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, \frac{1} { \in } ) can compute an \in-approximate Nash equilibrium in a two-player game, in which each player has n pure strategies, unless PPAD \subseteq 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 \subseteq P. --We prove that no algorithm for computing a Nash equilibrium in a two-player game can have smoothed complexity poly(n, \frac{1} {\sigma } ) under input perturbation of magnitude s, unless PPAD \subseteq RP. In particular, the smoothed complexity of the classic Lemke-Howson algorithm is not polynomial unless PPAD \subseteq 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 + 1 vertices of a unit hypercube. This lemma enables us to overcome the curse of dimensionality in reasoning about fixed-points in high dimensions.

Computing Nash Equilibria: Approximation and Smoothed Complexity

This paper describes recent developments in discrete convex analysis. Particular emphasis is laid on natural introduction of the classes of L-convex and M-convex functions in discrete and continuous variables. Expansion of the application areas is demonstrated by recent connections to submodular function maximization, finite metric space, eigenvalues of Hermitian matrices, discrete fixed point theorem, and matching games.

Recent Developments in Discrete Convex Analysis

This paper presents discrete convex analysis as a tool for use in economics and game theory. Discrete convex analysis is a new framework of discrete mathematics and optimization, developed during the last two decades. Recently, it has been recognized as a powerful tool for analyzing economic or game models with indivisibilities. The main feature of discrete convex analysis is the distinction of two convexity concepts, M-convexity and L-convexity, for functions in integer or binary variables, together with their conjugacy relationship. The crucial fact is that M-concavity in its variant is equivalent to the gross substitutes property in economics. Fundamental theorems in discrete convex analysis such as the M-L conjugacy theorems, discrete separation theorems and discrete fixed point theorems yield structural results in economics such as the existence of equilibria and the lattice structure of equilibrium price vectors. Algorithms in discrete convex analysis provide iterative auction algorithms for finding equilibria.

http://www.mechanism-design.org/arch/v001-1/p_05.pdf

Discrete convex analysis: A tool for economics and game theory

We study the problem of finding an envy-free allocation of a cake to d + 1 players using d cuts. Two models are considered, namely, the oracle-function model and the polynomial-time function model. In the oracle-function model, we are interested in the number of times an algorithm has to query the players about their preferences to find an allocation with the envy less than e. We derive a matching lower and upper bound of θ1/ed-1 for players with Lipschitz utilities and any d > 1. In the polynomial-time function model, where the utility functions are given explicitly by polynomial-time algorithms, we show that the envy-free cake-cutting problem has the same complexity as finding a Brouwer's fixed point, or, more formally, it is PPAD-complete. On the flip side, for monotone utility functions, we propose a fully polynomial-time algorithm FPTAS to find an approximate envy-free allocation of a cake among three people using two cuts.

Algorithmic Solutions for Envy-Free Cake Cutting

This paper develops three game-theoretical models to analyze shipping competition between two carriers in a new emerging liner container shipping market. The behavior of each carrier is characterized by an optimization model with the objective to maximize his payoff by setting optimal freight rate and shipping deployment (a combination of service frequency and ship capacity setting). The market share for each carrier is determined by the Logit-based discrete choice model. Three competitive game strategic interactions are further investigated, namely, Nash game, Stackelberg game and deterrence by taking account of the economies of scale of the ship capacity settings. Three corresponding competition models with discrete pure strategy are formulated as the variables in shipment deployment are indivisible and the pricing adjustment is step-wise in practice. A ɛ -approximate equilibrium and related numerical solution algorithm are proposed to analyze the effect of Nash equilibrium. Finally, the developed models are numerically evaluated by a case study. The case study shows that, with increasing container demand in the market, expanding ship capacity setting is preferable due to its low marginal cost. Furthermore, Stackelberg equilibrium is a prevailing strategy in most market situations since it makes players attain more benefits from the accommodating market. Moreover, the deterrence effects largely depend on the deterrence objective. An aggressive deterrence strategy may make potential monopolist suffer large benefit loss and an easing strategy has little deterrence effect.

Game-theoretical models for competition analysis in a new emerging liner container shipping market

Discrete fixed point theorem reconsidered

A discrete fixed point theorem and its applications

Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave

In this paper we show that a finite symmetric game has a pure strategy equilibrium if the payoff functions of players are integrally concave (the negative of the integrally convex functions due to Favati and Tardella [Convexity in nonlinear integer programming, Ricerca Operativa, 1990, 53:3-44]). Since the payoff functions of any two-strategy game are integrally concave, this generalizes the result of Cheng et al.\, [Notes on equilibria in symmetric games, Proceedings of the 6th Workshop On Game Theoretic And Decision Theoretic Agents, 2004, 23-28]. A simple algorithm to find an equilibrium is also provided.

Existence of a Pure Strategy Equilibrium in Finite Symmetric Games Where Payoff Functions are Integrally Concave

We consider the finite version of the weakly unilaterally competitive game (Kats and Thisse, in Int J Game Theory 21:291–299, 1992) and show that this game possesses a pure strategy Nash equilibrium if it is symmetric and quasiconcave (or single-peaked). The first implication of this result is that unilaterally competitive or two-person weakly unilaterally competitive finite games are solvable in the sense of Nash, in pure strategies. We also characterize the set of equilibria of these finite games. The second implication is that there exists a finite population evolutionarily stable pure strategy equilibrium in a finite game, if it is symmetric, quasiconcave, and weakly unilaterally competitive.

Takuya Iimura

Papers

Discrete fixed point theorem reconsidered

A discrete fixed point theorem and its applications

Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave

Existence of a Pure Strategy Equilibrium in Finite Symmetric Games Where Payoff Functions are Integrally Concave

Pure strategy equilibrium in finite weakly unilaterally competitive games