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 based on joint work with Sunil Simon CWI A Classification of Weakly Acyclic Games – p. 1/18

Preliminary Definitions Fix a game ( S 1 ,..., S n , p 1 ,..., p n ) . S : = S 1 ×···× S 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 s k = ( s ′ i � = s k − 1 i , s k − 1 ∀ k > 1 ∃ i ∃ s ′ − i ) . i A path is an improvement path if it is maximal and for all k > 1 , p i ( s k ) > p i ( s k − 1 ) , where i is the player who deviated from s k − 1 . Analogous concept: BR-improvement path. A Classification of Weakly Acyclic Games – p. 2/18

Finite Improvement Property G has the finite improvement property (FIP), if every improvement path is finite. Analogous concept: finite best response property (FBRP). Note: If G has the FIP , then it has a Nash equilibrium. G is weakly acyclic if for any joint strategy there exists a finite improvement path that starts at it. Analogous concept: BR-weakly acyclic game. A Classification of Weakly Acyclic Games – p. 3/18

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 # 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 ) . Theorem Every such game is weakly acyclic. A Classification of Weakly Acyclic Games – p. 4/18

Schedulers A scheduler, given a finite sequence ( s 1 ,..., s k ) of joint strategies not ending in 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. Analogous concept: a game G respects a BR-scheduler. A Classification of Weakly Acyclic Games – p. 5/18

Typology of Schedulers f is state-based if for some function g : S → R 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. 6/18

Local Schedulers: A Characterization 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 ] . Intuition: An improvement path respects a permutation π if the deviating player is always the π -first player who did not choose a best response. Note: A game respects a local scheduler if for some permutation π all improvement paths that respect π terminate. A Classification of Weakly Acyclic Games – p. 7/18

� � � � ��� ��� � � ��� ��� � ��� ��� � ��� � ��� ��� � � ��� ��� � � � � � ��� Dependencies FIP Local Set State Sched WA � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Local BR Set BR State BR Sched BR FBRP 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. 8/18

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 } , Y k + 1 : = Y k ∪{ s | ∃ i ∀ s ′ ( s i → s ′ ⇒ s ′ ∈ Y k ) } . For each s ∈ Y k + 1 \ Y k , let f State ( s ) : = i , where i is such that ∀ s ′ ( s i → s ′ ⇒ s ′ ∈ Y k ) . 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. 9/18

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. 10/18

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. 11/18

� ���� � � � � ���� � � ���� � ����� � ���� � � ����� � Example (ctd) 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 . Any improvement path that respects f ends in ( A , A ) . This game does not respect any set-based scheduler. If g ( { 1 , 2 } ) = 1 , then take (( B , B ) , ( A , B ) , ( A , C ) , ( B , C )) ∗ If g ( { 1 , 2 } ) = 2 , then take (( B , B ) , ( B , A ) , ( C , A ) , ( C , B )) ∗ . A Classification of Weakly Acyclic Games – p. 12/18

� � � � ��� ��� � � ��� ��� � � � ��� ��� � ��� ��� ��� � � ��� ��� � � � � � � ��� Final Classification FIP Local Set State � Sched WA � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Sched BR Set BR State BR � Local BR FBRP BRWA �� � �� � �� � � � �� � � � � � � A Classification of Weakly Acyclic Games – p. 13/18

� ��� � � ��� ��� � � ��� ��� � � � � ��� � � ��� ��� ��� � � ��� ��� � � � � � ��� Two Player Games Theorem Sched ⇒ FBRP . Note Set ⇒ Local . Final Classification: FIP Local � Set State � Sched WA � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Set BR � � State BR � � Sched BR Local BR � FBRP � BRWA � � � � � � � � �� � � � � � � � A Classification of Weakly Acyclic Games – p. 14/18

Potentials Given a game G and a scheduler f . F : S → R is called an f -potential if for every initial prefix of an improvement path ( s 1 ,..., s k , s k + 1 ) in G that respects f F ( s k + 1 ) > F ( s k ) . Theorem A finite game respects a scheduler f iff an f -potential exists. A Classification of Weakly Acyclic Games – p. 15/18

Example 1 Cyclic coordination games There is a special strategy t 0 ∈ � i ∈ N S i common to all the players, i ⊕ 1 and i ⊖ 1 : increment and decrement operations done in cyclic order within { 1 ,..., n } .  if s i = t 0 , 0   p i ( s ) : = if s i = s i ⊖ 1 and s i � = t 0 , 1  − 1 otherwise .  Theorem Each coordination game respects every local scheduler. Proof Idea. For every local scheduler f one can define an appropriate f -potential. A Classification of Weakly Acyclic Games – p. 16/18

Example 2 Theorem (Brokkelkamp and de Vries ’12) Each congestion game with player-specific payoff functions respects every BR-local scheduler. This does not hold for local schedulers. A Classification of Weakly Acyclic Games – p. 17/18

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