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Approximate Nash Equilibria in interesting games

Approximate Nash Equilibria in interesting games. Constantinos Daskalakis, U.C. Berkeley. If your game is interesting, its description cannot be astronomically long…. Game Species. e.g. bounded degree. graphical games. normal form games. interesting games.

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Approximate Nash Equilibria in interesting games

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  1. Approximate Nash Equilibriain interesting games Constantinos Daskalakis, U.C. Berkeley

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

  3. Game Species e.g. bounded degree graphical games normal form games interesting games e.g. constant number of players what else?

  4. Bad News… Computing a Mixed Nash Equilibrium ? - in normal form games is PPAD-complete [DGP ’05] even for 3 players [CD ’05, DP ’05] even for 2 players [CD ’06] - in graphical games PPAD-complete [DGP] even for 2 strategies per player and degree 3

  5. So what next? Computing approximate Equilibria (every player plays an approximate best response) Finding a better point in Christos’ cube correlated mixed Nash [DGP06, CD06] existence efficiency naturalness pure Nash Looking at other interesting games…

  6. 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] ( no FPTAS )  = constant??

  7. LMM ’04 log n - support is enough for all  2 [LMM ’03] take Nash equilibrium (x, y); take log n/ 2independent samples from x and y  subexponential algorithm for computing  - Nash

  8. A simple algorithm for .5 -approximate [DMP ’06] Column player finds: best response j to strategy i of row player Row player finds: best response k to strategy j of column player 1.0 j 0.5 i  0.5 approximate Nash! 0.5 k  [FNS ’06]: can’t do better with small supports! G = (R, C)

  9. Beyond Constant Support [DMP ’07] .38 can be achieved in polynomial time Generalization of Previous Idea: guess value of the eq. u sampling (similar to LMM) + LP +

  10. PTAS ?

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

  12. why interesting? • the succinctness argument : n players, s strategies, all interact, ns size! (the utility of a player depends on her strategy, and on how many other players play each strategy) • ubiquity: think of your favorite large game - is it anonymous? e.g. auctions, stock market, congestion, social phenomena, … "How many veiled women can we expect in Cairo ?" Characterization of equilibria in large anonymous games, [Blonski ’00]

  13. Pure Nash Equilibria Theorem [DP ’07]: In any anonymous game, there exists a 2Ls2-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?

  14. PTAS for anonymous gameswith two strategies Big Picture: • Discretize the space of mixed Nash equilibria. • 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. Big Question: what grid size is required to achieve approximation ? if function of  only PTAS if function of n  nothing

  15. PTAS (cont.) [Restrict attention to 2 strategies per player] Let p1 , p2 ,…,pnbe some mixed strategy profile. The utility of player 1 for playing pure strategy  is where the Xj’s are Bernoulli random variables with expectaion pj.

  16. PTAS (cont.) How is the utility affected if we replace the pi’s by another set of probabilities {qi}? Absolute Change in Utility where the Yj’s are Bernoulli random variables with expectaions qj.

  17. PTAS(cont.) Main Lemma: Given any constant k and any set of probabilities {pi}i, there exists a way to round the pi’s to qi’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 pi Q is the distribution of the sum of the Bernoullis qi no dependence on n  PTAS for anonymous games  approximation in time

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

  19. PTAS – complications (cont.) ii. Randomized Rounding Let the qi be random variables taking values which are multiples of 1/k so that E[qi] = pi. Then, for all t = 0,…, n, - Q[t] is a random variable which is a function of the qi’s e.g. - Q[t] has the correct expectation! E[Q[t]] = P[t] trouble: expectations are at most 1 and functions involve products

  20. PTAS(cont.) Our approach: Poisson Approximations Intuition: If pi’s were small  would be close to a Poisson dist’n of mean define the qi’s so that

  21. 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 pi’s approximation with translated Poisson distributions to achieve mean  and variance 2 define a Poisson(2 ) distribution; then shift it by  - 2

  22. Thank you for your attention!

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