Computing Nash Equilibria Research Articles

On the Convergence of the Lemke–Howson Algorithm for Bi-Matrix Games

We discuss conditions that guarantee the convergence of the computational Lemke–Howson algorithm for the solution of bi-matrix games and the computation of all Nash equilibria. An example is also presented.

Computing Nash Equilibria and Evolutionarily Stable States of Evolutionary Games

Stability analysis is an important research direction in evolutionary game theory. Evolutionarily stable states have a close relationship with Nash equilibria of repeated games, which are characterized by the folk theorem. When applying the folk theorem, one needs to compute the minimax profile of the game in order to find Nash equilibria. Computing the minimax profile is an NP-hard problem. In this paper, we investigate a new methodology to compute evolutionary stable states based on the level- ${k}$ equilibrium, a new refinement of Nash equilibrium in repeated games. A level- ${k}$ equilibrium is implemented by a group of players who adopt reactive strategies and who have no incentive to deviate from their strategies simultaneously. Computing the level- ${k}$ equilibria is tractable because the minimax payoffs and strategies are not needed. As an application, this paper develops a tractable algorithm to compute the evolutionarily stable states and the Pareto front of ${n}$ -player symmetric games. Three games, including the iterated prisoner’s dilemma, are analyzed by means of the proposed methodology.

Nash Equilibrium Computation in Subnetwork Zero-Sum Games With Switching Communications

In this paper, we investigate a distributed Nash equilibrium computation problem for a time-varying multi-agent network consisting of two subnetworks, where the two subnetworks share the same objective function. We first propose a subgradient-based distributed algorithm with heterogeneous stepsizes to compute a Nash equilibrium of a zero-sum game. We then prove that the proposed algorithm can achieve a Nash equilibrium under uniformly jointly strongly connected (UJSC) weight-balanced digraphs with homogenous stepsizes. Moreover, we demonstrate that for weighted-unbalanced graphs a Nash equilibrium may not be achieved with homogenous stepsizes unless certain conditions on the objective function hold. We show that there always exist heterogeneous stepsizes for the proposed algorithm to guarantee that a Nash equilibrium can be achieved for UJSC digraphs. Finally, in two standard weight-unbalanced cases, we verify the convergence to a Nash equilibrium by adaptively updating the stepsizes along with the arc weights in the proposed algorithm.

Analytical Approach in Computing Nash Equilibrium for Oligopolistic Competition of Transmission-Constrained GENCOs

This paper proposes an analytical solution for bidding strategy problem of transmission-constrained generating companies (GENCOs) in an hour-ahead electricity market. Here, for modeling a bidding strategy problem under a supply function equilibrium model, a bilevel programming (BLP) method is used. In the upper level, the profit of an intended GENCO is maximized, and in the lower level, the independent system operator clears the market via solving a bid-based security-constrained economic dispatch (SCED), in which SCED can be parameterized by GENCOs' bidding strategies. We apply an active-set method to solve a parametric SCED problem. The idea of implementing the active-set method is based on the fact that the optimal solution of a parametric quadratic programming (QP) problem with linear equality constraints can be determined directly. We also decompose generation scheduling, locational marginal prices, and consequently GENCOs' profits, in terms of GENCOs' bidding strategies. In this situation, the BLP transforms to a single-level inequality-constrained optimization, which maximizes GENCO's profit subject to inactive constraints of the lower level. In addition, GENCOs' strategic interactions in oligopolistic marketplace are investigated by using a game theoretic framework, and optimal bidding strategies are determined then by finding the Nash equilibrium of GENCOs' noncooperative competition. Two case studies are provided to illustrate the proposed method; in addition, a numerical method has been conducted to validate the results. The results clarify the impact of each GENCO's bidding strategy on the power flows of transmission lines.

Computation of Efficient Nash Equilibria for Experimental Economic Games

Nash equilibrium is the central solution concept with diverse applications for most games in game theory. For games with multiple equilibria, different equilibria can have different rewards for the players thus causing a challenge on their choice of strategies. This paper described and carried out an experiment on a game that was modelled as a three-player experimental economic game. The results were recorded and by the best response sets method we identified all the Pure Nash equilibria and computed the most efficient Nash equilibrium for the modelled game. Using the Brauwer's fixed point theorem we verified the existence of mixed Nash equilibrium in the game. An individual whose aim was to minimize risks played the risk dominant strategies whereas those aiming to maximize their profits, the payoff dominant strategies were played to achieve their most efficient Nash equilibrium. The computation of most efficient Nash Equilibrium in games can be applied to most situations in competitive Economic environment that are faced with multiple choices on which strategy is optimal.

Achieving fairness by sequential equilibrium in rational two‐party computation under incomplete information

AbstractFairness in secure two‐party computation ensures that either both of the communicating parties learn the output of some pre‐defined function or none of them does. Rational two‐party computation is an extension of two‐party computation that incorporates game theory into conventional two‐party (cryptographic) computation protocols for achieving fairness. From the standpoint of game theory, the strategies are designed for achieving equilibrium resulting in attaining fairness in rational two‐party computation protocols. Groce and Katz (Eurocrypt 2012) achieved fairness by computational Nash equilibrium under complete information. In this paper, protocols are considered in more practical scenarios of incomplete information, where fairness is achieved by sequential equilibrium. Our protocol has constant number of rounds as Groce and Katz do while achieving a stronger sequential equilibrium which also implies the computational Nash equilibrium. Copyright © 2015 John Wiley & Sons, Ltd.

Solution concepts for games with ambiguous payoffs

I consider games with ambiguous payoffs played by non-Expected Utility decision makers. Three equilibrium solutions are studied. Nash equilibrium in which equilibrium mixed strategies must be best responses, Crawford equilibrium in beliefs and pure equilibrium in beliefs in which equilibrium strategies are mixtures of best responses, with the latter restricting best responses to pure actions. I study the interactions between ambiguity preferences on one side and equilibrium properties on the other. I show how the equilibrium concepts differ, computing necessary and sufficient conditions for existence and equivalence. I also show how these solution concepts fare against two fundamental principles of Nash equilibrium in standard games: the principle of indifference and the reduction principle. Given both are central to the computation of Nash equilibrium in games with Expected Utility players, their failure indicates how relaxing the Expected Utility hypothesis may disrupt standard game theoretic results such as the characterization of equilibria in two-player games.

Applications of Algebra for Some Game Theoretic Problems

In this paper we present applications of polynomial algebra for the problem of computing Nash equilibria of a subclass of finite normal form games.We characterize Nash equilibria of a normal form game as solutions to a system of polynomial equations and define the subclass of games under consideration. We present an algebraic method for deciding membership decision to the subclass of games. A method based on group action to compute all Nash equilibria of the subclass of games is presented with examples to show working of the methods. We also present some related results and discuss properties of the subclass of games.

Computing Nash Equilibria in Bimatrix Games: GPU-Based Parallel Support Enumeration

Computing Nash equilibria is a very important problem in strategic analysis of markets, conflicts, and resource allocation. Unfortunately, computing these equilibria even for moderately sized games is computationally expensive. To obtain lower execution times it is essential to exploit the parallel processing capabilities offered by the currently available massively parallel architectures. To address this issue, we design a GPU-based parallel support enumeration algorithm for computing Nash equilibria in bimatrix games. The algorithm is based on a new parallelization method which achieves high degrees of parallelism suitable for massively parallel GPU architectures. We perform extensive experiments to characterize the performance of the proposed algorithm. The algorithm achieves significant speedups relative to the OpenMP and MPI-based parallel implementations of the support enumeration method running on a cluster of multi-core computers.

Electricity day-ahead markets: Computation of Nash equilibria

In a restructured electricity sector, day-ahead markets can be modeled as a game where some players - the producers - submit their proposals. To analyze the companies' behavior we have used the concept of Nash equilibrium as a solution in these multi-agent interaction problems. In this paper, we present new and crucial adaptations of two well-known mechanisms, the adjustment process and the relaxation algorithm, in order to achieve the goal of computing Nash equilibria. The advantages of these approaches are highlighted and compared with those available in the literature.

Accelerating Multiagent Reinforcement Learning by Equilibrium Transfer.

An important approach in multiagent reinforcement learning (MARL) is equilibrium-based MARL, which adopts equilibrium solution concepts in game theory and requires agents to play equilibrium strategies at each state. However, most existing equilibrium-based MARL algorithms cannot scale due to a large number of computationally expensive equilibrium computations (e.g., computing Nash equilibria is PPAD-hard) during learning. For the first time, this paper finds that during the learning process of equilibrium-based MARL, the one-shot games corresponding to each state's successive visits often have the same or similar equilibria (for some states more than 90% of games corresponding to successive visits have similar equilibria). Inspired by this observation, this paper proposes to use equilibrium transfer to accelerate equilibrium-based MARL. The key idea of equilibrium transfer is to reuse previously computed equilibria when each agent has a small incentive to deviate. By introducing transfer loss and transfer condition, a novel framework called equilibrium transfer-based MARL is proposed. We prove that although equilibrium transfer brings transfer loss, equilibrium-based MARL algorithms can still converge to an equilibrium policy under certain assumptions. Experimental results in widely used benchmarks (e.g., grid world game, soccer game, and wall game) show that the proposed framework: 1) not only significantly accelerates equilibrium-based MARL (up to 96.7% reduction in learning time), but also achieves higher average rewards than algorithms without equilibrium transfer and 2) scales significantly better than algorithms without equilibrium transfer when the state/action space grows and the number of agents increases.

A novel method to compute Nash equilibrium in non-cooperative n-person games based on differential evolutionary algorithm

Nash equilibrium constitutes a central solution concept in game theory. In this paper, a Nash equilibrium of a finite n-person non-cooperative game can be formulated to an non-linear optimal solution of the optimization model with zero optimal value. This paper investigates the effectiveness of differential evolutionary optimization to compute Nash equilibrium of non-cooperative n-person games, as global minimum of a real-valued function. The experiments demonstrate the feasibility of this approach for finding an equilibrium of an N-person game.

Non-cooperative games with minmax objectives

We consider noncooperative games where each player minimizes the sum of a smooth function, which depends on the player, and of a possibly nonsmooth function that is the same for all players. For this class of games we consider two approaches: one based on an augmented game that is applicable only to a minmax game and another one derived by a smoothing procedure that is applicable more broadly. In both cases, centralized and, most importantly, distributed algorithms for the computation of Nash equilibria can be derived.

Solving Mixed Strategy Nash-Cournot Equilibria under Generation and Transmission Constraints in Electricity Market

Generation capacities and transmission line constraints in a competitive electricity market make it troublesome to compute Nash Equilibrium (NE) for analyzing participants' strategic generation quantities. The NE can cause a mixed strategy NE rather than a pure strategy NE resulting in a more complicated computation of NE, especially in a multiplayer game. A two-level hierarchical optimization problem is used to model competition among multiple participants. There are difficulties in using a mathematical programming approach to solve a mixed strategy NE. This paper presents heuristics applied to the mathematical programming method for dealing with the constraints on generation capacities and transmission line flows. A new formulation based on the heuristics is provided with a set of linear and nonlinear equations, and an algorithm is suggested for using the heuristics and the newly-formulated equations.

Ranking games that have competitiveness-based strategies

An extensive literature in economics and social science addresses contests, in which players compete to outperform each other on some measurable criterion, often referred to as a player’s score, or output. Players incur costs that are an increasing function of score, but receive prizes for obtaining higher score than their competitors. In this paper we study finite games that are discretized contests, and the problems of computing exact and approximate Nash equilibria. Our motivation is the worst-case hardness of Nash equilibrium computation, and the resulting interest in important classes of games that admit polynomial-time algorithms. For games that have a tie-breaking rule for players’ scores, we present a polynomial-time algorithm for computing an exact equilibrium in the 2-player case, and for multiple players, a characterization of Nash equilibria that shows an interesting parallel between these games and unrestricted 2-player games in normal form. When ties are allowed, via a reduction from these games to a subclass of anonymous games, we give approximation schemes for two special cases: constant-sized set of strategies, and constant number of players.

Computing Nash Equilibrium in Wireless Ad Hoc Networks: A Simulation-Based Approach

This paper studies the problem of computing Nash equilibrium in wireless networks modeled by Weighted Timed Automata. Such formalism comes together with a logic that can be used to describe complex features such as timed energy constraints. Our contribution is a method for solving this problem using Statistical Model Checking. The method has been implemented in UPPAAL model checker and has been applied to the analysis of Aloha CSMA/CD and IEEE 802.15.4 CSMA/CA protocols.

Parameterized Two-Player Nash Equilibrium

We study the problem of computing Nash equilibria in a two-player normal form (bimatrix) game from the perspective of parameterized complexity. Recent results proved hardness for a number of variants, when parameterized by the support size. We complement those results, by identifying three cases in which the problem becomes fixed-parameter tractable. Our results are based on a graph-theoretic representation of a bimatrix game, and on applying graph-theoretic tools on this representation.

Nash equilibria and values through modular partitions in infinite games

Quotients through modular partitions are well adapted to the computation of Nash equilibria and values in the (zero-sum, two-player) game associated to a bounded skew-symmetric matrix. Applications to infinite oriented graphs are provided.

A survey on computation methods for Nash equilibrium

A dominant solution concept of non-cooperative game theory is the concept of Nash equilibrium. A Nash equilibrium is a strategy profile from where unilateral deviations do not pay. A nice property of this concept is the well known fact that every finite game has at least one Nash equilibrium. The proof given by Nash (1950) is based on Brouwer fixed point theorem which is very non-constructive. A natural question to ask is whether Nash equilibrium can be computed efficiently. This is still unknown in terms of complexity. Very recently (Daskalakis et al., 2006), it has been shown that the computation of Nash equilibrium is PPAD-complete; which is a new complexity class introduced by Papadimitriou (1994) mainly to capture the complexity of Nash equilibrium. This note tries to give a brief survey of the various algorithms that exist in the literature for computing the Nash equilibrium in finite games. Also, it will briefly touch upon the various existential conditions for Nash equilibrium for infinite games.

A new evolutionary approach for computing Nash equilibria in bimatrix games with known support

AbstractIn this paper, we present the application of the Differential Evolution (DE) algorithm to the problem of finding approximate Nash equilibria in matrix, non-zero sum games for two players with finite number of strategies. Nash equilibrium is one of the main concepts in game theory. It may be classified as continuous problem, where two probability distributions over the set of strategies of both players should be found. Every deviation from the global optimum is interpreted as Nash approximation and called ε-Nash equilibrium. The main advantage of the proposed algorithm is self-adaptive mutation operator, which direct the search process. The approach used in this article is based on the probability of chosing single pure strategy. In optimal mixed strategy, every strategy has some probability of being chosen. Our goal is to determine this probability and maximize payoff for a single player.