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 games normal 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 k approximate Nash! [DMP 06] [FNS 06]: cant 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 ?

Other Interesting Games? interesting games graphical games normal form games anonymous games Each player is different, but sees all other players as identical

why interesting? - the succinctness argument : n players, s strategies, all interact, n s 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 Equilibria Theorem [DP 07]: In any anonymous game, there exists a 2Ls 2 - approximate pure Nash equilibrium which can be found in polynomial time. (L = Lipschitz constant of the utility functions) how rapidly does the payoff change as players change strategy?

PTAS for anonymous games with two strategies Big Picture: Discretize the space of mixed Nash equilibria.Discretize the space of mixed Nash equilibria. Discrete set achieves some approximation which depends on the grid size.Discrete set achieves some approximation which depends on the grid size. Reduce the problem to computing a pure Nash equilibrium with a larger set of strategies.Reduce the problem to computing a pure Nash equilibrium with a larger set of strategies. Big Question: what grid size is required to achieve approximation ? if function of only PTAS if function of n nothing

PTAS (cont.) [Restrict attention to 2 strategies per player] Let p 1, p 2,…, p n be some mixed strategy profile. The utility of player 1 for playing pure strategy is where the X j s are Bernoulli random variables with expectaion p j.

PTAS (cont.) How is the utility affected if we replace the p i s by another set of probabilities {q i }? Absolute Change in Utility where the Y j s are Bernoulli random variables with expectaions q j.

PTAS(cont.) Main Lemma: Given any constant k and any set of probabilities {p i } i, there exists a way to round the p i s to q i s which are multiples of 1/k so that ||P - Q|| = O(k -1/2 ), where: P is the distribution of the sum of the Bernoullis p i P is the distribution of the sum of the Bernoullis p i Q is the distribution of the sum of the Bernoullis q i no dependence on n PTAS for anonymous games approximation in time

PTAS - complications Two natural approaches seem to fail: i. round to the closest multiple of 1/k suppose p i =1/n, for all i i = 0, for all i q i = 0, for all i Q [0] = 1, whereas Q [0] = 1, whereas variation distance 1-1/e variation distance 1-1/e

PTAS – complications (cont.) ii. Randomized Rounding Let the q i be random variables taking values which are multiples of 1/k so that E[q i ] = p i. Then, for all t = 0,…, n, - Q[t] is a random variable which is a function of the q i s - Q[t] is a random variable which is a function of the q i s e.g. - Q[t] - Q[t] has the correct expectation! E[Q[t]] = P[t] trouble: expectations are at most 1 and functions involve products

PTAS(cont.) Our approach: Poisson Approximations Intuition: If p i s were small If p i s were small would be close to a Poisson distn of mean define the q i s so that define the q i s so that

PTAS(cont.) Near the boundaries of [0,1] Poisson Approximations are sufficient Disadvantage of Poisson distribution: mean = variance This is disastrous for intermediate values of the p i s approximation with translated Poisson distributions approximation with translated Poisson distributions to achieve mean and variance 2 define a Poisson( 2 ) distribution; then shift it by - 2

Thank you for your attention!