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Unformatted text preview: Nau: Game Theory 1 Updated 10/5/11 CMSC 498T, Game Theory 5. ImperfectInformation Games Dana Nau University of Maryland Nau: Game Theory 2 Updated 10/5/11 Motivation So far, we’ve assumed that players in an extensiveform game always know what node they’re at Ø Know all prior choices • Both theirs and the others’ Ø Thus “perfect information” games But sometimes players Ø Don’t know all the actions the others took or Ø Don’t recall all their past actions Sequencing lets us capture some of this ignorance: Ø An earlier choice is made without knowledge of a later choice But it doesn’t let us represent the case where two agents make choices at the same time, in mutual ignorance of each other Nau: Game Theory 3 Updated 10/5/11 Definition An imperfectinformation game is an extensiveform game in which each agent’s choice nodes are partitioned into information sets Ø An information set = {all choice nodes an agent might be at} • The nodes in an information set are indistinguishable to the agent • So all have the same set of actions Ø Agent i ’s information sets are I i 1 , …, I im for some m, where • I i 1 ∪ … ∪ I im = {all nodes where it’s agent i ’s move} • I ij ∩ I ik = ∅ for all j ≠ k • For all nodes x,y ∈ I ij , › {all available actions at x } = {all available actions at y } A perfectinformation game is a special case in which each I ij contains just one node What is this? Nau: Game Theory 4 Updated 10/5/11 Example Below, agent 1 has two information sets: Ø I 11 = { w } Ø I 12 = { y,z } Ø In I 12 , agent 1 doesn’t know whether agent 2’s move was C or D Agent 2 has just one information set: Ø I 21 = { x } w Agent 2 x Agent 1 (1,1) z y Agent 1 (0,0) (2,4) (2,4) (0,0) A B C D E F E F Nau: Game Theory 5 Updated 10/5/11 Strategies A pure strategy for agent i is a function s i that selects an available action at each of i ’s information sets Ø s i ( I ) = agent i ’s action in information set I Thus {all pure strategies for i } is the Cartesian product Ø {actions available in I i 1 } × … × {actions available in I im } Agent 1’s pure strategies: {A,B} × {E, F} = {(A, E), (A, F), (B, E), (B, F)} Agent 2’s pure strategies: {C, D} w Agent 2 x Agent 1 (1,1) z y Agent 1 (0,0) (2,4) (2,4) (0,0) A B C D E F E F Nau: Game Theory 6 Updated 10/5/11 Extensive Form à Normal Form Any extensiveform imperfectinformation game can be transformed into an equivalent normalform game Same strategies and same payoffs Ø Thus same Nash equilibria, same Pareto optimal strategy profiles, etc. Just like we did it for perfectinformation games Ø Create an ndimensional payoff matrix in which the i ’th dimension corresponds to agent i ’s pure strategies C D (A,E) 0, 0 2, 4 (A,F) 2, 4 0, 0 (B,E) 1, 1 1, 1 (B,F) 1, 1 1, 1 w Agent 2 x Agent 1 (1,1) z y Agent 1 (0,0) (2,4) (2,4) (0,0) A B C D E F E F Nau: Game Theory 7 Updated 10/5/11...
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 Fall '11
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 Game Theory, NAU

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