Definitions Stability notions Other solution concepts Subclasses An Introduction to Cooperative Game Theory Maria Serna Fall 2016 AGT-MIRI Cooperative Game Theory Definitions Stability notions Other solution concepts Subclasses References G. Chalkiadakis, E. Elkind, M. Wooldridge Computational Aspects of Cooperative Game Theory Morgan & Claypool, 2012 Wikipedia. G. Owen Game Theory 3rd edition, Academic Press, 1995 AGT-MIRI Cooperative Game Theory Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses 1 Definitions 2 Stability notions 3 Other solution concepts 4 Subclasses AGT-MIRI Cooperative Game Theory Cooperative game theory model scenarios, where agents can benefit by cooperating, and binding agreements are possible. In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to the group, that has to divided it among its members: Transferable utility games. individuals: Non-transferable utility games. For the moment we focus on TU games Notation: N, set of players, C; S; X ⊆ N are coalitions. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. AGT-MIRI Cooperative Game Theory agents can benefit by cooperating, and binding agreements are possible. In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to the group, that has to divided it among its members: Transferable utility games. individuals: Non-transferable utility games. For the moment we focus on TU games Notation: N, set of players, C; S; X ⊆ N are coalitions. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. Cooperative game theory model scenarios, where AGT-MIRI Cooperative Game Theory In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to the group, that has to divided it among its members: Transferable utility games. individuals: Non-transferable utility games. For the moment we focus on TU games Notation: N, set of players, C; S; X ⊆ N are coalitions. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. Cooperative game theory model scenarios, where agents can benefit by cooperating, and binding agreements are possible. AGT-MIRI Cooperative Game Theory the group, that has to divided it among its members: Transferable utility games. individuals: Non-transferable utility games. For the moment we focus on TU games Notation: N, set of players, C; S; X ⊆ N are coalitions. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. Cooperative game theory model scenarios, where agents can benefit by cooperating, and binding agreements are possible. In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to AGT-MIRI Cooperative Game Theory individuals: Non-transferable utility games. For the moment we focus on TU games Notation: N, set of players, C; S; X ⊆ N are coalitions. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. Cooperative game theory model scenarios, where agents can benefit by cooperating, and binding agreements are possible. In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to the group, that has to divided it among its members: Transferable utility games. AGT-MIRI Cooperative Game Theory For the moment we focus on TU games Notation: N, set of players, C; S; X ⊆ N are coalitions. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. Cooperative game theory model scenarios, where agents can benefit by cooperating, and binding agreements are possible. In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to the group, that has to divided it among its members: Transferable utility games. individuals: Non-transferable utility games. AGT-MIRI Cooperative Game Theory Notation: N, set of players, C; S; X ⊆ N are coalitions. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. Cooperative game theory model scenarios, where agents can benefit by cooperating, and binding agreements are possible. In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to the group, that has to divided it among its members: Transferable utility games. individuals: Non-transferable utility games. For the moment we focus on TU games AGT-MIRI Cooperative Game Theory Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Non-Cooperative versus cooperative Games Non-cooperative game theory model scenarios where players cannot make binding agreements. Cooperative game theory model scenarios, where agents can benefit by cooperating, and binding agreements are possible. In cooperative games, actions are taken by groups of agents, coalitions, and payoffs are given to the group, that has to divided it among its members: Transferable utility games. individuals: Non-transferable utility games. For the moment we focus on TU games Notation: N, set of players, C; S; X ⊆ N are coalitions. AGT-MIRI Cooperative Game Theory for each subset of players C ⊆ N, v(C) is the amount that the members of C can earn by working together usually it is assumed that v is normalized: v(;) = 0, non-negative: v(C) ≥ 0, for any C ⊆ N, and monotone: v(C) ≤ v(D), for any C, D such that C ⊆ D A coalition is any subset of N N itself is called the grand coalition. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Characteristic Function Games A characteristic function game is a pair (N; v), where: N = f1; :::; ng is the set of players and v : 2N ! R is the characteristic function. AGT-MIRI Cooperative Game Theory usually it is assumed that v is normalized: v(;) = 0, non-negative: v(C) ≥ 0, for any C ⊆ N, and monotone: v(C) ≤ v(D), for any C, D such that C ⊆ D A coalition is any subset of N N itself is called the grand coalition. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Characteristic Function Games A characteristic function game is a pair (N; v), where: N = f1; :::; ng is the set of players and v : 2N ! R is the characteristic function. for each subset of players C ⊆ N, v(C) is the amount that the members of C can earn by working together AGT-MIRI Cooperative Game Theory normalized: v(;) = 0, non-negative: v(C) ≥ 0, for any C ⊆ N, and monotone: v(C) ≤ v(D), for any C, D such that C ⊆ D A coalition is any subset of N N itself is called the grand coalition. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Characteristic Function Games A characteristic function game is a pair (N; v), where: N = f1; :::; ng is the set of players and v : 2N ! R is the characteristic function. for each subset of players C ⊆ N, v(C) is the amount that the members of C can earn by working together usually it is assumed that v is AGT-MIRI Cooperative Game Theory A coalition is any subset of N N itself is called the grand coalition. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Characteristic Function Games A characteristic function game is a pair (N; v), where: N = f1; :::; ng is the set of players and v : 2N ! R is the characteristic function. for each subset of players C ⊆ N, v(C) is the amount that the members of C can earn by working together usually it is assumed that v is normalized: v(;) = 0, non-negative: v(C) ≥ 0, for any C ⊆ N, and monotone: v(C) ≤ v(D), for any C, D such that C ⊆ D AGT-MIRI Cooperative Game Theory N itself is called the grand coalition. Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Characteristic Function Games A characteristic function game is a pair (N; v), where: N = f1; :::; ng is the set of players and v : 2N ! R is the characteristic function. for each subset of players C ⊆ N, v(C) is the amount that the members of C can earn by working together usually it is assumed that v is normalized: v(;) = 0, non-negative: v(C) ≥ 0, for any C ⊆ N, and monotone: v(C) ≤ v(D), for any C, D such that C ⊆ D A coalition is any subset of N AGT-MIRI Cooperative Game Theory Definitions TU characteristic function games Stability notions Outcomes Other solution concepts Imputations Subclasses Characteristic Function Games A characteristic function game is a pair (N; v), where: N = f1; :::; ng is the set of players and v : 2N ! R is the characteristic function. for each subset of players C ⊆ N, v(C) is the amount that the members of C can earn by working together usually it is assumed that v is normalized: v(;) = 0, non-negative: v(C) ≥ 0, for any C ⊆ N, and monotone: v(C) ≤ v(D), for any C, D such that C ⊆ D A coalition is any subset of N N itself is called the grand coalition.