Introduction to Cooperative Games Mehdi Dastani BBL-521 M.M.Dastani@uu.nl
Cooperative Game Theory ◮ In non-cooperative games, decisions are made by individual players. ◮ Binding agreements between players is not possible ◮ Individual players maximise their own utilities ◮ In cooperative games, decisions are made by groups of players. ◮ Binding agreements between players is possible ◮ Individual players are utility maximisers but can benefit by cooperating Transferable utility games: actions are decided by groups of players and utilities are assigned to the groups. The group utility is then distributed among individual players.
Cooperative Game Theory ◮ Players form groups to perform tasks ◮ Each group of players receives a utility (to be distributed among themselves) ◮ Examples: ◮ Matching games based on weighted graphs D 9 A 7 E 6 4 7 5 3 C 5 6 B F ◮ Weighted voting game: ◮ A, B, C, and D have 45, 25, 15, and 15 votes. ◮ 51 votes are required to pass the $100 million bill.
Cooperative Game A cooperative game is G = ( N , v ) , where ◮ N is a set of players ◮ v : 2 N → R is the characteristic function of the game Example: G = ( { 1 , 2 } , v ) , where ◮ v ( ∅ ) = 0 ◮ v ( { 1 } ) = v ( { 2 } ) = 5 ◮ v ( { 1 , 2 } ) = 20 Note : ◮ v ( S ) is the value that is assigned to coalition S ⊆ N ◮ 2 N is the set of all possible coalitions ◮ A coalition structure CS is a partition on N ◮ Optimal coalition structure: max v ( S ) � CS S ∈ CS Exercise 1: Model weighted voting game as a cooperative game.
Cooperative Games Let G = ( N , v ) be a cooperative game. ◮ v is normalised: v ( ∅ ) = 0 ◮ v is non-negative: v ( S ) ≥ 0 for any S ⊆ N ◮ G is monotone if for any S 1 ⊆ S 2 ⊆ N : v ( S 1 ) ≤ v ( S 2 ) ◮ G is superadditive if for any disjoint S 1 , S 2 ⊆ N : v ( S 1 ) + v ( S 2 ) ≤ v ( s 1 ∪ S 2 ) An outcome x = � x 1 , . . . , x k � for a coalition S (consisting of k members in game G ) is a distribution of v ( S ) to its members such that � x i = v ( S ) i ∈ S . Given the cooperative game G = ( { 1 , . . . , n } , v ) , an outcome for the grand coalition is � x 1 , . . . , x n � such that � x i = v ( N ) i ∈ N Which coalitions can be formed? = ⇒ Which coalitions are stable?
The Core: Coalition Stability An outcome for a coalition is stable if no subcoalition can object to it. Is the grand coalition stable? An outcome � x 1 , . . . , x n � for the grand coalition is optimal when each coalition is getting at least what it can make on its own, i.e., no one has any incentive to deviate. The Core of a cooperative game G = ( { 1 , . . . , n } , v ) consists of all outcomes � x 1 , . . . , x n � for the grand coalition for which it holds: � v ( S ) , for all S ⊆ N x i ≥ i ∈ S In other words, the Core of a cooperative game G = ( { 1 , . . . , n } , v ) is the set of vectors in R n that satify the following constraints: ◮ � x i = v ( N ) i ∈ N ◮ � x i ≥ v ( S ) for all S ⊆ N i ∈ S Is the grand coalition stable? = ⇒ Is the Core non-empty? Exercise 2: Determine the Core of the weighted voting game.
Stability versus Fairness Outcomes in the Core are stable, but may not be fairly distributed. Example: G = ( { 1 , 2 } , v ) , where ◮ v ( ∅ ) = 0 ◮ v ( { 1 } ) = v ( { 2 } ) = 5 ◮ v ( { 1 , 2 } ) = 20 Every outcome between ( 15 , 5 ) , . . . , ( 5 , 15 ) is in the core, but some outcomes such as 15 , 5 are not fair. Which outcomes can be considered as fair?
Marginal Contribution The basic idea is to distribute the utility of a coalition based on the contribution of players in that coalition. Let G = ( N , v ) be a cooperative game. The marginal contribution of a player i to a coalition S ⊆ N \ { i } is denoted as µ i ( S ) and defined as follows: µ i ( S ) = v ( S ∪ { i } ) − v ( S ) The average marginal contribution of a player i in a game G is defined as follows: 1 � 2 n − 1 · µ i ( S ) S ⊆ N \{ i } Example: G = ( { 1 , 2 } , v ) , where ◮ v ( ∅ ) = 0 ◮ v ( { 1 } ) = v ( { 2 } ) = 5 ◮ v ( { 1 , 2 } ) = 20
Shapley Value In some cases, the marginal contribution of player i to coalition S depends on the order in which S is formed. Let G = ( N , v ) be a cooperative game where N = { 1 , . . . , n } . The set of possible permutations of players is Π( N ) . Note we have n ! permutations. Let C i ( π ) denote the set of predecessors of i in the permutation π ∈ Π( N ) . The Shapley value of player i in the game G is defined as follows: sh i = 1 � µ i ( C i ( π )) n ! π ∈ Π( N ) Exercise 3: determine the Shapley value for all players in G = ( { 1 , 2 , 3 } , v ) , where ◮ v ( ∅ ) = 0 ; v ( { 1 } ) = v ( { 2 } ) = v ( { 3 } ) = 5 ◮ v ( { 1 , 2 } ) = v ( { 1 , 3 } ) = 10 ; v ( { 2 , 3 } ) = 20 ◮ v ( { 1 , 2 , 3 } ) = 25
Shapley Value: Properties Let G = ( N , v ) be a cooperative game. ◮ Dummy player: player i is dummy if v ( S ) = v ( S ∪ { i } ) for any S ⊆ N ◮ Two players i and j are symmetric if v ( S ∪ { i } ) = v ( S ∪ { j } ) for any S ⊆ N ⊆ { i , j } . Properties of the Shapley value: 1. Efficiency: sh 1 + . . . + sh n = v ( N ) 2. Dummy: if i is a dummy player, sh i = 0 3. Symmetry: if i and j are symmetric, sh i = sh j 4. Additivity: sh i ( G 1 + G 2 ) = sh i ( G 1 ) + sh i ( G 2 ) for games G 1 and G 2 Theorem Shapley value is the only payoff distribution scheme that has properties 1- 4. Exercise 4: Check properties 1 - 4 for the game of previous slide. Exercise 5: Determine the Shapley values of the players in the Weighted voting game and check if their Shapely values is in the Core of the game.
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