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Cooperative Games (Von Neumann–Morgenstern Stable Sets)

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Computational Complexity

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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

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Abbreviations

Characteristic function form game:

 A characteristic function form game consists of a set of players and a characteristic function that gives each group of players, called a coalition, a value or a set of payoff vectors that they can gain by themselves. It is a typical representation of cooperative games. For characteristic function form games, several solution concepts are defined such as von Neumann – Morgenstern stable set, core, bargaining set, kernel, nucleolus, and Shapley value.

Abstract game:

An abstract game consists of a set of outcomes and a binary relation, called domination, on the outcomes. Von Neumann and Morgenstern presented this game form for general applications of stable sets.

Strategic form game:

A strategic form game consists of a player set, each player's strategy set, and each player's payoff function. It is usually used to represent non‐cooperative games.

Imputation:

An imputation is a payoff vector in a characteristic function form game that satisfies group rationality and individual rationality. Theformer means that the players divide the amount that the grand coalition of all players can gain, and the latter says that each player is assigned at least the amount that he/she can gain by him/herself.

Domination:

Domination is a binary relation defined on the set of imputations, outcomes, or strategy combinations, depending on the form of a given game. In characteristic function form games, an imputation is said to dominate another imputation if there is a coalition of players such that they can realize their payoffs in the former by themselves, and make each of them better off than in the latter. Domination given a priori in abstract games can be also interpreted in the same way. In strategic form games, domination is defined on the basis of commonly beneficial changes of strategies by coalitions.

Internal stability:

A set of imputations (outcomes, strategy combinations) satisfies internal stability if there is no domination between any two imputations in the set.

External stability:

A set of imputations (outcomes, strategy combinations) satisfies external stability if any imputation outside the set is dominated by some imputation inside the set.

von Neumann–Morgenstern stable set:

A set of imputations (outcomes, strategy combinations) is a von Neumann–Morgenstern stable set if it satisfies both internal and external stability.

Farsighted stable set:

A farsighted stable set is a more sophisticated stable set concept mainly defined for strategic form games. Given two strategy combinations x and y, we say that x indirectly dominates y if there exist a sequence of coalitions \( { S^1,\ldots,S^p } \) and a sequence of strategy combinations \( { y=x^0, x^1,\ldots,x^p=x } \) such that each coalition S j can induce strategy combination x j by a joint move from \( { x^{j-1} } \), and all members of S j end up with better payoffs at \( { x^p=x } \) compared to the payoffs at \( { x^{\kern1pt j-1} } \). A farsighted stable set is a set of strategy combinations that is stable both internally and externally with respect to the indirect domination. A farsighted stable set can be defined in abstract games and characteristic function form games.

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Author information

Authors and Affiliations

  1. Department of Economics, Gakushuin University, Tokyo, Japan

    Jun Wako

  2. Graduate School of Decision Science and Technology, Tokyo, Institute of Technology, Tokyo, Japan

    Shigeo Muto

Authors
  1. Jun Wako
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  2. Shigeo Muto
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Wako, J., Muto, S. (2012). Cooperative Games (Von Neumann–Morgenstern Stable Sets). In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_43

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