A Classification of Weakly Acyclic Games Krzysztof R. Apt CWI and - PowerPoint PPT Presentation

A Classification of Weakly Acyclic Games Krzysztof R. Apt CWI and University of Amsterdam joint work with Sunil Simon CWI A Classification of Weakly Acyclic Games – p. 1/29

Strategic Games Strategic game for n ≥ 2 players: For each player i : Strategies: a non-empty set S i , Payoff function: p i : S 1 ×···× S n → R , The players choose their strategies simultaneously. Notation: ( S 1 ,..., S n , p 1 ,..., p n ) . A Classification of Weakly Acyclic Games – p. 2/29

Finite Improvement Property (FIP) Fix a game ( S 1 ,..., S n , p 1 ,..., p n ) . s ′ i is a better response given s if p i ( s ′ i , s − i ) > p i ( s i , s − i ) . A path in S is a sequence ( s 1 , s 2 ,... ) of joint strategies such that i s k + 1 = ( s ′ ∀ k > 1 ∃ i ∃ s ′ i � = s k i , s k − i ) . A path is an improvement path if it is maximal and for all k > 1 , p i ( s k + 1 ) > p i ( s k ) , where i deviated from s k . G has the finite improvement property (FIP) if every improvement path is finite. Note If G has the FIP , then it has a Nash equilibrium. A Classification of Weakly Acyclic Games – p. 3/29

Weakly Acyclic Games (Young ’93, Milchtaich ’96) G is weakly acyclic if for any joint strategy there exists a finite improvement path that starts at it. Example H T E 1 , − 1 − 1 , − 1 , H 1 1 − 1 , 1 , − 1 − 1 , − 1 T 1 − 1 , − 1 − 1 , − 1 E 1 , 1 A Classification of Weakly Acyclic Games – p. 4/29

A Non-trivial Example (Milchtaich ’96) Congestion games with player-specific payoff functions. Each player has the same finite set of strategies (= resources), Each payoff function depends only on the chosen strategy and (negatively) on the number of players that chose it. So p i ( s ) = f i ( s i , k ) , where • k = |{ j | s j = s i }| , • k ≤ l → f i ( s i , k ) ≥ f i ( s i , l ) . Note Such games do not need to have the FIP . Theorem Every such game is weakly acyclic. A Classification of Weakly Acyclic Games – p. 5/29

Schedulers A scheduler, given a sequence ( s 1 ,..., s k ) of joint strategies s.t. s k is not a Nash equilibrium, selects a player who did not select in s k a best response. An improvement path ( s 1 , s 2 ,... ) respects a scheduler f if ∀ k s k + 1 = ( s ′ i , s k − i ) , where f ( s 1 ,..., s k ) = i . A game G respects a scheduler f if all improvement paths which respect f are finite. A Classification of Weakly Acyclic Games – p. 6/29

Analogous Concepts BR-improvement path. Finite best response property (FBRP). BR-weakly acyclic game. A game G respects a BR-scheduler. A Classification of Weakly Acyclic Games – p. 7/29

Typology of Schedulers f is state-based if for some function g : S →{ 1 ,..., n } f ( s 1 ,..., s k ) = g ( s k ) . g : P ( N ) → N is a choice function if for all A � = / 0 g ( A ) ∈ A . f is set-based if for some choice function g : P ( N ) → N f ( s 1 ,..., s k ) = g ( NBR ( s k )) , where NBR ( s ) : = { i | player i did not select a best response in s } . f is local if for such g , g ( A ) ∈ B ⊆ A implies g ( A ) = g ( B ) . A Classification of Weakly Acyclic Games – p. 8/29

Local Schedulers: A Characterization A scheduler f is local if for some choice function g : P ( N ) → N g ( A ) ∈ B ⊆ A implies g ( A ) = g ( B ) . Note: A scheduler is local iff for some permutation π of the players each time it selects the π -first unsatisfied player. More formally: Take a permutation π of 1 ,..., n . Let for A � = / 0 [ π ]( A ) := the first element from π ( 1 ) ,..., π ( n ) that belongs to A . Note: A scheduler is local iff it is of the form [ π ] . A Classification of Weakly Acyclic Games – p. 9/29

� � � � ✤✤✤ ✤✤✤ � � ✤✤✤ ✤✤✤ � ✤✤✤ ✤✤✤ � ✤✤✤ � ✤✤✤ ✤✤✤ � � ✤✤✤ ✤✤✤ � � � � � ✤✤✤ Dependencies FIP Local Set State Sched WA ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ FBRP Local BR Set BR State BR Sched BR BRWA ❴❴ ❴ ❴❴ ❴ ❴❴ ❴ ❴❴ ❴ ❴❴ ❴ ❴ ❴ ❴ ❴ ❴ FIP : the games that have the FIP , Local : games that respect a local scheduler, Set : games that respect a set-based scheduler, State : games that respect a state-based scheduler, Sched : games that respect a scheduler, WA : weakly acyclic games, FBRP : the games that have the FBRP , BRWA : BR-weakly acyclic games. A Classification of Weakly Acyclic Games – p. 10/29

Back to First Example H T E 1 , − 1 − 1 , − 1 , H 1 1 − 1 , 1 , − 1 − 1 , − 1 T 1 − 1 , − 1 − 1 , − 1 E 1 , 1 Does this game respect a scheduler? A Classification of Weakly Acyclic Games – p. 11/29

Schedulers versus State-based Schedulers Theorem 1 Sched ⇒ State . Proof Idea. Let Y : = ∪ k ∈ N Y k , where Y 0 : = { s ∈ S | s is a Nash equilibrium } , → s ′ ⇒ s ′ ∈ Y k ) } . Y k + 1 : = Y k ∪{ s | ∃ i ∀ s ′ ( s i For each s ∈ Y k + 1 \ Y k , let → s ′ ⇒ s ′ ∈ Y k ) . f State ( s ) : = i , where i is such that ∀ s ′ ( s i Claim 1 If G respects a scheduler, then Y = S . Claim 2 If Y = S , then G respects f State . Suppose now that G respects a scheduler. By Claim 1, Y = S , so f State is a state-based scheduler. By Claim 2, G respects f State . A Classification of Weakly Acyclic Games – p. 12/29

Schedulers versus State-based Schedulers Theorem 2 Sched BR ⇒ State BR . Proof. Analogous as for Theorem 1. Theorem 3 (For finite games) Sched BR ⇒ Sched . Proof Idea. Suppose a game respects a BR-scheduler f BR . We construct a scheduler f inductively by repeatedly scheduling the same player until he plays a best response, subsequently scheduling the player that f BR schedules. A Classification of Weakly Acyclic Games – p. 13/29

Remaining Implications Example State �⇒ Set . A B C A 2 , 2 2 , 0 0 , 1 B 0 , 2 1 , 1 1 , 0 C 1 , 0 0 , 1 0 , 0 A Classification of Weakly Acyclic Games – p. 14/29

� ✤✤✤✤ � ✤✤✤✤ � ✤✤✤✤ � ✤✤✤✤ � � � � � � ✤✤✤✤ � ✤✤✤✤ � The game respects state-based scheduler Better response graph: ( A , A ) ( A , B ) ( A , C ) � ❴❴❴❴❴❴❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ( B , A ) ( B , B ) ( B , C ) � ❴❴❴❴❴❴❴ � ❴❴❴❴❴❴❴ ( C , A ) ( C , B ) ( C , C ) ❴ ❴ ❴ ❴ ❴ ❴ � ❴❴❴❴❴❴ This game respects the state-based scheduler f ( A , C ) : = 2 , f ( C , A ) : = 1 , f ( B , B ) : = 1 . A Classification of Weakly Acyclic Games – p. 15/29

� ✤✤✤✤ � ✤✤✤✤ � ✤✤✤✤ � ✤✤✤✤ � � � � � � ✤✤✤✤ � ✤✤✤✤ � Set-based scheduler: case g ( { 1 , 2 } ) = 1 ( A , A ) ( A , B ) ( A , C ) � ❴❴❴❴❴❴❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ( B , A ) ( B , B ) ( B , C ) � ❴❴❴❴❴❴❴ � ❴❴❴❴❴❴❴ ( C , A ) ( C , B ) ( C , C ) ❴ ❴ ❴ ❴ ❴ ❴ � ❴❴❴❴❴❴ A Classification of Weakly Acyclic Games – p. 16/29

� ✤✤✤✤ � ✤✤✤✤ � ✤✤✤✤ � ✤✤✤✤ � � � � � � ✤✤✤✤ � ✤✤✤✤ � Set-based scheduler: case g ( { 1 , 2 } ) = 2 ( A , A ) ( A , B ) ( A , C ) � ❴❴❴❴❴❴❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ( B , A ) ( B , B ) ( B , C ) � ❴❴❴❴❴❴❴ � ❴❴❴❴❴❴❴ ( C , A ) ( C , B ) ( C , C ) ❴ ❴ ❴ ❴ ❴ ❴ � ❴❴❴❴❴❴ A Classification of Weakly Acyclic Games – p. 17/29

� � � � ✤✤✤ ✤✤✤ � � ✤✤✤ ✤✤✤ � � � ✤✤✤ ✤✤✤ � ✤✤✤ ✤✤✤ ✤✤✤ � � ✤✤✤ ✤✤✤ � � � � � � ✤✤✤ Final Classification FIP Local Set State � Sched WA ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Sched BR FBRP Local BR Set BR State BR � BRWA ❴❴ ❴ ❴❴ ❴ ❴❴ ❴ ❴ ❴ ❴❴ ❴ ❴ ❴ ❴ ❴ ❴ A Classification of Weakly Acyclic Games – p. 18/29

� ✤✤✤ � � ✤✤✤ ✤✤✤ � � ✤✤✤ ✤✤✤ � � � � ✤✤✤ � � ✤✤✤ ✤✤✤ ✤✤✤ � � ✤✤✤ ✤✤✤ � � � � � ✤✤✤ Two Player Games Theorem Sched ⇒ FBRP . Note Set ⇒ Local . Final Classification: FIP Local � Set State � Sched WA ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Set BR � ❴ State BR � ❴ Sched BR FBRP � Local BR � BRWA ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ A Classification of Weakly Acyclic Games – p. 19/29

IENBR by Example Consider X Y 2 , 1 0 , 0 A B 0 , 1 2 , 0 C 1 , 1 1 , 2 C is never a best response. Eliminating it we get X Y A 2 , 1 0 , 0 B 0 , 1 2 , 0 from which in two steps we get X 2 , 1 A A Classification of Weakly Acyclic Games – p. 20/29