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 Si, Payoff function: pi : S1 ×···×Sn →R, The players choose their strategies simultaneously. Notation: (S1,...,Sn, p1,..., pn).

A Classification of Weakly Acyclic Games – p. 2/29

Finite Improvement Property (FIP)

Fix a game (S1,...,Sn, p1,..., pn). 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 i sk+1 = (s′ i,sk −i).

A path is an improvement path if it is maximal and for all k > 1, pi(sk+1) > pi(sk), where i deviated from sk. 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 H 1,−1 −1, 1 −1, 1 T −1, 1 1,−1 −1,−1 E −1,−1 −1,−1 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 pi(s) = fi(si,k), where

• k = |{ j | s j = si}|,
• k ≤ l → fi(si,k) ≥ fi(si,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 (s1,...,sk) of joint strategies s.t. sk is not 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.

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(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. 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 = / [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

• ❴

❴❴ ❴

• ✤✤✤

✤✤✤

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. 10/29

Back to First Example

H T E H 1,−1 −1, 1 −1, 1 T −1, 1 1,−1 −1,−1 E −1,−1 −1,−1 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∈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.

A Classification of Weakly Acyclic Games – p. 12/29

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.

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

• ✤✤✤

✤✤✤

FBRP

• ❴

❴❴ ❴

• ✤✤✤

✤✤✤

LocalBR

• ❴

❴❴ ❴

SetBR

• ❴

❴❴ ❴

StateBR

• ❴

❴ ❴ ❴ SchedBR

• ❴

❴❴ ❴

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

• ✤✤✤

✤✤✤

FBRP

• ❴

❴ ❴ ❴

• ✤✤✤

✤✤✤

LocalBR

• ❴

❴ ❴ ❴ SetBR

• ❴

❴ ❴ ❴ StateBR

• ❴

❴ ❴ ❴ SchedBR

• ❴

❴❴ ❴

BRWA

A Classification of Weakly Acyclic Games – p. 19/29

IENBR by Example

Consider X Y A 2,1 0,0 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 A 2,1

A Classification of Weakly Acyclic Games – p. 20/29

IENBR

Theorem Suppose G′ is an outcome of applying IENBR to G. If s is a Nash equilibrium of G, then it is a Nash equilibrium of G′. If G is finite and s is a Nash equilibrium of G′, then it is a Nash equilibrium of G. If G is finite and is solved by IENBR, then the resulting joint strategy is a unique Nash equilibrium of G. (Apt ’05) Outcome of IENBR is unique (order independence).

A Classification of Weakly Acyclic Games – p. 21/29

Games that are solved by IENBR

Theorem Suppose a finite game is solved by IENBR. Then it is in StateBR. The scheduler used: always schedule the player who did not play a best response the longest, breaking ties in favour of the player with the smallest index. For a game solved by IENBR this scheduler f satisfies the following property. For all players i, for every BR-improvement path ρ = s0,s1,... that respects f and for all j ≥ 0, there exists k ≥ j such that i ∈ BR(sk). Informally: Every player eventually has or selects a best response.

A Classification of Weakly Acyclic Games – p. 22/29

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).

A Classification of Weakly Acyclic Games – p. 23/29

f-Potentials

Theorem A finite game respects a scheduler f iff an f-potential exists.

• Proof. (⇐)

f potential increases along every improvement path that respects f. (⇒) (Sketch). An improvement sequence: a prefix of an improvement path. Assign to each joint strategy s the number of improvement sequences that respect f and that terminate in it. Because the game respects f, by König’s Lemma this number is finite. This defines an f-potential.

A Classification of Weakly Acyclic Games – p. 24/29

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.

A Classification of Weakly Acyclic Games – p. 25/29

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. 26/29

Bounds on finding a Nash equilibrium

Theorem G: a weakly acyclic game G for n players such that each player has at most k strategies, in each joint strategy each player has at most one better response. T (G,s): a tree formed by the improvements paths that start in s. Then In each tree T (G,s) a Nash equilibrium can be found in O(nkn) steps. If G respects a scheduler, then in each tree T (G,s) a Nash equilibrium can be found in O(kn) steps.

A Classification of Weakly Acyclic Games – p. 27/29

Schedulers: an Assessment

Input size of various schedulers type of scheduler number of inputs general infinite state-based kn set-based 2n local n

A Classification of Weakly Acyclic Games – p. 28/29

Thank you for your attention

A Classification of Weakly Acyclic Games – p. 29/29