Recurrent certainty in games of partial information Anup Basil Mathew(IMSc) 1 joint work with Dietmar Berwanger(LSV, ENS-Cachan) GImInAL - 7,10 Dec 2013 10 Dec 2013 1 The author thanks LIA-INFORMEL for funding the visit to LSV during June-July 2013. 1 / 62
Extensive Form Games 1 Why finite imperfect games are hard. 2 Why infinite imperfect info games are hard. 3 Tractable classes 4 Conclusion 5 2 / 62
An ’Imperfect’ Scenario The Story Suppose a teacher gives as homework the following problems 1 α ∨ γ 2 β ∨ γ The next day a student is asked one of these questions. For some reason the student only hears ”.... ∨ γ ”. How should the student answer knowing that the teacher is asking one of the homework problems? How do we model this? 3 / 62
An ’Imperfect’ Scenario The Story Suppose a teacher gives as homework the following problems 1 α ∨ γ 2 β ∨ γ The next day a student is asked one of these questions. For some reason the student only hears ”.... ∨ γ ”. How should the student answer knowing that the teacher is asking one of the homework problems? How do we model this? 4 / 62
An ’Imperfect’ Scenario Model of the story T α ∨ γ β ∨ γ S S T F T F P F P F 5 / 62
Extensive Form Games Definitions A finite Extensive Form Game among ’[ n ]’ players is described by G := ( T , turn , ( I i ) i ∈ [ n ] , ( � i ) i ∈ [ n ] ) where ◦ T is a rooted action-labelled finite tree given by ( v 0 ,V,E, l ), l : E → A labels edges with actions from A ◦ turn : V → [ n ] gives the ownership of the nodes of the tree ◦ I i , the information partition of player i is a partiton of { v ∈ V | turn ( v ) = i } ◦ � i gives preferences of player i over maximal paths(or plays ) of T . 6 / 62
Extensive Form Games Definitions • A strategy σ i : I i → A for player i is a function that assigns an action to every information set I i ∈ I i . A strategy profile ( σ i ) i ∈ [ n ] is a tuple of strategies, one for each player. • A play v 0 a 0 v 1 a 1 ... a n − 1 v n is consistent with strategy σ i , if for every v i with turn ( v i ) = i σ i ( I i ) = a i where I i in the unique partition containing v i . 7 / 62
Extensive Form Games About preferred plays • A strategy( σ dom ) for player i is a dominant strategy if i , σ − i ) � i ( σ i , σ − i ) . ∀ σ i ∈ Σ i , ∀ σ − i ∈ Σ − i , ( σ dom i • If the preference relation is binary or win-loss , then dominant strategy is called winning strategy. • A team or coalition of players( S ⊂ [ n ] ) is said to have a dominant strategy if there exists a strategy σ dom for each i player i ∈ S such that ∀ ( σ i ) i ∈ S ∈ (Σ i ) i ∈ S , ∀ ( σ i ) i ∈ [ n ] \ S ∈ (Σ − i ) i ∈ [ n ] \ S , (( σ dom ) i ∈ S , ( σ i ) i ∈ [ n ] \ S ) � S (( σ i ) i ∈ S , ( σ i ) i ∈ [ n ] \ S ) . i Note that here we assume all players in the coalition have the same preference relation over plays . 8 / 62
Extensive Form Games Questions of interest • Does there exist a dominant strategy for player i ? • Does there exist a coordinated winning strategy for a coalition ? 9 / 62
Extensive Form Games Questions of interest • Does there exist a dominant strategy for player i ? • Does there exist a coordinated winning strategy for a coalition ? 10 / 62
Extensive Form Games Perfect Recall In this talk we are interested in determining the existense of coordinated winning strategy for a restricted class of games, namely games of perfect recall . Let X i ( v ) denote the sequence of information sets of player i that are encountered on the path from v 0 to v . A game is said to have perfect recall if for each player i , X i ( v )= X i ( v ′ ) whenever { v , v ′ } ⊆ I i for some I i ∈ I i . 11 / 62
Extensive Form Games Perfect recall Example of imperfect recall S T F S S T F T F 12 / 62
Extensive Form Games Perfect information A game is said to have perfect information if for every player i , ∀ I i ∈ I i it holds that | I i | = 1. Perfect information games are ’easy’. Why? Because winning strategies of subgames can be composed to give winning strategy of the constituent game. 13 / 62
Extensive Form Games Perfect information A game is said to have perfect information if for every player i , ∀ I i ∈ I i it holds that | I i | = 1. Perfect information games are ’easy’. Why? Because winning strategies of subgames can be composed to give winning strategy of the constituent game. 14 / 62
Extensive Form Games Perfect information A game is said to have perfect information if for every player i , ∀ I i ∈ I i it holds that | I i | = 1. Perfect information games are ’easy’. Why? Because winning strategies of subgames can be composed to give winning strategy of the constituent game. 15 / 62
Extensive Form Games Perfect information A game is said to have perfect information if for every player i , ∀ I i ∈ I i it holds that | I i | = 1. Perfect information games are ’easy’. Why? Because winning strategies of subgames can be composed to give winning strategy of the constituent game. 16 / 62
Extensive Form Games Imperfect information games Imperfect information games are ’hard’ because of the lack of compositionality. Example N C1 C3 C2 2 2 1 1 1 l1 l2 l3 l1 l2’ l3 l1 l2’ l3 2 2 2 2 2 2 2 2 2 2 2 2 0 1 0 1 0 1 T F T F T F 17 / 62
Extensive Form Games Infinite games Infinite games via a finite representation. Game graph A := ( V , A , δ, ( β i ) i ∈ [ n ] , turn , v 0 ) where • V is a finite set of graph nodes and A denotes the actions available to players. • δ : V × A → V is the transition function on V . • β i : V → B i gives the observables for player i at each state in V where B i is the set of observables for player i ∈ [ n ]. Additionally we assume that the structure of the arena is common knowledge to all players and that the turn functions are ’layered’. 18 / 62
Extensive Form Games Infinite games Infinite games via a finite representation. Game graph A := ( V , A , δ, ( β i ) i ∈ [ n ] , turn , v 0 ) where • V is a finite set of graph nodes and A denotes the actions available to players. • δ : V × A → V is the transition function on V . • β i : V → B i gives the observables for player i at each state in V where B i is the set of observables for player i ∈ [ n ]. Additionally we assume that the structure of the arena is common knowledge to all players and that the turn functions are ’layered’. 19 / 62
Extensive Form Games Example Figure : ⋆ Borrowed from [1] 20 / 62
Extensive Form Games Game graph to infinite game A → G • Game tree is given by the finite plays on the game graph. • For every play π := v 0 a 0 v 1 a 1 ... , we define an i-projection of a play π as follows β i ( π ) := β i ( v 0 , a 0 ) β i ( v 1 , a 1 ) ... where β i ( v i ) a i if turn ( v i ) = i β i ( v i , a i ) := β i ( v i ) � otherwise. This also gives us an equivalence relation on finite/infinite plays given by π 1 ∼ i π 2 iff β i ( π 1 ) = β i ( π 2 ). We’ll call this the uncertainty relation . • Information Partition of player i i.e I i is given by equivalence classes of ∼ i over finite plays. 21 / 62
Extensive Form Games Example Figure : ⋆ Borrowed from [1] 22 / 62
Extensive Form Games Winning conditions of infinite games Winning conditions i.e ’win-loss’ preference relation on plays for the team/coalition is given by W ⊆ Plays ( A ). We choose the following finite representatiion of winning conditions via γ : V → N . W := { π ∈ Plays ( A ) | lim inf i →∞ γ ( v i ) isodd } We additionally impose the restriction that the winning condition respects observational equivalence i.e ∀ π 1 ∈ W , if for some π ∈ Plays ( A ) , i ∈ [ n ] s.t β i ( π 1 ) = β i ( π ), then π ∈ W . 23 / 62
Extensive Form Games Why perfect information games are ’easy’. • Compositionality of strategies. • Analysis upto a certain finite level is enough. Memoryless determinacy of parity and mean payoff games: a simple proof : Henrik Bjorklund, Sven Sandberg, Sergei G. Vorobyov . 24 / 62
Extensive Form Games Why perfect information games are ’easy’. • Compositionality of strategies. • Analysis upto a certain finite level is enough. Memoryless determinacy of parity and mean payoff games: a simple proof : Henrik Bjorklund, Sven Sandberg, Sergei G. Vorobyov . 25 / 62
Extensive Form Games Imperfect information games are hard Existence of a winning strategy for a coalition is ’hard’. • Peterson-Reif • Pnueli-Rosner • Berwanger-Kaiser 26 / 62
Extensive Form Games Why imperfect information games are hard • Ever growing information sets. Not Really Since the number of states of the game graph are finite and states are all that determine future possible plays, any equivalence class can be ’shrunk’ upto uniqueness. But then they are easy. 27 / 62
Extensive Form Games Why imperfect information games are hard • Ever growing information sets. Not Really Since the number of states of the game graph are finite and states are all that determine future possible plays, any equivalence class can be ’shrunk’ upto uniqueness. But then they are easy. 28 / 62
Recommend
More recommend
