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

2. Special Classes of Games

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

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

5. Idea of these algorithms dynamic programming + discretization + TV bound 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 X i corresponding to player i ’s mixed strategy with another variable Y i 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 = (X 1, X 2, …, X n ) by Y = (Y 1, Y 2, …, Y n ) ? using independence

8. The TV Bound How much does the payoff of a player change if we replace X = (X 1, X 2, …, X n ) by Y = (Y 1, Y 2, …, Y n ) ? If I’m allowed to use discretization, I can make sure that degree #strategies strategy set of player i choose for approximation of. 2

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

12. Polymatrix Games - players are nodes of a graph G - player’s payoff is the sum of payoffs from all adjacent edges … … - edges are 2-player games 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 Theorem [Daskalakis-Papadimitriou ’09, Cai-Daskalakis’11] - a Nash equilibrium can be found efficiently with linear-programming; - 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. - the Nash equilibria comprise a convex set; strong indication that Nash eq. makes sense in this setting. i.e. payoffs approach equilibrium payoffs, and empirical strategies approach Nash equilibrium If the global game is zero-sum: essentially the broadest class of zero- sum games we could hope to solve

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. e.g. symmetry in auctions, congestion games, social phenomena, etc. ‘‘The women of Cairo: Equilibria in Large Anonymous Games.’’ Blonski, Games and Economic Behavior, 1999. “Partially-Specified Large Games.” Ehud Kalai, WINE, 2005. ‘‘Congestion Games with Player- Specific Payoff Functions.’’ Milchtaich, Games and Economic Behavior, 1996. - all players share the same set of strategies: S = {1,…, s} - payoff functions: u p = u p (σ ; n 1, n 2,…,n s ) number of the other players choosing each strategy in S choice of p Description Size: O(min {s n s, n s n })

16. PTAS There is a PTAS for anonymous games with a constant #strategies. Theorem [Daskalakis, Papadimitriou ’07, ’08]: Remarks: - exact computation is not known to be PPAD-complete for multi-player anonymous games with a constant number of strategies; - 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. sketch of algorithm for 2 strategies 0 1 0 1 p2p2 p1p1 discretize [0,1] n into multiples of δ, and restrict search to the discrete space pick best point in discrete space   since 2 strategies per player, Nash equilibrium lies in [0,1] n

18. sketch for 2 strategies (cont.) Basic Question: what grid size  is required for  - approximation? if function of  only  PTAS if function also of n  nothing 0 1 0 1 p2p2 p1p1   First trouble: size of search space 1  n but will deal with this later

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

20. The TV Bound How much does player p’s payoff from pure strategy σ change if we replace X = (X 1, X 2, …, X n ) with Y = (Y 1, Y 2, …, Y n ) ? 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 + discretization + TV bound complete this step (Exercise) assume that the players only use mixed strategies in probabilities that are multiples of. enough to guarantee a discretized - Nash equilibrium Resulting running time for 2 strategies.

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

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

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

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

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

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

28. Lemma: - The sum of m ≥ k 3 indicators X i 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 the corresponding symmetry… … i.e. close to a sum of indicators with the same expectation [tightness of parameters by Berry-Esséen]

29. proof of structural result round some of the X i ’s falling here to 0 and some of them to ε so that the total mean is preserved to within ε - if more than 1/ε 3 X i ’s are left here, appeal to previous slide (Binomial appx) similarly 0 ε 1-ε1 0 ε ε 1 - o.w. use Dask. Pap. ’07 (exists rounding into multiples of ε 2 )

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

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] constant #strategies per player: bad function of s is there an FPTAS?