Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users Values for graph-restricted games with coalition structure Anna Khmelnitskaya St. Petersburg Institute for Economics and Mathematics, Russian Academy of Sciences University of Eastern Piedmont, Alessandria April 14, 2008 Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users Aumann and Drèze (1974), Owen (1977) Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users Myerson (1977) Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users Vázquez-Brage, García-Jurado, and Carreras (1996) Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users model of the paper Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users case of the coincidence of both models Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users model of the paper Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users N 1 N 2 N k N m sharing an international river among multiple users without international firms Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users 1 Preliminaries TU games Games with coalition structure Games with cooperation structure 2 Graph games with coalition structure 3 CE G-values for games with cooperation structure The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values PG-values 4 CE values Stability Distribution of Harsanyi dividends 5 Generalization to games with level structure 6 Sharing a river with multiple users Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure TU games CE G-values for games with cooperation structure Games with coalition structure PG-values Games with cooperation structure Generalization to games with level structure Sharing a river with multiple users A cooperative TU game is a pair � N , v � where N = { 1 , . . . , n } is a finite set of n ≥ 2 players, v : 2 N → I R , v ( ∅ ) = 0, is a characteristic function. A subset S ⊆ N (or S ∈ 2 N ) of s players is a coalition, v ( S ) presents the worth of the coalition S . G N is the class of TU games with a fixed player set N . R 2 n − 1 of vectors { v ( S ) } S ⊆ N ( G N = I S � = ∅ ) A subgame of a game v is a game v | T with a player set T ⊂ N , T � = ∅ , and v | T ( S ) = v ( S ) for all S ⊆ T . A game v is superadditive if v ( S ∪ T ) ≥ v ( S ) + v ( T ) for all S , T ⊆ N such that S ∩ T = ∅ . Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure TU games CE G-values for games with cooperation structure Games with coalition structure PG-values Games with cooperation structure Generalization to games with level structure Sharing a river with multiple users The unanimity games { u T } T ⊆ N T � = ∅ , � 1 , T ⊆ S , u T ( S ) = for all S ⊆ N , 0 , T �⊆ S , create a basis for G N , i.e., every game v ∈ G N can be uniquely presented in the linear form � λ v v = T u T , T ⊆ N T � = ∅ � ( − 1 ) t − s v ( S ) , for all T ⊆ N , T � = ∅ . where λ v T = S ⊆ T λ v T is called the dividend of the coalition T in the game v . � λ v v ( S ) = T , for all S ⊆ N , S � = ∅ . T ⊆ S T � = ∅ Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure TU games CE G-values for games with cooperation structure Games with coalition structure PG-values Games with cooperation structure Generalization to games with level structure Sharing a river with multiple users For a permutation π : N → N , let π i = { j ∈ N | π ( j ) ≤ π ( i ) } be the set of players with rank number not greater than the rank number of i , including i itself. R n of a game v and a The marginal contribution vector m π ( v ) ∈ I permutation π is given by m π i ( v ) = v ( π i ) − v ( π i \ i ) , i ∈ N . By u we denote the permutation with natural ordering from 1 to n , i.e., u ( i ) = i , i ∈ N , and by l the permutation with reverse ordering n , n − 1 , . . . , 1, i.e., l ( i ) = n + 1 − i , i ∈ N . Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure TU games CE G-values for games with cooperation structure Games with coalition structure PG-values Games with cooperation structure Generalization to games with level structure Sharing a river with multiple users R n , For any G ⊆ G N , a value on G is a mapping ξ : G → I the real number ξ i ( v ) is the payoff to player i in the game v . The Shapley value of a game v λ v � T Sh i ( v ) = t , for all i ∈ N , T ⊆ N T ∋ i or Sh i ( v ) = 1 � m π i ( v ) , for all i ∈ N . n ! π ∈ Π The core of a game v R n | x ( N ) = v ( N ) , x ( S ) ≥ v ( S ) , for all S ⊆ N } . C ( v ) = { x ∈ I A value ξ is stable if for any superadditive game v ∈ G N , ξ ( v ) ∈ C ( v ) . Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure TU games CE G-values for games with cooperation structure Games with coalition structure PG-values Games with cooperation structure Generalization to games with level structure Sharing a river with multiple users A coalition structure or a system of a priori unions on N is a partition P = { N 1 , ..., N m } , N 1 ∪ ... ∪ N m = N , N i ∩ N j = ∅ , i � = j . A pair � v , P� presents a game with coalition structure (P-game). G P N is the set of all games with coalition structure with fixed N . R n that assigns a vector of payoffs A P-value is an operator ξ : G P N → I to any game with coalition structure. For a game with coalition structure � v , P� , following Owen we define a quotient game v P on the player set M = { 1 , . . . , m } : � v P ( Q ) = v ( N k ) , for all Q ⊆ M . k ∈ Q R n we denote x P = � � R m . For any payoff x ∈ I x ( N k ) k ∈ M ∈ I Notice that � v , { N }� coincides the game v itself. � N � = {{ 1 } , . . . , { n }} For i ∈ N let k ( i ) be the index such that i ∈ N k ( i ) . Anna Khmelnitskaya Values for GR-games with CS
Outline Preliminaries Graph games with coalition structure TU games CE G-values for games with cooperation structure Games with coalition structure PG-values Games with cooperation structure Generalization to games with level structure Sharing a river with multiple users A cooperation structure on N is specified by an undirected graph without loops L , L ⊆ L c = { { i , j } | i , j ∈ N , i � = j } , where L c is the complete graph on N while an unordered pair { i , j } is a link (eage) between players i , j ∈ N . A pair � v , L � constitutes a game with cooperation structure or, in other terms, a graph game (G-game). G L N is the set of all games with cooperation structure with fixed N . R n that assigns a vector of payoffs A G-value is an operator ξ : G L N → I to any game with cooperation structure. A subgraph of L on S ⊆ N is the graph L | S = {{ i , j } ∈ L | i , j ∈ S } . C L ( S ) is the set of all connected subcoalitions of S . S / L is the set of all components (maximally connected) of S ⊆ N . ( S / L ) i is the component of S containing player i ∈ S . Anna Khmelnitskaya Values for GR-games with CS
Recommend
More recommend
