Complexity Results about Nash Equilibria

Complexity Results about Nash Equilibria

Vincent Conitzer, Tuomas SandholmInternational Joint Conferences on Artificial Intelligence 2003 (IJCAI’03)

Presented by XU, JingFor COMP670O, Spring 2006, HKUST

2/18Complexity Results about Nash Equilibria (IJCAI’03)

Problems of interests

Noncooperative games Good Equilibria Good MechanismsMost existence questions are NP-hard for

general normal form games.Designing Algorithms depends on problem

structure.


Agenda

LiteratureA symmetric 2-player game and results o

n mixed-strategy NE in this gameComplexity results on pure-strategy Baye

s-Nash EquilibriaPure-strategy Nash Equilibria in stochasti

c (Markov) games


Literature

2-player zero-sum games can be solved using LP in polynomial time (R.D.Luce, H.Raiffa '57)

In 2-player general-sum normal form games, determining the existence of NE with certain properties is NP-hard (I.Gilboa, E.Zemel '89)

In repeated and sequential games (E. Ben-Porath '90, D. Koller & N. Megiddo '92, Michael Littman & Peter Stone'03, etc.) Best-responding Guaranteeing payoffs Finding an equilibrium


A Symmetric 2-player Game

Given a Boolean formula in conjunctive normal form, e.g. (x1Vx2)(-x1V-x2)

V={xi}, 's set of variables, let |V|=n

L={xi, -xi}, corresponding literals

C: 's clauses, e.g. x1Vx2, -x1V-x2

v: LV, i.e. v(xi)=v(-xi)= xi

G( ):=1=2= LVC{f}



Utility function



u1(a,b) =u2(b,a)P2

P1L V C f

L1, li-lj

-2, li=-lj-2 -2 -2

V2, v(l)x

2-n, v(l)=x -2 -2 -2

C2, lc

2-n, lc -2 -2 -2

f 1 1 1 0

x1 -x1 x2 -x2

x1 1 -2 1 1

-x1 -2 1 1 1

x2 1 1 1 -2

-x2 1 1 -2 1


Theorem 1

If (l1,l2,…,ln) satisfies and v(li) = xi, then There is a NE of G() where both players play li with pro

bability 1/n, with E(ui)=1. The only other Nash equilibrium is the one where both pl

ayers play f, with E(ui)=0.

Proof: If player 2 plays li with p2(li)=1/n, then player 1

Plays any of li, E(u1)=1

Plays –li, E(u1)=1-3/n<1

Plays v, E(u1)=1

Plays c, E(u1)≤1, since every clause c is satisfied.


Theorem 1

No other NE: If player 2 always plays f, then player 1 plays f. If player 1 and 2 play an element of V or C, then a

t least one player had better strictly choose f. If player 2 plays within L{f}, then player 1 plays f. If player 2 plays within L and either p2(l)+p2(-l) <1/

n, then player 1 would play v(l), with E(u1)>2*(1-1/n)+(2-n)*(1/n)=1.

Both players can only play l or -l simultaneously with probability 1/n, which corresponds to an assignment of the variables.

If an assignment doesn’t satisfy , then no NE.



u1(a,b) =u2(b,a)P2

P1L V C f

L1, li-lj

-2, li=-lj-2 -2 -2

V2, v(l)x

2-n, v(l)=x -2 -2 -2

C2, lc

2-n, lc -2 -2 -2

f 1 1 1 0

x1 -x1 x2 -x2

x1 1 -2 1 1

-x1 -2 1 1 1

x2 1 1 1 -2

-x2 1 1 -2 1


Corollaries

Theorem1: Good NE is satisfiable.


Corollaries


Corollaries

Hard to obtain summary info about a game’s NE, or to get a NE with certain properties.

Some results were first proven by I. Gilboa and E. Zemel ('89).


Corollaries

A NE always exists, but counting them is hard, while searching them remains open.


Bayesian Game

Set of types Θi , for agent i (iA)Known prior dist. over Θ1 Θ2…Θ|A|

Utility func. ui: Θi12…|A| RBayes-NE:

Mixed-strategy BNE always exists (D. Fudenberg, J. Tirole '91).

Constructing one BNE remains open.


Complexity results

SET-COVER ProblemS={s1,s2,…, sn}S1, S2, …, Sm S, Si=SWhether exist Sc1, Sc2, … , Sck s.t. Sci=S ?

Reduction to a symmetric 2-player gameΘ= Θ1= Θ2={1, 2,…, k,} (k types each) is uniform= 1= 2={S1, S2, …, Sm, s1,s2,…, sn}Omit type in utility functions


Complexity results

Theorem 2: Pure-Strategy-BNE is NP-hard, even in symmetric 2-player games where is uniform.

Proof:If there exist Sci, then

both player play Sci when

their type is i. (NE)If there is a pure-BNE,

No one plays si

{Si (for i)} covers S.

P2

P1Sj sj

Si 11, sjSi

2, sjSi

si

3, siSj

-3k,siSj

-3k


Theorem 3

PURE-STRATEGY-INVISIBLE-MARKOV-NE is PSPACE-hard, even when the game is symmetric, 2-player, and the transition process is deterministic. (PNPPSPACEEXPSPACE)

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Complexity Results about Nash Equilibria