Vasilis Syrgkanis Cornell University 4 10 12 10 7 2 2 10 5 5 10 4 Can we efficiently compute some Pure Nash Equilibrium of such games? Not in the “general case”: PLS-hard Can we efficiently compute a “good” Pure Nash Equilibrium so that we can propose it to the players? Yes if players choose Spanning Trees or, in general, a base of some player-specific Matroid Formal Definitions Algorithm for Computing a Good Pure Nash Equilibrium PLS-hardness Results N players and F facilities For each player i a set of strategies S i 2 F Given a strategy profile s iN Si , denote: n f s : the number of players using f r f n f : a decreasing cost function r n : the cost of player i SC ( s ) C s n r n : the social cost Ci ( s) f si iN f f i f F f f f Anshelevich et al., FOCS ’04 If rf (n f ) has the form c f (n f ) / n f and c f ( n f ) is a nondecreasing concave function then there exists a PNE with social cost H nOPT (the worst PNE can be as bad as nOPT ) Fabrikant et al., STOC ‘04 - Ackermann et al., J ACM ‘08 It is PLS-complete to compute any PNE in Network Congestion Games with rf (n f ) a f n f b f There exists a polynomial time greedy algorithm that computes a PNE with social cost at most H nOPT for Matroid Cost Sharing Games with rf (n f ) of the form c f (n f ) / n f , where c f ( n f ) is a non-decreasing concave function. Note: Computing the best PNE or the PNE minimizing the potential function is NP-hard even in very simple matroid costsharing games. For general decreasing cost functions the algorithm computes a PNE with social cost at most the potential of the optimal strategy profile. It is PLS-complete to compute any PNE in Undirected Network Cost Sharing Games with rf (n f ) of the form c f (n f ) / n f , where c f (n f ) is a non-decreasing concave function. If we extend to Directed Networks then it is PLS-complete even in the case when rf (n f ) have the form c f / n f Formal Definitions Algorithm for Computing a Good Pure Nash Equilibrium PLS-hardness Results The Price of Stability of a game is the fraction of the Social Cost of the best PNE over the Optimal Social Cost. If a game admits a Potential such that for some quantity a and for any strategy profile s : SC(s) s aSC(s) Then, the PoS is at most a Let PNEg .m. be the global minimum of and OPT the socially optimal outcome. Then: SC ( PNEg .m. ) PNEg .m. OPT aSC (OPT ) Cost Sharing Games are Congestion Games and admit Rosenthal’s Potential: s nf r k f F k 1 If rf (n f ) cf nf , then s f c f F f H n f , SC s c f SC ( s) s H n SC ( s) PoS H n f s Compute a PNE that is as good as the upper bound on the PoS produced by the Potential Method r1 n1 r2 n2 r3 n3 Recall the proof of the Potential Method Attempt 1: Compute Socially Optimal PNE [ADKTWR04]: NP-hard even when rf n f 1 / n f Attempt 2: Compute global Potential Minimizer PNEg .m. [CCLNO07]: NP-hard even when rf n f 1 / n f SC ( PNEg .m. ) PNEg .m. OPT aSC (OPT ) Consider the following generalization of the greedy set cover approximation algorithm: At each iteration pick the facility that has minimum cost if all currently unassigned players were assigned to it. Assign all possible players to the chosen facility. When rf (n f ) c f / n f , the above is exactly the greedy H n - approximation algorithm for Set Cover c2 20 c3 30 c1 12 c1 10 Theorem 1 The greedy algorithm computes a PNE with social cost at most the potential of the socially optimal solution Corollary The greedy algorithm computes a PNE with social cost at most the best upper bound on the PoS given by the Potential Method SC ( PNEg .m. ) SC(PNE PNEgreedy OPT OPT aSC aSC(OPT ) g .m. ) r1 …… ri Nj …… rj …… rk N k rk …… Nk rm rj N j rk N k Matroid Cost Sharing: Si is the set of bases of a matroid F, I i e.g. Spanning Trees on a set of nodes (possibly different nodes for each player) 10 4 12 10 7 Algorithm 2: Build player’s strategies incrementally starting from empty sets At each iteration pick the facility that has minimum cost if it is added to the strategy of all possible players Add the facility to the strategy of all possible players Player 1 9 Player 2 8 5 10 12 10 7 Player 3 Theorem 2 Algorithm 2 computes a PNE of any Matroid Cost Sharing Game with social cost at most the potential of the optimal strategy profile. Prove outcome is PNE: A base of a matroid is minimum iff it is locally minimum under (1,1) exchanges Prove efficiency guarantee: Construct a 1-1 mapping of the facilities of a player in the greedy strategy and those in the optimum: When algorithm adds a facility to a player its corresponding optimal facility was also an option Formal Definitions Algorithm for Computing a Good Pure Nash Equilibrium PLS-hardness Results Theorem 3 Computing a PNE in General Cost Sharing Games where rf (n f ) c f / n f is PLS-complete Proof: Reduction from MAX CUT MAX CUT: Given a weighted graph G find a cut that cannot be increased by switching a node from one side to the other. Given an instance of MAX CUT create a Cost Sharing Game such that any PNE is a locally maximum cut. Create a player for each node of the graph A B s , s Each player has two strategies: i i A s Given a strategy profile if a player is playing i then assign the corresponding node to partition A, else to B Create incentives for players to play opposite strategies s B j wi,j siB f i ,2j … f i1,j j s Aj ci1, j ci2. j 2wi , j … … siA i … s1A s 2B s1B f11, 2 2 w1, 2 1 w2 ,3 w1, 3 f1,22 s 2A s3B 3 s3A C s f 21,3 f11,3 f1,23 f 22,3 wi, B w Ci siA , si wi, A wi i B i , s i i Network Cost Sharing Games Given a directed graph G (V , E ) , Si is the set of paths in a graph between two nodes si , ti 10 2 5 5 2 4 10 Theorem 5 Computing a PNE in Network Cost Sharing Games where rf (n f ) c f / n f is PLS-complete. Theorem 6 Computing a PNE in Undirected Network Cost Sharing Games where rf (n f ) c f n f / n f and c f n f is an non decreasing concave function. s1 s1A s 2B s2 f11, 2 f1,22 s 2A s3B s3 s1B f11,3 f 21,3 f1,23 s3A t1 f 22,3 t2 t3 s1A 2 wij s1 s 2B s1B f11, 2 f1,22 s 2A f s2 f1,23 H 6H f11,3 f1,23 t1 f 21,3 f11,3 s3A f1,22 s3 s3B H 1 1, 2 6H f 21,3 6H f 22,3 t2 6H t3 f 22,3 D' s1 s1A s 2B D' D f D s3 D t1 f 21,3 f11,3 f1,23 f 22,3 H D 6H f1,23 D' s3B D f11,3 D' f1,22 s3A f1,22 D' f11, 2 s 2A H 1 1, 2 D' s2 D s1B f 21,3 6H D 6H f 22,3 D D t2 6H D D t3 D Despite NP-completeness of computing Social Cost Minimizer and Potential Minimizer we manage to compute a PNE with very good social cost. Computing equilibria with social cost at most the potential of the optimal might prove useful in other games too For Network Games our results show that the decreasing cost function case is not easier than the increasing one