If there is a sequence of z≥ 3 players, 4i11 i21 : : : 1 iz5 such that for each of the zneighbor-pairs 84i11 i251 4i21 i351 : : : 1 4iz1 i159, ∃T1 ∈´min

i1i21 T2 ∈´min

i2i31 : : : 1 Tz ∈´min

izi1and in each Tj , at least one of

the neighbors ij1 ij+1 gets a nonzero share, then the shares of these z players must satisfy (Lemma 10):

f T14i11 T15fT24i21 T25 · · · f Tz4iz1 Tz5= f T14i21 T15f

T24i31 T25 · · · f Tz4i11 Tz50

Section A.3.1 is devoted to the proofs. Since f T = f TGWSV6�

T 7 for some �T , the above translate into consistencyconditions on the sequence of weight systems ì= 8�T 9T∈´ (Corollaries 2 and 3, respectively). These conditionsare generalizations of those used to prove Proposition 1 in Chen et al. [8] and Marden and Wierman [29]—thewelfare function used, see (1), is such that ´= 2N\8�9, which is “rich” enough to further simplify the aboveconsistency conditions. In such cases, the distribution rule f W is fully determined by “pairwise shares” of the formf W 4i1 8i1 j95.

Example 5. Among the three budget-balanced distribution rules that have a basis representation, namely,f W

2 , f W3 , and f W

4 , only f W2 satisfies both consistency conditions. f W

3 fails the global consistency test sincef8i1 j9

3 4i1 8i1 j95f8i1 j1 l9

3 4j1 8i1 j1 l95 6= f8i1 j1 l9

3 4i1 8i1 j1 l95f8i1 j9

3 4j1 8i1 j95, so does not guarantee equilibrium existence inall games; see Counterexample 5(a) in the proof of Lemma 9. Similarly, f W

4 fails the cyclic consistency test sincef8i1 j9

4 4i1 8i1 j95f8j1 k9

4 4j1 8j1 k95f8i1 k9

4 4k1 8i1 k95 6= f8i1 j9

4 4j1 8i1 j95f8j1 k9

4 4k1 8j1 k95f8i1 k9

4 4i1 8i1 k95 and hence does notguarantee equilibrium existence in all games; see Counterexample 6 in the proof of Lemma 10.

6.5. Existence of a universal weight system. The last step of the proof is to show that there exists a universalweight system �∗ = 4Ë∗1è∗5 that is equivalent to all the weight systems in ì= 8�T 9T∈´. That is, replacing �T

with �∗ for any coalition T does not change the distribution rule f TGWSV6�

T 7:

Proposition 6. If f W = ∑

T∈´ qT fT

GWSV6�T 7 is a budget-balanced distribution rule that guarantees the

existence of an equilibrium in all games G ∈§4N 1 f W 1W5, then there exists a weight system �∗ such that

4∀T ∈´5 f TGWSV6�

T 7= f TGWSV6�

∗70

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In §A.3.2, we prove this proposition by explicitly constructing �∗, given a sequence of weight systemsì= 8�T 9T∈´ that satisfies the consistency Corollaries 2 and 3.

Example 6. The only budget-balanced distribution rule to have survived all the necessary conditions is f W2 .

Using the construction in §A.3.2, it can be shown that f W2 is equivalent to the generalized weighted Shapley value

distribution rule f WGWSV6�

∗7, where the weight system �∗ = 4Ë∗1è∗5 is given by Ë∗ = 4 12 1

12 111 a5, where a is any

strictly positive number and è∗ = 88i1 j1 k91 8l99.

Acknowledgments. This research was supported by AFOSR [Grants FA9550-09-1-0538, FA9550-12-1-0359], ONR [GrantN00014-12-1-0643], and NSF [Grants CNS-0846025, CCF-1101470].

Appendix A. Proof of Theorem 1. In this appendix, we present the complete proof of Theorem 1. It is our intent that thissection be self-contained and independent of the partial outline presented in §6 and therefore may contain some redundancies.

First, note that we only need to prove one direction since it is known that for any weight system � and any mapping gSV, allgames in §4N 1 8f

gSV4W5GWSV 6�79W∈�1�5 have an equilibrium (Hart and Mas-Colell [18]).12 Thus, we present the bulk of the

proof—the other direction—proving that for distribution rules f� that are not generalized weighted Shapley values on somewelfare function, there exists a game in §4N 1 f�1�5 for which no equilibrium exists.

The general technique of the proof is as follows. First, we present a quick reduction to characterizing only budget-balanceddistribution rules f� that guarantees equilibrium existence for all games in §4N 1 f�1�5. Then we establish several necessaryconditions that these rules must satisfy. Effectively, for each W ∈�, these necessary conditions eliminate any budget-balanceddistribution rule f W that is not a generalized weighted Shapley value on W and hence give us our desired result. We establisheach of these conditions by a series of counterexamples that amount to choosing a resource set R, the local welfare functions8Wr9r∈R, and the associated action sets 8¡i9i∈N , for which failure to satisfy a necessary condition would lead to nonexistence ofan equilibrium.

Most counterexamples involve multiple copies of the same resource. To simplify specifying such counterexamples, weintroduce a scaling coefficient vr ∈�++ for each resource r ∈R, which denotes the number of copies of r so that we have

4∀a ∈¡5 ·4a5=∑

r∈RvrWr48a9r5 and Ui4a5= ∑

r∈aivrf

r4i1 8a9r50

Therefore, to exhibit a counterexample, in addition to choosing R, 8Wr9r∈R and 8¡i9i∈N , we also choose 8vr9r∈R.Throughout, we work with the basis-representation of the welfare function that was introduced in §3.1. For each W ∈�, it

is useful to think of the sets in ´W as being “coalitions” of players that contribute to the welfare function W (also referred toas contributing coalitions) and the corresponding coefficients in QW as being their respective contributions. Also, for simplicity,we normalize W by setting W4�5= 0 and therefore, � y´W . Before proceeding, we introduce some notation below, which isalso summarized in Table A.1 for easy reference.

Notation. For any subset S ⊆N , let ´W 4S5 denote the set of contributing coalitions in S:

´W 4S5= 8T ∈´W � T ⊆ S90

Using this notation, and the definition of inclusion functions from (2) in (3), we have an alternative way of writing W , namely,

W4S5= ∑

T∈´W 4S5

qWT 0 (A1)

For any subset S ⊆N , let NW 4S5 denote the set of contributing players in S:

NW 4S5= ∪´W 4S50

Using this notation, and the alternative definition of W from (A1), we have

W4S5=W4NW 4S550 (A2)

For any two players i1 j ∈N , let ´Wij denote the set of all coalitions containing i and j:

´Wij = 8T ∈´W � 8i1 j9⊆ T 90 (A3)

Let ¢⊆ 2N denote a collection of subsets of N . The relation ⊆ induces a partial order on ¢. Let ¢max and ¢min denote theset of maximal and minimal elements of the poset 4¢1⊆5, respectively:

¢max = 8B ∈¢ � 4@B′ ∈¢5 s0t0 B( B′91

¢min = 8B ∈¢ � 4@B′ ∈¢5 s0t0 B′ ( B90

Example 7. Let N = 8i1 j1 k9 be the set of players, and W2 2N →� as defined in Table A.2(a). Table A.2(b) shows thebasis representation of W , and Table A.2(c) illustrates our notation for this W .

12 In fact, notice that � has no role to play as far as equilibrium existence of games §4N 1 8fgSV4W5

GWSV 6�79W∈�1�5 is concerned since it does notdirectly affect player utilities. This observation will prove crucial later.

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Table A.1. Summary of notation.

Symbol Definition Meaning

´W 4S5 8T ∈´W � T ⊆ S9 Set of contributing coalitions in S

NW 4S5⋃

´W 4S5 Set of contributing players in S

´Wij 8T ∈´W � 8i1 j9⊆ T 9 Set of coalitions containing both players i and j

¢max 8B ∈¢ � 4@B′ ∈¢5 s0t0 B( B′9 Set of maximal elements of the poset 4¢1⊆5

¢min 8B ∈¢ � 4@B′ ∈¢5 s0t0 B′ ( B9 Set of minimal elements of the poset 4¢1⊆5

Proof outline. The proof is divided into five sections—each section incrementally builds on the structure imposed on thedistribution rule f by previous sections:

1. Reduction to budget-balanced distribution rules. We reduce the problem of characterizing all distribution rules f� thatguarantee equilibrium existence for all G ∈§4N 1 f�1�5 to characterizing only budget-balanced distribution rules f� thatguarantee equilibrium existence for all G ∈§4N 1 f�1�5.

2. Three necessary conditions. We establish three necessary conditions that collectively describe, for any W ∈�, for anysubset S of players, which players get shares of W4S5 and how these shares are affected by the presence of the other players.

3. Decomposition of the distribution rule. We use these conditions to show that for each W ∈�, f W must be representableas a linear combination of generalized weighted Shapley value distribution rules (with possibly different weight systems) on theunanimity games corresponding to the coalitions in ´W , with corresponding coefficients from QW .

4. Consistency of basis distribution rules. We establish two important consistency properties (one global and one cyclic) thatthese “basis” distribution rules should satisfy and restate these properties in terms of their corresponding weight systems.

5. Existence of a universal weight system. We use the two consistency conditions on the weight systems of the basisdistribution rules to show that there exists a single universal weight system that can replace the weight systems of all the basisdistribution rules without changing the resulting shares of any welfare function. This establishes, for each W ∈�, theequivalence of f W to a generalized weighted Shapley value on W with this universal weight system.

A.1. Reduction to budget-balanced distribution rules. First, we reduce the problem of characterizing all distributionrules f� that guarantee equilibrium existence for all G ∈§4N 1 f�1�5 to characterizing only budget-balanced distributionrules f� that guarantee equilibrium existence for all G ∈§4N 1 f�1�5:

Proposition 7. Given any set of local welfare functions �, their corresponding local distribution rules f� guarantee theexistence of an equilibrium for all games in §4N 1 f�1�5 if and only if they guarantee the existence of an equilibrium for allgames in §4N 1 f�1 gSV4�55.

Proof. This proposition is actually a subtlety of our notation. For games G ∈§4N 1 f�1�5, the welfare functions � donot directly affect strategic behavior (they only do so through the distribution rules f�). Therefore, in terms of strategicbehavior and equilibrium existence, the classes §4N 1 f�1�5 and §4N 1 f�1�′5 are identical for any two sets of welfarefunctions �1�′. Therefore, a distribution rule f� guarantees equilibrium existence for all games in §4N 1 f�1�5 if and onlyif it guarantees equilibrium existence for all games in §4N 1 f�1�′5. To complete the proof, simply pick �′ = gSV4�5, theactual welfares distributed by f�, as defined in (10). �

Notice that f� are budget-balanced distribution rules for the actual welfares they distribute, namely, gSV4�5 as definedin (10). Hence, it is sufficient to prove that for budget-balanced distribution rules f� that are not generalized weighted Shapleyvalues, there exists a game in §4N 1 f�1�5 for which no equilibrium exists.

Table A.2. Tables for Example 7.

(a) Definition of W (b) Basis representation of W (c) Illustration of notation

S W4S5 Coalition T ∈´W Contribution qWT ∈QW Symbol Value

� 0 8i9 1 ´W 48i1 j95 88i91 8j99

8i9 1 8j9 2 ´W 48j1 k95 88j91 8k91 8j1 k99

8j9 2 8k9 3 NW 48i1 k95 8i1 k9

8k9 3 8j1 k9 −2 ´Wij 88i1 j1 k99

8i1 j9 3 8i1 k9 −1 4´Wjk5

max 88i1 j1 k99

8j1 k9 3 8i1 j1 k9 1 4´Wik 5

min 88i91 8k99

8i1 k9 38i1 j1 k9 4

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A.2. Constraints on individual distribution rules. In the next two sections, we establish common constraints thateach budget-balanced distribution rule f W ∈ f� must satisfy in order to guarantee equilibrium existence for all games in§4N 1 f�1�5 for any given set of local welfare functions �. To do this, we deal with one welfare function at a time—for eachW ∈�, we only focus on the corresponding distribution rule f W guaranteeing equilibrium existence for all games in theclass §4N 1 f W 1W5. Note that this is justified because §4N 1 f W 1W5⊆§4N 1 f�1�5 for all W ∈�; so if f� guaranteesequilibrium existence for all games in §4N 1 f�1�5, then each f W ∈ f� must guarantee equilibrium existence for all games in§4N 1 f W 1W5.

Since we are dealing with only one welfare function at a time, we drop the superscripts from f W , ´W , QW , qWT , etc., in

order to simplify notation.

A.2.1. Three necessary conditions. Our goal in this section is to establish that for every subset S ⊆N of players, anybudget-balanced distribution rule f must distribute the welfare W4S5 only among contributing players and do so as if thenoncontributing players were absent:

Proposition 8. If f is a budget-balanced distribution rule that guarantees the existence of an equilibrium in all gamesG ∈§4N 1 f 1W5, then

4∀S ⊆N5 4∀ i ∈ S5 f 4i1 S5= f 4i1N 4S550

We prove this proposition in incremental stages by establishing the following necessary conditions for any subset S ⊆N :(a) If no contributing coalition is formed in S, then f does not allocate any utility to the players in S. Formally, in Lemma 1,

we show that if ´4S5= �, then for all players i ∈ S, f 4i1 S5= 0.(b) f distributes the welfare only among the contributing players in S. Formally, in Lemma 2, we generalize Lemma 1 by

showing that for all players i yN4S5, f 4i1 S5= 0.(c) f distributes the welfare among the contributing players in S as if all other players were absent. Formally, in Lemma 3,

we show that for all players i ∈N4S5, f 4i1 S5= f 4i1N 4S55.

Lemma 1. If f is a budget-balanced distribution rule that guarantees the existence of an equilibrium in all gamesG ∈§4N 1 f 1W5, then

4∀S ⊆N s0t0 ´4S5= �5 4∀ i ∈ S5 f 4i1 S5= 00 (A4)

Proof. The proof is by induction on �S�. The base case, where �S� = 1, is immediate because from budget-balance, wehave that for any player i ∈N ,

f 4i1 8i95={

q8i91 8i9 ∈´1

01 otherwise0

Our induction hypothesis is that (A4) holds for all subsets S of size z, for some 0 < z< �N �. Assuming that this is true, weshow that (A4) holds for all subsets S of size z+ 1. The proof is by contradiction and proceeds as follows.

Assume to the contrary that f 4i1 S5 6= 0 for some i ∈ S, for some S ⊆N , where ´4S5= � and �S� = z+ 1. Since f isbudget-balanced and z+ 1 ≥ 2, it follows that there is some j ∈ S − 8i9 with f4j1 S5 6= 0, such that f4i1 S5 · f4j1 S5 < 0; i.e.,f 4i1 S5 and f 4j1 S5 have opposite signs. Without loss of generality, assume that f 4i1 S5 < 0 and f 4j1 S5 > 0.

Counterexample 1. Consider the game in Figure A.1(a), with resource set R= 8r11 r29 and local resource coefficientsvr1

= vr2= 1. Players i and j have the same action sets—they can each choose either r1 or r2. All other players in S have a

fixed action—they choose both resources. Formally,

¡k ={

88r191 8r2991 k ∈ 8i1 j91

88r11 r2991 k ∈ S − 8i1 j90

This is essentially a game between i and j , with the payoff matrix in Figure A.1(b).Since ´4S5= �, it follows that ´4S ′5= � for all S ′ ⊆ S. Therefore, by letting S ′ = S − 8i9, we can apply the induction

hypothesis to S ′ to obtain f 4j1 S− 8i95= 0. Similarly, by letting S ′ = S− 8j9, we get f 4i1 S− 8j95= 0. We now use this to showthat none of the four outcomes of Counterexample 1 is an equilibrium—this contradicts that f guarantees the existence of an

(a) The game

r1Value: 1

Fixed players:

r2Value: 1

Fixed players:

r1

r2

r1

r2

j

ji

f (i, S), f ( j, S )

f (i, S), f ( j, S )f (i, S – { j}), f ( j, S – {i})

f (i, S – { j}), f ( j, S – {i})

j

i i

(b) The payoff matrix

S – {i, j} S – {i, j}

Figure A.1. Counterexample 1.

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equilibrium in all games G ∈§4N 1 f 1W5. First, consider the outcome 48r191 8r195. Given that player j is in r1, player i obtains apayoff of f 4i1 S5 in r1, which, by our assumption in step (ii), is negative. By deviating to r2, player i would obtain a payoff off 4i1 S − 8j95= 0, which is strictly better for player i. Hence, 48r191 8r195 is not an equilibrium. By nearly identical arguments, itcan be shown that the other three outcomes are also not equilibria. This completes the inductive argument. �


4∀S ⊆N5 4∀ i ∈ S −N4S55 f 4i1 S5= 00 (A5)

Proof. For 0 ≤ p ≤ �´� and 0 ≤ q ≤ n, let °qp denote the collection of all nonempty subsets S for which �´4S5� =

p and �S − N4S5� = q; i.e., S has exactly p contributing coalitions and q noncontributing players in it. Then � =8°0

01°101 : : : 1°

n01°

011°

111 : : : 1°

n11 : : : 1°

0�´�1°

1�´�1 : : : 1°

n�´�9 is an ordered partition of all nonempty subsets of N . Note that

we have slightly abused the usage of the term “partition” since it is possible that °qp = � for some p1q.

We prove the lemma by induction on �= 88°qp99, i.e., the tuple 4p1 q5. Our base cases are twofold:

(i) When p = 0; i.e., for any subset S ∈⋃nq=0 °

q0 , ´4S5= �. Thus, (A5) is true from Lemma 1.

(ii) When q = 0; i.e., for any subset S ∈⋃np=0 °

0p, S =N4S5. Thus, (A5) is vacuously true.

Our induction hypothesis is the following statement:

(A5) holds for all S ∈z⋃

p=0

n⋃

q=0

°qp

y⋃

q=0

°qz+11 for some 0 ≤ z < �´�1 and for some 0 ≤ y < n0

Assuming that this is true, we prove that (A5) holds for all S ∈ °y+1z+1 . In other words, assuming that for all subsets

S ∈⋃8°001°

101 : : : 1°

n01 : : : 1°

0z1°

1z1 : : : 1°

nz 1 : : : 1°

0z+11°

1z+11 : : : 1°

yz+19, we have already proved the lemma, so we focus on

proving the lemma for S ∈°y+1z+1 , the next collection in �. The proof is by contradiction and proceeds as follows.

Assume to the contrary that f 4i1 S5 6= 0 for some i ∈ S −N4S5, for some S ∈°y+1z+1 . Since z+ 1 ≥ 1 and y+ 1 ≥ 1, it must be

that �S� ≥ 2; i.e., S has at least two players. Also, because i yN4S5, it follows that N4S5=N4S − 8i95; so from (A2), we haveW4S5=W4S − 8i95. Since f is budget-balanced and W4S5=W4S − 8i95, we can express f 4i1 S5 as

f 4i1 S5= ∑

k∈S−8i9

4f 4k1S − 8i95− f 4k1S55

Because f 4i1 S5 6= 0, it is clear that at least one of the difference terms on the right-hand side is nonzero and has the same signas f 4i1 S5. That is, there is some j ∈ S − 8i9 such that

f 4i1 S54f 4j1 S − 8i95− f 4j1 S55 > 00 (A6)

Also, f 4i1 S − 8j95= 0. To see this, we consider the following two cases, where, for ease of expression, we let S ′ = S − 8j9.(i) If j ∈N4S5, then �´4S ′5�< �´4S5� = z+ 1, so S ′ ∈⋃z

p=0

⋃nq=0 °

qp.

(ii) If j yN4S5, then �´4S ′5� = �´4S5� = z+ 1 and �S ′ −N4S ′5�< �S −N4S5� = y+ 1, so S ′ ∈⋃yq=0 °

qz+1.

In either case, we can apply the induction hypothesis to S − 8j9 to conclude that f 4i1 S − 8j95= 0 since i yN4S − 8j95.Therefore, (A6) can be rewritten as

4f 4i1 S5− f 4i1 S − 8j9554f 4j1 S5− f 4j1 S − 8i955 < 00

To complete the proof, let us first consider the case where f 4i1 S5−f 4i1 S−8j95 < 0 and f 4j1 S5−f 4j1 S−8i95 > 0. For this case,Counterexample 1 illustrated in Figure A.1 along with the arguments for nonexistence of equilibrium therein (the proof ofLemma 1) serves as a counterexample here, too. The proof for when f 4i1 S5−f 4i1 S− 8j95 > 0 and f 4j1 S5−f 4j1 S− 8i95 < 0 issymmetric. �


4∀S ⊆N5 4∀ i ∈N4S55 f 4i1 S5= f 4i1N 4S550 (A7)

Proof. Since this is a tautology when N4S5= S, let us assume that N4S5( S. We consider two cases below.Case 1. �N4S5� = 1. Without loss of generality, let N4S5= 8i9. Since f is budget-balanced, and W4S5=W4N4S55, we can

express f 4i1N 4S55 asf 4i1N 4S55=∑

k∈Sf 4k1S50

From Lemma 2, we know that f 4k1S5= 0 for all k ∈ S −N4S5. Accordingly, f 4i1 S5= f 4i1N 4S55.Case 2. �N4S5� 6= 1. For 0 ≤ p≤ �´�, let °p denote the collection of all nonempty subsets S such that �N4S5� 6= 1 and

N4S5( S, for which �´4S5� = p; i.e., S has exactly p contributing coalitions in it. Then �= 8°01°11 : : : 1°�´�9 is an orderedpartition of all nonempty subsets S such that �N4S5� 6= 1 and N4S5( S. Note that we have slightly abused the usage of the term“partition” since it is possible that °p = � for some p.

We prove the lemma by induction on �. The base case, where S ∈°0, is vacuously true since N4S5= �. Our inductionhypothesis is that (A7) holds for all subsets S ∈⋃z

p=0 °p, for some 0 ≤ z < �´�. Assuming that this is true, we show that (A7)holds for all subsets S ∈°z+1.

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(a) The game (b) The payoff matrix

r11Value: 1

Fixed players:

r12Value: 1

Fixed players:

r21Value: 1

Fixed players:

r22Value: 1

Fixed players:

j j

L Rj

i

T

B

L R

f ( j, N(S)) + f ( j, S – {i})f (i, N(S )) + f (i, S – { j}), f (i, N(S )) – { j}) + f (i, S),

f (i, N(S )) – { j}) + f (i, S),f (i, N(S )) – {i}) + f ( j, S)

f (i, N(S )) – {i}) + f ( j, S)

f ( j, N(S )) + f ( j, S – {i})f (i, N(S )) + f(i, S – { j}),

Ti

Bi

S – {i, j}

S – {i, j}N(S) – {i, j}

N(S) – {i, j}


Before proceeding with the proof, we point out the following observation. Since f is budget-balanced, and W4S5=W4N4S55,we have

∑

k∈N4S5

f 4k1N 4S55=∑

k∈Sf 4k1S5= ∑

k∈N4S5

f 4k1S50 (A8)

where the second equality comes from Lemma 2, which gives us f 4k1S5= 0 for all k ∈ S −N4S5.The proof is by contradiction and proceeds as follows. Assume to the contrary that f 4k1 S5 6= f 4k1N 4S55 for some k ∈N4S5,

for some S ∈°z+1. Since z+ 1 ≥ 1 and �N4S5� 6= 1, �N4S5� ≥ 2. Then from (A8), we can pick i1 j ∈N4S5 such that

f 4i1 S5 < f 4i1N 4S551 (A9)

f 4j1 S5 > f 4j1N 4S550 (A10)

Counterexample 2. Consider the game in Figure A.2(a), with resource set R= 8r111 r121 r211 r229 and local resourcecoefficients vr11

= vr12= vr21

= vr22= 1. Player i is the row player and player j is the column player. All other players in N4S5

have a fixed action—they choose all four resources. And all players in S −N4S5 also have a fixed action—they chooseresources r12 and r21. Formally,

¡k =

T = 8r111 r1291B = 8r211 r2291 k = i1

L= 8r111 r2191R= 8r121 r2291 k = j1

8r111 r121 r211 r2291 k ∈N4S5− 8i1 j91

8r121 r2191 k ∈ S −N4S50

This is essentially a game between players i and j , with the payoff matrix in Figure A.2(b). The set of joint action profiles cantherefore be represented as ¡= 8TL1TR1BL1BR9.

Because i ∈N4S5, �´4S−8i95� = �´4N4S5−8i95�< �´4S5�. Also note that N4S−8i95=N4N4S5−8i95. Now considertwo cases:

(i) If j yN4S − 8i95, then j yN4N4S5− 8i95, and so from Lemma 2, f 4j1 S − 8i95= f 4j1N 4S5− 8i95= 0.(ii) If j ∈N4S− 8i95, then j ∈N4N4S5− 8i95. If N4S− 8i95= 8j9, then applying our analysis in Case 1 to S− 8i9 and

N4S5− 8i9, we havef 4j1 S − 8i95= f 4j1N 4S − 8i9551

f 4j1N 4S5− 8i95= f 4j1N 4N4S5− 8i9550(A11)

If N4S − 8i95 6= 8j9, then we know that �N4S − 8i95� ≥ 2. Accordingly, we can apply our induction hypothesis to S − 8i9 andN4S5− 8i9 to obtain (A11).In either case, we have

f 4j1 S − 8i95= f 4j1N 4S5− 8i950 (A12)

By similar arguments, we obtainf 4i1 S − 8j95= f 4i1N 4S5− 8j950 (A13)

We use the four properties in (A9), (A10), (A12), and (A13) to show that Counterexample 2 does not possess an equilibrium,thereby contradicting the fact that f guarantees the existence of an equilibrium in all games G ∈§4N 1 f 1W5. We show this foreach outcome:

(i) TL is not an equilibrium since player j has an incentive to deviate from L to R:

f 4j1N 4S5− 8i95+ f 4j1 S5 > f 4j1 S − 8i95+ f 4j1N 4S550

This results from combining (A10) and (A12).

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(ii) TR is not an equilibrium since player i has an incentive to deviate from T to B:

f 4i1 S − 8j95+ f 4i1N 4S55 > f 4i1N 4S5− 8j95+ f 4i1 S50

This results from combining (A9) and (A13).(iii) BR and BL are also not equilibria because in these action profiles, players j and i respectively have incentives to

deviate—the arguments are identical to cases (a) and (b) above, respectively.This completes the inductive argument. �

A.2.2. Decomposition of the distribution rule. Our goal in this section is to use the necessary conditions above(Proposition 8) to establish that f must be representable as a linear combination of generalized weighted Shapley valuedistribution rules (see (8) in Table 2) on the unanimity games corresponding to the coalitions in ´, with correspondingcoefficients from Q:

Proposition 9. If f is a budget-balanced distribution rule that guarantees the existence of an equilibrium in all gamesG ∈§4N 1 f 1W5, then there exists a sequence of weight systems ì= 8�T 9T∈´ such that

f = ∑

T∈´qT f

TGWSV6�

T 7

Note that for now, the weight systems �T could be arbitrary and need not be related in any way. We deal with how theyshould be “consistent” later in §A.3.1.

Before proceeding, we define a useful abstract mathematical object. The min-partition of a finite poset 4´1⊆5, denoted by�min4´5= 8°11°21 : : : 1°l9, is an ordered partition of ´, constructed iteratively as specified in Algorithm 1.

Algorithm 1 (Construction of �min4´5).°1 =´min

z← 2while ´ 6=⋃

1≤p<z°p do °z = 4´−⋃1≤p<z°p5min; z← z+ 1;

end while.

Example 8. Let 4´1⊆5 be a poset, where ´= 88i91 8j91 8j1 k91 8k1 l91 8j1 l91 8i1 j1 k99. Then

�min4´5= {

88i91 8j91 8k1 l991 88j1 k91 8j1 l991 88i1 j1 k99}

0

Construction of basis distribution rules. Given a budget-balanced distribution rule f that guarantees the existence of anequilibrium in all games G ∈§4N 1 f 1W5, we now show how to construct a sequence of basis distribution rules 8f T 9T∈´ suchthat (4) is satisfied. Let �min4´5= 8°11°21 : : : 1°l9 be the min-partition of the poset 4´1⊆5, and let f be a distribution rulefor W . Starting with z= 1, recursively define f T for each T ∈°z as

4∀S ⊆N5 4∀ i ∈N5 f T 4i1 S5=

1qT

(

f 4i1 T 5− ∑

T ′∈´4T 5−8T 9

qT ′f T ′4i1 S5

)

1 T ⊆ S1

01 otherwise0

(A14)

At the end of this procedure, we obtain the basis distribution rules 8f T 9T∈´. Note that it is not obvious from this constructionthat these basis distribution rules satisfy (4) or that they are generalized weighted Shapley value distribution rules on theircorresponding unanimity games. The rest of this section is devoted to showing these properties; first, here is an example todemonstrate this recursive construction.

Example 9. Consider the setting in Example 7, where N = 8i1 j1 k9 is the set of players, and W2 2N →� is the welfarefunction defined in Table A.2(a). The basis representation of W is shown in Table A.2(b). The set of coalitions is thereforegiven by ´= 88i91 8j91 8k91 8j1 k91 8i1 k91 8i1 j1 k99. For the poset 4´1⊆5, we have

�min4´5= {

88i91 8j91 8k991 88j1 k91 8i1 k991 88i1 j1 k99}

0

Consider the following two distribution rules for W .(i) fSV, the Shapley value distribution rule (see §2.1.2).

(ii) fEQ, the equal share distribution rule (see §2.1.1).Table A.3(a) shows fSV and fEQ for this welfare function. The basis distribution rules f T

SV and f TEQ that result from applying our

construction (A14) above are shown in Table A.3(b). For simplicity, we show only f TSV4·1 T 5 and f T

EQ4·1 T 5.The proof of Proposition 9 consists of four lemmas, as outlined below:(a) In Lemma 4, we show that each f T , as constructed in (A14), mimics f locally for its corresponding unanimity game

W T ; i.e., f T satisfies Proposition 8 for W =W T .(b) Using this property, in Lemma 5, we show that each f T is a budget-balanced distribution rule for its corresponding

unanimity game W T .

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(c) In Lemma 7, we show that f , and the basis distribution rules 8f T 9T∈´, satisfy (4); i.e., f =∑

T∈´ qT fT .

(d) Finally, in Lemma 8, we show that for each T ∈´, there exists a weight system �T such that f T = f TGWSV6�

T 7.

Lemma 4. Each f T as defined in (A14) satisfies

4∀S ⊆N5 4∀ i ∈N5 f T 4i1 S5={

f T 4i1 T 51 i ∈ T and T ⊆ S1

01 otherwise0(A15)

Proof. The proof is by induction on �min4´5. The base case where T ∈°1 is immediate because from (A14), for anyT ∈°1,

4∀S ⊆N5 4∀ i ∈N5 f T 4i1 S5=

1qT

f 4i1 T 51 T ⊆ S1

01 otherwise0

Our induction hypothesis is that f T satisfies (A15) for all T ∈⋃zp=1 °p for some 1 ≤ z < l. Assuming that this is true, we prove

that f T satisfies (A15) for all T ∈°z+1. To evaluate f T 4i1 S5 for some i ∈ S ⊆N , we consider the following three cases:(i) T * S. In this case, from (A14), f T 4i1 S5= 0.

(ii) i y T ⊆ S. Here, we know that f 4i1 T 5= 0 by definition. Also, for all T ′ ∈´4T 5− 8T 9, we have, T ′ ∈⋃zp=1 °p and

i y T ′; so from the induction hypothesis, f T ′4i1 S5= 0. Therefore, evaluating (A14), we get f T 4i1 S5= 0.

(iii) i ∈ T ⊆ S. In this case, we need to show that f T 4i1 S5= f T 4i1 T 5. By (A14), we have

f T 4i1 S5= 1qT

(

f 4i1 T 5− ∑

T ′∈´4T 5−8T 9

qT ′f T ′4i1 S5

)

1

f T 4i1 T 5= 1qT

(

f 4i1 T 5− ∑

T ′∈´4T 5−8T 9

qT ′f T ′4i1 T 5

)

0

For each T ′ ∈´4T 5− 8T 9, we know that T ′ ∈⋃zp=1 °p; hence, from the induction hypothesis, we have f T ′

4i1 S5= f T ′4i1 T 5.

Therefore, f T 4i1 T 5= f T 4i1 S5, as desired.Hence, f T satisfies (A15). �

Lemma 5. If f is a budget-balanced distribution rule for W , then each f T as defined in (A14) is a budget-balanceddistribution rule for W T ; i.e.,

4∀T ∈´5 4∀S ⊆N5∑

i∈Sf T 4i1 S5=W T 4S50

Proof. Since f T is of the form (A15) from Lemma 4, to show (local) budget-balance, we need only show that

4∀T ∈´5∑

i∈Tf T 4i1 T 5= 11 (A16)

Once again, the proof is by induction on �min4´5. The base case, where T ∈°1 follows from the budget-balance of f . Ourinduction hypothesis is that f T satisfies (A16) for all T ∈⋃z

p=1 °p for some 1 ≤ z < l. Assuming that this is true, we prove thatf T satisfies (A16) for all T ∈°z+1. For any T ∈°z+1, using (A14), we have

∑

i∈Tf T 4i1 T 5 = 1

qT

(

∑

i∈Tf 4i1 T 5−∑

i∈T

∑

T ′∈´4T 5−8T 9

qT ′f T ′4i1 T ′5

)

Table A.3. Tables for Example 9.

(a) Definition of fSV and fEQ (b) Basis distribution rules for fSV and fEQ

S fSV4·1 S5 fEQ4·1 S5 Coalition T ∈´ f TSV4·1 T 5 f T

EQ4·1 T 5

8i9 415 415 8i9 415 415

8j9 425 425 8j9 415 415

8k9 435 435 8k9 415 415

8i1 j9 41125(

321 3

2

)

8j1 k9(

121 1

2

) (

141 3

4

)

8j1 k9 41125(

321 3

2

)

8i1 k9(

121 1

2

) (− 121 3

2

)

8i1 k9(

121 5

2

) (

321 3

2

)

8i1 j1 k9(

131 1

31 1

3

) (− 161− 1

61 4

3

)

8i1 j1 k9(

561 4

31 11

6

) (

431 4

31 4

3

)

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= 1qT

(

W4T 5− ∑

T ′∈´4T 5−8T 9

∑

i∈T ′qT ′f T ′

4i1 T ′5)

= 1qT

(

∑

T ′∈´4T 5

qT ′ − ∑

T ′∈´4T 5−8T 9

qT ′

)

= 1qT

4qT 5= 10

where we have used the budget-balance of f , followed by the induction hypothesis and (A1). This completes the inductiveargument and hence the proof. �

Example 10. Consider the decomposition of fSV and fEQ illustrated in Example 9. Both f TSV and f T

EQ are locallybudget-balanced for all T ∈´; that is, they satisfy (A16).

Before continuing with the proof, in the next lemma, we present the conditional inclusion-exclusion principle, an importantand useful property of the basis distribution rules 8f T 9T∈´.

Lemma 6 (Conditional Inclusion-Exclusion Principle). For any T ∈´, there exist integers 8nT 4T′59T ′∈´ such that

the basis distribution rules 8f T 9T∈´ defined in (A14) satisfy

4∀ i ∈ T 5 qT fT 4i1 T 5= ∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′50 (A17)

Furthermore, if 8f T 9T∈´ satisfies

4∀S ( T 5 4∀ i ∈ S5 f 4i1 S5= ∑

T ′∈´4S5

qT ′f T ′4i1 T ′51 (A18)

then4∀ i ∈ T 5 4∀ j ∈ T − 8i95 0 = qT f

T 4i1 T − 8j95= ∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′ − 8j950 (A19)

Proof. For any T ∈´, setting S = T in (A14), and using Lemma 4, we get

4∀ i ∈ T 5 qT fT 4i1 T 5= f 4i1 T 5− ∑

T ′∈´4T 5−8T 9

qT ′f T ′4i1 T ′50 (A20)

It follows that by unraveling the recursion above, i.e., by repeatedly substituting for the terms qT ′f T ′4i1 T ′5 that appear in the

summation, we obtain (A17), where 8nT 4T′59T ′∈´ are some integers.

Let í4T 5= 8T ′ ∈´4T 52 i ∈ T ′9 denote the set of coalitions contained in T that contain i. Before proving (A19), we makethe following observation. From (A17), we have

4∀ i ∈ T 5 qT fT 4i1 T 5 = ∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′5

= ∑

T ′∈í4T 5

nT 4T′5

∑

T ′′∈í4T′5qT ′′f T ′′

4i1 T ′′5 (from (A14) and Lemma 4)

= ∑

T ′∈í4T 5

mi1 T 4T′5qT ′f T ′

4i1 T ′51 (A21)

where 8mi1 T 4T′59T ′∈´ are some integer coefficients. We now exploit the fact that (A21) holds for all distribution rules f to

show that the unique solution for the coefficients mi1 T 4T′5 is given by

mi1 T 4T′5=

{

11 T ′ = T 1

01 otherwise0(A22)

To see this, we first prove that given T ∈´ and i ∈ T , mi1 T 4T′′5= 0 for all T ′′ ∈í4T 5−8T 9 by induction on �min4í4T 5−8T 95.

To do this, we focus on the family of generalized weighted Shapley value distribution rules 8fGWSV6�S79S∈´4T 5 with weight

systems �S = 4Ë1èS5, where Ë= 41111 : : : 115 and èS = 4N − S1S5. By definition (see (8) in Table 2), for each T ′ ∈í4T 5,

f T ′GWSV6�

S74i1 T ′5=

1�T ′� 1 T ′ ⊆ S1

01 otherwise0(A23)

(i) For the base case, when T ′′ ∈°1, it follows from (A23), with S = T ′′, that for all T ′ ∈í4T 5,

f T ′GWSV6�

T ′′74i1 T ′5=

1�T ′� 1 T ′ = T ′′1

01 otherwise(A24)

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This results because since T ′′ ∈ °1, T ′ ⊆ T ′′ if and only if T ′ = T ′′. Now, we evaluate (A21) for the distribution rulefGWSV6�

T ′′7 to get

qT fTGWSV6�

T ′′74i1 T 5= ∑

T ′∈í4T 5


GWSV6�T ′′74i1 T ′50

Using (A24) to simplify the above equation, we get

0 =mi1 T 4T′′5qT ′′

1�T ′′�

since for any T ′ 6= T ′′, T ′ ∩ 4N − T ′5 6= �. Therefore, mi1 T 4T′′5= 0.

(ii) Our induction hypothesis is that mi1 T 4T′′5= 0 for all T ′′ ∈⋃z

p=1 °p, for some 1 ≤ z < l. Assuming that this is true, weprove that mi1 T 4T

′′5= 0 for all T ′′ ∈°z+1. If T ′′ ∈°z+1, it follows from (A23), with S = T ′′, that for all T ′ ∈ 8T 9⋃l

p=z+1 °p,(A24) holds, from a similar reasoning as above. Now, we evaluate (A21) for the distribution rule fGWSV6�

T ′′74 · 1 · 5 to get

qT fTGWSV6�

T ′′74i1 T 5= ∑

T ′∈í4T 5


GWSV6�T ′′74i1 T ′5

By grouping together terms on the right-hand side, we can rewrite this as

qT fTGWSV6�

T ′′74i1 T 5=mi1 T 4T 5qT f

TGWSV6�

T ′′74i1 T 5+

l∑

p=1

∑

T ′∈°p


GWSV6�T ′′74i1 T ′50 (A25)

Using the induction hypothesis, we get that

z∑

p=1

∑

T ′∈°p


GWSV6�T ′′74i1 T ′5= 00 (A26)

Using (A24), we get thatl∑

p=z+1

∑

T ′∈°p


GWSV6�T ′′74i1 T ′5=mi1 T 4T

′′5qT ′′1

�T ′′� 1

f TGWSV6�

T ′′74i1 T 5= 01

(A27)

since for all T ′ ∈ 8T 9∪ 4⋃l

p=z+1 °p5, if T ′ 6= T ′′, then T ′ ∩ 4N − T ′5 6= �. Therefore, using (A26) and (A27) in (A25), we getmi1 T 4T

′′5= 0.This completes the inductive argument. From this, it is straightforward to see that mi1 T 4T 5= 1.We now return to proving the remainder of the lemma, (A19). The right-hand side of (A19) can be evaluated as

∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′ − 8j95 = ∑

T ′∈´4T 5

nT 4T′5

∑

T ′′∈´4T ′−8j95

qT ′′f T ′′4i1 T ′′5 (from (A18))

= ∑

T ′∈í4T 5

nT 4T′5

∑

T ′′∈í4T′5qT ′′f T ′′

4i1 T ′′ − 8j95 (from Lemma 4)

= ∑

T ′∈í4T 5


4i1 T ′ − 8j95 (from (A21))

= qT fT 4i1 T − 8j95= 00 (from (A22))

This completes the proof. �

Lemma 7. If f is a budget-balanced distribution rule that guarantees the equilibrium existence in all games G ∈§4N 1 f 1W5,then the basis distribution rules 8f T 9T∈´ defined in (A14) satisfy

4∀S ⊆N5 4∀ i ∈ S5 f 4i1 S5= ∑

T∈´qT f

T 4i1 S50 (A28)

Proof. From Lemma 4, (A28) is equivalent to

4∀S ⊆N5 4∀ i ∈ S5 f 4i1 S5= ∑

T∈´4S5

qT fT 4i1 T 50 (A29)

Let S ⊆N . We consider three cases:Case 1. S ∈´. The proof is immediate here because rearranging the terms in (A14), we get

f 4i1 S5= ∑

T ′∈´4S5

qT ′f T ′4i1 S5= ∑

T ′∈´4S5

qT ′f T ′4i1 T ′50 (A30)

where the last equality follows from Lemma 4.

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Case 2. S 6=N4S5. In this case, we can apply Proposition 8; i.e., f 4i1 S5= f 4i1N 4S55, to reduce it to the following case,replacing S with N4S5.

Case 3. S =N4S5. In other words, S is a union of one or more coalitions in ´. The remainder of the proof is devoted tothis case.

For any subset S ⊆N such that S =N4S5, i.e., S is exactly a union of one or more coalitions in ´, we prove this lemma byinduction on �´4S5�. The base case, where �´4S5� = 1 (and hence S ∈´) is true from (A30). Our induction hypothesis is that(A29) holds for all subsets S ⊆N such that S =N4S5, with �´4S5� ≤ z for some 1 ≤ z < �´�. Assuming that this is true, weprove that (A29) holds for all subsets S ⊆N such that S =N4S5, with �´4S5� = z+ 1. If S ∈´, then the proof is immediatefrom (A30), so let us assume S y´. Before proceeding with the proof, we point out the following observation. Since f isbudget-balanced, we have

∑

i∈Sf 4i1 S5=W4S5 = ∑

T∈´4S5

qT (from (A1))

= ∑

T∈´4S5

qT∑

i∈Tf T 4i1 T 5 (from Lemma 5)

= ∑

i∈S

∑

T∈´4S5

qT fT 4i1 T 50 (from Lemma 4) (A31)

The proof is by contradiction and proceeds as follows. Assume to the contrary, that f 4k1 S5 6=∑

T∈´4S5 qT fT 4k1 T 5 for some

k ∈ S, for some S ⊆N such that S =N4S5, with �´4S5� = z+ 1. Since z+ 1 ≥ 2, and S =N4S5, it must be that �S� ≥ 2; i.e., Shas at least two players. Then from (A31), it follows that we can pick i1 j ∈ S such that

f 4i1 S5 >∑

T∈´4S5

qT fT 4i1 T 51 (A32)

f 4j1 S5 <∑

T∈´4S5

qT fT 4j1 T 50 (A33)

Because S =N4S5, for any S ′ ( S, �´4S ′5�< �´4S5�. Hence, applying the induction hypothesis,

4∀S ′ ( S5 4∀ i ∈ S ′5 f 4i1 S ′5= ∑

T∈´4S′5qT f

T 4i1 T 50 (A34)

Since every coalition T ∈´4S5 is a subset of S, (A34) holds when S is replaced with any T ∈´4S5. Therefore, Lemma 6, theconditional inclusion-exclusion principle, can be applied to obtain, for any coalition T ∈´4S5,

4∀ i ∈ T 5 qT fT 4i1 T 5= ∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′51

4∀ i ∈ T 5 4∀ j ∈ T − 8i95 qT fT 4i1 T − 8j95= ∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′ − 8j950

Summing up these equations over all T ∈´4S5, we get

4∀ i ∈ S5∑

T∈´4S5

qT fT 4i1 T 5= ∑

T∈´4S5

nS4T 5f 4i1 T 51

4∀ i ∈ S5 4∀ j ∈ S − 8i95∑

T∈´4S5

qT fT 4i1 T − 8j95= ∑

T∈´4S5

nS4T 5f 4i1 T − 8j950(A35)

where the constants 8nS4T 59T∈³ are given bynS4T 5= ∑

T ′∈´4S5T⊆T ′

nT ′ 4T 50

Counterexample 3. Our goal is to exploit inequalities (A32) and (A33) to build a counterexample that mimicsCounterexample 2 illustrated in Figure A.2, leading to a similar best-response cycle involving just players i and j . Equation (A35)suggests the following technique for achieving precisely this. Consider the game in Figure A.3, which has the same underlying2 × 2 box structure of Counterexample 2. The resources in the top half are added as follows:

(i) Add a resource r1 to the top left box.(ii) Add resources in R+

1 = 8rT1 2 T ∈´4S5 and nS4T 5 > 09 to the top right box.(iii) Add resources in R−

1 = 8rT1 2 T ∈´4S5 and nS4T 5 < 09 to the top left box.Then the bottom half is symmetrically filled up as follows.

(i) Add a resource r2 to the bottom right box.(ii) Add resources in R+

2 = 8rT2 2 T ∈´4S5 and nS4T 5 > 09 to the bottom left box.(iii) Add resources in R−

2 = 8rT2 2 T ∈´4S5 and nS4T 5 < 09 to the bottom right box.The resource set R is therefore given by

R= 8r11 r29∪R+1 ∪R−

1 ∪R+2 ∪R−

2 0

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jj

RL

R1–

r1Value: 1

r2Value: 1

R2+ R2

–

R1+

i

i

T

B


The local resource coefficients are given by

vr1= vr2

= 1 and 4∀T ∈´4S55 vrT1 = vrT2 = �nS4T 5�0In resources r1 and r2, we fix players in S − 8i1 j9. For each T ∈´4S5, in resources rT1 and rT2 , we fix players in T − 8i1 j9.Effectively, all players other than i and j have a fixed action in their action set, determined by these fixtures. The action set ofplayer i is given by ¡i = 8T 1B9, where,

T = 8rT1 ∈R−1 2 i ∈ T 9∪ 8r19∪ 8rT1 ∈R+

1 2 i ∈ T 91

B = 8rT2 ∈R+2 2 i ∈ T 9∪ 8r29∪ 8rT2 ∈R−

2 2 i ∈ T 90

The action set of player j is given by ¡j = 8L1R9, where

L= 8rT1 ∈R−1 2 j ∈ T 9∪ 8r19∪ 8rT2 ∈R+

2 2 j ∈ T 91

R= 8rT1 ∈R+1 2 j ∈ T 9∪ 8r29∪ 8rT2 ∈R−

2 2 j ∈ T 90

This is essentially a game between players i and j . The set of joint action profiles can therefore be represented as¡= 8TL1TR1BL1BR9.

We use the four properties in (A32), (A33), (A34), and (A35) to show that Counterexample 3 does not possess anequilibrium, thereby contradicting that f guarantees the existence of an equilibrium in all games G ∈§4N 1 f 1W5. We show thisfor each outcome:

(i) TL is not an equilibrium since player j has an incentive to deviate from L to R. To see this, consider the utilities ofplayer j when choosing L and R,

Uj4T 1L5= − ∑

T∈´4S5nS 4T 5<0

nS4T 5f 4j1 T 5+ f 4j1 S5+ ∑

T∈´4S5nS 4T 5>0

nS4T 5f 4j1 T − 8i951

Uj4T 1R5= ∑

T∈´4S5nS 4T 5>0

nS4T 5f 4j1 T 5+ f 4j1 S − 8i95− ∑

T∈´4S5nS 4T 5<0

nS4T 5f 4j1 T − 8i950

The difference in utilities for j between choosing R and L is therefore given by

Uj4T 1R5−Uj4T 1L5 =(

∑

T∈´4S5

nS4T 5f 4j1 T 5− f 4j1 S5

)

+(

f 4j1 S − 8i95− ∑

T∈´4S5

nS4T 5f 4j1 T − 8i95

)

=(

∑

T∈´4S5

qT fT 4j1 T 5− f 4j1 S5

)

+(

f 4j1 S − 8i95− ∑

T∈´4S5

qT fT 4j1 T − 8i95

)

=(

∑

T∈´4S5

qT fT 4j1 T 5− f 4j1 S5

)

+(

f 4j1 S − 8i95− ∑

T∈´4S−8i95

qT fT 4j1 T 5

)

= ∑

T∈´4S5

qT fT 4j1 T 5− f 4j1 S5 > 00

This results from using (A35) first, followed by (A34), Lemma 4, and then (A33).

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(ii) TR is not an equilibrium since player i has an incentive to deviate from T to B. The proof is along the same lines asthe previous case. Using similar arguments, we get

Ui4B1R5−Ui4T 1R5= f 4i1 S5− ∑

T∈´4S5

qT fT 4i1 T 51

which is positive, from (A32).(iii) BR and BL are also not equilibria because in these action profiles, players j and i respectively have incentives to

deviate—the arguments are identical to cases (a) and (b) above, respectively.This completes the inductive argument. �

Example 11. Consider the decomposition of fSV and fEQ into their respective basis distribution rules, as illustrated inExample 9. Let S = 8i1 j9.

(i) fSV4i1 S5= 1, and∑

T∈´4S5 qT fTSV4i1 T 5= q8i9f

8i9SV 4i1 8i95= 1. Thus, fSV satisfies (A28).

(ii) fEQ4i1 S5= 32 , and

∑

T∈´4S5 qT fTEQ4i1 T 5= q8i9f

8i9EQ4i1 8i95= 1. Therefore, fEQ does not satisfy (A28) and hence does not

guarantee the existence of an equilibrium in all games G ∈§4N 1 f 1W5.

From Lemma 7, it follows that any budget-balanced distribution rule f that guarantees the existence of an equilibrium in allgames G ∈§4N 1 f 1W5 satisfies (A18) for all T ∈´. Thus for such f , condition (A18) can be stripped off of Lemma 6, leadingto the (unconditional) inclusion-exclusion principle, a powerful tool that we use extensively in proving several subsequentlemmas. We formally state this in the following corollary:

Corollary 1 (Inclusion-Exclusion Principle). If f is a budget-balanced distribution rule that guarantees theexistence of an equilibrium in all games G ∈§4N 1 f 1W5, and 8f T 9T∈´ are the basis distribution rules defined in (A14), then forevery T ∈´, there exist integers 8nT 4T

′59T ′∈´ such that the following equations hold:

4∀ i ∈ T 5 qT fT 4i1 T 5= ∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′51 (A36)

4∀ i ∈ T 5 4∀ j ∈ T − 8i95 0 = qT fT 4i1 T − 8j95= ∑

T ′∈´4T 5

nT 4T′5f 4i1 T ′ − 8j950 (A37)

Example 12. To illustrate the inclusion-exclusion principle, let the set of players be N = 8i1 j1 k1 l9, and let the set ofcontributing coalitions be ´= 88i91 8i1 j91 8i1 k91 8i1 l91 8i1 j1 k1 l99. Then for T = 8i1 j1 k1 l9, unraveling the recursion in (A20)gives the following inclusion-exclusion formula for isolating f T 4i1 T 5, in terms of f :

qT fT 4i1 T 5= f 4i1 8i1 j1 k1 l95− f 4i1 8i1 j95− f 4i1 8i1 k95− f 4i1 8i1 l95+ 2f 4i1 8i950

The corresponding coefficients are given by

nT 48i1 j1 k1 l95= 1nT 48i1 j95= nT 48i1 k95= nT 48i1 l95= −1nT 48i95= 20

Lemma 8. If f is a budget-balanced distribution rule that guarantees the existence of an equilibrium in all gamesG ∈§4N 1 f 1W5, then for each basis distribution rule f T defined in (A14), there exists a weight system �T such that

f T = f TGWSV6�

T 70 (A38)

Proof. First, we show that each basis distribution rule f T is nonnegative. The proof is by contradiction, and proceeds asfollows. Assume to the contrary that f T 4k1 T 5 < 0 for some k ∈ T . From (A16), this is possible only if �T � ≥ 2, and it followsthat we can pick i1 j ∈ T such that qT f

T 4i1 T 5 < 0 and qT fT 4j1 T 5 > 0.

Counterexample 4. Our goal is to exploit the inequalities qT fT 4i1 T 5 < 0 and qT f

T 4j1 T 5 > 0 to build a counterexamplethat mimics Counterexample 3 illustrated in Figure A.3, leading to a similar best-response cycle involving just players i and j .The inclusion-exclusion principle (Corollary 1) suggests the following technique for achieving precisely this. Consider the gamein Figure A.4, which is nearly identical to Counterexample 3 except that resources r1 and r2 are absent. The resources in thetop half are added as follows.

(i) Add resources in R+1 = 8rT

′1 2 T ′ ∈´4T 5 and nT 4T

′5 > 09 to the top right box.(ii) Add resources in R−

1 = 8rT′

1 2 T ′ ∈´4T 5 and nT 4T′5 < 09 to the top left box.

Then the bottom half is symmetrically filled up as follows.(i) Add resources in R+

2 = 8rT′

2 2 T ′ ∈´4T 5 and nT 4T′5 > 09 to the bottom left box.

(ii) Add resources in R−2 = 8rT

′2 2 T ′ ∈´4T 5 and nT 4T

′5 < 09 to the bottom right box.The resource set R is therefore given by

R=R+1 ∪R−

1 ∪R+2 ∪R−

1 0


4∀T ′ ∈´4T 55 vrT

′1

= vrT

′2

= �nT 4T′5�0

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i

j j

RL

R1–

R2+ R2

–

R1+

i

T

B


For each T ′ ∈´4T 5, in resources rT′

1 and rT′

2 , we fix players in T ′ − 8i1 j9. Effectively, all players other than i and j have afixed action in their action set determined by these fixtures.

The action set of player i is given by ¡i = 8T 1B9, where

T = 8rT′

1 ∈R−1 2 i ∈ T ′9∪ 8rT

′1 ∈R+

1 2 i ∈ T ′91

B = 8rT′

2 ∈R+2 2 i ∈ T ′9∪ 8rT

′2 ∈R−

2 2 i ∈ T ′90

The action set of player j is given by ¡j = 8L1R9, where

L= 8rT′

1 ∈R−1 2 j ∈ T ′9∪ 8rT

′2 ∈R+

2 2 j ∈ T ′91

R= 8rT′

2 ∈R+1 2 j ∈ T ′9∪ 8rT

′1 ∈R−

2 2 j ∈ T ′90

This is essentially a game between players i and j . The set of joint action profiles can therefore be represented as ¡=8TL1TR1BL1BR9.

We use the inclusion-exclusion principle (Corollary 1) to show that Counterexample 4 does not possess an equilibrium,thereby contradicting that f guarantees the existence of an equilibrium in all games G ∈§4N 1 f 1W5. We show this for eachoutcome:

(i) TL is not an equilibrium because player j has an incentive to deviate from L to R. To see this, consider the utilities ofplayer j when choosing L and R,

Uj4T 1L5= − ∑

T ′∈´4T 5nT 4T

′5<0

nT 4T′5f 4j1 T ′5+ ∑

T ′∈´4T 5nT 4T

′5>0

nT 4T′5f 4j1 T ′ − 8i951

Uj4T 1R5= ∑

T ′∈´4T 5nT 4T

′5>0

nT 4T′5f 4j1 T ′5− ∑

T ′∈´4T 5nT 4T

′5<0

nT 4T′5f 4j1 T ′ − 8i950

The difference in utilities for j between choosing R and L is therefore given by

Uj4T 1R5−Uj4T 1L5 = ∑

T ′∈´4T 5

nT 4T′5f 4j1 T ′5− ∑

T ′∈´4T 5

nT 4T′5f 4j1 T ′ − 8i95

= ∑

T ′∈´4T 5

nT 4T′5f 4j1 T ′5 (from (A37))

= qT fT 4j1 T 5 > 0 (from (A36))

(ii) TR is not an equilibrium because player i has an incentive to deviate from T to B. The proof resembles the previouscase. By using similar arguments, we get,

Ui4B1R5−Ui4T 1R5= qT fT 4i1 T 5 < 00

(iii) BR and BL are also not equilibria, because in these action profiles, players j and i respectively have incentives todeviate—the arguments are identical to cases (a) and (b) above, respectively.

This completes the inductive argument.Now since each f T is nonnegative, budget-balanced, and satisfies Lemma 4, it is completely specified by �T � nonnegative

values, 8f T 4i1 T 52 i ∈ T 9, that sum to 1. Let èT = 4ST1 1 S

T2 5 be an ordered partition of T , where ST

1 = 8i2 f T 4i1 T 5 > 09 andST

2 = T − ST1 . Define a weight vector ËT as follows:

�Ti =

{

f T 4i1 T 51 i ∈ ST1 1

arbitrary positive value1 i ∈ ST2 0

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Then it follows that f T satisfies (A38) with weight system �T = 4ËT 1èT 5 constructed above. (See (8) in Table 2 to recall thedefinition of the generalized weighted Shapley value distribution rule.) This completes the proof. �

Example 13. Consider the decomposition of fSV and fEQ into their respective basis distribution rules, from Example 9.Clearly, f T

SV is nonnegative for all T ∈´, whereas f TEQ is not.

A.3. Consistency of basis distribution rules. It follows from Proposition 9, that each budget-balanced distribution rulef W ∈ f� that guarantees the existence of an equilibrium in all games G ∈§4N 1 f�1�5 is completely specified by a sequenceof weight systems ìW = 8�W1T 9T∈´W . But these weight systems could be “inconsistent” across different coalitions and acrossdifferent welfare functions. Thus, our next steps focus on proving that all the weight systems �T 1W are consistent—in otherwords, there exists a universal weight system �∗ that is equivalent to all the �W1T (replacing �W1T with �∗ for any coalitionT ∈´W for any W ∈� does not affect the distribution rule f W1T = f T

GWSV6�W1T 7).

To address the consistency of �W1T across different coalitions T under the same welfare function W , it is sufficient to workwith one welfare function at a time, just like in the previous module. However, to address the consistency across differentwelfare functions, it is necessary to work with more than one welfare function at a time—for every subset of welfare functions�⊆�, we only focus on the corresponding distribution rules f � that guarantee equilibrium existence for all games in theclass §4N 1 f �1�5. The justification is similar—§4N 1 f �1�5⊆§4N 1 f�1�5 for all �⊆�; so if f� guarantees equilibriumexistence for all games in §4N 1 f�1�5, then for every subset �⊆�, f � ⊆ f� must guarantee equilibrium existence for allgames in §4N 1 f �1�5.

In what follows, we work with k > 1 welfare functions (not necessarily distinct) at a time, say W11W21 : : : 1Wk, to addressconsistency across welfare functions, and then use the special case of W1 =W2 = · · · =Wk =W to address consistency acrosscoalitions under the same welfare function W . In order to simplify notation, we drop W from the superscripts. That is, for1 ≤ j ≤ k, we write f j instead of f Wj , ´j instead of ´Wj , qj

T instead of qWj

T , njT 4T

′5 instead of nWj

T 4T ′5, etc.

A.3.1. Two consistency conditions. Our goal in this section is to establish the following two important consistency propertiesthat the basis distribution rules f W1T must satisfy in order for the budget-balanced distribution rules f W =∑

T∈´W qWT f W1T to

guarantee the existence of an equilibrium in all games G ∈§4N 1 f�1�5. Recall that ´Wij , defined in (A3), refers to the set of

coalitions in ´W containing both players i and j . In addition, let ´Wij 4S5= 8T ∈´W 4S5 � 8i1 j9⊆ T 9 denote the set of coalitions

in ´W 4S5 containing both players i and j .(a) Global consistency: If there is a pair of players common to two coalitions (under the same or different welfare functions),

then their shares from these two coalitions (given by the corresponding f W1T values) must be “consistent,” as formalized inLemma 9. Here, we deal with at most two welfare functions at a time.

(b) Cyclic consistency: If there is a sequence of k ≥ 3 players, 4i11 i21 : : : 1 ik5 such that for each of the k neighbor-pairs84i11 i251 4i21 i351 : : : 1 4ik1 i159, ∃T1 ∈ 4´1

i1i25min1 T2 ∈ 4´2

i2i35min1 : : : 1 Tk ∈ 4´k

ik i15min, and in each Tj , at least one of the neighbors

ij 1 ij+1 gets a nonzero share (given by the corresponding f j1 Tj value), then the shares of these k players from these k coalitionsmust satisfy a “cyclic consistency” condition, as formalized in Lemma 10. Here, we deal with an arbitrary number of welfarefunctions at a time.

Lemma 9. Given any two local welfare functions W11W2, if f 1 =∑T∈´1 q1T f

11T and f 2 =∑T∈´2 q2T f

21 T are correspondingbudget-balanced distribution rules that guarantee equilibrium existence in all games G ∈§4N 1 8f 11 f 291 8W11W295, then for anytwo players i1 j ∈N and any two coalitions T ′ ∈´1

ij and T ∈´2ij ,

f 11 T ′4i1 T ′5f 21 T 4j1 T 5= f 21 T 4i1 T 5f 11 T ′

4j1 T ′50 (A39)

Proof. Note that it is sufficient to show (A39) for only those coalitions in ´1ij and ´2

ij in which at least one among i and j

get a nonzero share. Formally, define the collections ´1+ij and ´2+

ij as

´1+ij = 8T ∈´1

ij 2 f11T 4i1 T 5 > 0 or f 11T 4j1 T 5 > 091

´2+ij = 8T ∈´2

ij 2 f21 T 4i1 T 5 > 0 or f 21 T 4j1 T 5 > 090

(A40)

Let S be a minimal element (coalition) in the poset 4´1+ij 1⊆5, and without loss of generality, assume f 11 S4i1 S5 > 0. Then we

need only show13 that for any coalition T ∈´2+ij ,

f 21 T 4i1 T 5f 11 S4j1 S5= f 11 S4i1 S5f 21 T 4j1 T 50 (A42)

The proof is by contradiction. Assume to the contrary that for some T ∈´2+ij , f 21 T 4i1 T 5f 11 S4j1 S5 6= f 11 S4i1 S5f 21 T 4j1 T 5. We

consider the following two cases:

Case 1. q1Sq

2T > 0.

13 It can be shown that (A42) implies (A39): for the special case when W2 =W1, (A42) implies that for all T ′ ∈´1+ij ,

f 11 T ′4i1 T ′5f 11 S4j1 S5= f 11 S4i1 S5f 11 T ′

4j1 T ′50 (A41)

Let T ′ ∈´1ij and T ∈´2

ij . By assumption, f 11 S4i1 S5 6= 0. If f 11 S4j1 S5= 0, then (A42) and (A41) imply that f 21 T 4j1 T 5= 0 and f 11 T ′4j1 T ′5= 0,

in which case both sides of (A39) are zero. If f 11 S4j1 S5 6= 0, then none of the four terms in Equations (A42) and (A41) are zero, and therefore,by eliminating f 11 S4i1 S5 and f 11 S4j1 S5 between them, we get (A39).

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R1–

r1

r2

R2+ R2

–

R1+

(v1)

(v1)

(v1)

(v1)

Value: v2

Value: v2

i

i

T

B

j

R

j

L

Figure A.5. Counterexample 5(a).

Counterexample 5. (a) Our goal is to build a counterexample that mimics Counterexample 4 illustrated in Figure A.4,leading to a similar best-response cycle involving just players i and j . As before, we use the inclusion-exclusion principle(Corollary 1) to isolate just f 21 T by appropriately adding resources and setting action sets. Consider the game in Figure A.5,which is identical to Counterexample 4, except for the following changes:

(i) There are two additional resources, r1 and r2, so the resource set is now

R= 8r11 r29∪R+1 ∪R−

1 ∪R+2 ∪R−

1 0

(ii) The welfare function at r1 and r2 is W1. At all other resources, the welfare function is W2.(iii) The local resource coefficients are given by

vr1= vr2

= v2 and 4∀T ′ ∈´24T 55 vrT

′1

= vrT

′2

= v1�n2T 4T

′5�

where v1 > 0 and v2 > 0. We will discuss the specific choice of v11 v2 later.(iv) In resources r1 and r2, we fix players in S − 8i1 j9.(v) The actions T , B, L, R are modified to accommodate the two new resources:

T = 8rT′

1 ∈R−1 2 i ∈ T ′9∪ 8r19∪ 8rT

′1 ∈R+

1 2 i ∈ T ′91

B = 8rT′

2 ∈R+2 2 i ∈ T ′9∪ 8r29∪ 8rT

′2 ∈R−

2 2 i ∈ T ′91

L= 8rT′

1 ∈R−1 2 j ∈ T ′9∪ 8r19∪ 8rT

′2 ∈R+

2 2 j ∈ T ′91

R= 8rT′

1 ∈R+1 2 j ∈ T ′9∪ 8r29∪ 8rT

′2 ∈R−

2 2 j ∈ T ′90

To complete the specification of Counterexample 5(a), we need to specify the values of v1 > 0 and v2 > 0. We now show thatif f 21 T 4i1 T 5f 11 S4j1 S5 6= f 11 S4i1 S5f 21 T 4j1 T 5, then these values can be picked carefully in such a way that Counterexample 5(a)does not possess an equilibrium, thereby contradicting the fact that f 1 and f 2 guarantee equilibrium existence in all gamesG ∈§4N 1 8f 11 f 291 8W11W295. Consider each of the four outcomes:

(i) In action profiles TL and BR, player j has an incentive to deviate if Uj4T 1R5−Uj4T 1L5=Uj4B1L5−Uj4B1R5 > 0.This happens if

v1

(

∑

T ′∈´24T 5

n2T 4T

′5f 24j1 T ′5− ∑

T ′∈´24T 5

n2T 4T

′5f 24j1 T ′ − 8i95

)

− v24f14j1 S5− f 14j1 S − 8i955 > 00

Using the inclusion-exclusion principle (Corollary 1) to simplify the terms in the first bracket, and the basis representation of f 1

to simplify the difference in the second bracket, this condition is equivalent to

v14q2T f

21 T 4j1 T 55− v2

(

∑

T ′∈´1ij 4S5

q1T ′f 11 T ′

4j1 T ′5)

> 00

Since S is minimal in ´1+ij , this reduces to

v1q2T f

21 T 4j1 T 5 > v2q1Sf

11 S4j1 S50 (A43)

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(ii) Similarly, in action profiles TR and BL, player i has an incentive to deviate if Ui4B1R5−Ui4T 1R5=Ui4T 1L5−Ui4B1L5 > 0. This happens if

−v1

(

∑

T ′∈´24T 5

n2T 4T

′5f 24i1 T ′5− ∑

T ′∈´24T 5

n2T 4T

′5f 24i1 T ′ − 8j95

)

+ v24f14i1 S5− f 14i1 S − 8j955 > 00

By similar arguments as above, this condition reduces to

v1q2T f

21 T 4i1 T 5 < v2q1Sf

11 S4i1 S50 (A44)

Without loss of generality, assume q2T > 0 and q1

S > 0 (for the symmetric case when q2T < 0 and q1

S < 0, the same argumentsapply, but the deviations in the best-response cycle are reversed). By assumption, f 11 S4i1 S5 > 0. Now we consider two cases forf 11 S4j1 S5:

(i) f 11 S4j1 S5= 0. In this case, f 21 T 4j1 T 5 > 0 (for otherwise (A42) would be satisfied). It follows then that (A43) and(A44) always have a solution in strictly positive integers v1 and v2.

(ii) f 11 S4j1S5 > 0. In this case, suppose f 21 T 4i1 T 5f 11 S4j1 S5 < f 11 S4i1 S5f 21 T 4j1 T 5 (for the other case whenf 21 T 4i1 T 5f 11 S4j1 S5 > f 11 S4i1 S5f 21 T 4j1 T 5, the same arguments apply, but the deviations in the best-response cycle are reversed).Combining (A43) and (A44), we get

q2T

q1S

f 21 T 4i1 T 5

f 11 S4i1 S5<

v2

v1<

q2T

q1S

f 21 T 4j1 T 5

f 11 S4j1 S50

This inequality has a solution in strictly positive integers v1 and v2, if and only if

q2T

q1S

f 21 T 4i1 T 5

f 11 S4i1 S5<

q2T

q1S

f 21 T 4j1 T 5

f 11 S4j1 S5⇐⇒ f 21 T 4i1 T 5f 11 S4j1 S5 < f 11 S4i1 S5f 21 T 4j1 T 51

which is true by assumption.Case 2. q1

Sq2T < 0.

(b) Our goal remains the same—to build a counterexample in which a best-response cycle involving just players i and jexists. This counterexample breaks from symmetry, and we use the inclusion-exclusion principle (Corollary 1) thrice here toisolate two more basis distribution rules in addition to f 21 T . We now present the formal details.

Consider the game in Figure A.6, where we have various boxes with labels on them indicating which resource or set ofresources is present. Let Ti and Tj be some coalitions that contain i and j , respectively. We will discuss the specific choice ofTi1 Tj later. As before, we use the resource sets 4R+

1 1R−1 5 (with W2 the welfare function at all these resources) for isolating f 21 T .

In addition, we use resource sets 4R+3 1R

−3 5 (with Wx as the welfare function at all these resources) and 4R+

4 1R−4 5 (with Wy as

the welfare function at all these resources) to isolate two more basis distribution rules, f x1Tj and f y1 Ti , respectively, where thechoice of x1 y ∈ 81129 will be discussed later. In addition to these six sets, we also have a single resource r2 whose welfarefunction is W1. Formally,

R+1 = 8rT

′1 2 T ′ ∈´24T 5 and n2

T 4T′5 > 091 R−

1 = 8rT′

1 2 T ′ ∈´24T 5 and n2T 4T

′5 < 091

R+3 = 8rT

′3 2 T ′ ∈´x4Tj5 and nx

Tj4T ′5 > 091 R+

4 = 8rT′

4 2 T ′ ∈´y4Ti5 and nyTi4T ′5 > 091

R−3 = 8rT

′3 2 T ′ ∈´x4Tj5 and nx

Tj4T ′5 < 091 R−

4 = 8rT′

4 2 T ′ ∈´y4Ti5 and nyTi4T ′5 < 090

D

R1+

R4+

j

i

i

B

T

(v4)

R3+

(v3)

(v1)

R1–

(v1)

R4–

(v4)

R3–

r2Value: v2

(v3)

U j

Figure A.6. Counterexample 5(b).

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The resource set R is therefore given by

R= 8r29∪R+1 ∪R−

1 ∪R+3 ∪R−

3 ∪R+4 ∪R−

4 0


vr2= v21

4∀T ′ ∈´24T 55 vrT

′1

= v1�n2T 4T

′5�04∀T ′ ∈´x4Tj55 v

rT′

3= v3�nx

Tj4T ′5�1 and 4∀T ′ ∈ý4Ti55 v

rT′

4= v4�ny

Ti4T ′5�0

where v1, v2, v3, v4 > 0. We will discuss the specific choice of v1, v2, v3, v4 later. In resource r2, we fix players in S − 8i1 j9.For each T ′ ∈´24T 5, in resource rT

′1 , we fix players in T ′ − 8i1 j9. For each T ′ ∈´x4Tj5, in resource rT

′3 , we fix players in

T ′ − 8j9. For each T ′ ∈ý4Ti5, in resource rT′

4 , we fix players in T ′ − 8i9. Effectively, all players other than i and j have a fixedaction in their action set, determined by these fixtures. In addition, these fixtures might also specify mandatory sets of resourcesRi and Rj that players i and j must always be present in. The action sets of players i and j are given by ¡i = 8T 1B9 and¡j = 8U 1D9, where

T = 8rT′

4 ∈R−4 2 i ∈ T ′9∪ 8r29∪ 8rT

′1 ∈R−

1 2 i ∈ T ′9∪Ri1

B = 8rT′

1 ∈R+1 2 i ∈ T ′9∪ 8rT

′4 ∈R+

4 2 i ∈ T ′9∪Ri1

U = 8rT′

1 ∈R−3 2 j ∈ T ′9∪ 8r29∪Rj 1

D = 8rT′

1 ∈R−1 2 j ∈ T ′9∪ 8rT

′1 ∈R+

3 2 j ∈ T ′9∪ 8rT′

1 ∈R+1 2 j ∈ T ′9∪Rj 0

This is essentially a game between players i and j . The set of joint action profiles can therefore be represented as¡= 8T U1TD1BU1BD9.

To complete the specification of Counterexample 5(b), we need to specify the values of v1, v2, v3, v4 > 0, x1 y ∈ 81129, andTi, Tj . We now show that if f 21 T 4i1 T 5f 11 S4j1 S5 6= f 11 S4i1 S5f 21 T 4j1 T 5, then these values can be picked carefully in such a waythat Counterexample 5(b) does not possess an equilibrium, thereby contradicting the fact that f 1 and f 2 guarantee equilibriumexistence in all games G ∈§4N 1 8f 11 f 291 8W11W295. Consider each of the four outcomes:

(i) In action profile T U , player i has an incentive to deviate if Ui4B1U5−Ui4T 1U5 > 0. This happens if

v4

(

∑

T ′∈ý 4Ti5

nyTi4T ′5f y4i1 T ′5

)

+ v1

(

∑

T ′∈´24T 5

n2T 4T

′5f 24i1 T ′ − 8j95

)

− v2f14i1 S5 > 00

Note that Ui4B1U5 and Ui4T 1U5 include utilities to player i from resources in Ri; but while taking the difference, this cancelsout since i is fixed in these resources, and between these two action profiles, all other players also have a fixed action. Nowusing the inclusion-exclusion principle (Corollary 1) to simplify the terms in the first two brackets, we get

v4qyTif y1 Ti 4i1 Ti5 > v2f

14i1 S50 (A45)

(ii) In action profile BD, player i has an incentive to deviate if Ui4T 1D5−Ui4B1D5 > 0. This happens if

−v4

(

∑

T ′∈ý 4Ti5

nyTi4T ′5f y4i1 T ′5

)

− v1

(

∑

T ′∈´24T 5

n2T 4T

′5f 24i1 T ′5)

+ v2f14i1 S − 8j95 > 00

As before, the utility to player i from resources in Ri cancels out. Using the inclusion-exclusion principle to simplify the terms,we get

v2f14i1 S − 8j95− v1q

2T f

21 T 4i1 T 5 > v4qyTif y1 Ti 4i1 Ti50 (A46)

(iii) In action profile TD, player j has an incentive to deviate if Uj4T 1U5−Uj4T 1D5 > 0. This happens if

−v3

(

∑

T ′∈´x4Tj 5

nxTj4T ′5f x4j1 T ′5

)

+ v2f14j1 S5

− v1

(

∑

T ′∈´24T 5

n2T 4T

′5>0

n2T 4T

′5f 24j1 T ′ − 8i95− ∑

T ′∈´24T 5

n2T 4T

′5<0

n2T 4T

′5f 24j1 T ′5

)

> 00

The utility to player j from resources in Rj cancels out. Using the inclusion-exclusion principle to simplify the first term, we get

v2f14j1 S5− v1

(

∑

T ′∈´24T 5

n2T 4T

′5>0

n2T 4T

′5f 24j1 T ′ − 8i95− ∑

T ′∈´24T 5

n2T 4T

′5<0

n2T 4T

′5f 24j1 T ′5

)

> v3qxTjf x1Tj 4j1 Tj50 (A47)

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(iv) In action profile BU , player j has an incentive to deviate if Uj4B1D5−Uj4B1U5 > 0. This happens if

v3

(

∑

T ′∈´x4Tj 5

nxTj4T ′5f x4j1 T ′5

)

− v2f14j1 S − 8i95

+ v1

(

∑

T ′∈´24T 5

n2T 4T

′5>0

n2T 4T

′5f 24j1 T ′5− ∑

T ′∈´24T 5

n2T 4T

′5<0

n2T 4T

′5f 24j1 T ′ − 8i95

)

> 00

As before, the utility to player j from resources in Rj cancels out. Using the inclusion-exclusion principle to simplify the firstterm, we get

v3qxTjf x1Tj 4j1 Tj5 > v2f

14j1 S − 8i95− v1

(

∑

T ′∈´24T 5

n2T 4T

′5>0

n2T 4T

′5f 24j1 T ′5− ∑

T ′∈´24T 5

n2T 4T

′5<0

n2T 4T

′5f 24j1 T ′ − 8i95

)

0 (A48)

Combining inequalities (A45) and (A46), we get

v1q2T f

21 T 4i1 T 5+ v24f14i1 S5− f 14i1 S − 8j955 < 00

Using the basis representation of f 1 to simplify the difference in the bracket, this condition is equivalent to

v1q2T f

21 T 4i1 T 5+ v2

(

∑

T ′∈´1ij 4S5

q1T ′f 11 T ′

4i1 T ′5)

< 00


v1q2T f

21 T 4i1 T 5+ v2q1Sf

11 S4i1 S5 < 00 (A49)

Combining inequalities (A47) and (A48), we get

v1

(

∑

T ′∈´24T 5

n2T 4T

′5f 24j1 T ′5− ∑

T ′∈´24T 5

n2T 4T

′5f 24j1 T ′ − 8i95

)

+ v24f14j1 S5− f 14j1 S − 8i955 > 00

Using the inclusion-exclusion principle to simplify the terms in the first bracket, and the basis representation of f 1 to simplifythe difference in the second bracket, this condition is equivalent to

v1q2T f

21 T 4j1 T 5+ v2

(

∑

T ′∈´1ij 4S5

q1T ′f 11 T ′

4j1 T ′5)

> 00


v1q2T f

21 T 4j1 T 5+ v2q1Sf

11 S4j1 S5 > 00 (A50)

Without loss of generality, assume q2T > 0 and q1

S < 0 (for the symmetric case when q2T < 0 and q1

S > 0, the same argumentsapply, but the deviations in the best-response cycle are reversed). By assumption, f 11 S4i1 S5 > 0. Now we consider two cases forf 11 S4j1 S5:

(i) f 11 S4j1 S5= 0. In this case, f 21 T 4j1 T 5 > 0 (for otherwise, (A42) would be satisfied). It follows then that (A49) and(A50) always have a solution in strictly positive integers v1 and v2.

(ii) f 11 S4j1S5 > 0. In this case, suppose f 21 T 4i1 T 5f 11 S4j1 S5 < f 11 S4i1 S5f 21 T 4j1 T 5. (For the other case whenf 21 T 4i1 T 5f 11 S4j1 S5 > f 11 S4i1 S5f 21 T 4j1 T 5, the same arguments apply, but the deviations in the best-response cycle are reversed.)Combining (A49) and (A50), we get

−q2T

q1S

f 21 T 4i1 T 5

f 11 S4i1 S5<

v2

v1<−q2

T

q1S

f 21 T 4j1 T 5

f 11 S4j1 S5

Therefore, this inequality has a solution in strictly positive integers v1 and v2, if and only if

q2T

q1S

f 21 T 4i1 T 5

f 11 S4i1 S5>

q2T

q1S

f 21 T 4j1 T 5

f 11 S4j1 S5⇐⇒ f 21 T 4i1 T 5f 11 S4j1 S5 < f 11 S4i1 S5f 21 T 4j1 T 5

which is true by assumption.

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Finally, we need to show that given these carefully chosen values for v1 and v2, it is possible to find v3 > 0, v4 > 0,x1 y ∈ 81129, Ti, and Tj such that the inequalities (A45)–(A48) are satisfied. These four inequalities can be consolidated as

LHSj < v3qxTjf x1Tj 4j1 Tj5 < RHSj 1

LHSi < v4qyTif y1 Ti 4i1 Ti5 < RHSi0

We describe the procedure to find v3 > 0, x ∈ 81129, and Tj here. Finding v4 > 0, y ∈ 81129, and Ti is analogous. Specifically,we consider the case where RHSj > 0 (we discuss the other case later). Consider the two quantities f 11 S4j1 S5 and f 21 T 4j1 T 5.They are not both zero (for otherwise (A42) would be satisfied). We consider two subcases:

(i) If f 21 T 4j1 T 5 > 0, choose x = 2, Tj = T . Then it is possible to find v3 > 0 such that LHSj < v3q2T f

21 T 4j1 T 5 < RHSj

because by assumption, q2T > 0.

(ii) If f 21 T 4j1 T 5= 0, then f 11 S4j1 S5 > 0. Here, we slightly alter Counterexample 5(b) by modifying player j’s action set¡j = 8U 1D9 as follows—we simply switch the resources in R−

3 and R+3 between action U and action D. Formally,

U ′ = 8r29∪ 8rT′

1 ∈R+3 2 j ∈ T ′9∪Rj 1

D′ = 8rT′

1 ∈R−3 2 j ∈ T ′9∪ 8rT

′1 ∈R−

1 2 j ∈ T ′9∪ 8rT′

1 ∈R+1 2 j ∈ T ′9∪Rj 0

This alteration does not affect any of the arguments above, except that we now need to find v3 > 0, x ∈ 81129, and Tj such thatLHSj <−v3qTj f

Tj 4j1 Tj5 < RHSj . (In particular, (A49) and (A50) remain unchanged.) Choose x = 1, Tj = S. Then it ispossible to find v3 > 0 such that LHSj <−v3q

1Sf

11 S4j1 S5 < RHSj because by assumption, q1S < 0.

The case when RHSj < 0 is symmetric—we just interchange S and T , and the choice of x, throughout in the two subcases.That is, if f 11 S4j1 S5 > 0, we choose x = 1, Tj = S. Otherwise, we choose x = 2, Tj = T and alter Counterexample 5(b) thesame way as above. This completes the proof.14 �

From Lemma 8, we inferred that each f W1T is a generalized weighted Shapley value for the corresponding unanimitygame W T , with weight system �W1T = 4ËW1T 1èW1T 5 that we constructed using f W1T . As a result, the conditions on88f W1T 9T∈´W 9W∈� imposed by Lemma 9 translate to equivalent conditions on the weight systems 88�W1T 9T∈´W 9W∈�. Thefollowing corollary restates Lemma 9 in terms of the weight systems.

Corollary 2. Given any set of local welfare functions �, if f� are budget-balanced distribution rules that guaranteeequilibrium existence in all games G ∈ §4N 1 f�1�5, where for each W ∈ �, f W 2=∑

T∈´W qWT f T

GWSV6�W1T 7, where

�W1T = 4ËW1T 1èW1T = 4SW1T1 1 SW1T

2 55, then for any two players i1 j ∈N ,

(a) 4∃W1T 5 i ∈ SW1T1 1 j ∈ SW1T

2 =⇒ 4∀W ′1 T ′5 j ∈ SW ′1 T ′2 1

(b) 4∃W1T 5 i ∈ SW1T1 1 j ∈ SW1T

1 =⇒ 4∀W ′1 T ′5

4i5 i ∈ SW ′1 T ′1 ⇐⇒ j ∈ SW ′1 T ′

1 1

4ii5 8i1 j9⊆ SW ′1 T ′1 =⇒ �W1T

i

�W ′1 T ′i

= �W1Tj

�W ′1 T ′j

0

Explanation. In essence, this corollary states consistency constraints that the various weight systems that define thedistribution rules must satisfy for every pair of players. It is obtained by applying Lemma 9 for all pairs of welfare functions�×�. Suppose W1W ′ ∈�, and let T ∈´W

ij and T ′ ∈´W ′ij . Then from Lemma 9, we have

f W1T 4i1 T 5f W ′1 T ′4j1 T ′5= f W ′1 T ′

4i1 T ′5f W1T 4j1 T 50 (A51)

The different parts of the corollary then follow directly from applying the definition of the generalized weighted Shapley value(see (8) in Table 2) and simplifying the above equation for the corresponding cases.

Lemma 10. For any integer k ≥ 3, given any k welfare functions W11W21 : : : 1Wk, if 8f j 2=∑

T∈´j qjT f

j1 T 9kj=1 arecorresponding budget-balanced distribution rules that guarantee equilibrium existence in all games G ∈§4N 1 8f 11 f 21 : : : 1 f k918W11W21 : : : 1Wk95, and i11 i21 : : : 1 ik ∈N are any k players such that ∃T1 ∈ 4´1+

i1i25min1 T2 ∈ 4´2+

i2i35min1 : : : 1 Tk ∈ 4´k+

ik i15min,

thenf 11 T14i11 T15f

21 T24i21 T25: : : fk1Tk 4ik1 Tk5= f 11 T14i21 T15f

21 T24i31 T25 · · · f k1Tk 4i11 Tk50 (A52)

Proof. Recall that ´j+xy denotes the collection of those coalitions from ´j

xy in which at least one of x1y obtains anonzero share (according to f j ). Refer to (A40) for the formal definition.

Index arithmetic. In the rest of this proof, the index set is 81121 : : : 1 k9, and when we add an integer l to an index j , j + ldenotes the index that is l positions away from index j (cycling around if necessary). For example, suppose k = 3. Then forindex j = 2, j + 2 = 1 and j − 2 = 3.

For simplicity, denote qjTj

by qj , fj1 Tj 4ij 1 Tj5 by aj , and f j1 Tj 4ij+11 Tj5 by bj . Note that aj ≥ 0 and bj ≥ 0. Now (A52) can

be written ask∏

j=1

aj =k∏

j=1

bj 0 (A53)

14 Note that v1 and v2 can be scaled by an arbitrary positive constant if necessary, in order to ensure that an integral solution to v31 v4 exists.

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v2

d3 u3

u2

d2

d1T k– {i k,

i 1}

T k– {i k,

i 1}

T1 – {i1 , i2}

T2 –

{i2 , i3 }T3

–{i3,i4}

T3–

{i3,i4}T

2 –{i2 , i3 }

T1 – {i1 , i2}

v1

v1

u1

uk

dk

vk

vk

v2

v3

v3


Our proof technique mirrors those in previous sections. Assuming (A53) is not satisfied, we present a game where noequilibrium exists. Without loss of generality, let

k∏

j=1

aj <k∏

j=1

bj 0 (A54)

We present a family of counterexamples, each corresponding to a specific sign profile of the coefficients qj . (Recall that for agiven choice of the local welfare functions, qj are fixed.) The proof is in three stages:

(i) We present the details of the counterexample game Counterexample 6).(ii) We present four validity conditions on the action profiles of Counterexample 6 and define those action profiles that

satisfy at least one of them as valid action profiles, observing that such action profiles are never equilibria.(iii) We show that every action profile in Counterexample 6 is valid.

Counterexample 6. Consider the game in Figure A.7 that involves only players i11 i21 : : : 1 ik. There are 2k resources,arranged in two circular rows of k resources each. For each column j , both resources uj and dj share the same resource-specificcoefficient vj > 0 and same local welfare function Wj , and in both these resources, we fix players in Tj − 8ij 1 ij+19. Effectively,all players other than i11 i21 : : : 1 ik have a fixed action in their action set determined by these fixtures. In addition, these fixturesmight also specify mandatory sets of resources R11R21 : : : 1Rk that the players i11 i21 : : : 1 ik must always be present in,respectively. However, for simplicity, we will not explicitly represent this since such actions do not affect strategic behavior(utilities from these resources cancel out as far as unilateral deviations are concerned, just like they did in the proof ofLemma 9). Next, we need to specify the action sets of the players i11 i21 : : : 1 ik, which will depend intimately on propertiesregarding the sequence of signs of the coefficients qj . We begin by relabeling the indices according to a cyclic transformationthat is without loss of generality, to ensure that the following properties will be satisfied after the transformation:

(i) The first coefficient, q1, is negative unless all coefficients are positive.(ii) The last coefficient, qk, is positive unless all coefficients are negative.

(iii) The penultimate coefficient, qk−1, is positive unless no two adjacent coefficients are both positive.In essence, we cut down on the different sequences of signs of the coefficients that we need to consider. Formally, we definetwo special index sets, J and J ∗, as follows:

J = 8j � 1 ≤ j ≤ k and qj−1 > 0 and qj < 091 J ∗ = 8j � j ∈ J and qj−2 > 090

Now we define a special index k∗ as follows. If J ∗ 6= �, pick any k∗ ∈ J ∗. Otherwise, if J 6= �, then pick any k∗ ∈ J . IfJ = J ∗ = �, set k∗ = 1. Now we perform a cyclic transformation of the indices that resets k∗ = 1, by rotating Figure A.7counterclockwise by k∗ − 1 columns. In other words, index j becomes index j − k∗ + 1. In the rest of the proof, we assume thatFigure A.7 represents the counterexample after this transformation.

Next, we observe that given any profile of the signs of the coefficients qj , the k columns of resources in Figure A.7 can begrouped into several segments, each of which can be classified as one the following three kinds:

(i) �l, a maximal plus segment of length l: This segment consists of l > 1 contiguous columns i1 i+ 11 : : : 1 i+ l− 1 suchthat qj > 0 for i ≤ j ≤ i+ l− 1. In addition, we require maximality; i.e., if l 6= k, then qi−1 < 0 and qi+l < 0.

(ii) l, a maximal minus segment of length l: This segment consists of l > 1 contiguous columns i1 i+ 11 : : : 1 i+ l− 1such that qj < 0 for i ≤ j ≤ i+ l− 1. In addition, we require maximality; i.e., if l 6= k, then qi−1 > 0 and qi+l > 0.

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(iii) �l, a maximal alternating minus-plus segment of length l: This segment consists of l > 1 contiguous columns (l beingeven) i1 i+ 11 : : : 1 i+ l− 1 such that qj < 0 and qj+1 > 0 for j ∈ 8i1 i+ 21 i+ 41 : : : 1 i+ l− 29. In addition, we requiremaximality; i.e., if l 6= k, then qi−1 > 0 ⇒ qi−2 > 0 and qi+l < 0 ⇒ qi+l+1 < 0. Note that this kind of segment may share itsfirst/last column with a preceding minus/succeeding plus segment.

Example 14. Consider the sign profile 4−1−1+1−1+1+1−5. Our special index sets are given by J = 84179, J ∗ = 879.Hence, k∗ = 7; so without loss of generality we transform this sign profile to 4−1−1−1+1−1+1+5. Now the first threecolumns constitute an 3 segment, and the last two columns constitute a �2 segment. In between, we have a 4+1−5 segmentthat doesn’t fit any of our three definitions above. But if the immediate neighbors on either side are taken into consideration, wehave 4−1+1−1+5, which is a �4 segment. Thus the (unique) decomposition of this sign profile is given by 3�4 �2, wherecolumns 3 and 6 are shared between two neighboring segments.

Technically, the above definition of a �l segment allows for a spurious �2 segment to be sandwiched between an and anadjacent � segment. For example, the sign profile 4−1−1+1+5 could be decomposed as either 2 �2 or 2�2 �2. We excludethis possibility by requiring that every �l segment have at least one column that is not shared with a neighboring segment. Notethat this requirement also guarantees that the above decomposition is always unique.

We now specify the action sets for the players i11 i21 : : : 1 ik. For any player ij , his actions involve only the resources inadjacent columns j − 1 and j and specifically in only one of two ways:

(i) Straight players have the following action set:

¡ij= 84uj−11 uj51 4dj−11 dj590 (A55)

(ii) Diagonal players have the following action set:

¡ij= 84uj−11 dj51 4dj−11 uj590 (A56)

Whether player ij is straight or diagonal is determined as follows. We consider two cases:Case 1. j = 1. Player i1 is straight if and only if one of the following conditions is satisfied:(i) q1 > 0.

(ii) Column 1 is at the beginning of an segment and column k is at the end of a � segment.Case 2. j 6= 1. Player ij is straight if and only if one of the following conditions is satisfied:

(i) Column j is at the end of a �2l+1, l or �l segment.(ii) qj < 0, and column j is in a �l segment.

(iii) qj > 0, qj−1 < 0, and column j is at the beginning of a �l segment.(iv) qj < 0, qj−1 > 0, qj−2 < 0, and column j is at the beginning of an segment.

A player is diagonal if and only if he is not straight.

Example 15. Consider the 3�4 �2 sign profile of Example 14, namely, 4−1−1−1+1−1+1+5. There are seven players,i11 : : : 1 i7. Their action sets are represented pictorially in Figure A.8. Formally, players i1, i3, i5, i6 are straight, whereas playersi2, i4, i7 are diagonal.

a3

a4a5

u5

d5

d6

d7

u7

a7

a1

u1

d1

u6

a6

u2

a2

d2

d3

d4

u4

u3

Figure A.8. Action sets of the players in Example 15. Each arrow connecting two resources in columns j − 1 and j denotes the action ofplayer j choosing those two resources.

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Counterexample 6 is essentially a game between the players i11 i21 : : : 1 ik. To complete its specification, we need to specifythe resource-specific coefficients vj . For this, we pick any vj > 0 satisfying the following set of inequalities:

v1�q1�b1 > v2�q2�a21

v2�q2�b2 > v3�q3�a31

000

vk�qk�bk > v1�q1�a10

(A57)

We argue that if (A54) is satisfied, then this set of inequalities has a solution in strictly positive integers vj . To see this, firstobserve that if (A54) is satisfied, then bj 6= 0 for all 1 ≤ j ≤ k. If aj = 0 for some j , say, for j = 1, then (A57) can be solvedrecursively as follows.

First, pick any integer vk > 0; this satisfies the last inequality. Then for k− 1 ≥ j ≥ 11 pick any integer vj > 0 that satisfiesthe jth inequality; i.e.,

vj > vj+1

�qj+1��qj �

aj+1

bj0

Next, consider the case where aj 6= 0 for all 1 ≤ j ≤ k. For simplicity, denote vi�qi� by v′i.

For i 6= 1, multiplying the first i− 1 inequalities results in an upper bound for the ratio v′i/v

′1, and multiplying the last

k− i+ 1 inequalities results in a lower bound for the ratio v′i/v

′1. These bounds are given by

4∀ i12 ≤ i ≤ k5

∏k+1j=i+1 aj∏k

j=i bj<

v′i

v′1

<

∏i−1j=1 bj

∏ij=2 aj

0 (A58)

It can be seen that (A58) is feasible if and only if (A54) is satisfied. Algorithm 2 describes a procedure for obtaining v′i > 0 that

solves (A57).

Algorithm 2 (Solving (A57)).v′

1 ← 1Pick v′

2 satisfying (A58) for i = 2i ← 3While i ≤ k do

Pick v′i satisfying the inequality given by

∏k+1j=i aj

∏kj=i bj

<aiv′i < bi−1v

′i−1 (A59)

i ← i+ 1end while.

Note that during the ith iteration, (A59) is feasible for v′i because by using the inequalities in (A58) for i and i− 1, it

follows that∏k+1

j=i aj∏k

j=i bj<aiv

′i1 bi−1v

′i−1 <

∏i−1j=1 bj

∏i−1j=2 aj

0

and once again this is feasible if and only if (A54) is satisfied.Now we verify that the v′

i > 0 obtained through this procedure satisfy (A57):(i) From (A59), it is clear that the second through 4k− 15th inequalities are satisfied.

(ii) The first inequality is satisfied because v′1 = 1 and we picked v2 satisfying (A58) for i = 2, from which we get

v′2 < b1/a2.

(iii) The last inequality is satisfied since v′1 = 1 and the vk that we picked satisfying (A59) for i = k also satisfies (A58) for

i = k, from which we get vk >a1/bk.From these v′

j , we obtain vj by dividing out �qj �. Note that it is always possible to choose v′j such that all vj are rational. If this

is done, then these vj can all be scaled by a single positive constant to make them integers while still satisfying (A57).We observe that our definition of the players’ action sets in (A55) and (A56) ensures that in any action profile, every

column j (consisting of resources uj and dj ) must have both players ij and ij+1. (There are four possible ways by which thiscan happen.) We call an action profile a= 4a11 : : : 1 ak5 a valid action profile if there exists some 1 ≤ j ≤ k (called a validindex) such that one of four validity conditions is true. Each validity condition involves a configuration consisting of twoadjacent columns. We now present the four validity conditions: (The valid configurations that are referenced in these conditionsare illustrated in Figure A.9.)

(i) qj < 0, qj+1 < 0, aj ∩ aj+1 6= �. Visually, this corresponds to V1 or V2.(ii) qj > 0, qj+1 > 0, aj ∩ aj+1 = �. Visually, this corresponds to V3 or V4.

(iii) qj < 0, qj+1 > 0, aj ∩ aj+1 6= �, aj+1 ∩ aj+2 6= �. Visually, this corresponds to V2.

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V1:

qj < 0

uj

uj uj

uj

dj

dj dj

dj

vj

vj

vj

vj

vj

vj

vj

vj

uj

djvj

vj

uj

dj vj

vj

uj

djvj

vj

uj

djvj

vj uj

djvj

vj

qj < 0

qj > 0

qj > 0

qj < 0

qj > 0

qj < 0

qj < 0

qj < 0

qj + 1 < 0

vj + 1

vj + 1

vj + 1

vj + 1

vj + 1 vj + 1dj + 1

dj + 1

uj + 1

uj + 1 vj + 1

vj + 1dj + 1

uj + 1 vj + 1

vj + 1dj + 1

uj + 1 vj + 1

vj + 1dj + 1

uj + 1

vj + 1

vj + 1dj + 1

uj + 1

vj + 1

vj + 1dj + 1

uj + 1

vj + 1

vj + 1dj + 1

uj + 1

dj + 1

uj + 1

qj + 1 > 0 qj + 1 > 0

qj + 1

qj + 1 qj + 1 qj + 1qj + 1

qj + 1 < 0

V2:

V3:

V4:

Figure A.9. Possible (anonymous) configurations for valid action profiles. Each circle denotes a player. Recall that each column j mustcontain two players, ij and ij+1. Hence, in every configuration, there are two circles per column. In every configuration, at the top of eachcolumn, the sign of its coefficient is indicated. If there is no sign indicated, then it could be either positive or negative.

(iv) qj > 0, qj+1 < 0, aj ∩ aj+1 = �, aj+1 ∩ aj+2 = �. Visually, this corresponds to V4.Figure A.10 is more detailed, and we enumerate all possible ways the black circles in Figure A.9 can correspond to players

of their respective columns. For each set of valid configurations, we also show the utility to player ij+1 for the action that he isshown taking (aj+1), as well as for the action he could otherwise have chosen (±aj+1).

We now show that a valid action profile cannot be an equilibrium. From (A57), we get

4∀1 ≤ j ≤ k5 vj �qj �f Tj 44ij+11 Tj5 > vj+1�qj+1�f Tj+14ij+11 Tj+150 (A60)

Using (A60), it can be seen that in any valid action profile with a valid index j , i.e., action profiles containing any of theconfigurations of V1 through V4, player ij+1 always has an incentive to deviate.

For example, consider any of the four configurations of V1, where qj < 0, qj+1 < 0. The difference in the utilities to playerij+1 between deviating to ±aj+1 and staying in aj+1 is given by

ãj+1 = Uj+14±aj+15−Uj+14aj+15

= −vj(

f 4ij+11 Tj5− f 4ij+11 Tj − 8ij95)+ vj+1

(

f 4ij+11 Tj+15− f 4ij+11 Tj+1 − 8ij+295)

0

Using the basis representation of f , this can be simplified as,

ãj+1 = −vj∑

T∈´ij ij+14Tj 5

qT fT 4ij+11 T 5+ vj+1

∑

T∈´ij+1 ij+24Tj+15

qT fT 4ij+11 T 50

But Tj and Tj+1 are minimal in ´+ij ij+1

and ´+ij+1ij+2

, respectively. Therefore, we get

ãj+1 = −vjqjfTj 4ij+11 Tj5+ vj+1qj+1f

Tj+14ij+11 Tj+15

= vj �qj �f Tj 4ij+11 Tj5− vj+1�qj+1�f Tj+14ij+11 Tj+151

which is strictly positive, from (A60). Hence, in configuration V1, player ij+1 has an incentive to deviate. Similar arguments canbe constructed for configurations of V2, V3, and V4.

The final step is to show that no invalid action profile exists in Counterexample 6. We do this by showing that any attempt toconstruct an invalid action profile by choosing actions from the players’ action sets must fail. Before presenting the formaldetails, we return to Example 15 to highlight the intuition behind our approach.

Example 16. In Example 15, we specified the action sets of the seven players involved in an 3�4 �2 sign profile,namely, 4−1−1−1+1−1+1+5. Here, we show that for this sign profile, every admissible action profile is valid; i.e., it satisfiesone of the four validity properties. We do this by showing that any attempt to construct an invalid action profile must fail:

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V1:

qj < 0

qj < 0 qj < 0 qj < 0 qj < 0

uj uj uj uj

dj djdj dj

vj

vj vj

vj

vj

vj vj

vj

qj < 0 qj < 0 qj < 0 qj + 1 < 0 qj + 1 < 0

vj + 1 vj + 1 vj + 1

vj + 1 vj + 1 vj + 1

vj + 1

dj + 1

uj + 1

dj + 1 dj + 1

uj + 1uj + 1 uj + 1ij + 1

ij + 2

vj + 1

vj + 1dj + 1

uj + 1

ij + 1 ij + 2 ij + 1 ij + 2

ij + 1 ij + 2ij + 1 ij + 2

vj + 1dj + 1ij + 1

ij + 2

vj + 1uj + 1

vj + 1dj + 1

ij + 2

ij + 2

ij + 2ij ij + 1

uj

dj

vj

vj

ij ij + 1

ij ij + 1

uj

dj vj

vj

ij ij + 1

ij + 1

vj + 1

vj + 1dj + 1

uj + 1

ij + 1

vj + 1

vj + 1

dj + 1

uj + 1

ij ij + 1

uj

dj

vj

vj

ij ij + 1

ij ij + 1

ujij

ij

ij

ij

ij

ij

ij

ij

uj uj uj

dj djdj dj

vj

vj vj

vj

vj

vj vj

vj

vj + 1 vj + 1 vj + 1

vj + 1 vj + 1 vj + 1

vj + 1

dj + 1

uj + 1

dj + 1 dj + 1

uj + 1uj + 1 uj + 1

vj + 1dj + 1ij + 1 ij + 1

ij + 1ij + 1ij + 1 ij + 1

ij + 1ij + 1ij + 1

ij + 1ij + 1

uj

dj vj

vj

ij ij + 1

vj + 1

vj + 1dj + 1

uj + 1ij + 1 ij + 2

ij + 2ij + 2

ij + 2

ij + 1 ij + 2 ij + 1 ij + 2

ij + 1 ij + 2 ij + 1 ij + 2

vj + 1uj + 1

vj + 1dj + 1

uj

dj

vj

vj

uj

djvj

vj vj + 1

vj + 1dj + 1

uj + 1

vj + 1

vj + 1

dj + 1

uj + 1uj

dj

vj

vj

uj

djvj

vj

qj + 1 < 0

qj + 1 qj + 1 qj + 1 qj + 1

qj > 0 qj > 0 qj > 0 qj > 0 qj + 1 qj + 1 qj + 1 qj + 1

qj + 1 < 0

qj > 0 qj > 0 qj > 0 qj > 0 qj + 1 > 0 qj + 1 > 0 qj + 1 > 0 qj + 1 > 0

V2:

V3:

V4:

Uj+1�aj+1�= vjqjf �ij+1� Tj �+ vj+1qj+1f �ij+1� Tj+1 − �ij+2��

Uj+1�aj+1�= vjqjf �ij+1� Tj − �ij ��+ vj+1qj+1f �ij+1� Tj+1�

Uj+1�aj+1�= vjqjf �ij+1� Tj �+ vj+1qj+1f �ij+1� Tj+1�

Uj+1�aj+1�= vjqjf �ij+1� Tj − �ij ��+ vj+1qj+1f �ij+1� Tj+1 − �ij+2��

Uj+1�aj+1�= vjqjf �ij+1� Tj − �ij ��+ vj+1qj+1f �ij+1� Tj+1�

Uj+1�aj+1�= vjqjf �ij+1� Tj �+ vj+1qj+1f �ij+1� Tj+1 − �ij+2��

Uj+1�aj+1�= vjqjf �ij+1� Tj − �ij ��+ vj+1qj+1f �ij+1� Tj+1 − �ij+2��

Uj+1�aj+1�= vjqjf �ij+1� Tj �+ vj+1qj+1f �ij+1� Tj+1�

�

�

�

�

Figure A.10. Possible configurations for valid action profiles. The action set of player ij+1 is given by ¡ij+1= 8aj+11 ±aj+19. The specific

resources that these two actions include will depend on whether the player is straight or diagonal, which is not important here. We do notexplicitly highlight the actions of all players other than players ij 1 ij+11 ij+2. In each configuration, action aj+1 corresponds to the action thatplayer ij+1 is shown taking. Uj+14aj+15 denotes the utility to player ij+1 in the configurations shown, and it only depends on whether it is V1,V2, V3, or V4. In every configuration, at the top of each column, the sign of its coefficient is indicated where relevant. If no sign is indicated, itcould be either positive or negative.

(i) First, consider the 3 segment. Recall that player i2 is diagonal and player i3 is straight. It can be seen that in anyinvalid action profile, there are only four possible ways in which this segment can be filled up—configurations M1, M2, or theirsymmetric counterparts, M ′

1, M ′2, as illustrated in Figure A.11(a). To see this, take M1, for example:

• If i2 switches, the first two columns form a valid configuration of V2.• If i3 switches, the second and third columns form a valid configuration of V2.• If both i2 and i3 switch, the first two columns form a valid configuration of V1.

(ii) Next, consider the �4 segment. Recall that player i4 is diagonal, and players i5 and i6 are straight. It can be seen that inany invalid action profile, there are only 10 possible ways in which this segment can be filled up—configurations Z1 throughZ4, or their symmetric counterparts, Z′

1 through Z′4, as illustrated in Figure A.11(b). Here, if any configuration other than these

10 occurs, there will be adjacent columns what would form valid configurations of either V2 or V4.(iii) Finally, consider the �2 segment. Recall that player i7 is diagonal. It can be seen that in any invalid action profile, there

are only four possible ways in which this segment can be filled up—configurations P1, P2, or their symmetric counterparts, P ′1,

P ′2, as illustrated in Figure A.11(c). Here, if any configuration other than these four occurs, the two columns would form valid

configurations of either V3 or V4.It follows that any invalid action profile must be constructed by picking one configuration from each of Figures A.11(a)–A.11(c)and “stringing” them together. Note that in doing so, two columns shaded with the same color must be identical to be strungtogether since they correspond to overlapping columns. Therefore, it can be seen that there are only four ways of gluing

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(a) Invalid configurations for an �3 segment

(b) Invalid configurations for a �4 segment

–

M1:i1

i2

i1

i2

i3

i2

i3

i2 i4

i4

i4

i3

i3

i3

i4i3

i7i6

i7i7i6

i6

i7

i6

i7

i6

i7

i7

i6

i6

i7i6

i7i6

i7i6

i7i6

i7

i7

i7

i6

i6

i6

i6

i6i6

i6

i4i4i3

i3

i4

i4

i4

i3

i4

i3

i3

i4

i3

i3i4i4

i4i3

i4 i5

i7 i1

i7 i1

i4 i5

i4 i5

i5 i6

i4 i5

i4 i5

i4 i5

i5 i6

i5 i6

i5 i6

i5 i6i5 i6i3 i4

i3 i4

i5

i1

i1i7

i7

i5

i5

i5

i5

i5

i4

i4

i5

i5

i2

i1

i2

i1

i3

i2

i3

i2M2:

Z1:

Z2:

Z3:

Z4:

P1:

P2:

– – – – –

– – ––

– –+ + – –+ +

– –+ +– –+ +

– –+ +

– –+ +

– –+ +

++ + +

++++

– –+ +

– –+ +

– –+ +

– –

(c) Invalid configurations for a �2 segment

P2:�

P1:�

Z4:�

Z3:�

Z2:�

Z1:�

M2:�

M1:�

Note. The verticle shade indicates that the third column of this �3 segment must match the first column ofthe succeeding �4 segment since they overlap.

Note. We have clubbed together two configurations as �3 (and similarly, �′3) since they have identical

boundaries (first and last columns)—only boundary compatibility matters when gluing together differentsegments. The vertical shade indicates that the first column of this �4 segment must match the third columnof the proceding �3 segment since they overlap.

Note. The horizontal shade indicates that the fourth column of this �4 segment must match the first columnof the succeeding � 2 segment since they overlap. The horizontal shade indicates that the first column ofthis � 2 segment must match the fourth column of the proceding �4 segment since they overlap.

Figure A.11. Possible configurations for 3, �4, �2 segments within an invalid action profile for Example 16.

together such configurations: M1Z4P1, M1Z4P2, and their symmetric counterparts, M ′1Z

′4P

′1, M ′

1Z′4P

′2. None of these four action

profiles are invalid:(i) M1Z4P2 and M ′

1Z′4P

′2 are illegal action profiles since player i1 is straight, whereas in these two action profiles, he

chooses a diagonal action.(ii) M1Z4P1 and M ′

1Z′4P

′1 are legal action profiles but are valid because the last column and the first column (when wrapped

around) form a valid configuration of V4.Hence, all action profiles are valid.

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We now build on the general intuition that was demonstrated in the example above to provide a complete proof. First, westate some necessary conditions that any invalid action profile must satisfy within a �l, l, and �l segment. The proofs are byinduction and involve arguing that in order to avoid any of the valid configurations V1 through V4, while still respecting how theaction sets are defined, such segments must satisfy these necessary conditions. The visual configurations that are referenced inthese conditions are illustrated in Figure A.12.

(i) Let columns j1 j + 11 : : : 1 j + l− 1 form an l segment. Then in any invalid action profile, aj ∩ aj+1 = �. In addition,• uj ∈ aj ⇒ dj+l−1 ∈ aj+l−1,• dj ∈ aj ⇒ uj+l−1 ∈ aj+l−1.

Visually, any l segment of an invalid action profile must match configurations M11M2 or their symmetric counterparts M ′11M

′2.

If not, there will be adjacent columns that would form valid configurations of either V1 or V2.The proof is by induction on l, the length of the segment. For the base case, when l = 2, the arguments are similar to those

for the 3 segment in Example 16. Our induction hypothesis is that every invalid l segment must match one of the fourconfigurations M11M21M

′11M

′2, for some l > 2. Assuming this is true, now consider an invalid l+1 segment. Keep in mind

that from the definition of action sets, player ij+l is required to be straight since he is at the end of this segment. Let l denotethe subsegment formed by its first l columns. From the induction hypothesis, l must match one of the four configurationsM11M21M

′11M

′2:

• l cannot match M2 or M ′2 because in either case, in l+1, columns l and l+ 1 together will form valid configurations

of either V1 or V2 (depending on how column l+ 1 is occupied by the players ij+l and ij+l+1).• If l matches M1 or M ′

1, then l+1 will match one of the four configurations M1, M2, M ′1, M ′

2 (depending on howcolumn l+ 1 is occupied by the players ij+l and ij+l+1).

(ii) Let columns j1 j + 11 : : : 1 j + l− 1 form a �l segment. Then in any invalid action profile, aj ∩ aj+1 6= �. In addition,• uj ∈ aj ⇒ dj+l−1 ∈ aj+l−1,• dj ∈ aj ⇒ uj+l−1 ∈ aj+l−1.

Visually, any �l segment of an invalid action profile must match configurations P11 P2 or their symmetric counterparts P ′11 P

′2.

If not, there will be adjacent columns that would form valid configurations of either V3 or V4. The proof is by a similarinductive argument as the l case above except that it is more complicated—we need to consider segments of odd and evenlengths separately because whether player ij+l−1 is straight or diagonal in a �l segment depends on whether l is even or odd.We omit the proof for brevity.

(iii) Let columns j1 j + 11 : : : 1 j + l− 1 form a �l segment. Then in any invalid action profile, one of the following threestatements must hold:

• aj ∩ aj+1 6= � and aj+l−1 ∩ aj+l = �. In addition,—uj ∈ aj ⇒ 4uj+l−1 ∈ aj+l−1 AND dj+l−1 ∈ aj+l5,—dj ∈ aj ⇒ 4dj+l−1 ∈ aj+l−1 AND uj+l−1 ∈ aj+l5.

Visually, this corresponds to configuration Z1 or its symmetric counterpart Z′1.

• aj ∩aj+1 = � and aj+l−1 ∩aj+l = �. Visually, this corresponds to configurations Z21Z3 or their symmetric counterpartsZ′

21Z′3.• aj ∩ aj+1 = � and aj+l−1 ∩ aj+l 6= �. In addition,

—uj ∈ aj ⇒ dj+l−1 ∈ aj+l−1 ∩ aj+l,—dj ∈ aj ⇒ uj+l−1 ∈ aj+l−1 ∩ aj+l.

Visually, this corresponds to configuration Z4 or its symmetric counterpart Z′4.

Note that if none of these conditions is satisfied, then there will be adjacent columns that would form valid configurations ofeither V2 or V4. Once again, the proof is by a similar inductive argument and is omitted for brevity.

It follows that any invalid action profile must somehow be constructed by “stringing” together different-length instantiationsof these 16 configurations. Figure A.12 illustrates how this is done. There are four possible configurations for an l segment,four possible configurations for a �l segment, and eight possible configurations for a �l segment. Note that by definition theseare maximal segments, so when stringing together two configurations, they cannot be of the same type of segment. We discussall possible ways of putting together an invalid action profile below:

(i) First, observe that an invalid action profile cannot be constructed using exactly one of these 16 configurations (j = 1,l = k in this case) because when wrapped around, at the boundary that is formed by the last column and the first column, eitherof the following two scenarios occur:

• The boundary player, i1, ends up making an illegal choice, rendering the whole action profile illegal.• The boundary configuration (the configuration formed by the last column and the first column) ends up being a valid

configuration.(ii) Now we need at least two configurations to be strung together to create an invalid action profile. We investigate

possible ways of stringing together two configurations. Observe that configurations Z1, Z2, Z3 and their symmetric counterpartsZ′

1, Z′2, Z′

3 cannot be hooked to any configurations of a �l segment (since the overlapping columns do not match) and cannotbe glued to any configurations of an l segment (since this results in a valid boundary configuration of V4). Thus these sixconfigurations cannot be used to construct an invalid action profile and can be eliminated.

(iii) Among the remaining 10 configurations, observe that P1 and its symmetric counterpart P ′1 cannot be glued to any

configurations of an l or any remaining configurations of a �l segment (since this results in a valid boundary configurationof V4). Also, M2 and its symmetric counterpart M ′

2 cannot be hooked to any remaining configurations of a �l segment (sincethe overlapping columns do not match) and cannot be glued to any configurations of a �l segment (since this results in a valid

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M1:ij

ijij + 1

ij + 1

ij + 1

ij + 1

ij + 1

ij + 1

ij + 1

ij

ij + 1

ij + 1

ij + 1

ij + l

ij + l

ij + l

ij + l

ij + l

ij + l

ij

ij + 1

ij + 1

ij + 1

ij + 1

ij + l – 1

ij + l – 1

ij ij + 1

ij ij + 1

ij ij + 1

ij ij + 1

ij ij + 1

ij ij + 1

ij + l – 1

ij + l – 1

ij + l – 1 ij + l

ij + l – 1 ij + l

ij + l – 1 ij + l

ij + l – 1

ij + l – 1

ij + l – 1

ij + l – 1

ij + l – 1

ij + l – 1

ij + l – 1 ij + l

ij + l – 1 ij + l

ij

ij

ij

ij

ij

ij

ij + l – 1 ij + l

M2:

Z1:

Z2:

Z3:

Z4:

P1:

P2:

– –

– –

– –

– –

+ +

+ +

+–

+–

+–

+– +–

+–

+–

+–

+ +

+ +

P2:�

P1:�

Z4:�

Z3:�

Z2:�

Z1:�

M2:�

M1:�

Figure A.12. Possible configurations for l, �l, and �l segments (spanning columns j through j + l− 1) within an invalid action profile.An action profile can be constructed by performing one of two operations repeatedly: (1) Two segments can be “hooked” together to form alarger segment if the last column of the first segment and the first column of the second segment are identical in both occupancy and shadingscheme. In this case, these two columns overlap to become one column. For example, M1 and Z2 can be hooked together. (2) Two segmentscan potentially be “glued” together to form a larger segment if the last column of the first segment and the first column of the second segmentdo not have matching shading schemes. In this case, these two columns would not overlap but sit next to each other. Such a gluing is permittedonly if the resulting action of the boundary player ij+l is legal (permissible according to the definition of the action sets). For example, Z2 andM1 can be glued together since it results in the boundary player ij+l being straight, which respects the definition of a straight player.

boundary configuration of V2). Thus these four configurations cannot be used to construct an invalid action profile and can beeliminated.

(iv) We are left with M1, P2, Z4, which can all be hooked or glued with each other, and their symmetric counterparts, M ′1,

P ′2, Z′

4, which can also all be hooked or glued with each other. Also, none of M1, P2, Z4 can be hooked or glued with any ofM ′

1, P ′2, Z′

4 and vice versa (since either overlapping columns, if any do not match or the boundary player ij+l ends up makingan illegal choice).

(v) Therefore, an invalid action profile must be constructed by using only M1, P2, Z4 or only M ′1, P ′

2, Z′4. Also, such invalid

action profiles must begin with a minus sign and end with a plus sign (because of our cyclic transformation at the beginning).There are now only three cases to be considered:

• Begin with M1 (respectively, M ′1) and end with Z4 (respectively, Z′

4).

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• Begin with M1 (respectively, M ′1) and end with P2 (respectively, P ′

2).• Begin with Z4 (respectively, Z′

4) and end with P2 (respectively, P ′2).

But in each of these cases, when wrapped around, the boundary player i1 ends up making an illegal choice, rendering the wholeaction profile illegal.Hence, there exist no invalid action profiles. This concludes the proof. �

As before, we now present a useful inference from Lemma 10 in terms of the weight systems:

Corollary 3. Given any set of local welfare functions �, let f� be budget-balanced distribution rules that guaranteeequilibrium existence in all games G ∈ §4N 1 f�1�5, where for each W ∈ �, f W 2=∑

T∈´W qWT f T

GWSV6�W1T 7, where

�W1T = 4ËW1T 1èW1T = 4SW1T1 1 SW1T

2 55. Let i11 i21 : : : 1 ik ∈ N be any k players (k ≥ 3) such that ∃T1 ∈ 4´1+i1i2

5min1 T2 ∈4´2+

i2i35min1 : : : 1 Tk ∈ 4´k+

ik i15min, for any k welfare functions W11W21 : : : 1Wk ∈�. Then

(a) 4∀1 ≤ j ≤ k5 8ij 1 ij+19⊆ Sj1 Tj1

(b)∏k

j=1 �j1 Tjij

/�j1 Tjij+1

= 1.

A.3.2. Existence of a universal weight system. In order to establish the global consistency of the sequence of weightsystems ì= 88�W1T 9T∈´W 9W∈�, we need to show that there exists a universal weight system �∗ = 4Ë∗1è∗5 that is equivalentto all the weight systems in ì; i.e., replacing �W1T with �∗ for any coalition T ∈´W for any W ∈� does not affect thedistribution rule f W1T = f T

GWSV6�W1T 7. We show this by explicitly constructing è∗ and Ë∗. Before doing so we use ì to define

two useful relations �ì and =ì on the set N , as follows. For any two elements i1 j ∈N ,

i �ì j ⇐⇒ 44∃W ∈�5 4∃T ∈´W+ij 5 s0t0 i ∈ SW1T

1 5 OR 4i = j51

i =ì j ⇐⇒ 4i �ì j5 AND 4j �ì i50(A61)

Using Corollary 2(a) and 2(b)(i), we can write down an equivalent set of definitions for these relations:

i �ì j ⇐⇒ 44∀W ∈�5 4∀T ∈´W+ij 5 i ∈ SW1T

1 5 OR 4i = j51

i =ì j ⇐⇒ 44∀W ∈�5 4∀T ∈´W+ij 5 8i1 j9⊆ SW1T

1 5 OR 4i = j50(A62)

We denote the transitive closures of these relations by �+ì and =+

ì, respectively.

Lemma 11. Given any set of local welfare functions �, if f� are budget-balanced distribution rules that guaranteeequilibrium existence in all games G ∈§4N 1 f�1�5, described completely by the sequence of weight systems ì, then �+

ì

constitutes a partial order on N .

Proof. By definition, �+ì is both reflexive and transitive. To prove that it is a partial order on N , we need only show

antisymmetry; i.e., we need to show that for any i1 j ∈N , if i �+ì j and j �+

ì i, then i =+ì j . This is equivalent to showing that if

there is a cycle in �ì, i.e., if there exists a sequence of k distinct players i11 : : : 1 ik ∈N such that i1 �ì i2 �ì · · · �ì ik �ì i1,then it must be that i1 =ì i2 =ì · · · =ì ik =ì i1. The case where k = 1 is trivial. For k = 2, the proof is vacuous by definitionof =ì. For k ≥ 3, suppose there is a cycle in �ì. Then using the definition of �ì from (A62), ∀1 ≤ j ≤ k ∀W ∈ �∀Tj ∈´W+

ij ij+1, ij ∈ S

W1Tj1 . Then from Corollary 3(a), ∀1 ≤ j ≤ k ∃W ∈� ∃Tj ∈´W+

ij ij+1s0t0 8ij 1 ij+19⊆ S

W1Tj1 . The conclusion

then follows by using the definitions from (A61). �

We now present the construction of a universal weight system, �∗ = 4Ë∗1è∗5.• Construction of è∗. From Lemma 11, the relation �+

ì constitutes a partial order on N . And the corresponding relation =+ì

is an equivalence relation. We let è∗ = 4S∗1 1 S

∗2 1 : : : 1 S

∗k 5 be an ordered partition of N into its equivalence classes according to

=+ì, ordered in any manner that does not violate �+

ì; i.e., for any 1 ≤ j < l ≤ k, any ij ∈ S∗j and il ∈ S∗

l , il �+ì ij .

• Construction of Ë∗. We construct Ë∗ = 4�∗i 5i∈N in a piecewise fashion as follows. For each equivalence class S∗

r ∈è∗,consider the following two cases:

(1) �S∗r � = 1. In this case, for i ∈ S∗

r , set �∗i to an arbitrary strictly positive number.

(2) �S∗r � = k > 1. Let S∗

r = 8i11 i21 : : : 1 ik9. S∗r is an equivalence class determined by the relation =+

ì, the transitiveclosure of =ì. Thus by definition it must be that for some permutation of its elements, without loss of generality the identitypermutation, i1 =ì i2 =ì · · · =ì ik. Using the definition of =ì from (A62), this means

4∀1 ≤ j < k5 4∀W ∈�5 4∀Tj ∈´W+ij ij+1

5 8ij 1 ij+19⊆ SW1Tj1 0

For each 1 ≤ j < k, pick any15 Wj ∈� and Tj ∈ 4´j+ij ij+1

5min. To begin with, set 4�∗i11 �∗

i25= 4�

11 T1i1

1 �11 T1i2

5. If k = 2, we are done.Otherwise, for 3 ≤ j ≤ k, recursively set

�∗ij

=�j−11 Tj−1ij

�j−11 Tj−1ij−1

�∗ij−1

0 (A63)

15 At least one such 4Wj 1 Tj5 pair is guaranteed by the definitions in (A61).

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Example 17. Let N = 8i1 j1 k1 l1m1n9 be the set of players, and let there be just one local welfare function W , ´W =8T W

1 = 8i1 j91 T W2 = 8j1 k1 l91 T W

3 = 8m1n91 T W4 = 8i1m1n99. Also, let f be a distribution rule that guarantees the existence of an

equilibrium in all games G ∈§4N 1 f 1W5, described by the sequence of weight systems ì= 8�W1Ti = 4ËW1Ti 1èW1Ti 594i=1, where

ËW1T1 =ËW1T3 = 411251 ËW1T2 =ËW1T4 = 41121351

èW1T1 = 88i1 j91 8 991 èW1T2 = 88j1 k91 8l991 èW1T3 = 88m1n91 8 991 èW1T4 = 88i91 8m1n990

Using the definitions in (A61) or (A62), it can be seen that the players are related as follows:

i =+ì j =+

ì k �+ì l and i �+

ì m=+ì n0

Using the construction above, it can be seen that for è∗, both 88i1 j1 k91 8m1n91 8l99 and 88i1 j1 k91 8l91 8m1n99 are admissibleorderings of the three equivalence classes of =+

ì (they do not violate �+ì). As for the weights, we get Ë∗ = 4112141 a11125,

where a can be any strictly positive number. Any strictly positive scaling of Ë∗ would also be admissible.

Before proceeding to show that �∗ as constructed above is equivalent to all the weight systems in ì, we prove an importantproperty of Ë∗ in a quick lemma:

Lemma 12. With Ë∗ as derived above, for any S∗r ∈è∗ with �S∗

r �> 1, for any two players i1 j ∈ S∗r , for any W ∈�, and

for any coalition T ∈´W+ij with 8i1 j9⊆ SW1T

1 ,

�W1Ti

�W1Tj

= �∗i

�∗j

0 (A64)

Proof. Let �S∗r � = k > 1. Equivalently, we show that for all m ∈ 81121 : : : 1 k − 19, for all l ∈ 81121 : : : 1 k − m9,

(A64) holds for players i = il and j = il+m. The base case, where m = 1 follows by construction (A63) and by usingCorollary 2(b)(ii). For m≥ 2, suppose there exists a welfare function W ∈� and a coalition T ∈´W+

ij , with 8i1 j9⊆ SW1T1 .

Recall the welfare functions and coalitions 4Wj 1 Tj5, 1 ≤ j < k, that were picked for constructing Ë∗. From Corollary 2(b),it is sufficient to prove this lemma for T ∈ 4´W+

ij 5min. Now, using the definitions in (A61), j =ì i. Therefore, it followsthat the players i = il1 il+11 : : : 1 il+m = j form a cycle in =ì, i.e., il =ì il+1 =ì · · · =ì il+m =ì il. This means thatil ∈ S

l1 Tl1 1 il+1 ∈ S

l+11 Tl+11 1 : : : 1 il+m−1 ∈ S

l+m−11 Tl+m−11 1 il+m ∈ SW1T

1 . Applying Corollary 3(b) and (A63), we have

(l+m−1∏

j=l

�j1 Tjij

�j1 Tjij+1

)

�W1Til+m

�W1Til

= 1 =⇒ �W1Ti

�W1Tj

=l+m−1∏

j=l

�j1 Tjij

�j1 Tjij+1

=⇒ �W1Ti

�W1Tj

=l+m−1∏

j=l

�∗ij

�∗ij+1

=⇒ �W1Ti

�W1Tj

= �∗i

�∗j

0

This concludes the proof. �

Now we present the final lemma that establishes the global consistency of ì.

Lemma 13. Given any set of local welfare functions �, if f� are budget-balanced distribution rules that guaranteeequilibrium existence in all games G ∈§4N 1 f�1�5, where for each W ∈�, f W 2=∑T∈´W qW

T f W1TGWSV6�

W1T 7, then there existsa weight system �∗, such that

4∀W ∈�5 4∀T ∈´W 5 f W1TGWSV6�

W1T 7= f TGWSV6�

∗7 (A65)

Proof. We prove that �∗ as constructed above satisfies (A65). Consider any welfare function W ∈� and any coalitionT ∈´W . Let k = min8r � S∗

r ∩ T 6= �9. Then we need only show the following:(i) SW1T

1 ⊆ S∗k ,

(ii) SW1T2 ∩ S∗

k = �1

(iii) 4∀ i1 j ∈ SW1T1 5 �W1T

i /�W1Tj = �∗

i /�∗j .

Of these, the first two are immediate from the construction of è∗, and the third follows from Lemma 12. This completesthe proof. �

Appendix B. Proof of Proposition 2. First, note that we only need to prove one direction, since from (8) and (9) inTable 2, it follows that

q′T =

(

∑

j∈±T�j

)

q′′T ⇐⇒ q′

T fTGWSV6�7= q′′

T fTGWMC6�70 (B1)

To prove the other direction, it suffices to show that


GWMC6�7 =⇒ 4∀T ⊆N5 q′T =

(

∑

j∈±T�j

)

q′′T 1

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with the understanding that qT = 0 whenever T y´, for ´=´′, ´′′. The proof is by contradiction. Suppose f W ′GWSV6�7=

f W ′′GWMC6�7, and let T be a smallest subset for which q′

T 6= 4∑

j∈±T �j5q′′T , and i ∈ ±T . Then we have


GWMC6�7 =⇒ f W ′GWSV6�74i1 T 5= f W ′′

GWMC6�74i1 T 5

=⇒ ∑

S⊆T

q′Sf

SGWSV6�74i1 S5= ∑

S⊆T

q′′S f

SGWMC6�74i1 S5

=⇒ q′T f

TGWSV6�74i1 T 5+ ∑

S(Tq′Sf

SGWSV6�74i1 S5= q′′

T fTGWMC6�74i1 T 5+ ∑

S(Tq′′S f

SGWMC6�74i1 S5

=⇒ q′T =

(

∑

j∈±T�j

)

q′′T (from (B1))1

which contradicts our assumption. This completes the proof. �

Appendix C. Generalized weighted potential games. In Hart and Mas-Colell [18], the authors show that weightedShapley values result in weighted potential games by explicitly constructing a potential function (in a recursive form, almostidentical to the one in (11)). The authors claim that their result extends to generalized weighted Shapley values but do notprovide a proof. In this section, we fill this gap by showing that generalized weighted Shapley values result in a slight variantof weighted potential games, which we call generalized weighted potential games, defined as follows:

Definition 1. A finite game G= 4N 1 8¡i9i∈N 1 8Ui9i∈N 5 is a generalized weighted potential game if there exists a potentialfunction ê2 ¡→�m (where m is some positive integer) and a positive weight wi > 0 for each agent such that for every agenti ∈N , for every a−i ∈¡−i, and for every a′

i, a′′i ∈¡i,

Ui4a′i1 a−i5−Ui4a

′′i 1 a−i5=wi4êk4i54a

′i1 a−i5−êk4i54a

′′i 1 a−i551 (C1)

where k4i5 denotes the index of the first nonzero term of ê4a′i1 a−i5−ê4a′′

i 1 a−i5.

Generalized weighted potential games are a special subclass of the weaker, more general class of ordinal potential games,where the difference in the potential function (right-hand side of (C1)) is merely required to be of the same sign as thedifference in the utility function (left hand side of (C1)). Note that weighted potential games are simply generalized weightedpotential games with a one-dimensional potential function (m= 1).

Definition 1 applies to any finite noncooperative game in normal form. However, recall from §2 that in our model the agentutility functions are separable, given by

Ui4a5= ∑

r∈aif r4i1 8a9r50 (C2)

Hence, in searching for a potential function for G, it is natural to seek a separable potential function ê2 ¡→�m (where m issome positive integer), given by

ê4a5=∑

r∈RÔr48a9r51 (C3)

where Ôr 2 2N →�m is the “local” potential function at resource r . Therefore, to show that ê is a potential function for G, it issufficient to show that for every agent i ∈N , there exists a positive weight wi > 0 such that for every resource r ∈R, for everyplayer subset S ⊆N containing i,

f r4i1 S5=wi44�r5k4i54S5− 4�r5k4i54S − 8i9551 (C4)

where k4i5 denotes the index of the first nonzero term of Ôr4S5−Ôr4S − 8i95. Verifying this is quite straightforward; use(C2)–(C4) to check that (C1) is satisfied.

We now state our formal result. Recall that a weight system �= 4Ë1è5 consists of a strictly positive vector of playerweights Ë ∈�N

++ and an ordered partition è= 4S11 S21 : : : 1 Sm5 of the set of players N .

Theorem 3. For any welfare sharing game G= 4N 1R1 8¡i9i∈N 1 8f r9r∈R1 8Wr9r∈R5 and any weight system �, if for everyr ∈R, the distribution rule f r = f

W ′r

GWSV6�7= fW ′′r

GWMC6�7 with W ′r , W ′′

r being any two ground welfare functions related accordingto (12), then G is a generalized weighted potential game, with player weights Ë and the local potential function at resource r ,Ôr 6�74S5= 44�r 6�7514S51 4�r 6�7524S51 : : : 1 4�r 6�75m4S55 where for all 1 ≤ k ≤m, 4�r 6�75k4S5 is given in terms of W ′

r in thefollowing recursive form:

4�r 6�75k4S5= 1∑

i∈S �i

(

W ′r 4±Sm−k+15+∑

i∈S�i4�r 6�75k4S − 8i95

)

1 (C5)

and in terms of W ′′r in the following closed form:

4�r 6�75k4S5=W ′′r 4±Sm−k+151 (C6)

where ±Sk = S −⋃k−1l=1 Sl.

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Proof. First, we use the closed form expression in (C6) to show that Ôr 6�7 satisfies (C4). This involves proving, for any1 ≤ k ≤m, for any subset S containing a player i ∈ Sk, the following two steps:

(a) k4i5=m− k+ 1; i.e., W ′′r 4±Sl5−W ′′

r 44S − 8i95l5= 0 for all k+ 1 ≤ l ≤m;(b) f

W ′′r

GWMC6�74i1 S5= �i4W′′r 4±Sk5−W ′′

r 44S − 8i95k55.Observe that 4S − 8i95l = ±Sl − 8i9 for all 1 ≤ l ≤m. Part (a) is straightforward since if i ∈ Sk, then by definition, i y ±Sl for allk+ 1 ≤ l≤m. We now focus on part (b), which is exactly the definition of f W ′′

rGWMC6�7 in Table 1. Hence, the following is

simply an exercise in verifying the equivalence of the two definitions of f W ′′r

GWMC6�7 from Tables 1 and 2, using the basisrepresentation discussed in §3.2. Evaluating the left-hand side, we get

�i4W′′r 4±Sk5−W ′′

r 44S − 8i95k55= �i

(

∑

T∈´′′4±Sk5q′′T − ∑

T∈´′′4±Sk−8i95

q′′T

)

1

where for any player subset S ⊆N , ´′′4S5 denotes the set of all coalitions T ∈´′′ that are contained in S (T ⊆ S). Notice that±Sk does not contain any players in S1 ∪ S2 ∪ · · ·Sk−1. Therefore, ´′′4±Sk5 consists of those coalitions contained in S that do notcontain any player in S1 ∪ S2 ∪ · · ·Sk−1. Similarly, ´′′4±Sk − 8i95 consists of those coalitions contained in S − 8i9 that do notcontain any player in S1 ∪ S2 ∪ · · ·Sk−1. Therefore, the collection ´′′4±Sk5−´′′4±Sk − 8i95 consists precisely of those coalitionsT ∈´′′4S5 that do not contain any player in S1 ∪ S2 ∪ · · ·Sk−1 but contain player i. Since i ∈ Sk, this is the same as saying thatthe collection ´′′4±Sk5−´′′4±Sk − 8i95 contains precisely those coalitions T ∈´′′4S5 for which i ∈ ±T . Thus we get

�i4W′′r 4±Sk5−W ′′

r 44S − 8i95k55 = �i

(

∑

T∈´′′4±Sk5q′′T − ∑

T∈´′′4±Sk−8i95

q′′T

)

= ∑

T∈´′′4S52 i∈±Tq′′T �i

= ∑

T∈´′′q′′T f

TGWMC6�74i1 S5 4from (9)5

= fW ′′r

GWMC6�74i1 S50

To complete the proof, observe that when W ′r and W ′′

r are related according to (12), then for all 1 ≤ k ≤m, and all S ⊆N , theexpression for 4�r 6�75k4S5 (C6) satisfies the recursion (C5). �

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