We advance significantly beyond the recent progress on the algorithmic complexity of Nash equilibria by solving two major open problems in the approximation of Nash equilibria and in the smoothed analysis of algorithms. • We show that no algorithm with complexity poly(n, 1 ɛ) can compute an ɛ-approximate Nash equilibrium in a two-player game, in which each player has n pure strategies, unless PPAD ⊆ P. In other words, the problem of computing a Nash equilibrium in a twoplayer game does not have a fully polynomial-time approximation scheme unless PPAD ⊆ P. • We prove that no algorithm for computing a Nash equilibrium in a two-player game can have smoothed complexity poly(n, 1 σ) under input perturbation of magnitude σ, unless PPAD ⊆ RP. In particular, the smoothed complexity of the classic Lemke-Howson algorithm is not polynomial unless PPAD ⊆ RP. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional hypergrid with constant side-length, and show that it can host the embedding of the proof structure of any PPAD problem. We prove a key geometric lemma for finding a discrete fixed-point, a new concept defined on n +1vertices of a unit hypercube. This lemma enables us to overcome the curse of dimensionality in reasoning about fixed-points in high dimensions. 1

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We consider the problem of routing n users on m parallel links, under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. Thus, the problem is equivalent to the correspondingly restricted problem of assigning n jobs to m parallel machines. In a pure Nash equilibrium, no user may improve its own individual cost (delay) by unilaterally switching to another link from its set of allowed links. As our main result, we introduce a polynomial time algorithm to compute from any given assignment a pure Nash equilibrium with non-increased makespan. The algorithm gradually changes a given assignment by pushing unsplittable user traffics through a network that is defined by the users and the links. Here, we use ideas from blocking flows. Furthermore, we use similar techniques as in the generic Preflow-Push algorithm to approximate a schedule with minimum makespan, gaining an improved approximation factor of 2 − 1 for identical links, where w1 is the largest user traffic. w1 We extend this result to related links, gaining an approximation factor of 2. Our approximation algorithms run in polynomial time. We close with tight upper bounds on the coordination ratio for pure Nash equilibria.

The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matousek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. The new algorithm, like the existing randomized subexponential algorithms, uses only polynomial space, and it is almost as fast as the randomized subexponential algorithms mentioned above.

We prove that computing a Nash equilibrium in a 3-player game is PPAD-complete, solving a problem left open in (2).

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the complexity of four-player Nash equilibria [21], settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems: • Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. • The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time. Our results also have a complexity implication in mathematical economics: • Arrow-Debreu market equilibria are PPAD-hard to compute.

We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any non-trivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as a more general arithmetic circuit decision problem that characterizes P-time in a unit-cost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point

Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. Based on these varying notions of symmetry, we investigate the computational complexity of pure Nash equilibria. It turns out that in all four classes of games equilibria can be found efficiently when only a constant number of actions is available to each player, a problem that has been shown intractable for other succinct representations of multi-player games. We further show that identical payoff functions simplify the search for equilibria, while a growing number of actions renders it intractable. Finally, we show that our results extend to wider classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.

We settle the complexity of a well-known problem in networking by establishing that it is PSPACE-complete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. We show that the concept of sink equilibria proposed recently in [11] is also PSPACE-complete to analyze and approximate for graphical games. Finally, we propose a new equilibrium concept inspired by game dynamics, unit recall equilibria, which we show to be close to universal (exists with high probability in a random game) and algorithmically promising. We also give a relaxation thereof, called componentwise unit recall equilibria, which we show to be both tractable and universal (guaranteed to exist in every game).

doi 10.1287/moor.1080.0322