Subgame Perfect Farsighted Stability∗

Daniel Granot† Eran Hanany‡

June 2016

Abstract

Farsighted game theoretic solution concepts have been shown to provide insightful results

in many applications. However, we provide simple examples demonstrating that the existing

solution concepts may fail to incorporate some important aspects of farsighted reasoning. This

failure stems apparently from the reliance of these concepts on constructs such as indirect

dominance which only provide limited foresight. Accordingly, we explore in this paper the

implications of replacing indirect dominance with subgame perfection. That is, we propose a new

farsighted solution concept, the Subgame Perfect Consistent Set (SPCS), based on consistency

in the spirit of the von Neumann Morgenstern solution and on subgame perfect equilibrium.

Surprisingly, the SPCS is shown to always lead to Pareto efficiency in farsighted normal form

games. This result is demonstrated in various oligopolistic settings, and is shown to imply, for

example, that players who follow the SPCS reasoning are always able to share the monopolistic

profit in farsighted settings based on Bertrand and Cournot competition, and are always able to

achieve coordination and Pareto efficiency in decentralized supply chain contracting, even when

they are unable to form coalitions.

Key Words: Dynamic games, normal form games, farsighted stability, largest consistent set,

oligopolistic competition

1 Introduction

The introduction of farsighted game theoretic solution concepts was motivated by criticism of

‘myopia’ raised against both cooperative and noncooperative solution concepts. For example, the

Cournot equilibrium model was already criticized by Chamberlin (1933) on the grounds that firms

in the model ignore reactions by rival firms to their actions, the von Neumann and Morgenstern

solution (vNM 1944) was criticized by Harsanyi (1974) for failing to incorporate foresight, and the

strong Nash equilibrium was criticized (see Chwe 1994) for not being consistent in the sense that it

may reject outcomes in view of the existence of a strictly profitable deviation to another outcome

even though this latter outcome is not a strong Nash equilibrium.

∗This research was supported by the Natural Sciences and Engineering Research Council of Canada.†Operations and Logistics Division, Sauder School of Business, University of British Columbia, Vancouver BC,

Canada V6T 1Z2. E-mail: daniel.granot@sauder.ubc.ca‡Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel. Email: hananye@post.tau.ac.il

1

As compared to ‘myopic’ solution concepts, such as the vNM solution, core, or Nash equi-

librium, farsighted solution concepts characterize stable outcomes that are immune to deviations

by farsighted players who recognize that their own deviations may induce multiple deviations by

other players. Indeed, motivated by the above shortcoming of myopic solution concepts, a variety

of farsighted solutions were proposed, starting with Harsanyi (1974) who pioneered the approach

that coalitional behavior should be analyzed using subgame perfect equilibrium strategies of an

extensive form game. His aim was to provide a non-cooperative foundation to the vNM solution

in cooperative game theory. Greenberg (1990) introduced the conservative and optimistic stable

standards of behavior, which possess vNM type consistency. Konishi and Ray (2003) introduced a

dynamic Markov process of coalition formation, and characterized history independent equilibria

based on coalition value functions representing infinite horizon discounted payoffs.

Differently from the above approaches to farsightedness, the vNM farsighted stable set (vNM

FSS, Chwe 1994) and the largest consistent set (LCS, Chwe 1994) take a cooperative view. The vNM

FSS was shown by Chwe (1994) to be contained in the LCS, and farsightedness in both concepts1

is achieved via the indirect dominance relation, introduced by Harsanyi (1974) and formalized by

Chwe (1994). The LCS is (internally) consistent in the sense that a defection by coalition from a

state 1 in the LCS to another state 2 is deterred, if subsequent deviations by other coalitions from

2 could lead to a state 3 in the LCS which indirectly dominates 2 and at which not all members

of are strictly better off with respect to 1. Moreover, such defection from a state outside the

LCS is not deterred, making the LCS (externally) consistent. When the set of states in the game

is finite or countably infinite, it was shown by Chwe (1994, Proposition 2) that the LCS is also

externally stable in the sense that every state not in the LCS is indirectly dominated by another

state in the LCS.

The above farsighted solution concepts have led to various refinements and modifications, and

were applied in a variety of settings. For example, they were employed in various studies on

coalition formation (e.g., Mauleon and Vannetelbosch (M&V) 2004, Granot and Sošic 2005, and

Nagarajan and Sošic 2007), and network stability (e.g., Page et al. 2005, and Herings et al. 2009).

Not surprisingly, farsighted stable solution concepts could yield quite different insights than those

derived from myopic stable solution concepts. For example, in coalition formation studies (e.g.,

M&V 2004, and Granot and Yin 2008), it is shown that under certain conditions, the unique

farsighted stable coalition structure is the grand coalition while the unique myopic stable coalition

is the stand alone structure. We will elaborate in Section 2 on the above as well as other related

solution concepts and the various settings at which they were applied.

We provide simple examples (see Section 3) demonstrating that the existing solution concepts

may fail to incorporate some important aspects of farsighted reasoning. Since this failure stems

from relying on concepts such as indirect dominance, we ask what are the implications of assuming

instead that players’ foresight is modelled via subgame perfection. To this end, in this paper we

introduce a new farsighted solution concept, referred to as the Subgame Perfect Consistent Set

1For precise definitions of both concepts see Section 2.

2

(SPCS), which allows players or coalitions to look arbitrarily far ahead when they consider the

consequences of their deviations from an existing state. Indeed, from a given initial state, we model

the (non-stationary) evolution of play resulting from players/coalitions’ deviations as an extensive

form game, infinite in size, and define2 the SPCS to consist of all states which can be supported by a

subgame perfect equilibrium, refined to additionally satisfy both internal and external consistency

in the spirit of the vNM stable sets. Thus, the SPCS shares with existing farsighted solution

concepts the use of vNM type of consistency, but differently, it uses the reasoning of subgame

perfection, which endows players with unlimited farsightedness, instead of alternative farsighted

constructs such as indirect dominance. The SPCS also does not necessarily impose the assumption

of pessimism prevalent in the existing literature. As in the case of almost all other farsighted

solution concepts, the SPCS is a natural solution concept in settings in which actions are public,

transient states have no payoff consequences, and payoffs are realized only when players reach a

final agreement.

We analyze and characterize in this paper the SPCS for the general case when coalition de-

viations are possible. However, we are particularly interested to study situations in which only

deviations by individual players are allowed. This would correspond to farsighted settings based

on Cournot and Bertrand competition, where players are not allowed to collude or coordinate their

actions. We investigate the structure of the SPCS for the class of farsighted normal form games.

In particular, we show that even when only single players can deviate, quite surprisingly, the SPCS

solution leads to (weak) Pareto efficiency in any normal form game having a pure Nash equilib-

rium. Thus, the new farsighted concept leads to efficiency in “farsighted” versions of, e.g., Bertrand

games, Cournot games, some instances of Decentralized Supply Chains coordination problems and

in the Prisoner’s Dilemma. For all these settings, a SPCS is a singleton set consisting of a Pareto

efficient outcome weakly dominating the myopic (pure Nash) equilibrium outcome. This is achieved

using strategies that are similar to ‘grim-trigger’ strategies commonly used in folk theorems within

the repeated games literature. However, contrary to repeated games in which efficiency is achieved

as a possibility, the SPCS achieves efficiency as a necessary implication. Moreover, we show that

the SPCS solution leads to efficiency also for normal form games having no myopic equilibrium,

given that one is willing to accept as a possible outcome of the game a ‘swinging’ behavior in which

some of the players keep on moving between states.

We show that all the above results continue to hold when coalition deviations are permitted,

where in this case a myopic equilibrium is a state from which no coalition strictly prefers a one-

step deviation. An important contribution of our approach is that, contrary to many cooperative

game models which assume Pareto efficiency from the start, we derive such efficiency within a non-

cooperative framework. This is undertaken using strategies that could be thought of as modelling

tacit negotiation.

The plan of this paper is as follows. In Section 2 we provide the formal definitions of relevant,

related solution concepts in the literature. In Section 3 we provide examples motivating our SPCS.

2Precise definition is given in Section 4.

3

In Section 4 we introduce the farsighted game, formally define the SPCS and investigate some of its

basic properties. In Section 5 we introduce farsighted normal form games, derived from standard

one-shot normal form games by allowing players, or coalitions, to publicly and repeatedly, in their

turn, change their previous actions. For this class of games we prove that any set of outcomes

weakly dominating a myopic equilibrium is a SPCS if, and only if, it consists of a single Pareto

efficient outcome (up to equivalence to all players). The analysis is extended to farsighted normal

form games without a myopic equilibrium. We also illustrate the significance of our findings by

demonstrating, for example, that by contrast with existing farsighted solutions such as the LCS

and the Largest Cautious Consistent Set (LCCS), introduced by M&V (2004), but similarly to

the vNM FSS, farsighted players who follow the SPCS reasoning are always able to share the

monopolistic profit in farsighted settings based on Bertrand and Cournot competition. We also

show that the SPCS always leads to full coordination and Pareto efficiency in decentralized supply

chain contracting even when they cannot form coalitions. Section 6 concludes with a summary and

suggestions for future research. Proofs are collected in Appendix A. Appendix B provides some

further comparison of the SPCS and LCS.

2 Examination of the Related Literature

We briefly discuss in this section the relevant literature on farsighted solution concepts. One of the

most prominent farsighted consistent solution concepts is the largest consistent set (LCS), due to

Chwe (1994), wherein farsightedness is exclusively embodied in the notion of indirect dominance.

Before introducing the relevant definitions, let denote the set of all possible outcomes, and for

1 and 2 in , let 1 → 2 mean that coalition can effect a change from state 1 to state 2.

Definition 2.1 [Chwe 1994, Harsanyi 1974]. We say that a state ∈ is indirectly dominated

by a state ∈ , or ¿ , if there exist states 0 1 ∈ , where 0 = and = , and

coalitions 0 1 −1, such that → +1 and () () for all = 0 1 − 1 and ∈ .

Formally, consistent sets and the LCS are defined as follows:

Definition 2.2 [Chwe 1994] We say that a set ⊂ is consistent if ∈ if, and only if, for

every coalition and state ∈ such that → , there exists a state ∈ , such that = or

¿ , for which it is not the case that () () for all ∈ . The largest consistent set (LCS)

is the union of all consistent sets.

Thus, the LCS is the union of all consistent sets satisfying: (i) for any state in , no subset

of players that can deviate from would elect to do so, since such a deviation could lead, via

indirect dominance, to another state in in which not all members of are strictly better off

(internal consistency), and (ii) for any state not in , there always exists a subset of players

that can deviate from and would elect to do so, since in all states in that such a deviation may

4

lead to via indirect dominance, all members of are strictly better off (external consistency). In

the context of stability, the interpretation for indirect dominance provided by Chwe is that if ¿

and is presumed to be stable, then it is possible, not certain, that the coalitions 0 −1will move from to .

Note the pessimistic criterion embedded in the definition of the LCS, which is somewhat mollified

in M&V’s (2004) definition of the Largest Cautious Consistent set (LCCS), described below. Chwe

was aware of a possible criticism of the LCS for being too inclusive and that it may contain possibly

non-intuitive outcomes. Explicitly, as he argues, “If is consistent and ∈ , the interpretation is

not that will be stable, but that it is possible for to be stable. If an outcome is not contained

in any consistent , the interpretation is that cannot possibly be stable: there is no consistent

story in which is stable” (Chwe 1994; p. 303). Nevertheless, as we show in Example 3.1 in Section

3, there are outcomes not contained in the LCS and for which there are very consistent stories of

stability.

It should be noted that, prior to Chwe, Greenberg (1990) has developed the theory of social

situations, wherein he has introduced the concept of a stable standard of behavior (SB), which is

claimed to capture perfect foresight by individuals or coalitions. Some precise relationships between

a stable SB and the LCS were investigated by Chwe (1994). Xue (1998) has argued that the LCS

captures only partial foresight, and by employing Greenberg’s (1990) framework, characterized a

set of paths which constitutes a stable SB. See also Mariotti (1997), wherein a farsighted solution

concept similar to a stable SB is shown to achieve partial efficiency in normal form games with

coalitions.

Aside for the LCS, the stream of literature more directly related to this paper is concerned

with various refinements of the LCS: M&V’s LCCS (2004), Chwe’s vNM FSS (1994), and, to some

extent, Konishi and Ray’s EPCF (2003) and the path dominance core (Page and Wooders, 2009).

Let us first consider the Largest Cautious Consistent Set (LCCS), introduced by M&V (2004). The

LCCS is a proper refinement of the LCS, which could possibly be empty, wherein players are less

conservative than in the LCS. Specifically, as explained by M&V, once a coalition deviates from

to 0, this coalition should contemplate the possibility of ending up with positive probability atany state 00 in the stable set via indirect dominance, i.e. 0 = 00 or 0 ¿ 00. A state is neverstable if a coalition can deviate from to 0, and by doing so there is no risk that some coalitionmembers will end up worse off. Thus the LCCS results from a modification of the LCS pessimistic

criterion. The LCCS does provide new insights in some instances, as shown, e.g., by M&V in their

study of coalition formation, and by Granot and Yin (2008) in their analysis of a single-period

two-stage supply chain problem.

The vNM FSS is derived from the classical vNM solution by replacing direct dominance with

indirect dominance. It was introduced by Chwe (1994), who showed that it is a refinement of

the LCS. As such, the vNM FSS was shown to produce sharper results than the LCS in several

instances, such as symmetric Cournot and Bertrand oligopoly markets (Suzuki and Muto (2006)

and Masuda et al. (2000)), as will be further elaborated on in the sequel. Similarly, it was shown

5

by Mauleon et al. (2011) that, for one-to-one matching games, while the LCS may contain more

matchings than those in the core (see also Diamantoudi and Xue, 2003), a set of matchings is a vNM

FSS if and only if it is a singleton set whose unique member is a core element. See also Herings et al.

(2009), who use a modification of the vNM FSS in their investigation of farsighted stable networks.

Notwithstanding these sharper results, the vNM FSS could yield not very insightful results. For

example, for a game based on the provision of a perfectly ‘lumpy’ public good, without coalitions,

it was shown by Kawasaki and Muto (2009) that the vNM FSS includes almost all individually

rational outcomes.

Another approach to farsightedness, which was employed in the analysis of supply chain models,

is due to Konishi and Ray (2003). Therein, the authors have introduced a dynamic process of coali-

tion formation (PCF), given by a transition probability : × → [0 1] such thatP

∈ ( ) = 1

for each ∈ . A PCF induces a value function for each player, which captures the infinite hori-

zon discounted payoff to each player starting from any initial point under the Markov process . A

PCF is an equilibrium PCF (EPCF) if a coalitional move to some other state can be supported by

the expectation of a higher future value.

Evidently, the EPCF approach to farsightedness and coalition formation is quite different from

the reasoning leading to the LCS, LCCS, and the SPCS, the solution concept introduced in this

paper. By contrast with these three concepts, in which payoffs to players are realized only at the

final state and wherein players may entertain different conjectures about other players’ behavior,

in the EPCF players get discounted state-dependent payoffs at each stage, and all players have

common beliefs about future moves. Note that in the EPCF the strategies are Markovian, i.e.,

they depend only on the current state and not on the history of arrival to that state. By contrast,

as it will be shown in the sequel, the strategies in the SPCS could be history-dependent.

One of the attractions of the EPCF is that the set of all absorbing states under all deterministic

(i.e., ( ) ∈ {0 1}∀ ∈ ) absorbing EPCFs is a refinement, typically strict, of the LCS when

the discount factor is large enough. This result was used by e.g., Sošic (2006) and Nagarajan and

Sošic (2007, 2009) in their analysis of several Operations Management models. We note that the

EPCF approach to coalition formation may lead to non-efficient outcomes in normal form games,

as is evident from Observation 2 and the ensuing discussion in Konishi and Ray (2003).

Page and Wooders (2009) have introduced a model of network formation whose primitives

consist of a set of networks, players’ preferences, rules of network formation, and a dominance

relation on feasible networks. As noted by the authors, their network formation game can be

viewed as an abstract game with a finite set of states, , and a dominance relation among states

which could be either the direct dominance, (Chwe’s) indirect dominance, or path dominance. The

main concept introduced by Page and Wooders is the path dominance core, which consists of all

states which are not path dominated by any other distinct state. The path dominance core could

be empty. However, if it is not empty, and if the game is such that for any two states, and ,

there exists a coalition that can move from to , then all states in the path dominance core are

Pareto efficient.

6

(1, 1, 100)

C

(2, 10, 1000)

(10, 20, 10)

A

B

D

{1}

{2}

{3}

(5, 1, 1)

Figure 3.1: Transition tree in Example 3.1

Brams (1994) theory of moves (TOM) (see also Brams and Wittman 1981) offers a dynamic the-

ory approach to normal form games that bears some resemblance to our farsighted game approach

to normal form games. Indeed, TOM attempts to characterize non-myopic equilibria, or stable

outcomes, when players in a normal form game think ahead, and the predictions TOM provides

depend, as in our case, on the starting point or status quo state. However, TOM is only developed

for two-person normal form games in which each player has only two strategies, the players ac-

cording to TOM play in a strict alternating order, and, more importantly, TOM makes simplifying

assumptions to ensure a finite game tree representation of the dynamic normal form game. Related

extensions to TOM’s non-myopic equilibria include Kilgour (1984).

3 Motivating Examples

Farsightedness in the LCS, LCCS, and the vNM FSS, stems exclusively from indirect dominance

(Definition 2.1) which, as demonstrated in Examples 3.1 and 3.2, provide players with limited

foresight. The lack of complete farsightedness in these examples is used to motivate the introduction

of our new solution concept, the SPCS, wherein players are endowed with complete farsightedness

stemming from subgame perfection.

Example 3.1 Consider the game associated with Figure 3.1 that has four states with transitions

between states as illustrated in the figure. Each node represents a possible state of the game, and

the outgoing arrows represent a possible move by some coalition to a different state. If the final

state is , the utilities to the three players are, respectively, (5 1 1). Player 1 can move from to

state , in which the utilities to the players, if it is the final state, are (2 10 1000). Then, from

state , player 2 can move to state , in which the utilities to the three players are (10 20 10).

Also at state , player 3 can move to state , in which the utilities are (1 1 100). No other moves

are possible.

Let us first characterize the LCS and LCCS. Since states and are terminal states, no deviation

therefrom are possible, and both belong to the LCS and the LCCS. State cannot be a final state,

since player 2 can move from to and strictly increase his utility. Thus, state is neither in

the LCS nor in the LCCS. We further note that according to the farsighted reasoning of both the

7

LCS and the LCCS, player 3 will not move from state to state , since at state his utility is

100, which is strictly lower than his utility at . Next, consider state . Since, as clarified above,

at state player 2 will move to state and player 3 will not move to state , and since the utility

to player 1 at state , 10, is strictly larger than his utility at state , player 1 will move to state .

Thus, LCS = LCCS = {}. Next, let us consider the vNM FSS. State is directly dominated

by , and state is indirectly dominated by . Further, no deviation from or is possible,

and thus, neither state indirectly dominates the other. So, {} is both internally and externallystable and forms the unique vNM FSS.

Thus, all three farsighted solution concepts do not view state as possibly stable. In particular,

it follows from Chwe that since state is not in the LCS, there is no consistent story in which

state is stable. However, the instability of state according to the LCS, and, in fact, according

also to the LCCS and the vNM FSS, is predicated on the reasoning that player 3 will not deviate

from state to state because he is strictly better off at state than at state . But according

to these three solution concepts, state is also not stable and cannot be a final state. Thus, this

comparison is irrelevant and, in fact, is not consistent with the spirit of the LCS according to which

only final outcomes matter. Indeed, while under indirect dominance, a deviation from state can

only lead to final state , it is quite compelling to conclude that upon realizing that state cannot

possibly be a final state, a “farsighted” player 3, given an opportunity, will deviate from state

to state , just to preempt a future deviation by player 2 from state to state at which player

3 is strictly worse off. Thus, by contrast with the implications of indirect dominance, under more

complete farsightedness, a deviation from state can end up in the final state . Therefore, even

though state in not a member of the LCS, there is evidently a consistent and compelling story

for state to be stable: player 1 will prefer not to move from state since he may end up at state

at which he is strictly worse off. Indeed, according to the farsighted solution concept, SPCS,

developed in this paper, it will be verified in the sequel that SPCS = {} if is viewed as

sufficiently likely to occur, or if players use a pessimistic criterion.

It can be shown that the path dominated core with indirect dominance relation coincides with the

LCS, LCCS and vNM FSS. With direct dominance relation, since state for player 1 does not

directly dominate state , the path dominance core is {}. In the absence of state , LCS =LCCS = vNM FSS = path dominance core with indirect dominance relation = SPCS = {}, whilethe path dominance core with direct dominance relation, unfortunately, is {}.

Next, consider Example 3.2, which illustrates that even in very simple settings one may need

to analyze an infinite game tree in order to determine the farsighted outcomes.

Example 3.2 Consider the game depicted in Figure 3.2. Since there is no specification which

coalition/player is moving at a state in which more than one coalition/player can move, in principle

it could happen that player 1 gets a turn to move at state for any finite number of times before

player 2 gets a turn to move at this state. Thus, the game tree associated with Example 3.2 is

infinite, wherein, player 1 will move from state to state and then for any finite number of times

8

(0, 12)(8, 8)(6, 6)

AB

C

{1}

{1}

{2}

Figure 3.2: Transition tree in Example 3.2

could stay at state or move from state to and back to . Notwithstanding the difficulty that

may arise from an infinite game tree, the analysis of the game is straightforward.

Let us first characterize the LCS and LCCS. Clearly, since there is no possible deviation from state

, it is a final state, belonging to both the LCS and LCCS. State cannot be a final state, since

player 1 can move from it to state , at which he is strictly better off. Next, consider state .

Since a move by player 1 to state can subsequently lead to a move by player 1 back to state ,

state is contained in the LCS. Furthermore, since player 2 is strictly better off at state than at

state , according to the farsighted reasoning of the LCS (and the LCCS), player 2 will not move

from state to state . Thus, a move by player 1 from state to state can only be followed by

a return move by player 1 from state to state . Therefore, state is also in the LCCS, and,

we have, LCS = LCCS = {}. Further, since state does not indirectly dominate state , and

state directly dominates state , the vNM FSS and the path dominance core (both with direct

and indirect dominance) coincide with the LCS and the LCCS.

Since the game tree of Example 3.2 is infinite, the formal analysis of the SPCS associated with

Example 3.2, carried out in Section 4, is slightly more involved. However, intuitively, state is

not a final state, and while player 2 is strictly better off at state than at state , the former is

just a “pie in the sky” for him. Indeed, notwithstanding player 2 reluctance to move from state

to state , he is strictly better off at state than at state , which will be the final state if he

refuses to deviate to state . Thus, it is optimal for player 2 to effect a deviation from state to

state once he has such an opportunity. Therefore, player 1 will deviate from state to state

having full confidence that player 2 will eventually deviate from state to state , and therefore,

state cannot be a final state.

The above analysis suggests that in Example 3.2 the only farsighted final state is and, indeed, as

we will demonstrate in the sequel, the SPCS for Example 3.2 consists of the unique state .

4 Model and Solution: The Subgame Perfect Consistent Set

Consider a dynamic game in which players can act repeatedly and publicly by moving between

states they care about. For example, consider a group of firms engaged in contract negotiation, in

which a state is a vector of contract parameters set by the firms, and each firm is responsible for

a different part of the vector. Players only care about the final state reached, irrespective of the

sequence of actions that lead to it. Thus actions have no significant costs, and the time frame in

which they are taken is short with no relevant discounting of the final state utility. Despite the

9

dynamic nature of the game, it can be thought of as a one period interaction between the players.

This setting can be described as a game in effectiveness form (Chwe 1994, see also Greenberg

1990), denoted by h ()∈ (→)⊆i, where: is a finite set of players; is a measurable

set of possible states; the utility function : → R for each ∈ determines player ’s utility

from a state ∈ (when it is a final state); and the binary relation →⊆ × for each ⊆

describes the players ability to change the current state, where 1 → 2 for 1 2 ∈ means that

coalition of players can alter the current state 1 by moving to a new state 2 (with the empty

coalition always having null effectiveness, i.e. →∅= ∅). A state ∈ is said to be terminal if there is

no 0 ∈ , 0 6= , and ⊆ , such that → 0. For example, in the contract negotiation setting, can be the set of all contract parameter vectors possibly set by the firms, and the effectiveness of

a singleton coalition = {}, consisting of a single player , allows the player to alter the currentstate by changing their own contract part without changing the parameters set by the other players.

Furthermore, in this example, the effectiveness of a coalition consisting of two or more players can

describe agreements between members of the coalition that allow particular simultaneous changes

in the parameters set by all members of the coalition (again without changing the parameters set

by players outside the coalition).

As mentioned above, we envision an extensive form game with perfect information, in which

all actions are public, which we call the Farsighted Game (FG). The complete description of the

game is given by the following sequence of events. First, an initial state in is publicly chosen

by nature. Then, an infinite sequence of stages initiates, each stage consisting of the following two

steps: (Step 1) some coalition is publicly selected by nature; and (Step 2) the selected coalition can

publicly keep the current state unchanged or move to a new state in according to its effectiveness.

Formally, define the set of all possible histories in the game, H, by the set of all possibly infinitesequences = ()

=0, including the empty sequence when = −1, such that 0 is some initialstate in , for all odd numbers ≥ 1, is some coalition ⊆ , and for all even numbers ≥ 2, ∈ , such that = −2 or −2 →−1 . For two finite histories

0 ∈ H with even,

positive cardinality such that 00 = −1 or −1 →00, denote by (

0) the history in Hobtained when is followed by 0. Denote the set of infinite histories by H∞. We say that a history ∈ H converges for player if there exist () ∈ and an even, positive number such that

() = [()] for every even such that ≤ ≤ . We define 0 to be the minimal such

. A history converges when it converges for all players, in which case we omit the player index

and just write () and 0. Convergence occurs, in particular, when 0is a terminal state, or

when is finite. Define the player function : H\H∞ → ∪{}, where denotes a nature move,by (∅) = , () = for every finite history with odd cardinality ||, and () =

⊆ for

every finite history with even, positive cardinality. Nature’s move at history for which () =

is determined by a full support probability measure, , defined over when = ∅ and over 2otherwise. We assume that this probability can depend only on the last state in the history, i.e.,

= 0 when = 00

for any non-empty 0 with () = .3

3All the results in the paper would continue to hold if this assumption was omitted, but the pessimistic criterion

10

Note that the choice of the initial state by nature is made only to reflect the fact that none

of the players can affect this choice — one could alternatively define the game with a prespecified

initial state instead of random selection.

Note also that in a setting where players can repeatedly observe the actions made by others

and adjust their own actions accordingly, full farsightedness necessitates unbounded play: actions

are made taking into account the fact that any move by a coalition may be counteracted with a

further move by another coalition, without limit. Such an approach is in the spirit of modelling

farsighted negotiation. It is natural therefore that players only care about the final outcome of the

negotiation process, with no intermediate payoffs.

A strategy of a coalition is a function : { ∈ H \ H∞ | () = } → such that

() ∈ { ∈ | = −1 or −1 → }, specifying the action taken by this coalition afterany history in the game where this coalition is selected to make a choice (a complete contingent

plan). A strategy profile is a vector = ()⊆ specifying the strategy of each coalition .

Denote by − the vector of strategies for all coalitions except for . We consider subgame perfectequilibria of the FG. For the purpose of defining such equilibria, it is sufficient to consider subgames

beginning with coalitions’ decisions, i.e. following finite histories with even, positive cardinality.

Following such , a strategy profile generates a probability measure over the infinite histories 0

beginning with that are dictated by . Let Σ denote the set of strategy profiles which, following

, generate a probability measure having support consisting only of infinite converging histories.

Whenever ∈ Σ, this strategy profile generates for player a distribution | over (0) ∈ .

This distribution has countable support because there are finitely many coalitions that may be

selected and convergence occurs after a finite history. Elements of this support are called final

states. To define |, let

Φ| =

((0)

00=0

¯¯ 0 ∈ H∞ 0 = ∀ ∈ 0 1 and

0+1 = 0[(0)

=0]∀ = + 2 }

).

The set of final states according to strategy profile following history , i.e. the support of |, is

Ψ| = {(0) | 0 ∈ Φ|},

and then |() =P{0∈Φ||(0)=}

Q=350−1 (0)−1=0

(0) for ∈ .

In a subgame that follows a finite history with () = , coalition has a strict preference

relation  over strategy profiles, where weak preference % and indifference ∼ are definedfrom the strict preference  in the usual way. When the coalition is a singleton player , we

assume the following.

Assumption 4.1 For each player , the preference relation Â, when restricted to Σ, is contin-

uous and satisfies  0 whenever | strictly first order stochastically dominates 0|, where is ordered according to .

in Definition 4.4 was additionaly assumed.

11

For our analysis, the important implication of this assumption arises when considering a strategy

profile which leads to a degenerate distribution over final states, i.e. some 0 ∈ for sure (with

probability 1). In this case, the strategy profile is strictly worse than any strategy profile leading

to a non-degenerate distribution over final states which assigns positive probability only to states

in at least as good as 0.

For general coalitions we assume the following.

Assumption 4.2 For each coalition , the preference relation Â, when restricted to Σ, satisfies

 0 whenever %

0 for all members ∈ , with strict preference for at least one member

of .

Given coalition preferences, we can define a strategy profile as a subgame perfect equilibrium

if % ( −) for every coalition , every finite history with () = , and every strategy

for this coalition.

In analyzing the farsighted game described above, we seek a solution concept that produces a set

of states considered farsighted stable. As mentioned in the Introduction, our proposed solution is

based on subgame perfection and a consistency notion in the spirit of vNM stable sets (von Neumann

and Morgenstern 1944). Roughly speaking, this consistency notion requires the solution set of

states to satisfy the following two properties: (i) farsighted internal consistency: for each state

∈ , no coalition prefers to move away from , anticipating that such a move would eventually

end up in ; and (ii) farsighted external consistency: for each state ∈ , there always exists a

coalition that prefers to move away from , again anticipating that such a move would eventually

end up in . To formalize where a move might potentially lead to under subgame perfection, we

say that a state 2 is reachable from a state 1 if for a subgame in which 1 is the initial state,

there exists a subgame perfect equilibrium strategy profile such that 2 is a final state (i.e., the

play converges to it with positive probability). Denote by (1) the set of states 2 reachable from

1. Clearly for a state to be a final state according to a subgame perfect equilibrium, it must be

reachable from itself. This leads to the following definition, which relies on the logic of subgame

perfection to reflect farsightedness with respect to some set of states.

Definition 4.1 Given a strategy profile and a set of states , say that is -farsighted, denoted

, if is a subgame perfect equilibrium that satisfies the following two requirement:

(a) for any history = ( 1 2 ) such that ∈ () and {}=1 ⊇ 2 \ ∅, () = ;

and

(b) for any history = (1 1 1 2 1 −1 2 ) such that 1 6= 2, the set of final states

is Ψ| = ∩(2).

Requirement (a) says that after choices by all coalitions to stay at an initial state ∈

reachable from itself, is the final state for sure. According to requirement (b), after an initial

move from an initial state by some coalition, all reachable states in from 2, and only them, are

final states. Both requirements hold on or off equilibrium path. Our solution concept relies on this

12

formulation of farsightedness, and involves an additional sense of a fixed point apart from the usual

one delivered by equilibrium: The set is exactly the set of states from which, on equilibrium path

of a -farsighted strategy profile , no coalition moves when selected as initial states. Formally:

Definition 4.2 A set of states ⊆ is a Subgame Perfect Consistent Set (SPCS) of a Farsighted

Game (FG) if there exists such that ∈ if, and only if, () = for any coalition and

any finite history = ( 1 2 ).

Whenever satisfies definition 4.2 with respect to , we say that supports as a SPCS.

It will be also useful to consider the final states that a SPCS generates after the selection of initial

states outside the consistent set:

Definition 4.3 For a SPCS and ∈ \, define the attraction set, (), as the set of allstates in that, according to some subgame perfect equilibrium supporting as a SPCS, are

final states following the selection of as an initial state.

We now investigate some basic properties of our solution concept. First, the following example

demonstrates that a SPCS may fail to exist. In this example the failure is due to the lack of a

subgame perfect equilibrium.

Example 4.1 There is one player, is the set of natural numbers, () = for each ∈ ,

and the player can only move from = 1 to any 1 (with no other possible moves included in

the effectiveness relation). In this example there is no SPCS because there is no subgame perfect

equilibrium: when = 1 is selected as an initial state, for any choice the player can make there

exists a strictly better choice.

Despite such examples, the following proposition provides sufficient conditions for existence of

a non-empty SPCS.

Proposition 4.1 Whenever is finite and the effectiveness relation is acyclic, there exists a non-

empty SPCS.

When analyzing farsighted normal form games in Section 5, we will show that the assumptions

that is finite and the effectiveness relation is acyclic are not necessary for existence. Indeed, a

non-empty SPCS can exist also for games with an infinite or when the effectiveness relation is

cyclic.

In general, the SPCS may potentially depend on the probability measures governing the

coalition selection process, e.g., when players use expected utility maximization. Nonetheless, it is

possible for our solution to be insensitive to the particular probabilities assumed. To allow such

invariance, we can assume, similarly to Chwe (1994) and (particularly to) M&V (2004), that players

use a pessimistic criterion to make their choices.

13

Definition 4.4 Pessimistic criterion for player : the preference relation Â, when restricted to

Σ, is continuous and satisfies  0 whenever a worst state in the support of | is strictlybetter than a worst state in the support of 0|, or if | strictly first order stochastically dominates0| where is ordered according to .

This pessimistic criterion implies that a sure outcome ∈ is strictly preferred to any non-

degenerate distribution that gives positive probability to some state strictly worse than .

Let us now formally investigate the SPCS of the motivating examples of Section 3. In Example

3.1, the attraction set () = {}. This is supported by the strategy profile in which player1 does not move from , player 2 moves from to and player 3 moves from to . Clearly

properties (a) and (b) in Definition 4.1 and the additional property in Definition 4.2 are satisfied

for this strategy profile. It forms a subgame perfect equilibrium because player 2 prefers to

and player 3 prefers , the final state after a move to , to , the final state after not moving

from and anticipating a later move of player 2 from to ; player 1 uses the pessimistic

criterion in Definition 4.4, thus prefers the degenerate distribution of utilities resulting from to

the distribution of utilities from and that gives positive probability to , which is a worse

outcome for player 1 than . Note that state is reachable from itself, but state is not. Note

also that in this example the SPCS implies that player 3 will move from to despite the fact

that is worse than , because player 3 perceives rather than as a final state.

Next we apply the SPCS to Example 3.2. Intuitively, it is not a subgame perfect equilibrium

strategy for player 1 to remain forever at state . Thus, state can never be a final outcome of

the game. Since transient states have no payoffs, there is no meaning for averaging the payoffs of

transient states, thus no player would be content with player 1 repeatedly moving forever between

states and . More specifically, the unique SPCS is {} with attraction sets () = {} foreach ∈ \. This is supported by the strategy profile in which player 1 moves from to

(unless previously stayed at , in which case player 1 stays at forever), stays at until player

2 plays, and then moves back from to ; player 2 moves from to (unless previously stayed

at , in which case player 2 stays at forever). This strategy profile forms a subgame perfect

equilibrium because player 2 prefers , the final state after a move to , to , which occurs after

not moving from and anticipating a later move of player 1 from to and staying there forever.

Furthermore, player 1 prefers a move from , the final state after not moving from forever, to

, the final state after moving from to and anticipating a later move of player 2 from to ;

in the case that player 2 does not move from , player 1 prefers , the final state after returning

to , to , the final state after remaining at , which leads to remaining there forever. Note that

state is reachable from itself, but state is not, thus player 1 is not restricted by property (a)

in Definition 4.1 to stay at after both of them did not move from . This example shows the

importance of non-stationary strategies in supporting the desirable efficient final state . Player

1’s choice depends not only on the current state but also on the history of play by player 2: if player

2 has not yet been selected to make a choice, player 1 gives player 2 the opportunity to move to the

efficient state ; but after player 2 was selected to make a choice and chose not to move, player 1

14

gives up and returns to . Note that the fact that this strategy does not allow player 2 more than

one opportunity to move is not crucial: one could imagine player 1 giving player 2 a finite number

of such opportunities to move until giving up and returning to ; in any such strategy, player 2

will eventually move to on the equilibrium path.

Finally, in the last section of his paper, Chwe (1994) noted that many details are left out of the

definition of the LCS, leaving important issues blurred and resulting in a limited ‘farsightedness’.

We now revisit the examples that Chwe introduced to demonstrate the shortcomings of the LCS,

and show that, in contrast, the SPCS solution overcomes these problems. In Examples 4.2-4.5,

even though the LCS and the SPCS coincide, they lead to different final states if the game starts

from an initial state outside these sets.

Example 4.2 Optimal rather than better choice. = {1} (i.e., there is a single player) with = { }, →{1} , →{1} and 1() 1() 1(). LCS = { }, implying that when

is the initial state, player 1 will move to either or , even though is strictly preferred to . As

Chwe has noted: “This example illustrates how the largest consistent set does not incorporate any

idea of “best response”: coalitions will move to any, not just the best, of the outcomes which are

better than the status quo” (Chwe, 1994: page 322). In this example, the unique SPCS = LCS,

since and are terminal states according to the effectiveness relation. However, the attraction

set {}() = {}, since, when is the initial state, is the unique outcome that can be supported

as a final state according to a subgame perfect equilibrium in the game tree corresponding to this

example. Thus the unique SPCS implies a move from to . Indeed, this demonstrates that in

contrast to the LCS, the SPCS is predicated on the notion of best response.

Example 4.3 Not joining a coalition if it is better otherwise. = {1 2} with = { }, →{12} , →{2} , 1() 1() 1() and 2() 2() 2(). Again, since and are

terminal states, and both are preferred by both players to state , the unique SPCS = LCS = { }.Thus from an initial state , the LCS implies that either coalition {1 2} will move to state or

player 2 will move to state . The difficulty with this conclusion, as Chwe noted, is that player

2 prefers state to state , thus should prefer not to join player 1 to move to , as he could do

better by himself moving to . Indeed, there exists a subgame perfect equilibrium in the game tree

corresponding to this example, in which the coalition {1 2} never moves following the selection ofstate as an initial state. This choice can be interpreted as resulting from player 2’s preference

not to join player 1 in such a move, and instead to wait until an opportunity arises to move alone

from state to state . Consequently the implication of the SPCS in this case is that player 2 will

not join player 1 in a move to state . Note, however, that the attraction set {}() = { }, asthere also exists a subgame perfect equilibrium in which the coalition {1 2} sometimes moves from to , in which case there is positive probability for to be a final state when the initial state is .

Example 4.4 Waiting and not moving until a better opportunity arises. = {1 2} with =

{ }, →{1} , →{2} , 1() 1() 1(), and 2() 2() 2(). We have LCS =

{ }, implying that from an initial state , either player 1 will move to state or player 2 will move

15

to state . The difficulty with this conclusion, as pointed out by Chwe, is that both players prefer

state to state , so that if player 1 is provided with the opportunity to be the first to move from

the initial state , he should rather forfeit this opportunity and wait until player 2 is able to move.

Although for this example the unique SPCS = LCS, according to the SPCS, if provided with the

opportunity to move, Player 1 would not move to state . Indeed, the attraction set {}() = {}because state is the unique outcome that can be supported by a subgame perfect equilibrium in the

game tree corresponding to this example following the selection of state as an initial state.

Example 4.5 Moving to preempt other moves that may lead to worse outcomes. = {1 2} with = { }, →{1} , →{2} , 1() 1() 1(), and 2() 2() 2(). Here, LCS

= { }, implying that from an initial state , player 1 will move to state . However, as noted

by Chwe, even though player 2 prefers state to state , he should deviate from to given an

opportunity to do so; otherwise player 1 will deviate to state , which for player 2 is even less

preferred than state . Although the unique SPCS = LCS, the attraction set {}() = { }, asboth these states can be supported as final states according to a subgame perfect equilibrium in the

game tree corresponding to this example following the selection of state as an initial state. Thus

the SPCS generates a move from state to either to or . Note that this example is somewhat

similar to Example 3.1.

5 Farsighted Normal Form Games

In this section we analyze normal form games when they are viewed as farsighted games.

Definition 5.1 A game in effectiveness form h ()∈ (→)⊆i is a -normal form game

if is integer with 1 ≤ ≤ | |, = ×∈, where is the set of alternatives available to player

, and for each 1 2 ∈ and each coalition ⊆ , 6= ∅, the effectiveness relation 1 → 2

holds if, and only if, || ≤ and 1 = 2 for each ∈ . A normal form game is a -normal form

game for some .

We view the concept of a pure Nash equilibrium as myopic, as it does not involve farsighted

reasoning. Formally, a myopic equilibrium is defined as follows.

Definition 5.2 In a normal form game, ∈ is a myopic equilibrium if () ≥ () for each

∈ ⊆ and ∈ such that → .4

We show that the SPCS solution provides a surprising and striking conclusion in the farsighted

analysis of normal form games: whenever the game possesses a myopic equilibrium, Pareto efficiency

4Our analysis also applies when considering Stackelberg games (and extensive form games with perfect information

in general) in their reduced normal form. This is justified by the view that in the farsighted perspective, the leader

and the follower are indistinguishable. Thus, in this case, the results concerning farsighted normal form games apply

when replacing pure Nash equilibria with pure Stackelberg equilibria.

16

is always and necessarily achieved. Moreover, Pareto efficiency is achieved even when coalition

deviations are not permitted.

When analyzing farsighted normal form games we assume that each player’s utility function is

continuous and that a SPCS is required to be a compact set. To state the main result we need

some additional pieces of terminology/notation. We say that a state ∈ is Pareto efficient if

there does not exist a state 0 ∈ such that (0) () for all ∈ . Further, we write that

≈ {∗} whenever a subset ⊆ is countable and consists of states that are all equivalent for

all players, i.e. there exists ∗ ∈ such that (0) = (

∗) for each player and 0 ∈ .

We can now state our main result.

Theorem 5.1 (1) For any normal form game, if ⊆ is a non-empty SPCS, then ≈ {∗}for some Pareto efficient ∗ ∈ ; and

(2) For any normal form game, if ≈ {∗} for some Pareto efficient ∗ ∈ such that (∗) ≥

() for some myopic equilibrium ∈ and all ∈ , then is a SPCS.

For a 1-normal form game, the strategies supporting a SPCS in Theorem 5.1 are similar to ‘grim-

trigger’ strategies commonly used in folk theorems within the repeated games literature (see, e.g.,

Osborne and Rubinstein 1994), adapted to our setting where a repeated stage game is not required.

The strategies involve a threat of reaching an undesirable outcome off equilibrium path in order to

create incentives to reach a good outcome on equilibrium path. These strategies could be thought of

as social norms that are publicly known to all players in the game. Since no player has an incentive

to deviate from acting according to these social norms, they are indeed implemented, leading to the

outcome of the corresponding SPCS. But note that contrary to folk theorems in repeated games,

which do not claim to achieve efficiency, we achieve efficiency as a necessary implication. Thus the

contribution of Theorem 5.1 is the conclusion that whenever the normal form game possesses a

pure Nash equilibrium, such social norms must lead to Pareto efficiency, which is achieved rather

than assumed.

Theorem 5.1 implies that a game with no Pareto efficient states cannot have a SPCS. Existence

of Pareto efficient states is a mild condition satisfied in many interesting settings as demonstrated

in the following Example.

Example 5.1 Prisoner’s Dilemma Negotiation. In this 1-normal form game there are two players,

player R (the row player) and player C (the column player), each having two available alternatives

and utilities as in the following matrix.

1 1 4 0

0 4 3 3

For this game there is a unique SPCS = {∗} with utilities (3 3), i.e., (∗) = 3 for both ,

and the attraction sets are () = for each ∈ \. This SPCS is supported by the followingstrategy profile : after a selection of any initial state, each player moves to the alternative

17

corresponding to ∗ and then does not move anymore, unless some player previously did not doso, in which case each player moves to the alternative corresponding to the Nash equilibrium (1 1)

and then does not move anymore (see also proof of Theorem 5.1). As a result, irrespective of the

initial state, the SPCS consists of the cooperative outcome in the game. Note that (1 1) and (3 3)

are reachable from themselves, but (0 4) and (4 0) are not because the player with 0 utility could

always deviate and force the Nash equilibrium (1 1). Also note that in this example it can be shown

that the LCS, the LCCS and the vNM FSS5 are equal to the top-left right-bottom diagonal of the

matrix.

Theorem 5.1 provides sufficient but not necessary conditions for a Pareto efficient state to form

a SPCS. Indeed, as demonstrated in the following example, we may have normal form games where

= {∗} is a SPCS for some Pareto efficient ∗ that does not weakly dominate any myopicequilibrium.

Example 5.2 Consider a farsighted 1-normal form game with two players (R and C), where each

player has two alternatives and the utilities are described by the following matrix.

1 1 1 2

0 2 2 2

For simplicity, we denote states in by their utility pairs according to the matrix. In this game

the unique myopic equilibrium is (2 2), and thus is reachable from itself. By Theorem 5.1, the

set = {(2 2)} is a SPCS. Note that (0 2) is not reachable from itself, as player R can move

and ensure the higher payoff 1. Similarly, (1 1) is not reachable from itself, as player C can move

and ensure the higher payoff 2. But note that (1 2) is reachable from itself, and moreover, the set

= {(1 2)} is also a SPCS. This is true due to the following strategy profile: at (1 2) both playersstay; at (2 2) R stays and C moves to (0 2), unless C previously stayed, in which case they both

stay and (2 2) becomes the final state; at (0 2) R moves to (1 1) and C stays; at (1 1) R stays and

C moves to (1 2). To see that indeed this strategy profile forms a subgame perfect equilibrium note

that an attempt of R to improve by moving to (2 2) fails because C punishes by moving to (0 2);

C does not mind moving to (0 2) because R would move to (1 1), in which case C would move and

end up at (1 2); if, on the other hand, C stays at (2 2) then R stays there also, leading to payoff 2

for C, which is not strictly better than the payoff at (1 2).

We now provide an extension of the analysis above that applies to normal form games having

no myopic equilibrium. When an infinite history does not converge for player , we say that

leads for that player to ‘swinging’, denoted by , and define () = and 0 = ∞. Let = ∪ {}. Our arguments up to now did not rely on swinging in a strategy profile supportinga SPCS. By allowing such swinging, we can strengthen Theorem 5.1. To this end, we extend the

utility function for each player to by defining () = −∞ for all (see e.g., Harsanyi 1974

5For formal definitions of the LCS, the LCCS and the vNM FSS see Section 2.

18

and Mariotti 1997). This assumption fits well with a negotiation setting, as swinging behavior is

similar to a disagreement outcome which all players prefer to avoid. Following a finite history

with even, positive cardinality, a strategy now generates for player a distribution | over(

0) ∈ . This distribution has countable support because there are finitely many coalitions

that may be selected and convergence to any 6= occurs after a finite history. Elements of this

support are called final states, now possibly including also swinging. For each player we define

Ψ| and | relying on (0) instead of (0), and add that |() = 1−P

∈Ψ|∩ |().Coalition preferences are also defined as before, except that the restriction to Σ is omitted, and

is extended to when ordered according to . For ∈ Σ we can still define Ψ| and | asbefore. There is no change in the definition of a SPCS.

A game is said to be generic if () 6= (0) for any player and any two distinct states 0.

The following proposition shows that our assumption on swinging implies that a SPCS cannot be

empty for a generic game.

Proposition 5.1 Consider a generic game with swinging assigned utility −∞ for all players. If a

SPCS exists, then it is non-empty, and includes all terminal states.

The following example demonstrates a non empty SPCS despite the non-existence of a myopic

equilibrium and the fact that the normal form game is not generic.

Example 5.3 Matching Pennies Negotiation. Consider a 1-normal form game with two players

(R and C), where each player has two alternatives and the utilities are described by the following

matrix.

1−1 −1 1−1 1 1−1

In this game there is no myopic equilibrium, so Theorem 5.1 does not apply. Still, any set consisting

of a single state in is a SPCS, supported by a strategy profile similar to the one used in Example

5.2, i.e., the two players always stay at the single state in the given SPCS, and at any other state

the player that receives utility −1 moves, unless both previously stayed there, in which case no onemoves anymore. In this case, the player that receives utility −1 in the single state of the givenSPCS prefers to accept this utility than to insist on moving, which would lead to swinging with

utility −∞. In this way, every state is reachable from itself. Each diagonal in the matrix is also

a SPCS, supported by a strategy profile in which the players move only from states not on the

diagonal. For comparison, note that it can be shown that the LCS = . But the outcome according

to the two solution concepts is somewhat different: the LCS implies no move from the initial state,

whereas the SPCS leads to the unique state, up to equivalence for both players, in any given SPCS.

Further note that in this example it can be shown that the LCCS is empty.

Given the above, Theorem 5.1 can be strengthened to an if and only if statement: for any

generic normal form game, ⊆ is a SPCS if, and only if, = {∗} for some Pareto efficientstate ∗ ∈ . Intuitively, regardless of the other players’ actions, any coalition can threaten with a

19

swinging final state in a farsighted normal form game if it so desires, simply by always electing to

change the current state whenever it is selected to make a choice. Such a threat can be used instead

of a myopic equilibrium within the strategy profile supporting a SPCS. We will now show that this

result can be strengthened even further: suppose that for player , swinging is always assigned some

utility (finite or −∞). Accordingly, consider an extension of Definition 4.2, allowing to be

a member of a SPCS. Indeed, as argued above, in a generic normal form game, regardless of the

other players’ actions, any coalition can force a swinging final state in a farsighted normal form

game if it so desires. Note that Proposition 5.1 still holds: a SPCS cannot be empty as this implies

that it includes a swinging final state. This leads to the following result.

Theorem 5.2 For any generic normal form game, ⊆ is a SPCS if, and only if, = {∗}for some Pareto efficient state ∗ ∈ such that (

∗) ≥ for all ∈ .

Theorem 5.2 provides us with necessary and sufficient conditions for the existence of a non-

empty SPCS when swinging is allowed and its normalized utility is zero for all players. Reflecting

back on Example 5.3 we can appreciate the effect of allowing a swinging behavior.

Example 5.4 Matching Pennies Negotiation with swinging assigned utility = 0. By Theorem

5.2, the singleton set {} consisting of a swinging final state is the unique SPCS. This is truebecause is the unique Pareto efficient state in providing all players with non-negative utilities

(note that any other state is also Pareto efficient but fails the non-negativity condition, thus it is

not reachable from itself).

5.1 Some Applications

In this section we briefly illustrate the application of the SPCS solution concept to analyze various

classical oligopolistic settings, and some further examples.

5.1.1 Bertrand and Cournot Competition Negotiation

In the Bertrand and Cournot examples discussed in this subsection, as is the case in, e.g., Xue

(1998), Masuda et al. (2000) and Suzuki and Muto (2006), the players “compete” only once, based

on the final quantities or prices reached. Intermediate quantities or prices are transient states in

a tacit negotiation. They are not offered to the consumers and they bear no utility consequences

whatsoever to the players. Thus, only the final values matter. Swinging should be avoided, since if

it occurs, revenues that could be realized by the players will be forfeited. In the following we show

that the SPCS extends the efficiency results obtained in the literature for symmetric settings using

vNM FSS (Masuda et al., 2000).

Bertrand Competition Negotiation Consider a market with sellers of a homogeneous prod-

uct that are engaged in price competition tacit negotiation, in which seller sets price . The sellers

face a downward sloping demand function (), where is the lowest among the prices they set.

20

Suppose that the sellers have equal sales power, so that when the same price is set by several sellers,

each of them sells the same quantity. Assume further that there are no fixed costs and the unit

cost of the product for seller is , with ≡ min . When the unit costs are not all equal thereis no real price competition because the seller with the lowest unit cost can always set the price

slightly below the second lowest unit cost and gain the entire market alone.

It is well known that when this situation is viewed as a normal form game, there is a unique

Nash equilibrium in prices: all sellers with cost set the price ∗ = and the market is split between

them equally with zero profit to each of them.

For this setting a SPCS is = {∗} where ∗ is a Pareto efficient state in which several (possiblyall) sellers with cost lower than the monopolistic price, i.e. the price that solves

max(− )(),

each set this price, which allows them to share the monopolistic profit equally between them. The

remaining sellers (if any) set some price higher than the monopolistic price and receive zero profits.

This follows from Theorem 5.1 because such states ∗ are the only Pareto efficient states in thatweakly Pareto dominate the unique Nash equilibrium state. For comparison, Masuda et al.(2000)

show for symmetric settings that the LCS contains all states with positive profits to all firms,

including Pareto non-efficient states.

Cournot Competition Negotiation Consider a market with sellers of a homogeneous prod-

uct that are engaged in quantity competition tacit negotiation, in which each seller sets quantity

. The sellers face a decreasing inverse demand function determining the market price as, for sim-

plicity, max{− 0}, where =P is the total quantity sold. Suppose that there are no fixed

costs and the unit cost of the product for seller is . Thus the profit to seller is ≡ (−−)when ≤ and zero otherwise.

It is well known that this normal form game has a unique Nash equilibrium in quantities, in

which player sells quantity

∗ = − − 1

+ 1

X(− )

and makes profit ∗ = (∗ )2 (this holds under the assumption that ≥ ( + 1)max −

P ,

otherwise the market is not large enough to include all sellers in equilibrium). In the symmetric

case in which = for all , each player sells in equilibrium −+1

and makes profit ( −+1

)2.

To find the SPCS solution for this setting, Theorem 5.1 says that we need to compute the

Pareto efficient states that weakly Pareto dominate the unique Nash equilibrium state. Since in

any state ∈ , the quantity sold =

−− for all , the profit of any seller can be writtenas a function of the total quantity sold and the other sellers’ profits, given by

[ () 6=] ≡ (−X

6=

− −)(− −).

21

Therefore a Pareto efficient vector of profits, for which ≥ ∗ for all 6= , must satisfy that

maximizes [ () 6=] over the variable and subject to the constraint ≥ ∗ .In the symmetric case, [ () 6=] simplifies to (−−)−

P 6= , for which the maximal

equals −2, independently of the other sellers’ profits. In this case seller ’s profit is =

(−2)2 −P 6= , as long as ≥ ∗ for all (including ), and the total profit is (−

2)2. Such

Pareto efficient states always exist because for all ≥ 2, the total profit is always higher than thetotal Nash equilibrium profit, ( −

+1)2. This allows the total profit increase to be shared in some

way between the sellers. Therefore the SPCS solution implies that the sellers would share between

them a monopolistic total quantity, allowing them to share a monopolistic total profit in a way that

improves for each of them on the Nash equilibrium profit. For comparison, Masuda et al.(2000)

show for symmetric settings that the LCS contains all states with non-negative profits to all firms,

including Pareto non-efficient states.

5.1.2 Decentralized Supply Chain Contracting

The SPCS can be used to analyze decentralized supply chains and can be shown to lead to full

coordination and Pareto efficiency. We demonstrate this general point in the classic setting of

a vertical decentralized supply chain, based on the newsvendor model, with a single supplier and

several competing retailers. As is well known, system coordination and Pareto efficiency are trivially

achieved in a cooperative bargaining/negotiation modelling of this setting (see, e.g., Nagarajan and

Bassok 2008). We demonstrate that they can also be achieved in a non-cooperative modelling

of this problem when players are assumed to be farsighted. Moreover, coordination is achieved

contrary to the inability of revenue sharing contracts to do so in general (see Cachon and Lariviere

2005).

The market consists of retailers that face uncertain demand for a differentiated product, where

retailer ’s demand (1 ) is decreasing in their own unit price , increasing in each of the

other retailers’ unit prices and depends on a common random variable . Each retailer decides

about the price and the inventory level to be ordered from a single supplier before demand

realization. The supplier charges each retailer a wholesale price per unit ordered and a share

of the retailer’s total revenue. The unit cost for the supplier is with no fixed costs.

When this arrangement is viewed as a revenue sharing contract signed between the supplier

and the retailers, the setting becomes a two stage model: first the supplier sets the charges ()

under equal6 contractual terms across the retailers, and then the retailers set the prices and

order the quantities . Solving for Nash equilibrium of the second stage, each retailer’s profit, (1−)min{(1 )}− (where is the expectation operator over the distribution

of the random variable ), is maximized given the other retailers’ choices, leading to equilibrium

prices and quantities as a function of the contract parameters (). In the first stage one seeks a

contract that coordinates the supply chain by achieving the system optimal quantities and prices.

Note that such revenue sharing contracts, when the retailers must be offered equal contractual

6Equal contractual terms are legally required in the USA by the Robinson—Patman Act of 1936.

22

terms by the supplier, are unable to coordinate the system when the retailers are asymmetric (see,

e.g., Cachon and Lariviere 2005 and Krishnan and Winter 2011).

The conclusion of the SPCS solution for this setting is very different. A state in the game

consists of a vector [ (1 1) ( )], so the wholesale price, the revenue share , the

inventory levels and the prices are viewed as alternatives than can be altered in the game tree

without limit. By Theorem 5.1, we look for the Pareto efficient states that weakly dominate the

Stackelberg equilibrium. The wholesale price and the revenue share do not affect the system

profit, but only determine how it is shared between the players. Therefore a SPCS is a single state

∗ consisting of optimal prices and quantities for the system, and wholesale price and revenue sharevalues that split the optimal system total profit such that each player is weakly better off than in the

equilibrium. Such wholesale price and revenue share values exist whenever the equilibrium system

profit is strictly lower than the optimal one. Therefore, in contrast to the traditional equilibrium

analysis, the SPCS solution leads to system coordination.

6 Summary

Farsighted players, by contrast with myopic players, recognize the possible reactions by other players

to their actions, and deviate from a current state only if they believe that such a deviation would

not lead to an inferior outcome. Thus, not surprisingly, the analysis of games could yield strikingly

different results depending on whether players are assumed to be farsighted or myopic. Indeed,

recent contributions to the literature reveal that in competitive situations, farsighted players are

more likely to cooperate than myopic players. For example, Sošic (2006) showed that farsighted

players will tend to cooperate and retain the grand coalition structure, while myopic players are

more likely to form sub-coalitions leading to suboptimal results. Similarly, Granot and Yin (2008)

showed that farsighted suppliers of complementary products will cooperate and form an alliance of

all suppliers for the purpose of contracting with a downstream assembler, which will make all players

best off. By contrast, myopic players will not cooperate at all, and each supplier will independently

contract with the assembler, resulting with the worst performance for all players.

In general, many applications analyzed in the literature have adopted Chwe’s (1994) approach to

farsightedness, wherein it is assumed that players follow the indirect dominance logic. Players will

deviate from a status quo state if subsequent deviations by motivated coalitions would eventually

result with a better outcome. Solution concepts taking this approach include the LCS and its

refinements, including, for example, the vMM FSS and the LCCS. However, as demonstrated in our

paper and, indeed, recognized by Chwe, indirect dominance endows players with limited foresight.

Note, however, that the vNM FSS was demonstrated recently to provide insightful outcomes (see,

e.g., Masuda et al. 2000, Suzuki and Muto 2006, and Mauleon et al. 2011), although not always

(Kawasaki and Muto 2009).

Accordingly, we have introduced in this paper a new approach to farsightedness, leading to

the Subgame Perfect Consistent Set (SPCS). Our SPCS retains a main feature of several existing

23

farsighted solution concepts, namely a vNM type of consistency. However, it replaces indirect

dominance with subgame perfection, which endows players with complete foresight, and it does not

impose the pessimism assumption. Surprisingly, the SPCS is shown to lead, e.g., to (weak) Pareto

efficiency, without coalitions, in any normal form game having a myopic equilibrium. Furthermore,

as compared to the LCS, (i) it leads to an optimal rather than a better choice, (ii) it allows players

not to join coalitions if they so desire, (iii) it lets players wait for a better opportunity rather than

move, and (iv) it allows preemptions of other moves that may lead to worse outcomes.

Moreover, we have shown that farsighted players who adopt the SPCS reasoning will achieve full

cooperation and overcome misaligned incentives in a variety of settings in which Chwe’s type players

may fail to do so. Specifically, our farsighted players will always share the monopolistic profit in

farsighted settings based on Bertrand and Cournot competition and will always achieve supply

chain coordination and Pareto efficiency in the decentralized setting of the classical newsvendor

model and its variants.

Future work may consider the extension of our analysis beyond ordinary normal form games to

include, for example, coalition and network formation and matching.

Appendix A: Proofs

Proof of Proposition 4.1. We construct a set and a supporting strategy profile satisfying

the requirements of Definition 4.2. In the subgame following a given finite history , the assumptions

of the proposition imply that any infinite history converges after a finite length, the maximal value

of which is referred to as the length of the subgame. Therefore, a subgame perfect equilibrium ∗ canbe found using backwards induction with respect to the length of subgames. Along the backwards

induction construction of ∗, for a finite history with () = , denote by ∗() = {}|∗()|=1

the set of optimal choices for coalition taking into account future ∗ choices. Let ∗() = ,

where = 1 + |{ | = 1 3 = ∅}| mod |∗()|. Let be the set of all states such

that is reachable from itself under this construction, i.e. after it is selected as an initial state

it is also a final state. Modify ∗ so that all coalitions always stay at any ∈ . The set is

non-empty because it includes in particular terminal states according to the effectiveness relation,

i.e. states which no coalition can alter, the existence of which is ensured because is finite and

the effectiveness relation is acyclic. Let = ∗. According to , after any move to any state,

all optimal choices realize depending on the number of times the empty coalition was previously

selected to play. By construction, satisfies Definition 4.2 supported by , where is a

subgame perfect equilibrium strategy profile that satisfies property (a) in Definition 4.1. Moreover,

it satisfies property (b) in Definition 4.1 because after some initial move from an initial state (on

or off play path), any final state is by subgame perfection reachable from itself and thus belongs

to , and any reachable state ∈ is a final state because realizes all optimal choices after

any move to any state.

The following notation is used in the proof of Theorems 5.1 and 5.2. For every two states 0

24

and coalition , 0 denotes the state 00 ∈ defined by 00 = for all ∈ and 00 = 0 otherwise.

Proof of Theorem 5.1. We first prove (1). Let ⊆ be a non-empty SPCS, and consider

some subgame perfect equilibrium strategy profile that supports as a SPCS. We first show

that any state 2 ∈ is reachable from any state 1 ∈ . Fix such 1 2. For any finite history

such that 0 = 1, i.e., 1 is the initial state, let 0 be the minimal even, positive number such

that (2)−1(−2) = 2, or let 0 = 0 if such does not exist. For the subgame in which 1 is

selected as the initial state, consider the strategy profile 0 defined for any finite history such that0 = 1 and () = as follows: if →= ∅ then 0() = −1; if → 6= ∅ and 0 = 0 then let

0() = (2)(−1), otherwise let

0() = [

2 ()

=0−1]. In this subgame, according to

0,each coalition , when selected to make a choice whether to keep the current state or to move to

a new state, moves by choosing the alternatives ∈ corresponding to 2 for each player ∈

and then does not move anymore, except after the first time a coalition 0 was selected which couldmove from the current state to 2, i.e., = 2 for all ∈ 0, and in fact moved to 0, in whichcase the strategy profile 0 continues exactly as does following the selection of 2 as an initial

state and after an initial move by 0 from 2 to 0. Consequently the state 2 is the final state forsure according to 0 following 1 as the initial state. It is also the final state for sure according to following the selection of 2 as an initial state, because in this case dictates no moves — see

the definition of a SPCS. Moreover, since is a subgame perfect equilibrium, no coalition strictly

prefers to deviate from it after any history, in particular after a history in which all coalitions

stayed at 2. Thus no coalition strictly prefers to deviate from 0 after any history, in particularafter a history involving a partial process of moves from 1 to 2, because all other coalitions are

dictated by 0 to move to 2, and deviation alone of this coalition would lead to exactly the sameweakly inferior distributions over final states as in the case of deviation of this coalition from 2

according to . Therefore 0 is also a subgame perfect equilibrium, proving the reachability of 2

from 1. Since (1) = for any 1 ∈ , it follows that is a countable set, as property (b) of a

-farsighted strategy profile requires that = Ψ |, where Ψ | is a countable set.Now suppose that there are at least two distinct states in and a player that is not indifferent

between them. Since a SPCS is compact and players’ utility functions are continuous, there is a

worst state ∈ for player . Consider the history = ( {}), i.e., is selected as an initial stateand player is selected to make a choice whether to keep or to move to a new state. According

to , since ∈ and is a SPCS, player is supposed to keep , anticipating it as a final

state for sure — see the definition of a SPCS. But note that in case of an initial move by player

, off the equilibrium path, from to a new state, by property (b) of a -farsighted strategy

profile, any final state must be in , and, since all states in are reachable following a move away

from , there is a positive probability that the final state according to will be strictly better

than for player . Therefore, player can either choose for sure, or alternatively, can choose a

first order stochastically dominating distribution, as is the worst state in for . Since the first

order stochastically dominating distribution is strictly preferred, the choice of player to keep

violates the assumption that is a subgame perfect equilibrium. Since this argument holds for

25

any subgame perfect equilibrium strategy profile , we derived a contradiction to the assumption

that is a SPCS. Therefore, since is non-empty, it must consist of states that are all equivalent

for all players, i.e. ≈ {∗} for some ∗ ∈ .

Suppose now that ∗ is not Pareto efficient. Then there exists ∈ \ such that () (∗)

for each ∈ . We first show that is reachable from itself. For the subgame in which is selected

as the initial state, consider the strategy profile 0 defined such that 0() = for any coalition

and any finite history = ( 1 2 ), and 0() = () for any other history . In this

subgame, according to 0, each coalition stays at , except when some coalition has previously

not done so, in which case the strategy profile 0 continues exactly as does. Therefore 0 leadson equilibrium path to the final state for sure. Moreover, 0 inherits its subgame perfection from after any history in this subgame with some move away from , and it is also subgame perfect

following a history with no previous such moves because property (b) of a -farsighted strategy

profile applied to implies that an initial move away from will lead for sure to a final state

equivalent to ∗, which is strictly worse than for any coalition. Therefore is reachable from itself.Furthermore, applying the same argument to , any coalition anticipates that coalitions selected

later to make a choice can either stay at , which, by property (a) of a -farsighted strategy profile

and since is reachable from itself, would lead to as the final state for sure, or to make an initial

move away from , which, by property (b) of a -farsighted strategy profile, would lead for sure to

a final state equivalent to the strictly worse ∗. Therefore, no coalition will move away from after

it was selected as an initial state, contradicting the definition of a SPCS because ∈ . Thus ∗

must be Pareto efficient.

Next we prove (2). Suppose that ≈ {∗} for some Pareto efficient ∗ ∈ such that (∗) ≥ ()

for some myopic equilibrium ∈ and all ∈ . Index with {}||=1. For any finite history ,

let 0 be the minimal even, positive number such that −1 6= ∅ and there exists an even, positivenumber 0 such that 0 6= 0, or let

0 = 0 if such does not exist; let = 1 +

120 mod ||;

let 1 = 1 if there exists an even, positive number 0 such that () 6= for ∈ −1 and

→−1 6= ∅, otherwise let 1 = 0. Consider the strategy profile defined for any finite history

such that () = as follows: if →= ∅ then () = −1, otherwise: (a) if 0 ∈ then

(a1) () = 0 when 0 = 0, and (a2) () = (

)(−1) when 0 0 and 1 = 0, and (a3)

() = (−1) when 1 = 1; and (b) if 0 ∈ then (b1) () = 0 when 0 = 0, 0 ∈ (0)

and {}=13⊇ { ⊆ |→ 6= ∅}, otherwise (b2) () = ()(−1) when 1 = 0, and

(b3) () = (−1) when 1 = 1. According to , if an initial state in is selected or if all

effective coalitions were selected and none moved from an initial state reachable from itself, then

no coalition moves and this is the final state for sure; otherwise, if an initial move is made away

from an initial state in , or if all previously selected coalitions stayed at an initial state not in

, then each ∈ , the choice of which is determined by the realization of the history length

until the first selection of a non-empty coalition following the initial move, becomes the final state

by each coalition moving by choosing the alternative corresponding to and then not moving

anymore; after any deviation from the above, each coalition moves to the alternative corresponding

26

to and then does not move anymore. This strategy profile clearly satisfies both properties (a)

and (b) in Definition 4.1 (note that since is countable, there is no problem with the condition

Ψ | = ∩ (2)) and the additional property in Definition 4.2. We now verify that it forms asubgame perfect equilibrium. First note that on equilibrium path, whether or not the initial state

is in , any final state is equivalent for all players to ∗, and this is also the final state after aninitial move away from an initial state in . Any deviation after an initial move leads to the final

state , and since (∗) ≥ () for all ∈ , no coalition prefers to deviate from the equilibrium

path, leading indeed to a final state in . Moreover, after choices by all effective coalitions to stay

off equilibrium path at an initial state reachable from itself, it becomes the final state, supported

in equilibrium by its own reachability. Similarly, is reachable from itself because it is a myopic

equilibrium, thus when reaching off equilibrium path, no coalition prefers to deviate, as this can

only lead to a final state at most as good as . Consequently the strategy profile is a subgame

perfect equilibrium.

Proof of Proposition 5.1. If there are terminal states and a SPCS exists, then it is non-

empty because it must include all of them, as no coalition can alter a terminal state when it is

selected as an initial state. Suppose then that there are no terminal states. An empty SPCS

implies the existence of a subgame perfect equilibrium strategy profile that, on equilibrium

path, dictates a move by some coalition at any state ∈ , i.e. leads to swinging for sure, as

the game is generic. We first show that there exists a state which is reachable from itself. If

there exists a subgame, off the equilibrium path, with an initial state which is also the final

state according to , then is reachable from itself using the restriction of to this subgame,

which by assumption is a subgame perfect equilibrium. Otherwise, dictates a move by some

coalition after any history. In this case we fix some state and define a strategy profile 0 such that0() = for any coalition and any finite history = ( 1 2 ), and 0() = ()

for any other history . Then under 0, no coalition ever moves from when it is the initial state,

thus making a final state for sure, and 0 is a subgame perfect equilibrium because a deviation

from it at any finite history, on or off equilibrium path, will result in swinging, which is strictly

worse than for the deviating coalition. Therefore is reachable from itself. By property (a) of

a -farsighted strategy profile, after a history off the equilibrium path in which all coalitions

stayed at , since is reachable from itself, any coalition will anticipate that coalitions selected

later to make a choice will continue to stay at . Since is a subgame perfect equilibrium, along

this history any coalition can therefore ensure a bounded from below distribution of payoffs. Thus,

immediately after is selected as an initial state, any coalition would prefer to deviate and stay at

than to follow the equilibrium path and reach swinging, a contradiction to the subgame perfection

of . Therefore must be non-empty.

Proof of Theorem 5.2. Suppose first that ⊆ is a SPCS supported by . Suppose that

(0) for some

0 ∈ and some player ∈ . Consider the case where 0 is selected as aninitial state and player is selected to make a choice whether to keep 0 or to move to a new state.Since 0 ∈ , by the definition of a SPCS, dictates that player will keep 0, anticipating it as

27

a final state. But this player could instead adopt a strategy {} in which he always changes thecurrent state whenever such an opportunity arises, thus forcing a swinging final state, as the game

is generic. Since the utility, , of a swinging final state is strictly higher than (0), the choice of

player to keep 0 violates the subgame perfect equilibrium, a contradiction. Therefore (0) ≥

for all ∈ and ∈ . By an argument similar to the one used in the proof of Proposition 5.1,

is non-empty. We can now use an argument similar to the one used in the proof of Theorem 5.1,

modified only in the part proving uniqueness up to equivalence for all players, which would instead

conclude that ∩ = {∗}. Then using the remaining argument concerning Pareto efficiency,swinging can be eliminated from when it is not Pareto efficient. Therefore we conclude that

= {∗} for some Pareto efficient ∗ ∈ such that (∗) ≥ for all ∈ .

For the other direction, suppose that = {∗} for some Pareto efficient ∗ ∈ such that (∗) ≥

for all ∈ . We use an argument similar to the one used to prove (2) in Theorem 5.1. For

any finite history , let 0 be the minimal even, positive number such that −1 6= ∅ and thereexists an even, positive number 0 such that 0 6= 0, or let

0 = 0 if such does not exist;

let 1 = 1 if there exists an even, positive number 0 such that () 6= ∗ for ∈ −1 and→−1 6= ∅, otherwise let 1 = 0. Consider the strategy profile defined for any finite history suchthat () = as follows: if →= ∅ then () = −1, otherwise: (a) if 0 ∈ ∩ then (a1)

() = 0 when 0 = 0, and (a2)

() = (

∗)(−1) when 0 0,

1 = 0 and

∗ 6= , and (a3)

() = (−1) when 1 = 1 or

∗ = , for some ∈ such that (−1) 6= −1; and (b) if0 ∈ then (b1) () = 0 when

0 = 0, 0 ∈ (0)∩ and {}=13

⊇ { ⊆ |→ 6= ∅},otherwise (b2) () = (

∗)(−1) when 1 = 0 and ∗ 6= , and (b3) () = (−1) when1 = 1 or ∗ = , for some ∈ such that (−1) 6= −1. According to , if an initial

state in ∩ is selected or if all effective coalitions were selected and none moved from an initial

state reachable from itself, then no coalition moves and this is the final state for sure; otherwise, if

an initial move is made away from an initial state in ∩ , or if all previously selected coalitionsstayed at an initial state not in ∩, then ∗ becomes the final state by each coalition moving bychoosing the alternatives corresponding to ∗ and then not moving anymore, or by keep changingthe current state whenever it is selected to make a choice in the case of swinging; after any deviation

from the above, each coalition conducts a swinging behavior. This strategy profile clearly satisfies

both properties (a) and (b) in Definition 4.1 and the additional property in Definition 4.2. We now

verify that it forms a subgame perfect equilibrium. First note that on equilibrium path, whether

or not ∗ is the initial state, ∗ is the final state, and this is also the final state after an initialmove away from the initial state ∗. Any deviation after an initial move leads to a swinging finalstate, in which case each player has utility , as the game is generic, and since (

∗) ≥ for all

∈ , no coalition prefers to deviate from the equilibrium path, leading indeed to the final state

∗. Moreover, after choices by all effective coalitions to stay off equilibrium path at an initial state

reachable from itself, it becomes the final state, supported in equilibrium by its own reachability.

Similarly, a swinging final state is reachable from itself because it is forced by the other coalitions

when they are selected to make a choice. Consequently the strategy profile is a subgame perfect

28

equilibrium.

Appendix B: Further Comparison of the SPCS and the LCS

The apparent similarity between Chwe’s consistent set and the SPCS definitions is clear. In both

cases, if is the stable set then a deviation from any ∈ will not occur, a deviation from

∈ will occur and only states in can be final states. The main difference between the two

concepts is the reasoning that governs the sequence of deviations the players follow after a deviation

from a stable or a non-stable state. In the LCS, the reasoning is exclusively indirect dominance,

while in the SPCS solution, the sequence of moves leads to a final outcome in the SPCS which is

supported by a subgame perfect equilibrium in the Farsighted Game, providing the players with

more complete foresight.

An additional similarity between the SPCS and the LCS concerns the belief players are assumed

to have on other players’ future actions. In the LCS, all players believe in the same final state, which

provides a reasonable justification for their collaboration to carry out the corresponding sequence

of moves that leads to the final state. Similarly, all players believe in the same strategy profile

supporting a SPCS, leading to the subgame perfect reasoning that justifies their part in moving to

the final state. Both kinds of reasoning can be viewed as resulting from social norms that players

possess when entering the game (as further explained in the discussion following Theorem 5.1).

One of the differences between the LCS and the SPCS solutions concerns their existence, unique-

ness and non-emptiness. In view of the isotonicity of , the LCS always exists uniquely, simply

being the set of all outcomes which can possibly be stable (Proposition 1 in Chwe 1994, proved

by employing Tarski’s fixed point argument on the function ). However, the LCS may be empty.

Sufficient conditions for the non-emptiness of the LCS were provided by Chwe for games with a

finite or countably infinite set of states . These sufficient conditions include non-existence of

certain infinite sequences of indirect dominance. Xue (1997) provided weaker sufficient conditions

that also do not require a countably infinite . By contrast, a SPCS may not exist (Example 4.1

— note that in this example the LCS exists and consists of all states 1), but a SPCS must be

non-empty when it exists (Proposition 5.1), and there may be multiple non-empty SPCSs, each of

which is a largest set in the sense of set inclusion (Theorem 5.1). Theorems 5.1 and 5.2 provide nec-

essary conditions and sufficient conditions (that are also identical in Theorem 5.2) for the existence

of a non-empty SPCS for normal form games. The following is another example (adapted from

an example by Chwe 1994) that highlights the difference between the LCS and SPCS regarding

existence.

Example 6.1 There is one player, is the set of natural numbers, () = for each ∈ , and

the player can move from any to +1 (with no other possible moves included in the effectiveness

relation). In this example, LCS = ∅, but the SPCS does not exist because there is no subgame perfectequilibrium: the player strictly prefers to keep moving to higher and higher states indefinitely, but

this means swinging that is assigned utility −∞, thus the player strictly prefers not moving from

29

any state. This example highlights a difference between the two kinds of farsighted reasoning: in the

LCS, the player continues to move to higher states although this does not lead to a good outcome,

whereas such moves cannot be considered as part of subgame perfect equilibrium strategies supporting

a SPCS.

References

Brams, S.J. 1994. Theory of Moves. Cambridge University Press, New York, NY.

Brams, S.J., D. Wittman. 1981. Nonmyopic Equilibria in 2x2 Games. Conflict Management and

Peace Science, 6, 39-62.

Cachon, G.P., M.A. Lariviere. 2005. Supply Chain Coordination with Revenue-Sharing Contracts:

Strengths and Limitations, Management Science, 51 (1), 30—44.

Chamberlin, E.H. 1933. The theory of monopolistic competition. Harvard University Press,

Cambridge, MA.

Chwe, M.S-Y. 1994. Farsighted Coalitional Stability. Journal of Economic Theory, 63, 299-325.

Diamantoudi, E., L. Xue. 2003. Farsighted Stability in Hedonic Games. Social Choice and

Welfare, 21(1), 39-61.

Granot, D., G. Sošic. 2005. Formation of Alliances in Internet Based Supply Exchanges. Man-

agement Science 51 (1), 92-105.

Granot, D., S. Yin. 2008. Competition and Cooperation in Decentralized Push and Pull Assembly

Systems. Management Science, 54 (4), 733-747.

Greenberg, J.H. 1990. The Theory of Social Situations. Cambridge University Press, New York,

NY.

Harsanyi, J.C. 1974. An Equilibrium-Point Interpretation of Stable Sets and a Proposed Alterna-

tive Definition. Management Science, 20 (11), 1472-1495.

Herings P.J-J., A. Mauleon, V. Vannetelbosch. 2009. Farsightedly stable networks. Games and

Economic Behavior, 67(2), 526—541.

Kawasaki and Muto. 2009. Farsighted stability in provision of perfectly “Lumpy” public goods.

Mathematical Social Sciences, 58, 98-109.

Kilgour, D.M. 1984. Equilibria for far-sighted players. Theory and Decision, 16, 135-157.

Konishi, H., D. Ray. 2003. Coalition formation as a dynamic process. Journal of Economic

Theory, 110, 1—41.

30

Krishnan, H., R.A. Winter. 2011. On the Role of Revenue-Sharing Contracts in Supply Chains.

Operations Research Letters, 39 (1), 28-31.

Mariotti, M. 1997. A Model of Agreements in Strategic Form Games. Journal of Economic Theory,

74, 196-217.

Masuda, T., A. Suzuki, S. Muto. 2000. Farsighted von Neumann-Morgenstern Stability Leads to

Efficiency in Oligopoly Markets. Working paper.

Mauleon, A., V. Vannetelbosch. 2004. Farsightedness and Cautiousness in Coalition Formation

Games with Positive Spillovers. Theory and Decision, 56, 291—324.

Mauleon, A., V. Vannetelbosch, W. Vergote. 2011. von Neumann—Morgenstern Farsightedly

Stable Sets in Two-Sided Matching. Theoretical Economics 6, 499—521.

Nagarajan, M., Y. Bassok. 2008. A Bargaining Framework in Supply Chains: The Assembly

Problem. Management science, 54 (8), 1482-1496.

Nagarajan, M., G. Sošic. 2007. Stable Farsighted Coalitions in Competitive Markets. Manage-

ment Science, 53 (1), 29-45.

Nagarajan, M., G. Sošic. 2009. Coalition Stability in Assembly Models. Operations Research, 57

(1), 131—145.

Osborne, M.J., A. Rubinstein. 1994. A Course in Game Theory. MIT Press, Cambridge, MA.

Page F.H., M.H. Wooders, S. Kamat. 2005. Networks and Farsighted Stability. Journal of

Economic Theory, 120, 257—269.

Page F.H., M.H. Wooders. 2009. Strategic basins of attraction, the path dominance core, and

network formation games. Games and Economic Behavior, 66, 462—487.

Sošic, G. 2006. Transshipment of Inventories Among Retailers: Myopic vs. Farsighted Stability.

Management Science, 52 (10), 1493-1508.

Suzuki, A., S. Muto. 2006. Farsighted Behavior Leads to Efficiency in Duopoly Markets. Annals

of the International Society of Dynamic Games, 8(6), 379-395.

von Neumann, J., O. Morgenstern. 1944. Theory of Games and Economic Behavior. New York:

John Wiley and Sons.

Xue, L. 1997. Nonemptiness of the Largest Consistent Set. Journal of Economic Theory, 73 (2),

453-459.

Xue, L. 1998. Coalitional Stability under Perfect Foresight. Economic Theory, 11 (3), 603-627.

31