values of games with probabilistic graphs

Mathematical Social Sciences 37 (1999) 79–95

Values of games with probabilistic graphsa , b c*Emilio Calvo , Javier Lasaga , Anne van den Nouweland

a ` `Departament d’Analisi Economica, Universitat de Valencia, Av. dels Tarongers, s /n, EdificioDepartamental Oriental, 46022 Valencia, Spain

b ´Departamento de Economıa Aplicada I, Avda. Lehendakari Agirre, 83, 48015 Bilbao, SpaincDepartment of Economics, 435 PLC, 1285 University of Oregon, Eugene, OR 97403-1285, USA

Received 10 June 1997; received in revised form 10 December 1997; accepted 6 January 1998

Abstract

In this paper we consider games with probabilistic graphs. The model we develop is anextension of the model of games with communication restrictions by Myerson (1977). In theMyerson model each pair of players is joined by a link in the graph if and only if these two playerscan communicate directly. The current paper considers a more general setting in which each pairof players has some probability of direct communication. The value is defined and characterized inthis context. It is a natural extension of the Myerson value and it turns out to be the Shapley valueof a modified game. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Probabilistic graphs; Communication restrictions; Direct communication; Myersonvalue; Shapley value

1. Introduction

The way cooperative games with side payments in characteristic function form aredefined is to assign a real number to each coalition of agents. This number may beinterpreted as the economic possibilities presented to the particular coalition at hand,regardless of the actions of the other agents in the group. It is generally assumed thatthere are no restrictions on communication between players and, hence, every subgroupof players can effectively cooperate. But there are many economic and political contextsin which restrictions in communication arise, and then full cooperation fails.

One of these possible settings appears when direct communication is not necessarilytransitive. A simple example of this kind is found in the administrative machinery of auniversity, where one has to know the proper channels in order to get things done. To

*Corresponding author. Tel.: 1349-63-82-82-46; fax: 1349-63-82-82-49; e-mail: [email protected]

0165-4896/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved.PI I : S0165-4896( 98 )00013-4

80 E. Calvo et al. / Mathematical Social Sciences 37 (1999) 79 –95

capture nontransitive communications structures, Myerson (1977) introduced graphs tomodel communication channels between players. In such a communication graph theplayers are the nodes and the presence of a link between two players indicates that thesetwo players can communicate directly. The graph induces a partition on the set ofplayers into connected components which are the coalitions of players that can negotiateeffectively in the game. For an overview of the line of research on games withcommunications restrictions readers are referred to Borm et al. (1994) and van denNouweland (1993).

In this paper we will extend the model of Myerson (1977). In general it will not be thecase that two players either are able to communicate and hence cooperate or are not ableto do so, but rather that they will be able to cooperate up to a certain degree. Forexample, in many markets we observe the existence of intermediaries who try to bringbuyers and sellers in contact with one another. These intermediaries play an importantrole in clearing the market, and they are paid for their intermediation. We could modelthis situation assuming that there are communication links only between buyers andintermediaries, and between intermediaries and sellers. But a more accurate descriptionof this situation should allow us to incorporate into the model the possibility that sellersand buyers reach agreements directly, without intermediation. The probability that directtransactions will take place depends on several factors, one of the most significant beingthe cost of gaining information about different offers and demands in the market.Introducing these probabilities into the communication graph enables us to find the valueof the players in the market as a function of the uncertainties involved in the bilateralrelationships.

Another typical example appears in voting games, where parties in a parliament areconsidered. In order to get a measure of the power of different agents in majority votingsituations, these situations can be modeled as cooperative (voting) games and then somesolution concept can be used to assign an index of power to each party in the votingsituation. Well known power indices are the Shapley–Shubik index (cf. Shapley andShubik, 1954) and the Banzhaf index (cf. Banzhaf, 1965). Both aforementioned papersshow that simply counting the number of votes of each party does not provide a reliableindication of the power of the parties in general. In order to get a good indication of thepower of each party it is important to consider so called winning coalitions, i.e.,coalitions of parties that hold a majority of the votes. Then, instead of merely countingvotes, the power of a party is the probability that it has of being pivotal to the success ofa wining coalition. This approach, however, fails to take into account sociological,political, and ideological aspects that influence the degree of compatibility of theobjectives of different parties. These aspects may obviously influence the power of aparty. A party that has sociological, political, or ideological objectives that are oppositeto the objectives of most other parties may often vote differently from most parties. Thiswill diminish the chances this party has of being critical to the success of a winningcoalition. We can associate with each pair of parties a probability that reflects the a prioridegree of compatibility of both parties. An extension of the Shapley–Shubik index tothis type of model is then defined, and this generalized index provides an a prioriindication of the power of political parties when these kinds of sociological, political,and ideological aspects are taken into consideration.

The structure of the paper is as follows. Following this introduction, Section 2 is

E. Calvo et al. / Mathematical Social Sciences 37 (1999) 79 –95 81

devoted to the introduction of Myerson’s (1977) model of games with deterministicgraphs. We will discuss this model by means of two typical examples in which we showhow to introduce bilateral probabilities of communication into the model. In Section 3we formally define probabilistic graphs, we show how characteristic functions aremodified by such graphs, and we define a generalization of the value for this model. InSection 4 we offer an axiomatic characterization of this value and Section 5 containsconcluding remarks.

2. Deterministic graphs

Let N 5 h1,2, . . . ,nj be a set of players and let L(N) 5 hhi, jjui [ N, j [ N, i ± jj be theset of unordered pairs of distinct players of N. We will refer to these unordered pairs ascommunication links. A communication graph on N is any subset of links L of L(N). LetL be the set of all communication graphs on N. For any such graph L, any coalition S #N

N, and any players i and j in S, we say that i and j are connected by L within S if andonly if there exists some chain of communication links within S such that i and j areconnected through the chain, i.e. if i 5 j or there exist hh , h , . . . ,h j # S such that1 2 k

h 5 i, h 5 j and hh , h j [ L for every t 5 1, . . . ,k 2 1. Let S / denote the unique1 k t t11 L

partition of S into the groups of players which are connected by L within S. S / can beL

interpreted as the collection of smaller coalitions into which S would break up, if playerscould only coordinate along the links in L. We say that S is internally connected by L ifand only if S / 5 hSj. Given any communication graph L, we can think of the internallyL

connected coalitions as the set of coalitions that can negotiate effectively in a game.A coalitional game is a pair (N, v), where N is the set of players and v is a

Ncharacteristic function v:2 →R, with v([)50, where each number v(S) is interpretedas the wealth of transferable utility which the members of S would have to divide amongthemselves if they were to cooperate together and with no one outside S. Let G be theN

set of all coalitional games with player set N.The introduction of communication restrictions (modeled by means of a graph)

changes the payoffs attainable by coalitions. Given any communication graph L, and acoalitional game (N, v), define v to be the characteristic function that would result if weL

altered the situation represented by v, requiring that players can only communicate alonglinks in L, so that (cf. Myerson, 1977)

;S # N, v (S): 5 O v(T ).LT [S /L

A game with a communication graph is a triple (N, v, L), where (N, v)[G andN1L[L . An allocation rule for games with communication graphs is a functionN

Nc :G 3L →R , where c (N, v, L) should be interpreted as the utility payoff whichN N i

player i would expect in game (N, v) if L represents the restrictions on cooperativeagreements between the players. Allocation rules can be found by applying solution

1We will slightly abuse notation and write (N, v, L)[G 3L .N N


concepts such as the Shapley value (cf. Shapley, 1953), the nucleolus (cf. Schmeidler,1969), and others to the game (N, v ) associated with a game with a communicationL

2graph (N, v, L). In this paper we will restrict ourselves to the Shapley value (6*) and,following Aumann and Myerson (1988), we will refer to the allocation rule that weobtain in this way as the Myerson value.

Definition 1. Let (N, v, L) be a game with a communication graph. Then the MyersonNvalue of (N, v, L), }(N, v, L)[R , is defined by

}(N, v, L): 5 6*(N, v ).L

Next, we show by means of two examples a drawback of this deterministic approachand a way to overcome it.

2.1. Example 1

Consider an agent who wants to sell his house and another agent who would beinterested in buying the house. It is quite usual for agents who want to buy or sell ahouse to go to a broker, because a broker is someone who has a lot of informationconcerning possible buyers and sellers of houses. So, the broker can act as anintermediary between the buyer and the seller. However, if the broker acts as anintermediary, then he or she has to be paid for this intermediation.

We consider a small example. There are three agents, a seller (s), a buyer (b), and abroker (i), so N5hs, b, ij. If the house of the seller can be sold to the buyer, then asurplus of say 1 unit is created. The characteristic function is v(S)51 if and only if S isequal to hs, bj or hs, b, ij, and v(S)50 otherwise. Suppose now that both the seller andthe buyer know the broker, but they cannot interact directly. Hence the communicationgraph in this case is L5hhs, ij, hb, ijj and the modified characteristic function isv (S)50, for all S#hs, b, ij, and v (hs, b, ij)51. Computing the Myerson value we findL L

that }(N, v, L) assigns 1 /3 to each of the three players.If we assume that the seller and the buyer can also communicate directly, without the

intermediation of the broker, then the graph coincides with the complete graph L(hs, b,ij) and the modified characteristic function v coincides with the original characteris-L(N )

tic function v. Hence }(N, v, L(N))56*(N, v), which in our case gives 6* (N,s

v)56* (N, v)51/2 and 6* (N, v)50, where 6* (N, v) denotes the payoff to playerb i k

k[N.Both ways of modelling seem not to capture the situation realistically. To be more

realistic the model must take into account the fact that the buyer and the seller may notknow of each other but may try to find each other without the intermediation of thebroker to avoid the brokerage fee. Then there is some probability, possibly very small,that they can make the transaction without the intermediation of the broker. In order tointroduce this fact into the model, suppose that the probability that the seller and the

2We refer the reader to Shapley (1953) for a definition of the Shapley value.


buyer can make a transaction without the intermediation of the broker is some p[[0, 1].Now the expected worth of the coalition of the seller and the buyer is the probability thatthey can make the transaction multiplied by the surplus generated if the transaction ismade, so the expected value is p. The probability of communication between either theseller and the broker or the buyer and the broker is taken to be equal to 1, as before.Denote by v the characteristic function that takes into account these probabilities ofp

communication. Then v (hs, bj)5p and v (S)5v(S) otherwise. If we compute thep p

Shapley value for v , then we find that the broker gets a payoff of 6* (N, v )5(12p) /p i p

3 and that the seller and the buyer each get 6* (N, v )56* (N, v )5(21p) /6. Hence,s p b p

we see that the payoff of the broker decreases when the probability of direct cooperationbetween the seller and the buyer increases.

Notice that in this example it holds that v is a convex combination of the originalp

characteristic function v, which is the characteristic function we find with the completegraph, and the characteristic function v that corresponds to the graph with linksL

between the buyer and the broker and between the seller and the broker only. In formula:v 5pv1(12p)v , which corresponds to the fact that p is the probability that thep L

complete graph L(hs, b, ij) is actually formed and (12p) is the probability that L5hhs,ij, hb, ijj is formed. Therefore, using linearity of the Shapley value, we have

6*(N, v ) 5 p6*(N, v) 1 (1 2 p)6*(N, v ).p L

2.2. Example 2

Voting situations in a parliament can be modeled by a weighted majority game, that is,assume there are n parties, where party i has s seats, and a coalition of parties needsi

more than half the number of votes to win a ballot. This is a cooperative game (N, w)with player set N5h1,2, . . . ,nj and characteristic function w defined by w(S)51 if

1]o s . o s , and w(S)50 otherwise, for all S#N. A coalition of parties S is calledi[S i i[N i2

winning if w(S)51 and losing if w(S)50. Coalition S is said to be minimal winning ifw(S)51 and w(T )50 for all strict subcoalitions T of S. We denote by MW the set of allminimal winning coalitions of (N, w). Note that the characteristic function of a weightedmajority game is completely determined by the set MW.

Measures of power in such games count how often a member of a winning coalition ispivotal, i.e. how often this member’s defection from such a coalition would cause it tobe losing. We consider here the Shapley–Shubik index. Suppose that a bill is to bedecided upon by an assembly, and that the bill under consideration aligns the votingparties in order of their enthusiasm for the proposal. Given any such alignment, therewill be a single pivotal voting party, i.e. one who, by joining the more enthusiasticparties, brings the coalition up to winning strength. Assuming a priori that all n!orderings of the voting parties are equally likely, the Shapley–Shubik index for eachparty is precisely the probability that it is pivotal. Mathematically, the Shapley–Shubikindex is the Shapley value of the weighted majority game.

This approximation takes into account only the number of seats that each party has,but fails to recognize that certain parties are ideologically closer together than others.This fact influences the power of each party, for those who are ideologically closer to


each other are more likely to cooperate than parties which are extreme and isolated intheir aims.

A way to include compatibilities of parties in the model is to define a graph for the setof parties in which each pair of parties i and j is joined by a link if and only if these twoparties are ideologically compatible. Parties can only cooperate if they are connected onthe graph directly by a link, or indirectly through other parties. Now we have a triple (N,w, L), where (N, w) is a weighted majority game and L is a communication graph on N.The introduction of L modifies the voting game, and we denote by MW the set of allL

minimal winning coalitions for the game w . The modified power index is then theL

Shapley value of (N, w ).L

As an illustration, consider a fictitious parliament, in which there are four parties;N5he, l, c, rj, where e denotes the ‘‘extreme left’’ party, l the ‘‘left’’ party, c the‘‘center’’ party, and r the ‘‘right’’ party. Suppose that the set of minimal winningcoalitions, which is determined by the number of seats each party holds, is MW5hhe, lj,he, rj, hl, rjj. In this case c is a null party and the remaining three parties are symmetricin the voting game. Hence, the power index corresponding to this situation is

1]6* (N, w) 5 , for i 5 e, l, r, and 6* (N, w) 5 0.i c3

Suppose now that we place the parties in the socioeconomic left–right dimension andconsider the communication graph L 5hhe, lj, hl, cj, hc, rjj. Here, in L , parties can1 1

communicate directly only if they are placed face to face in the left–right line.Therefore, as an example, l and r can only reach some kind of agreement if c is includedin this agreement: this can be interpreted by the fact that both l and r, given theirideological differences, could only justify an agreement before their respective voters, ifit were also accepted by an intermediate party, c, in the ideological spectrum.

In the game modified according to L , (N,w ), the set of minimal winning coalitions1 L1

is MW 5hhe, lj, hl, c, rjj. Note that here c becomes pivotal for some orders. TheL1

modified power index is

1]} (N, w, L ) 5 } (N, w, L ) 5 ,c 1 r 1 12

3 7] ]} (N, w, L ) 5 , and } (N, w, L ) 5 .e 1 l 112 12

Alternatively, we can consider the graph L 5hhe, lj, hl, cj, hl, rj, hc, rjj, which is less2

extreme than L , because it only excludes direct communication between extreme left1

party e and both parties c and r. Now the set of minimal winning coalitions isMW 5hhe, lj, hl, rjj and the modified power index isL2

1]} (N, w, L ) 5 } (N, w, L ) 5 ,e 2 r 2 6

4]} (N, w, L ) 5 , and } (N, w, L ) 5 0.l 1 c 16


Graphs L and L differ only in link hl, rj. Its absence yields party c a positive power1 2

given its intermediation role between parties l and r.However, in a parliament two parties are not generally wholly compatible or

incompatible, but they are usually compatible to a certain degree. Therefore, it is moreappealing to associate with each pair of parties i and j a number p between 0 and 1 thatij

reflects their degree of compatibility. This number p should be interpreted as theij

probability that the two parties will agree on a particular issue. For example, supposethat link hl, rj has a probability of p[[0, 1]. Then, graphs L and L happen with1 2

probabilities (12p) and p respectively. Defining w 5(12p)w 1pw , we obtain asp L L1 2

power indices for the new situation

3 2 p 7 1 p]] ]]6* (N, w ) 5 , 6* (N, w ) 5 ,e p l p12 12

1 2 p 1 1 p]] ]]6* (N, w ) 5 , and 6* (N, w ) 5 .c p r p12 12

Here, the power of party c is obtained now as a function of the degree ofincompatibility between parties l and r. As these parties become more compatible, thepower of party c declines.

3. Probabilistic graphs

In this section we will extend the model of Myerson (1977) in order to be able towork with probabilistic graphs. A game with a probabilistic graph is a triple (N, v, p),where (N, v) is a coalitional game and p:hhi, jjui, j[N, i±jj→[0, 1] is a function thatassigns to each pair of agents i and j the probability that these two agents cancommunicate directly. The probabilities are assumed to be independent. Sometimes wewill refer to the function p as a system of probabilities. Further, we will often denote pij

instead of p(hi, jj).Let (N, v, p) be a game with a probabilistic graph. With this game we will associate a

new coalitional game (N, v ), called the communication game, that incorporates both thep

economic possibilities of the agents described by the coalitional game (N, v) and theprobabilities of bilateral communication described by the system of probabilities p. Sincewe are dealing with probabilities of communication, we will consider expected profits inthe new game.

Let i, j[N, i±j. Then, with probability p agents i and j are able to communicate. Ifij

this is so, then they can cooperate and obtain v(hi, jj). But with probability 12p theij

agents cannot communicate and in this case they cannot obtain more than v(hij)1v(h jj).Therefore, the expected profit of agents i and j is

v ( i, j ): 5 p v( i, j ) 1 (1 2 p )(v( i ) 1 v( j )).h j h j h j h jp ij ij

Generalizing the idea that is at the basis of this definition, we can define the expectedprofit of arbitrary coalitions of agents. Let S#N be a fixed coalition of agents and defineL(S):5hhi, jjui, j[S, i±jj, the set of all possible communication links between agents in


S. We will often denote typical links in L(S) by l. For each set of links L#L(S), theprobability that L is the communication graph that is realized among the agents in S is:

Sp (L): 5P p P (1 2 p ).l ll[L l[L(S )L

Now, suppose L#L(S) is the set of communication links that is realized. Note that thegraph L induces a partition of S into communication components S / . Correspondingly,L

the worth obtainable by coalition S if L#L(S) is realized is v (S). Now, we can defineL

the expected profit of coalition S, namely

Sv (S): 5 O p (L)v (S).p LL#L(S )

The procedure described above is a generalization of the procedure followed by3Myerson (1977). To see this, note that a deterministic communication graph L on N can

be identified with a function p:hhi, jjui, j[N, i±j→[0, 1], defined by p(hi, jj)51 if hi,jj[L and p(hi, jj)50 if hi, jj[⁄ L. It is easily seen that for this p it holds that v 5v .p L

We want to define allocation rules for games with probabilistic graphs, i.e., rules thatassociate a vector of payoffs with each game with a probabilistic graph. Formally,denoting by G the space of coalitional games with player set N and by P the set ofN N

systems of probabilities for player set N, an allocation rule is a function c :G 3NNP →R . In this paper we will extend the Shapley value to games with probabilisticN

graphs and we will refer to the allocation rule that we obtain in this way as the Myersonvalue.

Definition 2. Let (N, v, p) be a game with a probabilistic graph. Then the Myerson valueNof (N, v, p), }(N, v, p)[R , is defined by

}(N, v, p): 5 6*(N, v ).p

4. An axiomatic characterization

We show in this section that the Myerson value can be axiomatically characterizedusing two of its properties, component efficiency and fairness.

Consider a coalition that has probability 0 of communicating with any player outsidethe coalition and that is minimal with respect to this property. Component efficiency isan axiom that states that the total payoff to the players in such a coalition should beequal to the expected profit of the coalition. In order to introduce component efficiencyformally, we need some notations. Let (N, v, p) be a game with a probabilistic graph.With this game we associate a deterministic graph (N, L ) defined as follows: l5hi,p

jj[L if and only if p .0. The graph (N, L ) induces a partition of N intop ij p

3Although the formal expression of v bears some similarity with the formula of Owen’s (1972) multilinearp

extension, we point out that the game v is not the linear extension of the game v.p


communication components. We will refer to this partition as N /p. Now, we are ready tointroduce the property component efficiency formally.

NDefinition 3. An allocation rule c :G 3 P →R is component efficient if for all gamesN N

with a probabilistic graph (N, v, p) and all communication components C[N /p it holdsthat

O c (N, v, p) 5 v (C).i pi[C

In the context of weighted majority games, where it can never be the case that thereare two disjoint components of parties that are both winning, component efficiencysimply states that the total power is divided among the parties that are in the componentconsisting of a winning coalition of parties.

The fairness axiom states that when the possibility for direct communication betweentwo players is destroyed, other things being equal, then the payoffs of both these playerschange by the same amount, so either they both lose the same amount or they both gainthe same amount.

NDefinition 4. An allocation rule c :G 3P →R is fair if for all games with aN N

probabilistic graph (N, v, p) and all i, j[N it holds that

c (N, v, p) 2 c (N, v, p ) 5 c (N, v, p) 2 c (N, v, p ),i i 2ij j j 2ij

4where p (hk, lj)5p(hk, lj) if hk, lj±hi, jj, and p (hi, jj)50 .2 ij 2 ij

In the context of weighted majority games the fairness axiom states that when a partydecides to try and oppose another party to diminish its power, then this will cause anequal loss in power for this party and its opponent.

NTheorem 1. The Myerson value is the only allocation rule c :G 3P →R satisfyingN N

component efficiency and fairness.

Proof. To prove that the Myerson value satisfies component efficiency, let (N, v, p) be agame with a probabilistic graph and let C[N /p. We split up (N, v ) into two games, (N,p

C N•C C N•Cv ) and (N, v ), where for all S#N v (S):5v (S>C) and v (S):5v (S•C).p pC N•CSince C is a component of (N, L ) we know that v 5v 1v . It follows from thep p

5 Cdummy property of the Shapley value that 6* (N, v )50 for all i[C and that 6* (N,i iCv )50 for all i[N•C. Hence,

4An equivalent definition of the fairness property is obtained when replacing ‘p (hi, jj)50’ by ‘p (hi,2 ij 2 ij

jj)[[0, 1]’. The statement of the axiom would then be as follows. When the probability for directcommunication between two players changes, then the payoffs to both players change by the same amount.

5We refer the reader to Shapley (1953) for the properties of the Shapley value used in this proof.


C N•C CO } (N, v, p) 5O 6* (N, v ) 1O 6* (N, v ) 5O 6* (N, v )i i i ii[C i[C i[C i[C

C C5O 6* (N, v ) 5 v (N) 5 v (C),i p

i[N

6where the first and the fourth equality follow from the additivity and efficiency of theShapley value.

To show fairness, let (N, v, p) be a game with a probabilistic graph and let i, j[N,i±j, with p .0. Set w:5v 2v , where p is identical to p except for p (hi, jj)50.ij p p 2 ij 2 ij2ij

Note that v (S)5v (S) for all S#N with hi, jj#⁄ S. So, if S#N such that i[⁄ S or j[⁄ S,p p2ij

then w(S)50. So, the only coalitions with nonzero worth in the game (N, w) arecoalitions containing both i and j. Hence, it follows from symmetry of the Shapley valuethat 6* (N, w)56* (N, w). Using linearity of the Shapley value, we obtaini j

6* (N, v ) 2 6* (N, v ) 5 6* (N, v ) 2 6* (N, v ).i p i p j p j p2ij 2ij

1 2To prove uniqueness, suppose that c and c are two allocation rules that are1 2component efficient and fair. We will prove that c and c must be identical.

1 2 1 2First, note that component efficiency of c and c implies that c and c areidentical for games with probabilistic graphs where the probabilities of communication

1 2are zero for all pairs of players. Now, suppose that c and c are not identical. Then, let(N, v, p) be a game with a probabilistic graph with a minimum number of links with

1 2positive probability such that c (N, v, p)±c (N, v, p). By the minimality of (N, v, p),1 2we know that for any link hi, jj with p .0, it holds that c (N, v, p )5c (N, v, p ).ij 2 ij 2 ij

1 2Hence, fairness of both c and c implies that

1 1 1 1c (N, v, p) 2 c (N, v, p) 5 c (N, v, p ) 2 c (N, v, p )i j i 2ij j 2ij

2 2 25 c (N, v, p ) 2 c (N, v, p ) 5 c (N, v, p)i 2ij j 2ij i

22 c (N, v, p).j

From this we see that

1 2 1 2c (N, v, p) 2 c (N, v, p) 5 c (N, v, p) 2 c (N, v, p),i i j j

whenever i and j are in the same communication component C[N /p. Thus, we can find1 2numbers d , C[N /p, such that c (N, v, p)2c (N, v, p)5d for all i[C and allC i i C

1 2C[N /p. Now, we use component efficiency of both c and c , which implies that forall C[N /p it holds that

6We do indeed use additivity here. As is well-known, additivity does not usually make sense in a context ofvoting games. Note, however, that in the instance where we use it here there will be no problem, because the

C N\Cdecomposition of v is done in such a way that one of the two games v and v is the zero game, and hencep

the sum of the two games is a well defined voting game.


1 2O c (N, v, p) 5O c (N, v, p) 5 v (C).i i pi[C i[C

1 2Hence, we have 05o c (N, v, p)2o c (N, v, p)5uCud , and so d 50.i[C i i[C i C C1 2Therefore, we conclude that c (N, v, p)5c (N, v, p). ji i

Remark 4.1. In this theorem it is possible to replace the axiom of fairness by the7requirement of balanced contributions . An allocation rule has balanced contributions if

the loss (or gain) that the isolation of a player i inflicts on a player j is equal to theeffect that the isolation of player j has on player i. Formally, an allocation rule

Nc :G 3P →R has balanced contributions if for all games with a probabilistic graphN N

(N, v, p) and all i, j[N it holds that

c (N, v, p) 2 c (N, v, p ) 5 c (N, v, p) 2 c (N, v, p ),i i 2j j j 2i

where p (hk, lj)5p(hk, lj) if i[⁄ hk, lj, and p (hk, lj)50 if k5i or l5i. The Myerson2 i 2 iNvalue is the unique allocation rule c :G 3P →R satisfying component efficiency andN N

balanced contributions.

Remark 4.2. For practical purposes it may be useful to decompose the Myerson valueas shown in the following proposition.

Proposition 2. Let (N, v, p) be a game with a probabilistic graph. Then

N}(N, v, p) 5 O p (L)}(N, v, L).L#L(N )

NProof. We show that o p (L)}(N, v, L) verifies component efficiency andL#L( N)

fairness. Then theorem 1 can be invoked to prove the proposition.To prove component efficiency, let C be a component of N /p. If i[C and j[N•C

Nthen p 50, and therefore p (L)50 for all L such that L#⁄ (L(C)<L(N•C)). Considerij

L#(L(C)<L(N•C)). Then C is a union of components in N / . Using this and theL

component efficiency of the deterministic Myerson value, we obtain

O } (N, v, L) 5 O O } (N, v, L) 5 O v(K) 5 O v(K)i iF Gi[C K[N / :K#C i[K K[N / :K#C K[C /L L H(L )

5 v (C),H(L )

where H(L)5L>L(C). Defining H9(L)5L>L(N•C), it follows that

7The proof of this statement is quite similar to the proofs given in Myerson (1980) and van den Nouweland(1993) (theorem 3.2.1) and we refer the reader to these papers.


N NO O p (L)}(N, v, L) 5O O p (L)}(N, v, L)F G F Gi[C i[CL#L(N ) L#(L(C )<L(N•C ))

N 85 O p (L) O } (N, v, L) 5F Gi

i[CL#(L(C )<L(N•C ))

C N•C CO p (H(L)) ? p (H9(L))[v (C)] 5 O p (H )H(L )L#(L(C )<L(N•C )) H #(C )

N•C CO p (H9) [v (C)] 5 O p (H )v (C) 5 v (C).H H pS DH 9#L(N•C ) H #L(C )

To show fairness, first note that

N NO p (L)} (N, v, L) 5 O p (L)} (N, v, L)2ij i 2ij iL#L(N ) L#L(N )• i, jhh jj

N N5 O ( p (L) 1 p (L ))} (N, v, L)1ij i

L#L(N )• i, jhh jj

N5 O p (L)} (N, v, L ),i 2ij

L#L(N )

where L 5L•hhi, jjj and L 5L<hhi, jjj. Using this and fairness of the determinis-2 ij 1 ij

tic Myerson value we obtain

N N NO p (L)} (N, v, L) 2 O p (L)} (N, v, L) 5 O p (L)(} (N, v, L)i 2ij i iL#L(N ) L#L(N ) L#L(N )

N2 } (N, v, L )) 5 O p (L)(} (N, v, L) 2 } (N, v, L ))i 2ij j j 2ij

L#L(N )

N N5 O p (L)} (N, v, L) 2 O p (L)} (N, v, L),j 2ij j

L#L(N ) L#L(N )

which proves fairness. j

Next we show that the Myerson value for games with a probabilistic graph is stable inthe sense of Myerson (1977). He defined stability to be the property that two players

9always (weakly) benefit from reaching a bilateral agreement whenever the game (N, v)10is superadditive. In a context of games with probabilistic graphs stability states that two

players always (weakly) benefit when the underlying game is superadditive and theprobability of communication between the two players increases, all else being equal.

8Note that 12p 51 for all l[⁄ (L(C)<L(N\C)).l9A formal statement of the stability property is: c (N, v, L<hhi, jjj)$c (N, v, L) for all L#L(N).i i10A game (N, v) is superadditive if v(S<T )$v(S)1v(T ) for all disjoint coalitions S and T.


NDefinition 5. An allocation rule c :G 3P →R is stable if for all games with aN N

probabilistic graph (N, v, p) where the coalitional game (N, v) is superadditive and forall i, j[N it holds that

c (N, v, p) $ c (N, v, q) and c (N, v, p) $ c (N, v, q),i i j j

where q[P is such that q(hk, lj)5p(hk, lj) if hk, lj±hi, jj, and p(hi, jj)$q(hi, jj).N

Proposition 3. The Myerson value is stable.

Proof. Let (N, v) be a superadditive game and let p and q be two systems ofprobabilities such that q(hk, lj)5p(hk, lj) if hk, lj±hi, jj, and p(hi, jj)$q(hi, jj). Firstnote that it follows from proposition 2 that

N}(N, v, p) 5 O p (L)}(N, v, L)L#L(N )

N N5 O [ p (L)}(N, v, L) 1 p (L < i, j )}(N, v, L < i, j )].h j h j

L#(L(N )• i, j )hh jj

(1)

For all L#(L(N)•hhi, jjj) define

NP (L): 5P p P (1 2 p ).2ij l ll[L l[L(N )•(L< i, j )hh jj

Then (1) can be rewritten as

NO P (L)[ p (}(N, v, L < i, j ) 2 }(N, v, L)) 1 }(N, v, L)].h j2ij ijL#(L(N )• i, j )hh jj

N NNote that P (L)5Q (L) for all L#(L(N)•hhi, jjj). Therefore, for k5i, j we now2 ij 2 ij

have

N} (N, v, p) 2 } (N, v, q) 5 O P (L)[( pk k 2ij ijL#(L(N )• i, j )hh jj

2 q )(} (N, v, L < i, j ) 2 M (N, v, L))].h jij k k

This last expression is nonnegative, because p 2q $0 by hypothesis and } (N, v,ij ij k

L<hhi, jjj)2} (N, v, L)$0 by the stability of the deterministic Myerson value. jk

It is possible to define a potential function for games with probabilistic graphs alongthe lines of Hart and Mas-Colell (1989) (Shapley value) and Winter (1992) (Myersonvalue). Since it is a straightforward generalization, we will not include it in this paper.


5. Concluding remarks

Remark 5.1. Some analysts may be unwilling to use the independence assumption inorder to determine the probabilities involved in each communication graph. Recall thatby using this assumption we only need the value of p ’s to calculate the probability ofij

each graph L#L(S). It is possible to drop independence and still have a way to measurethe expected value of the players: One need only use as the primitive a probability

Sfunction configuration ( p ) defined directly over the set of all possible communica-S#NS Stion graphs, that is, for every S#N, p :hLuL#L(S)j→[0, 1], such that each p is a

S Sprobability distribution ( p (L)$0 for all L#L(S) and o p (L)51). GivenL#L(S )S( p ) , the formula of v applies for the calculation of the expected characteristicS#N p

function.Note that with the independence assumption, if we have n parties we only need know

nS DuL(N)u5( ) probabilities, while without this assumption we need to know oS#N2uL(S )uuhLuL#L(S)ju5o 2 probabilities.S#N

Remark 5.2. The model in this paper can be applied to voting situations in order to takeinto account the (in)compatibilities of the parties in such situations. Our model offers analternative to the traditional spatial voting models that are used to model ideologicalconsiderations of political parties. When applying the model in this paper to votingsituations, the critical point is clearly the correct calculation of degrees of compatibility.Many factors come together to determine these. The main one is the location of theparties in political space. This space is made up of various dimensions: the socio-economic left–right dimension, the religious vs. lay dimension, the center vs. peripherydimension, etc. The traditional approach to modelling ideological factors is a geometricmodel where voters are situated in some ideological space, where distance betweenvoters represents ideological difference. Owen (1971) and Shapley (1977) proposedadapting the Shapley–Shubik index to such spatially placed voters. For a detailedexplanation of this approach as well as for further references, see the survey by Straffin(1994). Once the parties have been placed in such an ideological space, a system ofprobabilities can be derived from the distances between parties. There have been seriousattempts to measure the positions of parties in some policy spaces. A useful source ofreferences regarding this approach can be found in Laver and Schofield (1990).

Nevertheless, we wish to warn against any naive direct translation of distances inpolitical space into degrees of compatibility. It is often the case that parties which areclose in this space have more difficulties to reach agreements with each other than withparties that are further away. The explanation is that parties compete for voters in theneighborhood of their ideological territory. In this competition the closest parties arethe greatest enemies. This is similar to a pattern sometimes observed in interactionsbetween countries. If neighboring countries fight to determine their borders, they oftenform alliances with the neighbors of their neighbors (the enemy of my enemy is myfriend). In politics similar behavior prevails when a party splits into two. At first the twonew parties stress their differences more than their similarities and are thus less likely toreach agreements. This is also typical behavior between radical factions in the extremeleft-wing spectrum. However, when parties reach an agreement about how to share their


voter territory and they are very close, it is also usual for them to form a stable coalitionfor elections in order to increase their number of seats, thus acting as a single party.

Though parties’ spatial location is an important determinant of the bilateralprobabilities of communication, there are also other factors which play a role. One ofthe most important is the leader’s psychological profile. Character compatibilities donot necessarily correspond to ideological affinities, and good or bad ‘‘chemistry’’between negotiators affects the course of negotiations. Finally, we would add to the listthe influence of power distribution in collateral institutions, such as regional parlia-ments and town halls, in which parties are also involved. The predisposition to reachagreements with another party is also a strategic consideration based on the overallrepercussions that such an agreement might have for other institutions. So the values ofthe p ’s also form part of the interaction strategies of parties. The model presented inij

this paper allows us to study the power impact of such changes.In spatial voting models suitable dimensions have to be chosen and parties have to be

identified with a particular point in the ideological space. In models of games withprobabilistic graphs degrees of compatibility have to be chosen. It seems that thedegrees of compatibility are more general because they encompass more than ideologi-cal considerations. On the other hand, degrees of compatibility might be more sensitiveto changes in the political environment (in a broad sense). All this might imply thatspatial voting models are more appropriate when one wants to investigate long-termrelations and that games with probabilistic graphs are more appropriate to studyshort-term relations.

Remark 5.3. For practical applications the critical point in the process is the correctcalculation of the probabilities p . This is a delicate issue and each context has its ownij

difficulties. As an example, Calvo and Lasaga (1997) calculated a system of prob-abilities of bilateral communication between the political parties in the SpanishParliament of June 1993. To obtain these estimates, before the elections a survey wascarried out among political commentators from press, radio and television. The surveyconsisted of a questionnaire in which the journalists were asked to give a numberbetween 0 and 100 for each pair of parties i and j. This was to reflect the probabilitythat parties i and j would come to an agreement in parliament. The system ofprobabilities was worked out by simply taking the average of all of the reactions to thesurvey.

Remark 5.4. There are several ways in which the model of Section 3 can begeneralized. One direction is to consider a priori unions of the parties. Doing sogenerates the possibility to consider government coalitions as part of the description ofthe voting situation. Since the government will generally consist of more than onepolitical party, and since these parties are more or less committed to some governmentalpolicy, one can imagine that by making the a priori union of the governmental coalitionspart of the model one will get even more accurate indices of the power of the parties inthe parliament. Generalizing the model in this direction means extending the models ofgames with a priori unions as defined by Owen (1977), and of games with a priori

´unions and deterministic communication graphs as studied by Vazquez-Brage et al.


(1996). One can also extend the model of Hart and Kurz (1983, 1984), in order to studyendogenous formation of government coalitions. Some of the possible extensionsdescribed in this paragraph are considered in Calvo and Lasaga (1997). An interestingresult in the paper by Calvo and Lasaga is that they are able to explain the fact thatsometimes government coalitions are not minimal winning coalitions (as might beexpected according to Riker’s (1962) size principle or the minimal winning coalitionprinciple), but rather include more parties than would be necessary to have a majority ofthe votes. This result stems from the incorporation of probabilities of cooperation intothe model: it often happens that if the parties which form the alliance have lowprobability communication values one with another, the introduction of a newcomer intothe alliance who has high probabilities of cooperation with these parties has a positiveeffect and increases the expected power of all of them. In the stability analysis of theSpanish parliament performed in Calvo and Lasaga (1997) an example of this fact willbe found. This communication gain effect can be seen as an additional theoreticalsupport for the empirical evidence of the occurrence of surplus coalition (i.e.nonminimal winning) governments.

Acknowledgements

´This research was partly supported by the Universidad del Paıs Vasco (project UPV036.321-HA 127/93 and 127/94). The authors thank Yakar Kannai and two anonymousreferees for their constructive comments.

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values of games with probabilistic graphs

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