6.853: Topics in Algorithmic Game Theory

1. 6.853: Topics in Algorithmic Game Theory Lecture 13 Fall 2011 Constantinos Daskalakis

2. Special Classes of Games

3. Special Classes of 2-player Games poly-time solvable (lecture 2) rank-1 two-player games ( R + C has rank 1 ) zero-sum two-player games ( ) poly-time solvable [Adsul-Garg-Mehta-Sohoni ’10] low-rank two-player games ( ) PTAS [Kannan, Theobald ’09] sparse two-player games ( constant number of non-zero entries in each row, column) PTAS [Daskalakis, Papadimitriou ’09] note : exact is PPAD-complete [Chen,Deng,Teng 06] small probability games ( equilibrium with non-trivial support on a linear number of strategies) [Daskalakis, Papadimitriou ’09] PTAS note : exact is PPAD-complete win-lose games (all payoff entries in {0,1}) no PTAS is known.. exact is PPAD-complete [follows from Abbott, Kane, Valiant ’05] also no FPTAS [Chen, Teng, Valiant ’07]

4. Special Classes of Graphical Games limitations on the graph structure: line / cylic graphical games (many players, 2 strategies per player) exact algorithm [Elkind, Goldberg, Goldberg ’06] the only class of graphs where exact equilibria can be computed trees (many players, constant #strategies) FTPAS [Kearns, Littman, Singh ’01] bounds on the cyclicity of the graph: Theorem [Daskalakis, Papadimitriou ’08] An -Nash equilibrium of a graphical game with n players, maximum degree d, treewidtht, and at most s strategies per player can be computed in time polynomial in n and . 2 e.g. if d, s are bounded, and t = O(logn), the above algorithm is a PTAS, since the input size is .

5. Idea of these algorithms dynamic programming TV bound discretization + + assume that the players only use mixed strategies in probabilities that are multiples of a fixed fraction find the best discretized collection of mixed strategies What is the loss in approximation due to the discretization?

6. [ Total Variation Distance Def: The total variation (TV) distance between two random variables X and Y is the L1 distance of their PDFs. ]

7. The TV Bound In a game, the mixed strategy of each player is a random variable independent of the random variables of the other players. The effect of the discretization is to replace the random variable Xi corresponding to player i’s mixed strategy with another variable Yi whose probability for every pure strategy is an integer multiple of the discretization parameter . How much does the payoff of a player change if we replace X = (X1, X2, …, Xn) by Y = (Y1, Y2, …, Yn) ? using independence

8. The TV Bound If I’m allowed to use discretization , I can make sure that strategy set of player i How much does the payoff of a player change if we replace X = (X1, X2, …, Xn) by Y = (Y1, Y2, …, Yn) ? degree #strategies 2 choose for approximation of .

9. Idea of these algorithms dynamic programming TV bound discretization + + assume that the players only use mixed strategies in probabilities that are multiples of . because of TV bound, the best discretized collection of mixed strategies is guaranteed to be an .-Nash equilibrium 2 runtime: 2

10. multiplayer zero-sum games (review from lecture 3)

11. Multiplayer Zero-Sum, wha? Take an arbitrary two-player game, between Alice and Bob. Add a third player, Eve, who does not affect Alice or Bob’s payoffs, but receives payoff The game is zero-sum, but solving it is PPAD-complete. intractability even for 3 player, if three-way interactions are allowed. What if only pairwiseinteractions are allowed?

12. Polymatrix Games … - players are nodes of a graph G - edges are 2-player games - player’s payoff is the sum of payoffs from all adjacent edges … N.B. finding a Nash equilibrium is PPAD-complete for general games on the edges [D, Gold, Pap ’06] What if the total sum of players’ payoffs is always 0?

13. Polymatrix Games essentially the broadest class of zero-sum games we could hope to solve Theorem [Daskalakis-Papadimitriou ’09, Cai-Daskalakis’11] If the global game is zero-sum: - a Nash equilibrium can be found efficiently with linear-programming; - the Nash equilibria comprise a convex set; - if every node uses a no-regret learning algorithm (such as the MWU method from Lecture 4), the players’ behavior converges to a Nash equilibrium. i.e. payoffs approach equilibrium payoffs, and empirical strategies approach Nash equilibrium strong indication that Nash eq. makes sense in this setting.

14. Anonymous Games

15. anonymous games Every player is (potentially) different, but only cares about how many players (of each type) play each of the available strategies. - all players share the same set of strategies: S = {1,…, s} - payoff functions: up = up (σ ; n1, n2,…,ns ) number of the other players choosing each strategy in S choice of p Description Size: O(min {s ns, nsn}) e.g. symmetry in auctions, congestion games, social phenomena, etc. ‘‘Congestion Games with Player- Specific Payoff Functions.’’ Milchtaich, Games and Economic Behavior, 1996. ‘‘The women of Cairo: Equilibria in Large Anonymous Games.’’ Blonski, Games and Economic Behavior, 1999. “Partially-Specified Large Games.” Ehud Kalai, WINE, 2005.

16. PTAS Theorem [Daskalakis, Papadimitriou ’07, ’08]: There is a PTAS for anonymous games with a constant #strategies. - exact computation is not known to be PPAD-complete for multi-player anonymous games with a constant number of strategies; Remarks: - on the flip side, if n is small and s is large (few players, many strategies) then trivially PPAD-complete, since general 2-player games can be reduced to this.

17. 1  0 1 0  sketch of algorithm for 2 strategies • since 2 strategies per player, Nash equilibrium lies in [0,1]n • discretize [0,1]n into multiples of δ, and restrict search to the discrete space • pick best point in discrete space p2 p1

18. n 1  1  0 1 0  sketch for 2 strategies (cont.) First trouble: size of search space p2 but will deal with this later p1 Basic Question: what grid size  is required for  - approximation? if function of  only PTAS if function also of n nothing

19. sketch for 2 strategies (cont.) Theorem [Daskalakis, Papadimitriou ’07]: Given - nind. Bernoulli’s Xi with expectations pi,i =1,…, n - a constant  independent of n there exists another set of Bernoulli’s Yi with expectations qi such that qi’s are integer multiples of  N.B. argument from last lecture gives in fact:

20. The TV Bound How much does player p’s payoff from pure strategy σ change if we replace X = (X1, X2, …, Xn) with Y = (Y1, Y2, …, Yn) ? Given previous theorem, can guarantee that there exists a discretized point making the above difference at most by selecting .

21. Completing the algorithm dynamic programming TV bound discretization + + assume that the players only use mixed strategies in probabilities that are multiples of . enough to guarantee a discretized - Nash equilibrium complete this step (Exercise) Resulting running time for 2 strategies.

22. The first probabilistic approximation theorem Theorem [Daskalakis, Papadimitriou ’07]: Given - nind. Bernoulli’s Xi with expectations pi,i =1,…, n - a constant  independent of n there exists another set of Bernoulli’s Yi with expectations qi such that qi’s are integer multiples of  argument from last time gives in fact:

23. proof of approximation result Law of Rare Events + CLT - roundingpi’s to the closest multiple of  gives total variation n - probabilistic rounding up or down quickly runs into problems - what works: Poisson Approximations (Stein’s Method) Berry-Esséen

24. proof of approximation result Intuition: If pi’s were small  would be close to a Poisson with mean  define the qi’s so that

25. proof of approximation result Poisson approximation is only good for small values of pi’s. (LRE) For intermediate values of pi’s, Normals are better. (CLT) Berry-Esséen Berry-Esséen

26. Anonymous Games Summary 2-strategies per player: [DP ’07] constant #strategies per player: [DP ’08] bad function of s

27. is there a faster PTAS? Theorem [Daskalakis ’08]: There is an oblivious PTAS with running time the underlying structural result… Theorem [D’08]: In every anonymous game there exists an ε-approximate Nash equilibrium in which • - either all players who mix play the same mixed strategy • - or, at most mix, and they choose mixed strategies which are integer multiples of

28. the corresponding symmetry… Lemma: - The sum of m ≥ k3 indicators Xi with expectations in [1/k,1-1/k] is O(1/k)-close in total variation distance to a Binomial distribution with the same mean and variance … i.e. close to a sum of indicators with the same expectation [tightness of parameters by Berry-Esséen]

29. proof of structural result 1-ε 1-ε 1-ε 1 1 ε ε ε 0 0 similarly round some of the Xi’s falling here to 0 and some of them toεso that the total mean is preserved to within ε • if more than 1/ε3Xi’s are left here, appeal to previous slide (Binomial appx) - o.w. use Dask. Pap. ’07 (exists rounding into multiples of ε2)

30. Anonymous Games Summary 2-strategies per player: [DP ’07] [D ’08] constant #strategies per player: [DP ’08] bad function of s

31. Is there an even faster PTAS? Theorem [Daskalakis, Papadimitriou ’08]: There is a non-oblivious PTAS with running time the underlying probabilistic result [DP ’08]: If two sums of indicators have equal moments up to moment k then their total variation distance is O(2-k).

32. Anonymous Games Summary 2-strategies per player: [DP ’07] [D ’08] [DP ’09] is there an FPTAS? constant #strategies per player: bad function of s