What is the sum of the following infinite series 1 x x 2 x3 xn where 0 x 1 A B C D E Infinity 1 2x 1 3x 1 x 1 1 x Cooperative Game Theory Coalitional Games Focus on what groups can accomplish if they work together Contrast to Nash equilibrium which focuses on what individuals can do acting alone sometimes known as non cooperative game theory Coalitional Game with transferable payoffs A set of players N A coalition S is a subset of N Grand coalition is N itself Coalitional game with transferable payoffs assigns a value v S to every subset of S An action for the coalition S is a distribution of Its total value to its members Think of v S as an amount of money that the coalition can earn on its own and can divide this money in any way that adds to v S Two Player unanimity game An Almost Trivial Example Players 1 and 2 Non empty subsets are 1 2 1 2 Let v 1 v 2 0 and v 12 1 Set of actions available to coalition 1 2 are distributions of payoffs x1 x2 such that x1 0 x2 0 and x1 x2 1 Three player majority game A slightly more interesting example Players 1 2 and 3 Non empty subsets of N 1 2 3 are N 12 13 23 1 2 3 A cake whose total value is 1 is to be divided Any coalition that is a majority can choose how to divide it Then v 12 v 23 v 13 v 123 1 and v 1 v 2 v 3 0 Actions available to any coalition are possible divisions of the cake For example coalition 12 can choose any division such that x1 0 x2 0 and x1 x2 1 Landowner workers example Player 1 owns all of the land There are k landless workers Total food produced is given by a production function f x where x is the number of people who work on the land Landless workers can produce nothing on their own Landowner can only produce f 1 with his own labor Coalition values for landowner worker model Player 1 is the landowner By himself he gets v 1 f 1 Workers get nothing without land so any coalition S to which the landowner does not belong has v S 0 A coalition S to which the landowner does belong gets V S f k 1 where k is the number of workers in the coalition The Core The core of a coalitional game is the set of outcomes x actions by the grand coalition such that no coalition has an action that all of its members prefer to x Examples Two person unanimity game Core consists of all possible divisions Why Three person majority game Core is the empty set Why Five people must decide how to divide a pie of fixed size Any 4 of them can agree to overturn the status quo agreement and redistribute the pie among themselves A One outcome in the core of this game is that in which each player gets 1 5 of the pie B One outcome in the core of this game is that in which 4 players each get of the pie and one player gets nothing C The core of this game is empty Landowner worker game 2 workers Let x1 x2 x3 be an allocation of the output f 3 from 3 people working on landowner s land If this in the core then it must be that x1 f 1 otherwise coalition 1 could do better by self x1 x2 f 2 otherwise 12 could do better by themselves x1 x3 f 2 otherwise 13 could do better by themselves x1 x2 x3 f 3 Conclusions x1 x3 f 2 and x1 x2 x3 f 3 imply that x2 f 3 f 2 Similarly x1 x2 f 2 and x1 x2 x3 f 3 imply that x3 f 3 f 2 Thus no worker gets more than the marginal product of a third worker Example Suppose f 1 3 f 2 5 f 3 6 Then we must have x1 3 x2 1 and x3 1 Any allocation where 2 and 3 each gets a nonnegative amount and 1 gets more than 3 units will be in the core One owner two possible buyers Owner person 1 has an object that is worthless to him worth 1 to either of two possible buyers persons 2 and 3 Persons 2 and 3 each start out with more than 1 Trade is possible Two outcomes are in the core Person 1 sells object to 2 for 1 Person 1 sells object to 3 for 1 Why is nothing else in the core Previous example except that Person 2 values object at 1 Person 3 values it at v 1 What is in core Person 2 gets the object and pays person 1 a price p that is between v and 1