Approximate Nash Equilibria

in interesting games

Constantinos Daskalakis, U.C. Berkeley

If your game is interesting, its description cannot be astronomically long…

Game Species

interesting

games

graphical gamesnormal form

games

e.g. bounded degree

e.g. constant number of players

what else?

Bad News…

Computing a Mixed Nash Equilibrium ?

is PPAD-complete [DGP ’05]

even for 3 players [CD ’05, DP ’05]

even for 2 players [CD ’06]

- in normal form games

PPAD-complete [DGP]

even for 2 strategies per player and degree 3

- in graphical games

So what next?Computing approximate Equilibria

(every player plays an approximate best response)

Finding a better point in Christos’ cube

efficiency

existence

naturalness

correlated

pure Nash

mixed Nash[DGP06, CD06]

Looking at other interesting games…

Approximate Equilibria for 2-players?

Compute a point at which each player has at most - regret..

for = 2-n PPAD-Complete [ DGP, CD]

for = n- PPAD-Complete for any [CDT ’06]

= constant ??

( no FPTAS )

LMM ’04

log n

2- support is enough for all

[LMM ’03] take Nash equilibrium (x, y); take log n/ 2 independent samples from x and y

subexponential algorithm for computing - Nash

A simple algorithm for .5 -approximate

Column player finds: best response j to strategy i of row player

Row player finds: best response k to strategy j of column player

G = (R, C)

i

j

k0.5

0.5

1.0

0.5 approximate Nash!

[DMP ’06]

[FNS ’06]: can’t do better with small supports!

Beyond Constant Support [DMP ’07]

.38 can be achieved in polynomial time

Generalization of Previous Idea:

sampling(similar to LMM)

LP+guess value of the eq. u +

PTAS ?PTAS ?

Other Interesting Games?Other Interesting Games?

interesting

games

graphical gamesnormal form

games

anonymous

games“Each player is different, but sees all other players as identical”

why interesting?why interesting?

- the succinctness argument :

n players, s strategies, all interact, ns size!

Characterization of equilibria in large anonymous games, [Blonski ’00]

e.g. auctions, stock market, congestion, social phenomena, …

"How many veiled women can we expect in Cairo ?"

- ubiquity: think of your favorite large game - is it anonymous?

(the utility of a player depends on her strategy, and on how many other players play each strategy)

Pure Nash EquilibriaPure Nash EquilibriaTheorem [DP ’07]Theorem [DP ’07]: :

In any anonymous game, there exists a 2In any anonymous game, there exists a 2LsLs22--approximate approximate pure Nash equilibriumpure Nash equilibrium which can which can be found in polynomial time.be found in polynomial time.

((LL = Lipschitz constant of the utility functions) = Lipschitz constant of the utility functions)

how rapidly does the payoff change as how rapidly does the payoff change as players change strategy?players change strategy?

PTAS for anonymous gamesPTAS for anonymous gameswith two strategieswith two strategies

Big Picture:Big Picture:• Discretize the space of mixed Nash equilibria.Discretize the space of mixed Nash equilibria.• Discrete set achieves some approximation which Discrete set achieves some approximation which

depends on the grid size.depends on the grid size.• Reduce the problem to computing a pure Nash Reduce the problem to computing a pure Nash

equilibrium with a larger set of strategies.equilibrium with a larger set of strategies.

Big Question:Big Question:

what grid size is required to achieve approximation what grid size is required to achieve approximation ??if function of if function of only only PTAS PTASif function of n if function of n nothing nothing

PTAS (cont.)PTAS (cont.)

[Restrict attention to 2 strategies per player][Restrict attention to 2 strategies per player]

Let Let pp11 , , pp2 2 ,…,,…, ppnn be some mixed strategy profile.be some mixed strategy profile.

The utility of player 1 for playing pure strategy The utility of player 1 for playing pure strategy is is

1

10

( , ) Prn

jjt

u t X t

where the where the XXjj’s are Bernoulli random variables with expectaion ’s are Bernoulli random variables with expectaion ppjj..

PTAS (cont.)PTAS (cont.)

How is the utility affected if we replace the pHow is the utility affected if we replace the p ii’s by another set of probabilities {q’s by another set of probabilities {q ii}?}?

Absolute Change in UtilityAbsolute Change in Utility

1

1 10

( , ) Pr Prn

j jj jt

u t X t Y t

where the where the YYjj’s are Bernoulli random variables with expectaions ’s are Bernoulli random variables with expectaions qqjj..

1 1j jj jTV

X Y

PTAS(cont.)PTAS(cont.)

Main Lemma:Main Lemma: Given any constant Given any constant kk and any set of probabilities and any set of probabilities {{ppii}}ii , there exists a way to round the , there exists a way to round the ppii’s to ’s to qqii’s which are ’s which are multiples of 1/multiples of 1/kk so that so that

||P - Q|| = O(k||P - Q|| = O(k-1/2-1/2),),

where:where: P is the distribution of the sum of the Bernoullis pP is the distribution of the sum of the Bernoullis pii

Q is the distribution of the sum of the Bernoullis qQ is the distribution of the sum of the Bernoullis q ii

no dependence on n no dependence on n PTAS for anonymous games PTAS for anonymous games

approximation in time2(1/ )On

PTAS - complicationsPTAS - complications

Two natural approaches seem to fail:Two natural approaches seem to fail:

i. round to the closest multiple of 1/i. round to the closest multiple of 1/kksuppose suppose ppii =1/=1/n ,n , for all i for all i

qii = 0, for all = 0, for all ii

QQ [0] = 1, whereas [0] = 1, whereas1 1

[0] 1n

nPn e

variation distance variation distance 1-1/e 1-1/e

PTAS – complications (cont.)PTAS – complications (cont.)ii. Randomized Roundingii. Randomized Rounding

Let the Let the qqii be random variables taking values which be random variables taking values which

are multiples of 1/are multiples of 1/kk so that so that

E[E[qqii] = ] = ppii..

Then, for all Then, for all t t = 0,…, = 0,…, nn, ,

- - QQ[[tt] is a random variable which is a function of the ] is a random variable which is a function of the qqii’s’s

[ ] ii

Q n qe.ge.g..

- - QQ[[tt]] has the correct expectation!

E[E[QQ[[tt]] = ]] = PP[[tt]] troubletrouble: expectations are : expectations are at most 1 and functions at most 1 and functions involve productsinvolve products

PTAS(cont.)PTAS(cont.)Our approach: Our approach: Poisson ApproximationsPoisson Approximations

Intuition:Intuition:

If If ppii’s were small ’s were small ii

X would be close to a Poisson dist’n would be close to a Poisson dist’n of meanof mean i

i

p

ii

X

define the define the qqii’s so that’s so that i ii i

q p

ii

Y

ii

Poisson p i

i

Poisson q

PTAS(cont.)PTAS(cont.)

Near the boundaries of [0,1] Poisson Approximations are sufficient Near the boundaries of [0,1] Poisson Approximations are sufficient

Disadvantage of Poisson distribution: Disadvantage of Poisson distribution:

mean = variancemean = variance

This is disastrous for intermediate values of the This is disastrous for intermediate values of the ppii’s’s

approximation with approximation with translated Poisson distributionstranslated Poisson distributions

to achieve mean to achieve mean and variance and variance 2 2

define a Poisson(define a Poisson(22 ) distribution; then ) distribution; then shift it by shift it by - - 22

Thank you for your attention!