Complexity Results about Nash Equilibria Vincent Conitzer, Tuomas Sandholm International Joint Conferences on Artificial Intelligence 2003 (IJCAI’03) Presented

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)

Documents

Complexity Results about Nash Equilibria Vincent Conitzer, Tuomas Sandholm International Joint Conferences on Artificial Intelligence 2003 (IJCAI’03) Presented