Computational Complexity

2012 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Cooperative Games (Von Neumann–Morgenstern Stable Sets)

  • Jun Wako
  • Shigeo Muto
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1800-9_43

Article Outline

Glossary

Definition of the Subject

Introduction

Stable Sets in Abstract Games

Stable Set and Core

Stable Sets in Characteristic Function form Games

Applications of Stable Sets in Abstract and Characteristic Function Form Games

Stable Sets and Farsighted Stable Sets in Strategic Form Games

Applications of Farsighted Stable Sets in Strategic Form Games

Future Directions

Bibliography

Keywords

Nash Rabie Monopoly 

Definition of the Subject

The von Neumann–Morgenstern stable set for solution (hereafter stable set) is the first solution concept in cooperative game theorydefined by J. von Neumann and O. Morgenstern. Though it was defined cooperative games in characteristic function form, von Neumann and Morgenstern gavea more general definition of a stable set in abstract games. Later, J. Greenberg and M. Chwe cleared a way to apply the stable set conceptto the analysis of non‐cooperative games in strategic and extensive forms. Though general existence of stable sets in characteristic function formgames was denied by a 10‐person game presented by W.F. Lucas, stable sets exist in many important games. In voting games, for example, stablesets exist, and they indicate what coalitions can be formed in detail. The core, on the other hand, can be empty in voting games, though it is one of thebest known solution concept in cooperative game theory. The analysis of stable sets is not necessarily straightforward, since it can reveal a varietyof possibilities. However, stable sets give us deep insights into players' behavior in economic, political and social situations such as coalitionformation among players.

Introduction

For studies of economic or social situations where players can take cooperative behavior, the stable set was defined by von Neumann andMorgenstern [31] as a solution concept for characteristic function form cooperative games. Theyalso defined the stable set in abstract games so that one can apply the concept to more general games including non‐cooperativesituations. Greenberg [9] and Chwe [3] cleareda way to apply the stable set concept to the analysis of non‐cooperative games in strategic and extensive forms.

The stable set is a set of outcomes satisfying two stability conditions: internal and external stability. The internal stability means thatbetween any two outcomes in the set, there is no group of players such that all of its members prefer one to the other and they can realize the preferredoutcome. The external stability means that for any outcome outside the set, there is a group of players such that all of its members havea commonly preferred outcome in the set and they can realize it. Though general existence was denied by Lucas [18] and Lucas and Rabie [21], the stable set has revealed manyinteresting behavior of players in economic, political, and social systems.

Von Neumann and Morgenstern (also Greenberg) assumed only a single move by a group of players. Harsanyi [13] first pointed out that stable sets in characteristic function form games mayfail to cover “farsighted” behavior of players. Harsanyi's work inspired Chwe's [3]contribution to the formal study of foresight in social environments.

Chwe paid attention to a possible chain of moves such that a move of a group of players will bring about a new move of anothergroup of players, which will further cause a third group of players to move, and so on. Then the group of players moving first should take intoaccount a sequence of moves that may follow, and evaluate their profits obtained at the end. By incorporating such a sequence of moves,Chwe [3] defined a more sophisticated stable set, which we call a farsighted stable set inwhat follows. Recent work by Suzuki and Muto [51,52] showed that the farsighted stable set provides more reasonable outcomes than the original (myopic) stable set inimportant classes of strategic form games.

The rest of the chapter is organized as follows. Section “Stable Sets in AbstractGames” presents the definition of a stable set in abstract games. Section “Stable Setand Core” shows basic relations between the two solution concepts of stable set and core. Section “Stable Sets in Characteristic Function form Games” gives the definition of a stable set in characteristicfunction form games. Section “Applications of Stable Sets in Abstract and Characteristic Function FormGames” first discusses general properties of stable sets in characteristic function form games, and then presents applications ofstable sets in abstract and characteristic function games to political and economic systems. Section “StableSets and Farsighted Stable Sets in Strategic Form Games” gives the definitions of a stable set and a farsighted stable setin strategic form games. Section “Applications of Farsighted Stable Sets in Strategic FormGames” discusses properties of farsighted stable sets and their applications to social and economicsituations. Section “Future Directions” ends the chapter withremarks. Section “Bibliography” offers a list of references.

Stable Sets in Abstract Games

An abstract game is a pair \( {(W,\succ) } \) of a set of outcomes W and an irreflexive binaryrelation \( { \succ } \) on W, where irreflexivity means that \( { x\succ x }\) holds for no element \( { x\in W }\). The relation \( { \succ }\) is interpreted as follows: if \( { x\succ y }\) holds, then there must exist a set of players such that they can induce x from y by themselves and all of them are better off in x.

A subset K of W is called a  stableset of abstract game \( { (W,\succ) } \) if the following two conditions are satisfied:

  1. 1.

    Internal stability : For any two elements \( { x,y\in K, x\succ y } \) never holds.

     
  2. 2.

    External stability: For any element \( { z\notin K } \), there must exist \( { x\in K } \) such that \( { x\succ z } \).

     

We explain more in detail what the external and internal stability conditions imply in the definition of a stable set. Suppose players havecommon understanding that each outcome inside a stable set is “stable” and that each outcome outside the set is“unstable”. Here the “stability” means that no group of players has an incentive to deviate from it, and the“instability” means that there is at least one group of players that has an incentive to deviate from it. Then the internal and externalstability conditions guarantee that the common understanding is never disproved, and thus continues to prevail.

In fact, suppose the set is both internally and externally stable, and pick any outcome in the set. Then by internal stability, no group of playerscan be better off by deviating from it and inducing an outcome inside the set. Thus no group of players reaches an agreement to deviate, which makes eachoutcome inside the set remain stable. Deviating players may be better off by inducing an outcome outside the set; but outcomes outside the set arecommonly considered to be unstable. Thus deviating players can never expect that such an outcome will continue. Next pick any outcome outside theset. Then by external stability, there exists at least one group of players who can become better off by deviating from it and inducing an outcome insidethe set. The induced outcome is considered to be stable since it is in the set. Hence the group of players will deviate. Hence each outcome outside theset remains unstable.

Stable Set and Core

Another solution concept that is widely known is a core. For a given abstract game \( { G=(W,\succ) } \), a subset Cof W is called the core of G if\( C = \{ x \in W|\,{\rm there}\,{\rm is}\,{\rm no}\,y \in W\,{\rm with}\,y \mathcal{>} x\} \). From the definition, the core satisfies internal stability. Thus the core C of G is contained in any stable set of G if thelatter exists. To see this, suppose that \( { C\not\subset K }\) for a stable set K, and C isnon-empty, i. e., \( { C\ne \emptyset }\). (If \( { C=\emptyset }\), then clearly \( { C\subset K }\).) Pick any element \( { x\in C\backslash K }\). Since \( { x\notin K }\), by external stability there exists \( y \in K \) with \( { y\succ x } \), which contradicts \( { x\in C } \).

When the core of a game satisfies external stability, it has very strong stability. It is called the stable core. The stable core is the unique stable set of the game.

Stable Sets in Characteristic Function form Games

An n‑person game in characteristic function form with transferableutility is a pair \( { (N,v) } \) ofa player set \( { N=\{1,2, \ldots,n\} } \)and a characteristic function  v on the set \( { 2^N } \) of all subsets of N such that\( { v(\emptyset)=0 } \). Each subsetof N is called a  coalition . The game \( { (N,v) } \) is often calleda TU- game. A characteristic function form game without transferable utility is called an NTU- game: its characteristic function gives eachcoalition a set of payoff vectors to players. For NTU-games and their stable sets, refer to Aumann and Peleg [1] and Peleg [35]. In this section, hereafter, we deal with onlyTU characteristic function form games, and refer to them simply as characteristic function form games.

Let \( { (N,v) } \) bea characteristic function form game. The characteristic function v assigns a real number \( { v(S) } \) to each coalition \( { S\subseteq N } \). The value \( { v(S) } \) indicates the worth that coalition S can achieve by itself.

An n‑dimensional vector \( { x=(x_1,x_2,\ldots,x_n) }\) is called a payoff vector. A payoff vector x is called an imputation if the following two conditions are satisfied:

  1. 1.

    Group rationality : \( { \sum\nolimits_{i=1}^n{x_i} =v(N) } \),

     
  2. 2.

    Individual rationality: \( { x_i \ge v(\{i\}) } \) for each \( { i\in N } \).

     

The first condition says that all players cooperate and share the worth \( { v(N) } \) that they can produce. The second condition says that each player must receiveat least the amount that he/she can gain by him/herself. Let A be the set of all imputations.

Let \( { x, y } \) be any imputationsand S be any coalition. We say that x dominates y via S andwrite this as \( { x\operatorname{dom}_S y } \)if the following two conditions are satisfied:

  1. 1.

    Coalitional rationality: \( { x_i >y_i } \) for each \( { i\in S } \),

     
  2. 2.

    Effectivity : \( { \sum\nolimits_{i\in S} {x_i} \le v(S) } \).

     

The first condition says that every member of coalition S strictlyprefers x to y. The second condition says that coalition S can guarantee the payoff x i foreach member \( { i\in S } \) by themselves. Wesay that x dominates y (denoted by\( { x\operatorname{dom}y } \)) if there existsat least one coalition S such that \( {x\operatorname{dom}_S y } \).

It should be noted that a pair (A, dom) is an abstract game defined in Sect. “Stable Sets in Abstract Games”. It is easily seen that “dom” is an irreflexive binary relationon A. A stable set and the core of game \( { (N,v) } \) are defined to be a stable set and the core of the associated abstractgame (A, dom), respectively.

Since von Neumann and Morgenstern defined the stable set, its general existence had been one of the most important problems in game theory. Theproblem was eventually solved in a negative way. Lucas [18] found the following 10‐personcharacteristic function form game in which no stable set exists.

A game with no stableset: Consider the following 10‐person game:
$$ \begin{aligned}N&=\{1,2,\ldots,10\}\:,\\ v(N)&=5\:,\enskip v(\{1,3,5,7,9\})=4\:,\\ v(\{3,5,7,9\})&=v(\{1,5,7,9\})=v(\{1,3,7,9\})=3\:,\\ v(\{1,2\})&=v(\{3,4\})=v(\{5,6\})=v(\{7,8\})\\ &=v(\{9,10\})=1\:,\\ v(\{3,5,7\})&=v(\{1,5,7\})=v(\{1,3,7\})=v(\{3,5,9\}) \\ &=v(\{1,5,9\})=v(\{1,3,9\})=2\:,\\ v(\{1,4,7,9\})&=v(\{3,6,7,9\})=v(\{5,2,7,9\})=2\enskip\text{and}\\ v(S)&=0\enskip\text{for all other}\enskip S\subset N\:.\end{aligned} $$

Though this game has no stable set, it has a nonempty core. A game with no stable set and an empty core was also found by Lucas andRabie [21].

We remark on a class of games in which a stable core exists. As mentioned before, if a stable set exists, it always contains thecore, which is of course true also in characteristic function form games. Furthermore, in characteristic function form games, there is an interestingclass, called convex games, in which the core satisfies external stability. That is, the core is a stable core. A characteristic function formgame \( { (N,v) } \) is a  convex game if for any \( { S,T\subseteq N }\) with \( { S\subset T }\) and for any \( i\notin T, v(S\cup\{i\})-v(S)\le v(T\cup \{i\})-v(T) \), i. e., the bigger coalition a player joins, the larger the player'scontribution becomes. In convex games, the core is large and satisfies external stability. For the details, refer to Shapley [45].

Though general existence is denied, the stable set provides us with very useful insights into many economic, political, and social issues. In thefollowing, we will present some stable set analyzes applied to those issues.

Applications of Stable Sets in Abstract and Characteristic Function Form Games

Symmetric Voting Games

This section deals with applications of stable sets to voting situations. Let us start with a simple example.

Example 1

Suppose there is a committee consisting of three players 1, 2 and 3. Each player has one vote. Decisions are done according to a simple majority rule. That is, to pass a bill, at least two votes are necessary.

Before analyzing players' behavior, we first formulate the situation as a characteristic function form game. Let the player set be \( { N=\{1,2,3\} } \). Since a coalition of a simple majority of players can pass any bill, we give value 1 to such coalitions. Other coalitions can pass no bill. We thus give them value 0. Hence the characteristic function is given by
$$ v(S)=\begin{cases} 1 & \text{if}\enskip \vert S\vert \ge 2\:,\\ 0 & \text{if}\enskip \vert S\vert \le 1\:, \end{cases} $$
where \( { \vert S\vert } \) denotes the number of players in coalition S. The set of imputations is given by
$$ A=\{x=(x_1 ,x_2 ,x_3)\vert x_1 +x_2 +x_3 =1,x_1 ,x_2 ,x_3 \ge 0\}\:.$$
One stable set of this game is given by the set K consisting of three imputations, \( (1/2,1/2,0),(1/2,0,1/2),(0,1/2,1/2) \). A brief proof is the following. Since each of the three imputations has only two numbers \( { 1/2 } \) and 0, internal stability is trivial. To show external stability, take any imputation \( { x=(x_1,x_2,x_3) } \) from outside K. Suppose first \( { x_1 < 1/2 } \). Since \( { x\notin K } \), at least one of x 2 and x 3 is less than \( { 1/2 } \). We assume \( { x_2 < 1/2 } \). Then \( { (1/2,1/2,0) } \) dominates x via coalition \( { \{1,2\} } \). Next suppose \( { x_1 = 1/2 } \). Since \( { x\notin K, 0<x_2 ,x_3 <1/2 } \). Thus \( { (0,1/2,1/2) } \) dominates x via coalition \( { \{2,3\} } \). Finally suppose \( { x_1 > 1/2 } \). Then \( { x_2 ,x_3 <1/2 } \), and thus \( { (0, 1/2, 1/2) } \) dominates x via coalition \( { \{2,3\} } \). Thus the proof of external stability is complete. This three-point stable set indicates that a two‐person coalition is formed, and that players in the coalition share equally the outcome obtained by passing a bill.

This game has another three types of stable sets. First, any set \( { K_c^1 =\{x\in A\vert x_1 =c\} } \) with \( { 0\le c<1/2 } \) is a stable set. The internal stability of each K c 1 is trivial. To show external stability, take any imputation \( x=(x_1,x_2,x_3)\notin K_c^1 \). Suppose \( x_1 > c \). Define by \( y_1 =c, y_2 =x_2 +(x_1 -c)/2, y_3 =x_3 +(x_1-c)/2 \). Then \( { y\in K_c^1 } \) and \( { y \operatorname{dom}_{\{2,3\}} x } \). Next suppose \( { x_1<c } \). Notice that at least one of x 2 and x 3 is less than \( 1-c \) since \( c < 1/2 \). Suppose without loss of generality \( x_2 < 1-c \). Since \( c < 1/2 \), we have \( (c, 1-c, 0)\in K_c^1 \) and \( (c, 1-c, 0)\operatorname{dom}_{\{1,2\}}x \). Thus external stability holds. This stable set indicates that player 1 gets a fixed amount c and players 2 and 3 negotiate for how to allocate the rest \( 1-c \). Similarly, any sets \( K_c^2 = \{x\in A\vert x_2 =c\} \) and \( K_c^3 = \{x\in A\vert x_3 =c\} \) with \( 0\le c < 1/2 \) are stable sets.

The three‐person game of Example 1 has no other stable set. See von Neumann and Morgenstern [31]. The former stable setis called a  symmetric (or objective ) stable set,while the latter types are called discriminatory stable sets.

As a generalization of the above result, symmetric stable sets are found in general n‑person simple majority voting games. An n‑person characteristic function form game \( { (N,v) } \) with \( { N=\{1,2,\ldots,n\} } \) is called a  simple majority voting game if
$$ v(S)=\begin{cases} 1 & \text{if}\enskip \vert S\vert >n/2\:, \\ 0 & \text{if}\enskip \vert S\vert \le n/2\:.\end{cases} $$
A coalition S with \( { v(S)=1 } \), i. e., with \( { \vert S\vert > n/2 } \), is called a  winning coalition. A winning coalition including no smaller winning coalitions is called a  minimal winning coalition. In simple majority voting games, a minimal winning coalition means a coalition of \( { (n+1)/2 } \) players if n is odd, or \( { (n+2)/2 } \) players if n is even. The following theorem holds. See Bott [2] for the proof.

Theorem 1

Let \( { (N,v) } \) be a simple majority voting game. Then the following hold.
  1. (1)
    If n is odd, then the set
    $$ K=\langle\underbrace{2/(n+1),\dots,2/(n+1)}_{\frac{n+1}{2}},\underbrace{0,\ldots,0}_{\frac{n-1}{2}}\rangle $$
    is a stable set where the symbol \( { \langle x\rangle } \) denotes the set of all imputations obtained from x through permutations of its components.
     
  2. (2)
    If n is even, the set
    $$ K=\langle\{x\in A\vert\underbrace{x_1 =\ldots=x_{n/2}}_{\frac{n}{2}}\ge\underbrace {x_{(n/2)+1} =\ldots=x_n}_{\frac{n}{2}}\}\rangle $$
    is a stable set, where
    $$ A=\bigg\{x=(x_1 ,\ldots ,x_n)\bigg|\sum_{i=1}^n {x_i} =1,x_1 ,\ldots ,x_n \ge 0\bigg\} $$
    and \( { \langle Y\rangle=\mathop \cup \limits_{x\in Y} \langle x\rangle } \) .
     

It should be noted from (1) of Theorem 1 that when the number of players is odd, a minimal winning coalition is formed. The members of the coalition share equally the total profit. On the other hand, when the number of players is even, (2) of Theorem 1 shows that every player may gain a positive profit. This implies that the grand coalition of all players is formed. In negotiating for how to share the profit, two coalitions, each with \( { n/2 } \) players, are formed and profits are shared equally within each coalition. Since at least \( { n/2+1 } \) players are necessary to win when n is even, an \( { n/2 } \)-player coalition is the smallest coalition that can prevent its complement from winning. Such a coalition is called a  minimal blocking coalition. When n is odd, an \( { (n+1)/2 } \)-player minimal winning coalition is also a minimal blocking coalition.

General Voting Games

In this section, we present properties of stable sets and cores in general (not necessarily symmetric) voting games. A characteristic function from game \( { (N,v) } \) is called a simple game if \( { v(S)=1\enskip\text{or}\enskip 0 } \) for each nonempty coalition \( { S\subseteq N } \). A coalition S with \( { v(S)=1 } \) (resp. \( { v(S)=0 } \)) is a winning coalition (resp. losing coalition). A simple game is called a voting game if it satisfies (1) \( { v(N)=1 } \), (2) if \( { S\subseteq T } \), then \( { v(S)\le v(T) } \), and (3) if S is winning, then \( { N-S } \) is losing. The first condition implies that the grand coalition N is always winning. The second condition says that a superset of a winning coalition is also winning. The third condition says that there are no two disjoint winning coalitions. It is easily shown that the simple majority voting game studied in the previous section satisfies these conditions. A player has a veto if he/she belongs to every winning coalition. As for cores of voting games, the following theorem holds.

Theorem 2

Let \( { (N,v) } \) be a voting game. Then the core of \( { (N,v) } \) is nonempty if and only if there exists a player with veto.

Thus the core is not a useful tool for analyzing voting situations with no veto player. In simple majority voting games, no player has a veto, and thus the core is empty. The following theorem shows that stable sets always exist.

Theorem 3

Let \( { (N,v) } \) be a voting game. Let S be a minimal winning coalition and define a set K by
$$ K=\bigg\{x\in A\bigg| \sum_{i\in S} {x_i =1, x_i} =0 \forall i\notin S\bigg\}\:.$$
Then K is a stable set.

Thus in voting games, a minimal winning coalition is always formed, and they gain all the profit. For the proofs of these theorems, see Owen [34]. Further results on stable sets in voting games are found in Bott [2], Griesmer [12], Heijmanns [16], Lucas et al. [20], Muto [26,28], Owen [32], Rosenmüller [36], Shapley [43,44].

Production Market Games

Let us start with a simple example.

Example 2

There are four players, each having one unit of a raw material. Two units of the raw material are necessary for producing one unit of an indivisible commodity. One unit of the commodity is sold at p dollars.

The situation is formulated as the following characteristic function form game. The player set is \( { N=\{1,2,3,4\} } \). Since two units of the raw material are necessary to produce one unit of the commodity, the characteristic function v is given by
$$ v(S)=2p\enskip{\text{if}}\enskip \vert S\vert =4\:,\quad v(S)=p\enskip{\text{if}}\enskip\vert S\vert=3,2\:,\\ v(S)=0\enskip{\text{if}}\enskip \vert S\vert =1,0\:. $$
The set of imputations is
$$ A=\{x=(x_1,x_2,x_3,x_4)\vert x_1 +x_2 +x_3 +x_4 =2p,\;\\ x_1,x_2,x_3,x_4 \ge 0\}\:. $$
The following set K is one of the stable sets of the game:
$$ K=\langle\{x=(x_1,x_2,x_3,x_4)\in A\vert x_1 =x_2 =x_3\ge x_4 \}\rangle\:.$$
To show internal stability, take two imputations \( x=(x_1, x_2,x_3,x_4) \) with \( x_1=x_2=x_3\ge x_4 \) and \( y=(y_1,y_2,y_3,y_4) \) in K. Suppose x dominates y. Since \( x_1=x_2=x_3 \ge p/2 \ge x_4 \), the domination must hold via coalition \( { \{i,4\} } \) with \( { i=1,2,3 } \). Then we have a contradiction \( 2p=\sum\nolimits_{i=1}^4 {x_i} > \sum\nolimits_{i=1}^4 {y_i} = 2p \), since \( { y\in K } \) implies that the largest three elements of y are equal. To show external stability, take \( z=(z_1,z_2,z_3,z_4)\notin K \). Suppose \( z_1 \ge z_2 \ge z_3 \ge z_4 \). Then \( z_1 > z_3 \). Define \( y=(y_1,y_2,y_3,y_4) \) by
$$ y_i =\begin{cases} z_3 +\displaystyle\frac{z_1 +z_2 -2z_3}{4} & \text{for}\enskip i=1,2,3\:, \\ z_4 +\displaystyle\frac{z_1 +z_2 -2z_3}{4} & \text{for}\enskip i=4\:.\end{cases} $$
Then \( { y\in K } \) and \( { y \operatorname{dom}_{\{3,4\}} z } \), since \( { y_3 >z_3, y_4 >z_4 } \) and \( { y_3 +y_4 \le p = v(\{ 3,4\}) } \).

This stable set shows that in negotiating for how to share the profit of \( { 2p } \) dollars, three players form a coalition and share equally the gain obtained through collaboration. At least two players are necessary to produce the commodity. Thus a three‐player coalition is the smallest coalition that can prevent its complement from producing the commodity, i. e., a minimal blocking coalition. We would claim that in the market a minimal blocking coalition is formed and that profits are shared equally within the coalition.

An extension of the model was given by Hart [14] and Muto [27]. Hart considered the following production market with n players, each holding one unit of a raw material. To produce one unit of an indivisible commodity, k units of raw materials are necessary. The associated production market game is defined by the player set \( { N=\{1,2,\ldots,n\} } \) and the characteristic function ν given by
$$ \nu(S)=\begin{cases} 0 & \text{if}\enskip 0\le |S|<k\:,\\ p & \text{if}\enskip k\le |S|<2k\:,\\ \vdots\\ {jp} & \text{if}\enskip jk\le |S| < (j+1)k\:,\\ \vdots\\ hp & \text{if}\enskip hk\le |S|<n\:, \end{cases}\:, $$
where \( { n=hk+r } \) and \( { h,r } \) are integers such that \( { h\ge 1 } \) and \( { 0\le r\le k-1 } \). When \( { h=1 } \),
$$ \nu(S)=\begin{cases} 0 & \text{if}\enskip |S|<k\:, \\ p & \text{if}\enskip |S|\ge k\:.\end{cases} $$
The following theorem holds.

Theorem 4

Suppose \( { h=1 } \). Let \( { t=n-k+1 } \) and \( n=tu+w \) where \( { u,w } \) are integers such that \( { u\ge 1 } \) and \( { 0\le w\le t-1 } \). Then the following set K is a stable set.
$$ K=\langle\{x=(x_1,\ldots,x_n)\in A |\\ x_1 =\ldots=x_t \ge x_{t+1}=\ldots=x_{2t}\\ \ge\ldots\ge x_{tu+1} =\ldots=x_n =0\rangle\:, $$
where
$$ A=\bigg\{x=(x_1,\ldots,x_n)\bigg|\sum_{i=1}^n x_i =p,x_1,\ldots,x_n \ge 0\bigg\} $$
is the set of imputations.

The theorem shows that in negotiating for how to share the profit, minimal blocking coalitions, i. e., coalitions of \( { n-k+1 } \) players, are formed and within each coalition, profits are shared equally. Players failing to form a coalition gain nothing. When \( { h\ge 2 } \), the following theorem holds.

Theorem 5

Suppose \( { h\ge 2 } \). Let
$$ K=\Big\langle\Big\{x=(x_1,\dots,x_n)\in A \Big|\\ x_1=\dots=x_{n-k+1} \ge {\small\frac{p}{k}}\ge x_{n-k+2} =\dots=x_n \Big\}\Big\rangle\:, $$
where
$$ A=\bigg\{x=(x_1,\dots,x_n)\bigg| \sum_{i=1}^n x_i =hp,x_1,\dots,x_n \ge 0\bigg\}\:. $$
Then K is a stable set if and only if
$$ n\ge (h+1)(k-1)\:. $$
Therefore if n is large or k is small, then a minimal blocking coalition is formed and the rest of the players also form a coalition. Within each coalition, profits are shared equally.

The next example deals with the case in which more than one raw materials are necessary to produce a commodity.

Example 3

Two types of raw materials P and Q are needed, one unit each, to produce one unit of an indivisible commodity, which is sold at p dollars. Player 1 holds one unit of raw material p, and each of players 2 and 3 holds one unit of raw material Q.

This situation is formulated as the following characteristic function form game. The player set is \( { N=\{1,2,3\} } \). Since one unit of raw materials P and Q are necessary to produce the commodity, the characteristic function ν is given by
$$ \begin{aligned} \nu(N)&=p\:,\\ \nu(\{1,2\})&=\nu(\{1,3\})=p\:,\enskip\nu(\{2,3\})=0\:,\\ \nu(\{1\})&=\nu(\{2\})=\nu (\{3\})=0\:,\enskip\nu(\emptyset)=0\:.\end{aligned} $$
The set of imputations is
$$ A=\{x=(x_1,x_2,x_3) | x_1+x_2+x_3 =p,\; x_1,x_2,x_3 \ge 0\}\:.$$
The following set K is one of the stable sets in this game:
$$ K=\{x=(x_1,x_2,x_3)\in A| x_2=x_3\}\:.$$
To show internal stability, take two imputations \( x=(x_1,x_2,x_3) \) and \( { y=(y_1,y_2,y_3) } \) in K and suppose x dominates y. Then the domination must hold via coalitions \( { \{1,2\}, \{1,3\} } \) since values of other coalitions (except \( { \{1,2,3\} } \)) are 0. If \( { x \operatorname{dom}_{\{1,2\}} y } \), then \( { x_1 > y_1 } \) and \( { x_2 > y_2 } \) hold. Thus we have a contradiction \( p=\sum\nolimits_{i=1}^3 {x_i} > \sum\nolimits_{i=1}^3 {y_i}=p \). The domination via \( { \{1,3\} } \) leads to the same contradiction. To show external stability, take any imputation \( { z=(z_1,z_2,z_3)\notin K } \). Then \( z_2 \ne z_3 \). Without loss of generality, let \( { z_2 < z_3 } \). Define \( y=(y_1,y_2,y_3) \) by
$$ y_i =\begin{cases} z_1 +\displaystyle\frac{z_3 -z_2}{3} & \text{for}\enskip i=1\:, \\ z_2 +\displaystyle\frac{z_3 -z_2}{3} & \text{for}\enskip i=2\:, \\ z_2 +\displaystyle\frac{z_3 -z_2}{3} & \text{for}\enskip i=3\:.\end{cases} $$
Then \( { y\in K } \) and \( { y\operatorname{dom}_{\{1,2\}} z } \), since \( { y_1 > z_1, y_2 > z_2 } \) and \( { y_1 +y_2 < \nu (\{1,2\}) } \). This stable set shows that in negotiating for how to share the profit p dollars, players 2 and 3 form a coalition against player 1 and share equally the gain obtained through collaboration.
There exist other stable sets in which players 2 and 3 collaborate but they do not share equally the profit. More precisely, the following set
$$ K=\{x=(x_1,x_2,x_3)\in A|\\ x_2\enskip\text{and}\enskip x_3\enskip {\text{move towards the same direction}}\} $$
is a stable set, where “move towards the same direction” means that if x 2 increases then x 3 increases, and if x 2 decreases then x 3 decreases.

A generalization of the results above is given by the following theorem due to Shapley [42]. Shapley's original theorem is more complicated and holds in more general markets.

Theorem 6

Suppose there are m players, \( { 1,\dots,m } \), each holding one unit of raw material P, and n players, \( { m+1,\dots,m+n } \), each holding one unit of raw material Q. To produce one unit of an indivisible commodity, one unit of each of raw materials Pand Q is necessary. One unit of commodity is sold at p dollars. In this market, the following set K is a stable set.
$$ K=\{x=(x_1,x_2,\dots,x_{m+n})\in A|\\ x_1 =\dots=x_m,x_{m+1} =\dots=x_{m+n}\}\:.$$
where
$$ A=\bigg\{x=(x_1,\dots,x_m,x_{m+1},\dots,x_{m+n}) \bigg|\\ \sum\limits_{i=1}^{m+n} x_i =p \times \min(m,n),\; x_1,\dots,x_{m+n} \ge 0\bigg\}\:, $$
is the set of imputations of this game.

This theorem shows that players holding the same raw material form a coalition and share equally the profit gained through collaboration.

For further results on stable sets in production market games, refer to Hart [14], Muto [27], Owen [34]. Refer also to Lucas [19], Owen [33], Shapley [41], for further general studies on stable sets.

Assignment Games

The following two sections deal with markets in which indivisible commodities are traded between sellers and buyers, or bartered among agents. The first market is the assignment market originally introduced by Shapley and Shubik [47].

An assignment market consists of a set of \( { n(\ge 1) } \) buyers \( { B=\{1,\dots,n\} } \) and a set of n sellers \( { F=\{1^{\prime},\dots,n^{\prime}\} } \). Each seller \( { k^{\prime}\in F } \) is endowed with one indivisible commodity to sell, which is called object k′. Thus F also denotes the set of n objects in the market. The objects are differentiated. Each buyer \( { i\in B } \) wants to buy at most one of the objects, and places a nonnegative monetary valuation \( { u_{ik^{\prime}}(\ge 0) } \) for each object \( { k^{\prime}\in F } \). The matrix \( { U=(u_{ik^{\prime}})_{(i,k^{\prime})\in B\times F} } \) is called the valuation matrix. The sellers place no valuation for any objects. An assignment market is denoted by \( { M(B,F,U) } \). We remark that an assignment market with \( { \vert B\vert \ne \vert F\vert } \) can be transformed into the market with \( { \vert B\vert =\vert F\vert } \) by adding dummy buyersresp. sellers, and zero rows resp. columns correspondingly to valuation matrix U.

For each coalition \( { S\subseteq B\cup F } \) with \( { S\cap B\ne \emptyset } \) and \( { S\cap F\ne \emptyset } \), we define assignment problem \( { P(S) } \) as follows:
$$ \begin{aligned} P(S)\colon \overline{m}(S)&=\mathop{\max}\limits_x \sum\limits_{(i,k^{\prime})\in (S\cap B)\times (S\cap F)} u_{ik^{\prime}} x_{ik^{\prime}}\\ s.t. & \sum\limits_{k^{\prime}\in S\cap F} x_{ik^{\prime}} \le 1\quad \text{for all}\enskip i\in S\cap B\:,\\ & \sum\limits_{i\in S\cap B} x_{ik^{\prime}} \le 1 \quad\text{for all}\enskip k^{\prime}\in S\cap F\:,\\ & x_{ik^{\prime}} \ge 0 \quad\text{for all}\enskip(i,k^{\prime})\in (S\cap B)\times (S\cap F)\:.\end{aligned} $$
Assignment problem \( { P(S) } \) has at least one optimal integer solution (see Simonnard [49]), which gives an optimal matching between sellers and buyers in S that yields the highest possible surplus in S. Without loss of generality, we assume that the rows and columns of valuation matrix U are arranged so that the diagonal assignment \( { x^\ast } \) with \( { x^\ast_{ii^{\prime}} =1, i=1,\dots,n } \), is one of the optimal solutions to \( { P(B\cup F) } \).
For a given assignment market \( { M(B,F,U) } \), we define the associated assignment game G to be the characteristic function form game \( { (B\cup F,v) } \). The player set of G is \( { B\cup F } \). The characteristic function v is defined as follows: \( { v(S)=\overline{m}(S) } \) for each \( { S\subseteq F\cup B } \) with \( { S\cap B\ne \emptyset } \) and \( { S\cap F\ne \emptyset } \). For coalitions only of sellers or buyers, they cannot produce surplus from trade. Thus \( { v(S)=0 } \) for each S with \( { S\subseteq B, S\subseteq F } \), or \( { S=\emptyset } \). The imputation set of G is
$$ A=\Big\{(w,p)\in \Re_+^B \times \Re_+^F \Big| \sum_{i\in B} w_i +\sum_{k^{\prime}\in F} p_{k^{\prime}} = v(B\cup F)\Big\}\:.$$
Shapley and Shubik [47] proved that for any assignment game G, the core C is given by the set of optimal solutions to the dual problem of assignment problem \( { P(B\cup F) } \), i. e.,
$$ \begin{aligned}C&=\{(w,p)\in A\vert w_i +p_{k^{\prime}}\ge u_{ik^{\prime}} =v(\{i,k^{\prime}\})\\ \quad &\text{for each}\enskip (i,k^{\prime})\in B\times F\}\:, \end{aligned} $$
and thus the core is nonempty. They also showed that sellers' core payoff vector p gives market prices of the respective objects at which the demand and supply equilibrates for each object.

The general existence of stable sets in assignment games is still unsolved. However, as mentioned in Sect. “Stable Set and Core”, if a game has the stable core, i. e., the core with external stability, then it is the unique stable set of the game. Thus we consider when an assignment game has the stable core.

Given an assignment market \( { M(B,F,U) } \), we say that valuation matrix U satisfies the dominant diagonal condition if all of its diagonal entries are row and column maximums, i. e.,
$$ \begin{aligned} u_{ii^{\prime}} &= \max \{u_{ik^{\prime}} \vert k^{\prime}\in F\}\\ &=\max \{u_{ji^{\prime}} \vert j\in B\}\quad\text{for each}\enskip i=1,\dots,n\:.\end{aligned} $$
The dominant diagonal condition implies that each buyer i can yield the maximum surplus by purchasing the object of seller i′. Thus theplayers do not have to compete for partners. They come to be more concerned with the bargaining with his/her best matched partner. Then it is proved that valuation matrix U satisfies the dominant diagonal condition if and only if the core of G includes the imputations \( { \big(\underline{w} ,\overline{p}\big) } \) and \( { \big(\overline{w},\underline{p}\big) } \), where \( { \underline{w}_i =0, \overline{p}_{i^{\prime}} =u_{ii^{\prime}}, \overline{w}_i = u_{ii^{\prime}} } \), and \( { \underline{p}_{i^{\prime}} = 0 } \) for each \( { i\in N } \). Furthermore, the following theorem holds.

Theorem 7

Let \( { M(B,F,U) } \) be any assignment market with \( { \vert B\vert =\vert F\vert } \). The associated assignment game G has the stable core if and only if valuation matrix U satisfies the dominant diagonal condition.

It is also proved that an assignment game G is convex if and only if valuation matrix U satisfies that \( { u_{ik^{\prime}}=0 } \) for each \( { (i,k^{\prime})\in B\times F } \) with \( { i\ne k } \). This implies that the core has the von Neumann–Morgenstern stability in a larger class of assignment games including convex assignment games. For more details, refer to Solymosi and Raghavan [50].

House Barter Games

In this section, we consider a market in which only indivisible commodities are bartered. This market was originally considered by Shapley and Scarf [46].

The market we consider has \( { n(\ge 2) } \) players, each endowed with one indivisible commodity, e. g., a house. Let \( { N=\{1,\dots,n\} } \) be the set of players. The n indivisible commodities are differentiated. The commodity initially owned by player i is called house i. Thus N also denotes the set of houses in the market. We assume that each player wants to own exactly one house, and no player disposes any house. Each player i has a complete, reflexive, and transitive preference relation R i on N. Here, \( { jR_i h } \) denotes that player i prefers house j at least as well as house h. Let \( { jP_i h } \) denote that player i strictly prefers j to h, and \( { jI_i h } \) denote that player i is indifferent between j and h. Defining players' preferences this way, we assume that each player strictly prefers owning a house to owning no house. The bundle \( { R=(R_i)_{i\in N} } \) of players' preference relations is called a preference profile.

There is no divisible good such as money in the market. The players only exchange their houses to make a mutually beneficial trade. Thus an allocation of the market is defined to be a bijection x from N onto N, where \( { x(i) } \) denotes the house assigned to player i in x. An allocation can be regarded as a permutation of N. An allocation x is also indicated by the vector \( { x=(x_1,\dots,x_n) } \) with \( { x_i=x(i) } \) for each \( { i\in N } \). Let A be the set of allocations. The market above is referred to as house barter market \( { M(N,R) } \), or briefly market M.

Let \( { x,y } \) be any allocations of market M. For each coalition \( { S\subseteq N } \), let \( { x(S) } \) be the set of houses assigned to the members of S in x, i. e.,
$$ x(S)=\{j\in N\vert j=x(i)\quad \text{for some}\enskip i\in S\}\:.$$
We say that x weakly dominates y (denoted by \( { x \operatorname{wdom}y } \)) if there exists a coalition S satisfying the following conditions:
  1. 1.

    \( { x(i)R_i y(i) } \) for each \( { i\in S } \) with P i holding for at least one \( { i\in S } \),

     
  2. 2.

    \( { x(S)=S } \).

     
The second condition is the effectivity condition, which requires that each player i in S can obtain house \( { x(i) } \) by exchanging their own endowments. We say that x strongly dominates y (denoted by \( { x \operatorname{sdom}y } \)) if \( { x(i)P_i y(i) } \) for each \( { i\in S } \), and \( { x(S)=S } \). We use the notations \( { x\operatorname{wdom}_S y } \) and \( { x\operatorname{sdom}_S y } \) to indicate the associated coalition S.

An allocation xis said to be individually rational if \( { x(i)R_i i } \) for each player \( { i\in N } \). An allocation x is Pareto efficient if there exists no allocation \( { y\in A } \) with \( { y\operatorname{wdom}_N x } \). If there is no \( { y\in A } \) with \( { y\operatorname{sdom}_N x } \), then x is weakly Pareto efficient . The three sets of individually rational, Pareto efficient, and weakly Pareto efficient allocations are denoted by \( { IR, PA } \), and \( { WPA } \), respectively.

We define cores and stable sets of market M by cores and stable sets of the associated house barter games \( { (A,\operatorname{wdom}) } \) and \( { (A,\operatorname{sdom}) } \), which are the abstract games with the outcome sets given by the allocation set A, and the binary relations on A given by the weak and strong dominations, respectively.

A nonempty subset of A is referred to asa  wdom stable set of market M if it is a stable set of abstract game \( { (A,\operatorname{wdom}) } \). A nonempty subset of A is referred to as a  sdom stable set stable set of market M if it is a stable set of abstract game \( { (A,\operatorname{sdom}) } \). The wdom and sdom stable sets are the stable sets defined by the weak and strong dominations, respectively. A subset of A, which may be empty, is called the strict core of market M if it is the core of abstract game \( { (A,\operatorname{wdom}) } \). The core of market M is the core of abstract game \( { (A,\operatorname{sdom}) } \). The strict core and the core of market M are the cores defined by the weak and strong dominations, respectively.

From the definitions above, a wdom stable set is a subset of PA, and an sdom stable set is a subset of WPA. However, both wdom and sdom stable sets may not be subsets of IR. The strict core is a subset of \( { PA \cap IR } \). The core is a subset of \( { WPA \cap IR } \).

Shapley and Scarf [46] proved that the core is nonempty for any house barter market \( { M(N,R) } \). However, since external stability is not imposed on the core, the core does not necessarily coincide with an sdom stable set. In fact, the following example shows that there is a house barter market with no sdom stable set.

Example 4

Let M 1 be the market with the player set \( { N=\{1,2,3\} } \) and the following preference profile:
$$ \begin{aligned} &1)\quad 2\enskip P_1\enskip 3\enskip P_1\enskip 1\:,\\ &2)\quad 3\enskip P_2\enskip 1\enskip P_2\enskip 2\:,\\ &3)\quad 1\enskip P_3\enskip 2\enskip P_3\enskip 3\:,\\ \end{aligned} $$

Market M 1 has six allocations: \( x^1=(2,3,1), x^2=(2,1,3), x^3=(1,3,2), x^4=(3,2,1), x^5=(3,1,2), x^6=(1,2,3) \). From the preference profile, x 1 is clearly a core allocation. Allocation x 1 strongly dominates x 5 and x 6. Let \( { X=\{x^2,x^3,x^4\} } \). We note that x 1 does not strongly dominate any \( { x^k\in X } \). Here, suppose market M 1 has an sdom stable set K. Then the external stability of K implies \( { x^1\in K } \) and \( { K\cap X\ne \emptyset } \). Note that \( { x^2\operatorname{sdom}_{\{1,2\}} x^4, x^4\operatorname{sdom}_{\{1,3\}} x^3 } \), and \( { x^3\operatorname{sdom}_{\{2,3\}} x^2 } \). This together with the internal stability of K implies that K can contain only one allocation \( { x^k\in X } \). However, the allocation x k strongly dominates only one allocation in \( { X\backslash \{x^k\} } \). This means that K does not have external stability, which is a contradiction. Thus, there exists no sdom stable set in market M 1.

Market M 1 however has a nice feature: the singleton \( { \{x^1\} } \) is the strict core, and x 1 weakly dominates all the other allocations \( { x^2,\dots,x^6 } \). In addition, every player shows only strict preferences. Noticing these facts, Roth and Postlewaite [38] proved that for any house barter market \( { M(N,R) } \), if each player has a strict preference relation, then the strict core is a singleton, and it is the unique wdom stable set, i. e., the wdom stable core. Wako [54] proved that this property is extended as follows:

Theorem 8

For any house barter market \( { M(N,R) } \), if the strict core \( { SC } \) is nonempty, then it is the unique wdom stable set. Furthermore, for any \( { x,y\in SC, x(i)I_i y(i) } \) for each \( { i\in N } \).

Even if some players do not have strict preference relations, the strict core is characterized as the wdom stable core as far as it is nonempty. The wdom stable core \( { SC } \) of market M has nice properties. First, each allocation \( { x\in SC } \) is individually rational, Pareto efficient, and not weakly dominated by any other allocations since x is a strict core allocation. Secondly, any allocation outside \( { SC } \) is weakly dominated by some allocation in \( { SC } \), since \( { SC } \) is a wdom stable set. Thirdly, even if \( { SC } \) contains different allocations, we may choose any one of them, since they are indifferent for each player.

However, the strict core can be empty when indifferences are allowed in preference relations. Shapley and Scarf [46] already discussed this point by the following example.

Example 5

Let M 2 be the market with the player set \( { N=\{1,2,3\} } \) and the following preference profile:
$$ \begin{aligned} &1)\quad 2\enskip P_1\enskip 3\enskip I_1\enskip 1\:,\\ &2)\quad 1\enskip I_2\enskip 3\enskip P_2\enskip 2\:,\\ &3)\quad 2\enskip P_3\enskip 1\enskip I_3\enskip 3\:,\\ \end{aligned} $$
It can be verified that the strict core of M 2 is empty, and that the sets \( V^1=\{(2,3,1),(2,1,3)\} \) and \( V^2=\{(1,3,2),(3,1,2)\} \) are both wdom stable sets of M 2. Thus Theorem 8 does not carry over to the cases with the strict core being empty.
Quint and Wako [40] then gave a necessary and sufficient condition for the strict core to be nonempty. For each player \( { i\in N } \) and each nonempty coalition\( { S\subseteq N } \), let \( { B_i (S) } \) be the set of player i's most‐preferred house in S, i. e., \( { B_i (S)=\{h\in S\vert hR_i j } \) for each \( { j\in S\} } \). We call a partition \( { T=\{T_1,\dots,T_m \} } \) of Npartition by minimal self‐mapped sets (PMSS) if each \( { T_k \in T } \) satisfies the following conditions:
$$ T_k =\bigcup\limits_{i\in T_k} {B_i \big(N\backslash \cup _{l=1}^{k-1} T_l\big)}\enskip\text{and there}\\ \text{is no}\enskip S\subset T_k\enskip\text{with}\enskip S=\bigcup\limits_{i\in S} {B_i \big(N\backslash \cup _{l=1}^{k-1} T_l\big)}\:. $$

There exists at least one PMSS for any market M. When there are more than one PMSSs, each PMSS consists of the same sets with only the order of some sets being different. We say that group \( { T_k \in T } \) is a lower (resp. higher) group of \( { T_l \in T } \) if \( { k>l } \) (resp. \( { k<l } \)). The fact that \( { T=\{T_1 ,\dots,T_m \} } \) is a PMSS means that for each player i of any given group \( { T_k \in T } \), the houses most preferred by i among the endowments of groups \( { T_k } \) and lower are all owned in his/her group \( { T_k } \). The following theorem holds.

Theorem 9

Let \( { T=\{T_1,\dots,T_m \} } \) be a PMSS of a given house barter market \( { M(N,R) } \). Then the strict core is nonempty if and only if there exists an allocation \( { x\in A } \) such that
$$ x(T_k)=T_k\enskip\text{and}\enskip x(i)\in B_i \big(N\backslash \cup _{l=1}^{k-1} T_l\big)\\\text{for each}\enskip i\in T_k \enskip\text{and each}\enskip T_k \in T\:.$$

The necessary and sufficient condition above requires that in any group \( { T_k \in T } \), each player \( { i\in T_k } \) can obtain his/her most‐preferred houses (among those owned in groups \( { T_k } \) and lower) through a feasible exchange within his/her own group T k . We refer to this condition as segmentability . Suppose a house barter market has segmentability, and that some players in a group T k of PMSS T have more preferable houses in higher groups. However, such houses can be exchanged within the higher groups in a mutually beneficial way, and by the definition of a PMSS, no player in the higher groups has an incentive to trade with lower groups. In this sense, the market is segmented into distinct groups. It follows from Theorems 8 and 9 that in house barter markets, segmentability is necessary and sufficient for the existence of the wdom stable core. Quint and Wako [40] gave a polynomial‐time algorithm to examine segmentability of a given house barter market. The following example shows a house barter market with segmentability.

Example 6

Let M 3 be the market with the player set \( { N=\{1,2,3,4,5,6\} } \) and the following preference profile:
$$ \begin{aligned} &1)\quad 2\enskip P_1\enskip 3\enskip P_1\enskip 5\enskip P_1\enskip 4\enskip P_1\enskip 1\enskip P_1\enskip 6\:,\\ &2)\quad 1\enskip I_2\enskip3\enskip P_2\enskip 4\enskip P_2\enskip 6\enskip P_2\enskip\, 5\enskip P_2\enskip 2\:,\\ &3)\quad 1\enskip P_3\enskip 2\enskip P_3\enskip 3\enskip P_3\enskip 4\enskip P_3\enskip 5\enskip P_3\enskip 6\:,\\ &4)\quad 2\enskip P_4\enskip 5\enskip P_4\enskip 6\enskip P_4\enskip 3\enskip P_4\enskip 4\enskip P_4\enskip 1\:,\\ &5)\quad 1\enskip I_5\enskip 4\enskip P_5\enskip 5\enskip P_5\enskip 3\enskip P_5\enskip 6\enskip P_5\enskip 2\:,\\ &6)\quad 3\enskip P_6\enskip 6\enskip P_6\enskip 1\enskip P_6\enskip 2\enskip P_6\enskip 4\enskip P_6\enskip 5\:.\end{aligned} $$
Although market M 3 has two PMSSs, \( T=\{T_1 =\{1, 2, 3\},T_2 =\{4,5\},T_3 =\{6\}\} \) and \( T^{\prime}=\{T^{\prime}_1 =\{1, 2, 3\},T^{\prime}_2 =\{6\},T^{\prime}_3 =\{4,5\}\} \), the differences are only the order of the groups. The wdom stable core of M 3 is the set \( K=\{(2, 3, 1, 5, 4, 6)\} \).

The house barter market was also discussed by Moulin [25] from a wide perspective of cooperative microeconomics and game theory. Ehlers [5] initiated a study on stable sets of two-sided matching games, which were originally studied by Gale and Shapley [8]. For a comprehensive study on two-sided matching games, refer to Roth and Sotomayor [39].

Stable Sets and Farsighted Stable Sets in Strategic Form Games

We first defined a stable set in an abstract game in Sect. “Stable Sets in AbstractGames”. This means that we can also define a stable set of a strategic form game. In this section we introduce a moresophisticated stable set concept: a  farsighted stable set of a strategic form game.

Let \( { G=(N,\{X_i\}_{i\in N},\{u_i\}_{i\in N}) }\) be an n‑person strategic form game, where \( { N=\{1,2,\dots,n\} } \) is the set of players, X i is the set of strategies ofplayer i, and u i isplayer i's payoff function , \( { u_i\colon X=X_1 \times X_2 \times\dots\times X_n \to \Re } \) (the set of realnumbers).

For any two strategy combinations \( x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)\in X \), we say that x is induced from y via coalition \( { S\subseteq N } \) if \( { y_i=x_i } \) for each \( { i\in N\backslash S } \), that is, thecombination x is reached from y by moves only of players in S. It is easily seen from the definition that if x is induced from y via coalition \( { S, y }\) is induced from x via coalition S. Thus we write this inducement relation as \( {x\leftrightarrow_S y } \). We say that x indirectlydominates y (denoted by \( { x\succ y} \)) if there exist a sequence of strategy combinations \( { y=x^0,x^1,\dots,x^{p-1},x^p=x } \) and a sequence of coalitions \( { S^1,\dots,S^p } \) such that for each \( { j=1,\dots,p, x^{j-1}\leftrightarrow_{S^j} x^j } \) and\( { u_i (x)>u_i (x^{j-1}) } \) for each\( { i\in S^j } \). We sometimes say thatx indirectly dominates y starting with S 1 (denoted by\( { x\succ _{S^1} y } \)) to specify the setof players which deviates first from y.

We hereupon remark that in the definition of indirect domination we implicitly assume that joint moves by groups of players are neitheronce-and-for-all nor binding, i. e., some players in a deviating group may later make another move with players within or even outside thegroup. It should be noted that the indirect domination defined above is borrowed from Chwe [3]. Though Harsanyi [13] first proposed the notion of indirectdomination, his definition was given in characteristic function form games.

When \( { p=1 } \) in the definition ofindirect domination, we simply say that x directly dominates y, which is denoted by \( { x\succ^d y } \). When we want to specify a deviatingcoalition, we say that x directly dominates y via coalition S, which is denoted by \( { x\succ_S^d y }\). This direct domination in strategic form games was defined by Greenberg [9].

Let pairs \( { (X,\succ) } \) and\( { (X,\succ^d) } \) be the abstract gamesassociated with game G. A farsighted stable set of G is defined to be a stable set of abstract game \( {(X,\succ) } \) with indirect domination. A stable set of abstract game \( { (X,\succ^d) } \) with direct domination is simply calleda stable set of G .

Applications of Farsighted Stable Sets in Strategic Form Games

Existence of farsighted stable sets in strategic form games remains unsolved. Nevertheless, it has turned out through applications that farsightedstable sets give much sharper insights into players' behavior in economic, political, and social situations than (myopic) stable sets with directdominations. In what follows, we show the analyses of farsighted stable sets in the prisoner's dilemma and two types of duopoly market games in strategicform.

Prisoner's Dilemma

To make discussion as clear as possible, we will focus on the following particular version of the prisoner' s dilemma shown below. Similar results hold in general prisoner' s dilemma games.
$$ \begin{aligned}&\text{\textbf{Prisoner's Dilemma:}}\\[-1mm]&\begin{array}{ccp{13mm}p{13mm}}& & \multicolumn{2}{c}{\text{Player 2}}\\& & \text{Cooperate} & \text{\;\;Defect} \\ \cline{3-4}\ifx\letex\relax\multirow{2}{*}{\text{Player 1}}\else\text{Player 1}\fi & \text{Cooperate} & \multicolumn{1}{|c}{4,4} & \multicolumn{1}{|c|}{0, 5}\\\cline{3-4}& \text{Defect} & \multicolumn{1}{|c}{5, 0} & \multicolumn{1}{|c|}{1, 1}\\\cline{3-4}\end{array}\end{aligned} $$
We first present a farsighted stable set derived when two players use only pure strategies. In this case, the set of strategy combinations is X = {(Cooperate, Cooperate), (Cooperate, Defect), (Defect, Cooperate), (Defect, Defect)}, where in each combination, the former (resp. the latter) is player 1's (resp. 2's) strategy.

The direct domination relation of this game is summarized as in Fig. 1.

Figure 1

Here → denotes a direct domination. For example, “(Cooperate, Cooperate)\( {\xrightarrow{2} } \)(Cooperate,Defect) means (Cooperate, Defect) \( { \mathop \succ_{\{2\}}^d } \) (Cooperate,Cooperate). “(Cooperate, Cooperate) \( { \mathop \leftarrow \limits^{12} } \) (Defect, Defect)” means (Cooperate, Cooperate) \( { \succ _{\{1,2\}}^d } \) (Defect, Defect)

From Fig. 1, no stable set (with direct domination) exists in the prisoner's dilemma. However, since (Cooperate, Cooperate) \( { \succ } \) (Cooperate, Defect), (Defect,Cooperate) and there is no other indirect domination, the singleton {(Cooperate, Cooperate)} is the unique farsighted stable set with respect to \( { \succ } \). Hence if the two players are farsighted and make a joint but not binding move, the farsighted stable set succeeds in showing that cooperation of the players results in the unique stable outcome.

We now study stable outcomes in the mixed extension of the prisoner's dilemma, i. e., the prisoner's dilemma with mixed strategies played. Let \( { X_1=X_2=[0,1] } \) be the sets of mixed strategies of players 1 and 2, respectively, and let \( { t_1 \in X_1 } \) (resp. \( { t_2 \in X_2 } \)) denote the probability that player 1 (resp. 2) plays “Cooperate”. It is easily seen that the minimax payoffs to players 1 and 2 are both 1 in this game. We call a strategy combination that gives both players at least (resp. more than) their minimax payoffs, an individually rational (resp. a strictly individually rational) strategy combination. We then have the following theorem.

Theorem 10

Let
$$ T=\{(t_1,t_2)\vert 1/4 < t_1 \le 1\:, t_2 =1\}\cup\\ \{(t_1,t_2)\vert t_1 =1,1/4<t_2 \le 1\}\:, $$
and define the singleton \( { K^1(t_1,t_2)=\{(t_1,t_2)\} } \) for each \( (t_1,t_2)\in T \). Let \( K^2=\{(0,0),(1,1/4)\} \) and \( K^3=\{(0,0), (1/4,1)\} \). Then the sets \( { K^2, K^3 } \), and any \( { K^1(t_1,t_2) } \) with \( { (t_1,t_2)\in T } \) are farsighted stable sets of the mixed extension of the prisoner's dilemma. There are no other types of farsighted stable sets in the mixed extension of the prisoner's dilemma.

This theorem shows that if the two players are farsighted and make a joint but not binding move in the prisoner's dilemma, then essentially a single Pareto efficient and strictly individually rational strategy combination results as a stable outcome. i. e., \( { K^1(t_1,t_2) } \). We, however, have two exceptional cases \( { K^2, K^3 } \) that (Defect, Defect) could be stable together with one Pareto efficient point at which one player gains the same payoff as in (Defect, Defect).

n‑Person Prisoner's Dilemma

We consider an n‑person prisoner's dilemma. Let \( N=\{1,\dots,n\} \) be the player set. Each player i has two strategies: C (Cooperate) and D (Defect). Let \( { X_i =\{C,D\} } \) for each \( { i\in N } \). Hereafter we refer to a strategy combination as a state. The set of states is \( { X=\prod\nolimits_{i\in N} {X_i} } \). For each coalition \( { S\subset N } \), let \( { X_S =\prod\nolimits_{i\in S} {X_i} } \) and \( { X_{-S} =\prod\nolimits_{i\in S^c} {X_i} } \), where S c denotes the complement of S with respect to N. Let \( { x_s } \) and \( { x_{-s} } \) denote generic elements of X S and \( { X_{-S} } \), respectively. Player i's payoff depends not only on his/her strategy but also on the number of other cooperators. Player i's payoff function \( { u_i\colon X\to \Re } \) is given by \( { u_i(x)=f_i(x_i,h) } \), where \( { x\in X,x_i \in X_i } \) (player i's choice in x), and h is the number of players other than i playing C. We call the strategic form game thus defined an n‑person prisoner's dilemma game.

For simplifying discussion, we assume that all players are homogeneous and each player has an identical payoff function. That is, f i 's are identical, and simply written as f unless any confusion arises. We assume the following properties on the function f

Assumption 1

  1. (1)

    \( { f(D,h) > f(C,h) } \) for all \( { h=0,1,\dots,n-1 } \)

     
  2. (2)

    \( { f(C,n-1) > f(D,0) } \)

     
  3. (3)

    \( { f(C,h) } \) and \( { f(D,h) } \) are strictly increasing in h.

     

Property (1) means that every player prefers playing D to playing C regardless of which strategies other players play. Property (2) means that if all players play C, then each of them gains a payoff higher than the one in \( { (D,\dots,D) } \). Property (3) means that if the number of cooperators increases, every player becomes better off regardless of which strategy he/she plays.

It holds from Property (1) that \( { (D,\dots,D) } \) is the unique Nash equilibrium of the game. Here for \( { x,y\in X } \), we say that y is Pareto superior to x if \( { u_i (y)\ge u_i (x) } \) for all \( { i\in N } \) and \( { u_i(y) > u_i(x) } \) for some \( { i\in N } \). The state \( { x\in X } \) is said to be Pareto efficient if there is no \( { y\in X } \) that is Pareto superior to x. By Property (2), \( { (C,\dots,C) } \) is Pareto superior to \( { (D,\dots,D) } \). Together with Property (3), \( { (C,\dots,C) } \) is Pareto efficient.

Given a state x, we say that x is individually rational if for all \( { i\in N, u_i (x)\ge \mathop {\min}\nolimits_{y_{-i} \in X_{-i}} \mathop {\max}\nolimits_{y_i \in X_i} u_i (y) } \). If a strict inequality holds, we say that x is strictly individually rational. It holds from (1), (3) of Assumption 8.1 that \( { \mathop{\min}\nolimits_{y_{-i} \in X_{-i}} \mathop{\max}\nolimits_{y_i \in X_i} u_i (y)=f(D,0) } \).

The following theorem shows that any state that is strictly individually rational and Pareto efficient is itself a singleton farsighted stable set. That is, any strictly individually rational and Pareto efficient outcome is stable if the players are farsighted. Refer to Suzuki and Muto [51,52] for the details.

Theorem 11

For n‑person prisoner's dilemma game, if x is a strictly individually rational and Pareto efficient state, then \( { \{x\} } \) is a farsighted stable set.

Duopoly Market Games

We consider two types of duopoly: Cournot quantity‐setting duopoly and Bertrand price‐setting duopoly. For simplifying discussion, we will consider a simple duopoly model in which firms' cost functions and a market demand function are both linear. Similar results, however, hold in more general duopoly models.

There are two firms 1,2, each producing a homogeneous good with the same marginal cost \( { c > 0 } \). No fixed cost is assumed.

(1) Cournot duopoly: Firms' strategic variables are their production levels. Let x 1 and x 2 be production levels of firms 1 and 2, respectively. The market price \( { p(x_1,x_2) } \) for x 1 and x 2 is given by
$$ p(x_1,x_2)=\max(a-(x_1 +x_2),0)\:, $$
where \( { a>c } \). We restrict the domain of production of both firms to \( { 0\le x_i \le a-c\:,i=1,2 } \). This is reasonable since a firm would not overproduce to make a nonpositive profit. When x 1 and x 2 are produced, firm i's profit is given by
$$ \pi_i(x_1,x_2)=(p(x_1,x_2)-c)x_i\:.$$
Thus Cournot duopoly is formulated as the following strategic form game
$$ G^C=(N,\{X_i\}_{i=1,2}\:,\{\pi_i\}_{i=1,2})\:, $$
where the player set is \( { N=\{1,2\} } \), each player's strategy set is a closed interval between 0 and \( { a-c } \), i. e., \( X_1 = X_2 = [0,a-c] \), and their payoff functions are \( \pi_i,i=1,2 \). Let \( X=X_1 \times X_2 \). The joint profit of two firms is maximized when \( x_1 + x_2 =(a-c)/2 \).
(2) Bertrand duopoly: Firms' strategic variables are their price levels. Let
$$ D(p)=\max\;(a-p,0) $$
be the market demand at price p. Then the total profit at p is
$$ \prod(p)=(p-c) D(p)\:. $$
We restrict the domain of price level p of both firms to \( { c\le p\le a } \). This assumption is also reasonable since a firm would avoid a negative profit. The total profit \( { \prod(p) } \) is maximized at \( { p=(a+c)/2 } \), which is called a monopoly price.
Let p 1 and p 2 be prices of firms 1 and 2, respectively. We assume that if firms' prices are equal, then they share equally the total profit, otherwise all sales go to the lower pricing firm of the two. Thus firm i's profit is given by
$$ \begin{aligned} \rho_i(p_i,p_j)=\begin{cases} \prod(p_i) & \text{if}\enskip p_i <p_j \\ \prod(p_i)/2 & \text{if}\enskip p_i = p_j \\ 0 & \text{if}\enskip p_i > p_j \end{cases}\\\text{for}\enskip i,j=1,2, i\ne j\:.\end{aligned} $$
Hence Bertrand duopoly is formulated as the strategic form game
$$ G^B=\left(N,\{Y_i\}_{i=1,2},\{\rho_i\}_{i=1,2}\right)\:, $$
where \( { N=\{1,2\}, Y_1 = Y_2 = [c,a] } \), and \( { \rho_i(i=1,2) } \) is i's payoff function. Let \( { Y=Y_1 \times Y_2 } \).

It is well-known that a Nash equilibrium is uniquely determined in either market: \( { x_1 =x_2 =(a-c)/3 } \) in the Cournot market, and \( { p_1 =p_2 =c } \) in the Bertrand market.

The following theorem holds for the farsighted stable sets in Cournot duopoly.

Theorem 12

Let \( { (x_1,x_2)\in X } \) be any strategy pair with \( { x_1 + x_2 =(a-c)/2 } \). Then the singleton \( { \{(x_1,x_2)\} } \) is a farsighted stable set. Furthermore, every farsighted stable set is of the form \( { \{(x_1,x_2)\} } \) with \( { x_1 +x_2 =(a-c)/2 } \) and \( { x_1,x_2 \ge 0 } \).

As mentioned before, any strategy pair \( { (x_1,x_2) } \) with \( { x_1 +x_2 =(a-c)/2 } \) and \( { x_1,x_2 \ge 0 } \) maximizes two firms' joint profit. This suggests that the von Neumann–Morgenstern stability together with firms' farsighted behavior produce joint profit maximization even if firms' collaboration is not binding.

As for Bertrand duopoly, we have the following theorem, which claims that the monopoly price pair is itself a farsighted stable set, and no other farsighted stable set exists. Therefore the von Neumann–Morgenstern stability together with firms' farsighted behavior attain efficiency (from the standpoint of firms) also in Bertrand duopoly. Refer to Suzuki and Muto [53] for the details.

Theorem 13

Let \( { p=(p_1,p_2) } \) be the pair of monopoly prices, i. e., \( { p_1 =p_2 =(a+c)/2 } \). Then the singleton \( { \{p\} } \) is the unique farsighted stable set.

For studies of stable sets with direct domination in duopoly market games, refer to Muto and Okada [29,30]. Properties of stable sets and Harsanyi's original farsighted stable sets in pure exchange economies are investigated by Greenberg et al. [10]. For further studies on stable sets and farsighted stable sets in strategic form games, refer to Kaneko [17], Mariotti [24], Xue [55,56], Diamantoudi and Xue [4].

Future Directions

In this paper, we have reviewed applications of vonNeumann–Morgenstern stable sets in abstract games,characteristic function form games, and strategic form games to economic, political and social systems.

Stable sets give us insights into coalition formation among players in the systems in question. Farsighted stable sets, especially applied to someeconomic systems, show that players' farsighted behavior leads to Pareto efficient outcomes even though their collaboration is not binding. The stable setanalysis is also applicable to games with infinitely many players. Those analyses show us new approaches to large economic and social systems withinfinitely many players. For the details, refer to Hart [15], Einy et al. [7], Einy and Shitovitz [6], Greenberg et al. [11], Shitovitz and Weber [48], and Rosenmüller andShitovitz [37]. There is also a study on the linkage between common knowledge of Bayesianrationality and achievement of stable sets in generalized abstract games. Refer to Luo [22,23] for the details.

Analyses of social systems by applying the concepts of farsighted stable sets as well as stable sets must further advance theoretical studies ongames in which players inherently take both cooperative and non‐cooperative behavior. Those studies will in turn have impacts on developments ofeconomics, politics, sociology, and many applied social sciences.

Bibliography

Primary Literature

  1. 1.
    Aumann R, Peleg B (1960) Von Neumann–Morgenstern solutions to cooperative games without side payments. Bull Am Math Soc 66:173–179MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bott R (1953) Symmetric solutions to majority games. In: Kuhn HW, Tucker AW (eds) Contribution to the theory of games, vol II. Annals of Mathematics Studies, vol 28. Princeton University Press, Princeton, pp 319–323Google Scholar
  3. 3.
    Chwe MS-Y (1994) Farsighted coalitional stability. J Econ Theory 63:299–325MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Diamantoudi E, Xue L (2003) Farsighted stability in hedonic games. Soc Choice Welf 21:39–61MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ehlers L (2007) Von Neumann–Morgenstern stable sets in matching problems. J Econ Theory 134:537–547MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Einy E, Shitovitz B (1996) Convex games and stable sets. Games Econ Behav 16:192–201MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Einy E, Holzman R, Monderer D, Shitovitz B (1996) Core and stable sets of large games arising in economics. J Econ Theory 68:200–211MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69:9–15MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Greenberg J (1990) The theory of social situations: an alternative game theoretic approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  10. 10.
    Greenberg J, Luo X, Oladi R, Shitovitz B (2002) (Sophisticated) stable sets in exchange economies. Games Econ Behav 39:54–70MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Greenberg J, Monderer D, Shitovitz B (1996) Multistage situations. Econometrica 64:1415–1437MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Griesmer JH (1959) Extreme games with three values. In: Tucker AW, Luce RD (eds) Contribution to the theory of games, vol IV. Annals of Mathematics Studies, vol 40. Princeton University Press, Princeton, pp 189–212Google Scholar
  13. 13.
    Harsanyi J (1974) An equilibrium‐point interpretation of stable sets and a proposed alternative definition. Manag Sci 20:1472–1495MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hart S (1973) Symmetric solutions of some production economies. Int J Game Theory 2:53–62MATHCrossRefGoogle Scholar
  15. 15.
    Hart S (1974) Formation of cartels in large markets. J Econ Theory 7:453–466CrossRefGoogle Scholar
  16. 16.
    Heijmans J (1991) Discriminatory von Neumann–Morgenstern solutions. Games Econ Behav 3:438–452MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kaneko M (1987) The conventionally stable sets in noncooperative games with limited observations I: Definition and introductory argument. Math Soc Sci 13:93–128MATHCrossRefGoogle Scholar
  18. 18.
    Lucas WF (1968) A game with no solution. Bull Am Math Soc 74:237–239MATHCrossRefGoogle Scholar
  19. 19.
    Lucas WF (1990) Developments in stable set theory. In: Ichiishi T et al (eds) Game Theory and Applications, Academic Press, New York, pp 300–316Google Scholar
  20. 20.
    Lucas WF, Michaelis K, Muto S, Rabie M (1982) A new family of finite solutions. Int J Game Theory 11:117–127MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Lucas WF, Rabie M (1982) Games with no solutions and empty cores. Math Oper Res 7:491–500MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Luo X (2001) General systems and φ‑stable sets: a formal analysis of socioeconomic environments. J Math Econ 36:95–109MATHCrossRefGoogle Scholar
  23. 23.
    Luo X (2006) On the foundation of stability. Academia Sinica, Mimeo, available at http://www.sinica.edu.tw/~xluo/pa14.pdf
  24. 24.
    Mariotti M (1997) A model of agreements in strategic form games. J Econ Theory 74:196–217MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Moulin H (1995) Cooperative Microeconomics: A Game‐Theoretic Introduction. Princeton University Press, PrincetonGoogle Scholar
  26. 26.
    Muto S (1979) Symmetric solutions for symmetric constant‐sum extreme games with four values. Int J Game Theory 8:115–123MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Muto S (1982) On Hart production games. Math Oper Res 7:319–333MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Muto S (1982) Symmetric solutions for (n,k) games. Int J Game Theory 11:195–201MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Muto S, Okada D (1996) Von Neumann–Morgenstern stable sets in a price‐setting duopoly. Econ Econ 81:1–14Google Scholar
  30. 30.
    Muto S, Okada D (1998) Von Neumann–Morgenstern stable sets in Cournot competition. Econ Econ 85:37–57Google Scholar
  31. 31.
    von Neumann J, Morgenstern O (1953) Theory of Games and Economic Behavior, 3rd ed. Princeton University Press, PrincetonMATHGoogle Scholar
  32. 32.
    Owen G (1965) A class of discriminatory solutions to simple n‑person games. Duke Math J 32:545–553MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Owen G (1968) n‑Person games with only 1, n-1, and n‑person coalitions. Proc Am Math Soc 19:1258–1261MathSciNetMATHGoogle Scholar
  34. 34.
    Owen G (1995) Game theory, 3rd ed. Academic Press, New YorkGoogle Scholar
  35. 35.
    Peleg B (1986) A proof that the core of an ordinal convex game is a von Neumann–Morgenstern solution. Math Soc Sci 11:83–87Google Scholar
  36. 36.
    Rosenmüller J (1977) Extreme games and their solutions. In: Lecture Notes in Economics and Mathematical Systems, vol 145. Springer, Berlin Google Scholar
  37. 37.
    Rosenmüller J, Shitovitz B (2000) A characterization of vNM‐stable sets for linear production games. Int J Game Theory 29:39–61Google Scholar
  38. 38.
    Roth A, Postlewaite A (1977) Weak versus strong domination in a market with indivisible goods. J Math Econ 4:131–137MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Roth A, Sotomayor M (1990) Two-Sided Matching: A Study in Game‐Theoretic Modeling and Analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  40. 40.
    Quint T, Wako J (2004) On houseswapping, the strict core, segmentation, and linear programming, Math Oper Res 29:861–877MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Shapley LS (1953) Quota solutions of n‑person games. In: Kuhn HW, Tucker TW (eds) Contribution to the theory of games, vol II. Annals of Mathematics Studies, vol 28. Princeton University Press, Princeton, pp 343–359Google Scholar
  42. 42.
    Shapley LS (1959) The solutions of a symmetric market game. In: Tucker AW, Luce RD (eds) Contribution to the theory of games, vol IV. Annals of Mathematics Studies, vol 40. Princeton University Press, Princeton, pp 145–162Google Scholar
  43. 43.
    Shapley LS (1962) Simple games: An outline of the descriptive theory. Behav Sci 7:59–66MathSciNetCrossRefGoogle Scholar
  44. 44.
    Shapley LS (1964) Solutions of compound simple games. In Tucker AW et al (eds) Advances in Game Theory. Annals of Mathematics Studies, vol 52. Princeton University Press, Princeton, pp 267–305Google Scholar
  45. 45.
    Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Shapley LS, Scarf H (1974) On cores and indivisibilities. J Math Econ 1:23–37MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Shapley LS, Shubik M (1972) The assignment game I: The core. Int J Game Theory 1:111–130MathSciNetCrossRefGoogle Scholar
  48. 48.
    Shitovitz B, Weber S (1997) The graph of Lindahl correspondence as the unique von Neumann–Morgenstern abstract stable set. J Math Econ 27:375–387MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Simonnard M (1966) Linear programming. Prentice‐Hall, New JerseyMATHGoogle Scholar
  50. 50.
    Solymosi T, Raghavan TES (2001) Assignment games with stable core. Int J Game Theory 30:177–185MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Suzuki A, Muto S (2000) Farsighted stability in prisoner’s dilemma. J Oper Res Soc Japan 43:249–265MathSciNetMATHGoogle Scholar
  52. 52.
    Suzuki A, Muto S (2005) Farsighted stability in n‑person prisoner’s dilemma. Int J GameTheory 33:431–445MathSciNetMATHGoogle Scholar
  53. 53.
    Suzuki A, Muto S (2006) Farsighted behavior leads to efficiency in duopoly markets. In: Haurie A, et al (eds) Advances in Dynamic Games. Birkhauser, Boston, pp 379–395CrossRefGoogle Scholar
  54. 54.
    Wako J (1991) Some properties of weak domination in an exchange market with indivisible goods. Jpn Econ Rev 42:303–314Google Scholar
  55. 55.
    Xue L (1997) Nonemptiness of the largest consistent set. J Econ Theory 73:453–459MATHCrossRefGoogle Scholar
  56. 56.
    Xue L (1998) Coalitional stability under perfect foresight. Econ Theory 11:603–627MATHCrossRefGoogle Scholar

Books and Reviews

  1. 57.
    Lucas WF (1992) Von Neumann–Morgenstern stable sets. In: Aumann RJ, Hart S (eds) Handbook of Game Theory with Economic Applications, vol 1. North‐Holland, Amsterdam, pp 543–590CrossRefGoogle Scholar
  2. 58.
    Shubik M (1982) Game theory in the social sciences: Concepts and solutions. MIT Press, BostonMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Jun Wako
    • 1
  • Shigeo Muto
    • 2
  1. 1.Department of EconomicsGakushuin UniversityTokyoJapan
  2. 2.Graduate School of Decision Science and Technology, TokyoInstitute of TechnologyTokyoJapan