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 (S1,...,Sn, p1,..., pn). S := S1 ×···×Sn. s′

i is a better response given s if pi(s′ i,s−i) > pi(si,s−i).

A path in S is a sequence (s1,s2,...) of joint strategies such that ∀k > 1∃i∃s′

i = sk−1 i

sk = (s′

i,sk−1 −i ).

A path is an improvement path if it is maximal and for all k > 1, pi(sk) > pi(sk−1), where i is the player who deviated from sk−1. Analogous concept: BR-improvement path.

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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 pi(s) = fi(si,k), where

• k = |{ j | s j = si}|,
• k ≤ l → fi(si,k) ≥ fi(si,l).

Theorem Every such game is weakly acyclic.

A Classification of Weakly Acyclic Games – p. 4/18

Schedulers

A scheduler, given a finite sequence (s1,...,sk) of joint strategies not ending in a Nash equilibrium, selects a player who did not select in sk a best response. An improvement path (s1,s2,...) respects a scheduler f if ∀k sk+1 = (s′

i,sk −i),

where f(s1,...,sk) = 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(s1,...,sk) = g(sk). g : P(N)→N is a choice function if for all A = / g(A) ∈ A. f is set-based if for some choice function g : P(N)→N f(s1,...,sk) = g(NBR(sk)), 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 = / [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
• FBRP
• LocalBR
• SetBR
• StateBR
• SchedBR
• 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∈NYk, where Y0 := {s ∈ S | s is a Nash equilibrium}, Yk+1 := Yk ∪{s | ∃i∀s′(s i →s′ ⇒s′ ∈ Yk)}. For each s ∈ Yk+1 \Yk, let fState(s) := i, where i is such that ∀s′(s i →s′ ⇒s′ ∈ Yk). Claim 1 If G respects a scheduler, then Y = S. Claim 2 If Y = S, then G respects fState. Suppose now that G respects a scheduler. By Claim 1, Y = S, so fState is a state-based scheduler. By Claim 2, G respects fState.

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Schedulers versus State-based Schedulers

Theorem 2 SchedBR ⇒ StateBR.

• Proof. Analogous as for Theorem 1.

Theorem 3 (For finite games) SchedBR ⇒ Sched. Proof Idea. Suppose a game respects a BR-scheduler fBR. We construct a scheduler f inductively by repeatedly scheduling the same player until he plays a best response, subsequently scheduling the player that fBR schedules.

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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
• FBRP
• LocalBR
• SetBR
• StateBR
• SchedBR
• 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
• FBRP
• LocalBR
• SetBR
• StateBR
• SchedBR
• 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

• f an improvement path (s1,...,sk,sk+1) in G that

respects f F(sk+1) > F(sk). 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 t0 ∈

i∈N Si common to all

the players, i⊕1 and i⊖1: increment and decrement operations done in cyclic order within {1,...,n}. pi(s) :=      if si = t0, 1 if si = si⊖1 and si = t0, −1

• therwise.

Theorem Each coordination game respects every local scheduler. Proof Idea. For every local scheduler f one can define an appropriate f-potential.

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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.

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THANK YOU

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