Constantinos Daskalakis

The Complexity of Computing a Nash Equilibrium

In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwers fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwers problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489-532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwers problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof.

A note on approximate Nash equilibria

In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [Richard J. Lipton, Evangelos Markakis, Aranyak Mehta, Playing large games using simple strategies, in: EC, 2003], and no approximation better than 14 is possible by any algorithm that examines equilibria involving fewer than logn strategies [Ingo Althofer, On sparse approximations to randomized strategies and convex combinations, Linear Algebra and its Applications (1994) 199]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a 12-approximate Nash equilibrium in any 2-player game. For the more demanding notion of approximately well supported Nash equilibrium due to [Constantinos Daskalakis, Paul W. Goldberg, Christos H. Papadimitriou, The complexity of computing a Nash equilibrium, SIAM Journal on Computing (in press) Preliminary version appeared in STOC (2006)] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0 or 1), and that an approximation of 56 is possible, contingent upon a graph-theoretic conjecture. Subsequent work extends the 14 impossibility result of Ingo Althofers paper, as mentioned above, to 12 [Tomas Feder, Hamid Nazerzadeh, Amin Saberi, Approximating nash equilibria using small-support strategies, in: EC, 2007], making our 12-approximate Nash equilibrium algorithm optimal among the algorithms that only consider mixed strategies of sublogarithmic size support. Moreover, techniques similar to our techniques for approximately well supported Nash equilibria are used in [Spyros Kontogiannis, Paul G. Spirakis, Efficient algorithms for constant well supported approximate equilibria in bimatrix games, in: ICALP, 2007] for obtaining an efficient algorithm for 0.658-approximately well supported Nash equilibria, unconditionally.

Optimal Multi-dimensional Mechanism Design: Reducing Revenue to Welfare Maximization

We provide a reduction from revenue maximization to welfare maximization in multidimensional Bayesian auctions with arbitrary - possibly combinatorial - feasibility constraints and independent bidders with arbitrary - possibly combinatorial-demand constraints, appropriately extending Myersons single-dimensional result [21] to this setting. We also show that every feasible Bayesian auction - including in particular the revenue-optimal one - can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidders type ti is transformed into a virtual type fi(ti), via a bidder-specific function. Then, the allocation maximizing virtual welfare is chosen. Using this characterization, we show how to find and run the revenue-optimal auction given only black-box access to an implementation of the VCG allocation rule. We generalize this result to arbitrarily correlated bidders, introducing the notion of a second-order VCG allocation rule. Our results are computationally efficient for all multidimensional settings where the bidders are additive, or can be efficiently mapped to be additive, albeit the feasibility and demand constraints may still remain arbitrary combinatorial. In this case, our mechanisms run in time polynomial in the number of items and the total number of bidder types, but not type profiles. This is polynomial in the number of items, the number of bidders, and the cardinality of the support of each bidders value distribution. For generic correlated distributions, this is the natural description complexity of the problem. The runtime can be further improved to polynomial in only the number of items and the number of bidders in itemsymmetric settings by making use of techniques from [15].

Progress in approximate nash equilibria

It is known [5] that an additively ε-approximate Nash equilibrium (with supports of size at most two) can be computed in polynomial time in any 2-player game with ε=.5. It is also known that no approximation better than .5 is possible unless equilibria with support larger than logn are considered, where n is the number of strategies per player. We give a polynomial algorithm for computing an ε-approximate Nash equilibrium in 2-player games with ε ≈ .38; our algorithm computes equilibria with arbitrarily large supports.

Computing Equilibria in Anonymous Games

We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s2lambda), where s is the number of strategies and lambda is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an isin-approximate Nash equilibrium when the number of strategies is two.

A note on approximate nash equilibria

In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than is possible by any algorithm that examines equilibria involving fewer than logn strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a -approximate Nash equilibrium in any 2-player game. For the more demanding notion of well-supported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0–1), and that an approximation of is possible contingent upon a graph-theoretic conjecture.

Probabilistic analysis of linear programming decoding

We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of ldquoprobabilistic expansionrdquo in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting.

Computing pure nash equilibria in graphical games via markov random fields

We present a reduction from graphical games to Markov random fields so that pure Nash equilibria in the former can be found by statistical inference on the latter. Our result, when combined with the junction tree algorithm for statistical inference, yields a unified proof of all previously known tractable cases of the NP-complete problem of finding pure Nash equilibria in graphical games, but also implies efficient algorithms for new classes, such as the games with O(log n) treewidth. Furthermore, this important problem becomes susceptible to a wealth of sophisticated and empirically successful techniques from Machine Learning.

Message passing algorithms and improved LP decoding

Linear programming (LP) decoding for low-density parity-check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance-coming close to that of iterative decoding algorithms-and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05 fraction of correctable errors for rate-½ codes; this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Our analysis exploits an explicit connection between LP decoding and message-passing algorithms and, unlike other techniques, directly works with the primal linear program. An interesting byproduct of our method is a notion of a “locally optimal” solution that we show to always be globally optimal (i.e., it is the nearest codeword). Such a solution can in fact be found in near-linear time by a “reweighted” version of the min-sum algorithm, obviating the need for LP. Our analysis implies, in particular, that this reweighted version of the min-sum decoder corrects up to a 0.05 fraction of errors.

Extreme-Value Theorems for Optimal Multidimensional Pricing

We provide a Polynomial Time Approximation Scheme for the multi-dimensional unit-demand pricing problem, when the buyers values are independent (but not necessarily identically distributed.) For all epsilon>0, we obtain a (1+epsilon)-factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasi-polynomial, when sampled from regular distributions, and polynomial in n^{poly(log r)}, when sampled from general distributions supported on a set [u_min, r u_min]. We also provide an additive PTAS for all bounded distributions. Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all epsilon >0, g(1/epsilon) distinct prices suffice to obtain a (1+epsilon)-factor approximation to the optimal revenue for MHR distributions, where g(1/epsilon) is a quasi-linear function of 1/epsilon that does not depend on the number of items. Similarly, for all epsilon>0 and n>0, g(1/epsilon \cdot log n) distinct prices suffice for regular distributions, where n is the number of items and g() is a polynomial function. Finally, in the i.i.d. MHR case, we show that, as long as the number of items is a sufficiently large function of 1/epsilon, a single price suffices to achieve a (1+epsilon)-factor approximation. Our results represent significant progress to the single-bidder case of the multidimensional optimal mechanism design problem, following Myersons celebrated work on optimal mechanism design [Myerson 1981].