lecture8 - CMSC 498T, Game Theory 8. Coalitional Game...

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Nau: Game Theory 1 Updated 11/30/11 CMSC 498T, Game Theory 8. Coalitional Game Theory Dana Nau University of Maryland
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Nau: Game Theory 2 Updated 11/30/11 Coalitional Games with Transferable Utility Given a set of agents, a coalitional game defines how well each group (or coalition ) of agents can do for itself—its payoff Ø Not concerned with how the agents make individual choices within a coalition, how they coordinate, or any other such detail Transferable utility assumption: the payoffs to a coalition may be freely redistributed among its members Ø Satisfied whenever there is a universal currency that is used for exchange in the system Ø Implies that each coalition can be assigned a single value as its payoff
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Nau: Game Theory 3 Updated 11/30/11 Coalitional Games with Transferable Utility A coalitional game with transferable utility is a pair G = ( N,v ), where Ø N = {1, 2, …, n } is a finite set of players Ø v : 2 N associates with each coalition S ˧ N a real-valued payoff v ( S ), that the coalition members can distribute among themselves v is the characteristic function Ø We assume v ( ) = 0 A coalition’s payoff is also called its worth Coalitional game theory is normally used to answer two questions: (1) Which coalition will form? (2) How should that coalition divide its payoff among its members? The answer to (1) is often “the grand coalition” (all of the agents) Ø But this answer can depend on making the right choice about (2)
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Nau: Game Theory 4 Updated 11/30/11 Example: A Voting Game Consider a parliament that contains 100 representatives from four political parties: Ø A (45 reps.), B (25 reps.), C (15 reps.), D (15 reps.) They’re going to vote on whether to pass a $100 million spending bill| (and how much of it should be controlled by each party) Need a majority ( 51 votes) to pass legislation Ø If the bill doesn’t pass, then every party gets 0 More generally, a voting game would include Ø a set of agents N Ø a set of winning coalitions W 2 N In the example, all coalitions that have enough votes to pass the bill Ø v ( S ) = 1 for each coalition S W Or equivalently, we could use v ( S ) = $100 million Ø v ( S ) = 0 for each coalition S W
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Nau: Game Theory 5 Updated 11/30/11 Superadditive Games A coalitional game G = ( N,v ) is superadditive if the union of two disjoint coalitions is worth at least the sum of its members’ worths Ø for all S, T N , if S T = , then v ( S ˫ T ) v ( S ) + v ( T ) The voting-game example is superadditive Ø If S T = , v ( S ) = 0, and v ( T ) = 0, then v ( S ˫ T ) 0 Ø If S T = and v ( S ) = 1, then v ( T ) = 0 and v ( S ˫ T ) = 1 Ø Hence v ( S ˫ T ) v ( S ) + v ( T ) If G is superadditive, the grand coalition always has the highest possible payoff Ø For any S N , v ( N ) v ( S ) + v ( N–S ) v ( S )
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Nau: Game Theory 6 Updated 11/30/11 Interference The book says that for superadditive games, coalitions can always work together without interfering with one another Ø In the spending-bill example, I think this ignores the question of how
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lecture8 - CMSC 498T, Game Theory 8. Coalitional Game...

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