Nash Equilibria In Bimatrix Games Research Articles

Code and Data Repository for Identifying Socially Optimal Equilibria Using Combinatorial Properties of Nash Equilibria in Bimatrix Games

Code and Data Repository for Identifying Socially Optimal Equilibria Using Combinatorial Properties of Nash Equilibria in Bimatrix Games

A Polynomial-Time Algorithm for 1/2-Well-Supported Nash Equilibria in Bimatrix Games

A Polynomial-Time Algorithm for 1/2-Well-Supported Nash Equilibria in Bimatrix Games

A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games

Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [ 38 ], with an approximation guarantee of (0.3393+δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a \((\frac{1}{3}+\delta)\) -Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [ 38 ], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.

On tightness of Tsaknakis-Spirakis descent methods for approximate Nash equilibria

This article explores the minimum approximation ratio for Nash equilibrium in bi-matrix games, focusing on the Tsaknakis and Spirakis (TS) methods. The previous SOTA, TS algorithm, achieved an approximation ratio of 0.3393, but efforts to improve the analysis of the TS algorithm have been unsuccessful. This work demonstrates that the bound of 0.3393 is tight for the TS algorithm and presents a theoretical worst-case analysis. A condition for identifying tight instances is provided, along with a generator. While most generated instances are unstable, indicating potential improvements, stable instances exist where perturbations cannot enhance the 0.3393 bound. Other approximate algorithms, such as regret-matching and fictitious play, achieve better ratios on these instances. The generated instances can serve as benchmarks for approximate Nash equilibrium algorithms. The article also mentions progress in the TS algorithm, achieving an approximation ratio of 1/3, which can be further studied using the presented techniques.

Search for Nash Equilibria in Bimatrix Games with Probability and Quantile Payoff Functions

Search for Nash Equilibria in Bimatrix Games with Probability and Quantile Payoff Functions

Automatic verification of concurrent stochastic systems

Automated verification techniques for stochastic games allow formal reasoning about systems that feature competitive or collaborative behaviour among rational agents in uncertain or probabilistic settings. Existing tools and techniques focus on turn-based games, where each state of the game is controlled by a single player, and on zero-sum properties, where two players or coalitions have directly opposing objectives. In this paper, we present automated verification techniques for concurrent stochastic games (CSGs), which provide a more natural model of concurrent decision making and interaction. We also consider (social welfare) Nash equilibria, to formally identify scenarios where two players or coalitions with distinct goals can collaborate to optimise their joint performance. We propose an extension of the temporal logic rPATL for specifying quantitative properties in this setting and present corresponding algorithms for verification and strategy synthesis for a variant of stopping games. For finite-horizon properties the computation is exact, while for infinite-horizon it is approximate using value iteration. For zero-sum properties it requires solving matrix games via linear programming, and for equilibria-based properties we find social welfare or social cost Nash equilibria of bimatrix games via the method of labelled polytopes through an SMT encoding. We implement this approach in PRISM-games, which required extending the tool’s modelling language for CSGs, and apply it to case studies from domains including robotics, computer security and computer networks, explicitly demonstrating the benefits of both CSGs and equilibria-based properties.

An experimental study of a DC optimization algorithm for bimatrix games

This paper presents a nonconvex approach for computing a Nash equilibrium in bimatrix games. It relies on the Linear Complementarity Problem (LCP) formulation of the game and follows the Difference of Convex Algorithm (DCA) general scheme. To measure the performance of the proposed approach, numerical experiments as well as a comparative study with the Lemke Howson (LH) algorithm are carried out.

Semidefinite Programming and Nash Equilibria in Bimatrix Games

We explore the power of semidefinite programming (SDP) for finding additive ɛ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact Nash equilibrium can be recovered. We show that, for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on the iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most 2 and ɛ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a [Formula: see text]-Nash equilibrium can be recovered for any game, or a [Formula: see text]-Nash equilibrium for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any Nash equilibrium, and testing whether there exists a Nash equilibrium where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.

An algorithm for finding approximate Nash equilibria in bimatrix games

This paper describes an algorithm for calculating approximately mixed Nash equilibria (NE) in bimatrix games. The algorithm fuzzifies the strategies with normalized possibility distributions. The fuzzification takes advantage of the piecewise linearity of possibility distributions and transforms the NE problem of bimatrix games into a reduced form. The algorithm is guaranteed to find approximate NE in bimatrix games and ensures that the approximate NE is the saddle point of expected payoff functions in the reduced form. The algorithm provides a method of determining how close an approximate NE is to a solution during computation. Numerical results show that the new algorithm is approximately seven-time faster than the Lemke–Howson (LH) algorithm when the game size is 96, and the value of approximation deviation can be as small as 0.1.

An Improved Quasi-Polynomial Algorithm for Approximate Well-Supported Nash Equilibria

We focus on the problem of computing approximate Nash equilibria in bimatrix games. In particular, we consider the notion of approximate well-supported equilibria, which is one of the standard approaches for approximating equilibria. It is already known that one can compute an ε-well-supported Nash equilibrium in time nO (log n/ε2), for any ε > 0, in games with n pure strategies per player. Such a running time is referred to as quasi-polynomial. Regarding faster algorithms, it has remained an open problem for many years if we can have better running times for small values of the approximation parameter, and it is only known that we can compute in polynomial-time a 0.6528-well-supported Nash equilibrium. In this paper, we investigate further this question and propose a much better quasi-polynomial time algorithm that computes a (1/2 + ε)-well-supported Nash equilibrium in time nO(log logn1/ε/ε2), for any ε > 0. Our algorithm is based on appropriately combining sampling arguments, support enumeration, and solutions to systems of linear inequalities.

Justifying optimal play via consistency

Developing normative foundations for optimal play in two‐player zero‐sum games has turned out to be surprisingly difficult, despite the powerful strategic implications of the minimax theorem. We characterize maximin strategies by postulating coherent behavior in varying games. The first axiom, called consequentialism, states that how probability is distributed among completely indistinguishable actions is irrelevant. The second axiom, consistency, demands that strategies that are optimal in two different games should still be optimal when there is uncertainty regarding which of the two games will actually be played. Finally, we impose a very mild rationality assumption, which merely requires that strictly dominated actions will not be played. Our characterization shows that a rational and consistent consequentialist who ascribes the same properties to his opponent has to play maximin strategies. This result can be extended to characterize Nash equilibrium in bimatrix games whenever the set of equilibria is interchangeable.

Pure Strategy Nash Equilibria Refinement in Bimatrix Games by Using Domination Efficiency along with Maximin and the Superoptimality Rule

Background. Multiple Nash equilibria bring a new problem of selecting amongst them but this problem is solved by refining the equilibria. However, none of the existing refinements can guarantee a single refined Nash equilibrium. In some games, Nash equilibria are nonrefinable.Objective. For solving the nonrefinability problem of pure strategy Nash equilibria in bimatrix games, the goal is to develop an algorithm which could facilitate in refining the equilibria as much further as possible.Methods. A Nash equilibria refinement is suggested, which is based on the classical refinement by selecting only efficient equilibria that dominate by their payoffs. The suggested refinement exploits maximin subsequently. The superoptimality rule is involved if maximin fails to produce just a single refined equilibrium.Results. An algorithm of using domination efficiency along with maximin and the superoptimality rule has been developed for refining Nash equilibria in bimatrix games. The algorithm has 10 definite steps at which the refinement is progressively accomplished. The developed concept of the equilibria refinement does not concern games with payoff symmetry and mirror-like symmetry.Conclusions. The suggested pure strategy Nash equilibria refinement is a contribution to the equilibria refinement game theory field. The developed algorithm allows selecting amongst nonrefinable Nash equilibria in bimatrix games. It partially removes the uncertainty of equilibria, without going into mixed strategies. There are only two negative cases when the refinement fails. For a case when more than a single refined equilibrium is produced, the superoptimality rule may be used for a player having multiple refined equilibrium strategies but the other player has just a single refined equilibrium strategy.

Distributed Methods for Computing Approximate Equilibria

We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players’ payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players. Our method gives improved bounds on the complexity of computing approximate Nash equilibria in a number of different settings. Firstly, it gives a polynomial-time algorithm for computing approximate well supported Nash equilibria (WSNE) that always finds a 0.6528-WSNE, beating the previous best guarantee of 0.6608. Secondly, since our algorithm solves the two LPs separately, it can be applied to give an improved bound in the limited communication setting, giving a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE, which beats the previous best known guarantee of 0.732. It can also be applied to the case of approximate Nash equilibria, where we obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and always finds a 0.382-approximate Nash equilibrium, which improves the previous best guarantee of 0.438. Finally, the method can also be applied in the query complexity setting to give an algorithm that makes O(n log n) payoff queries and always finds a 0.6528-WSNE, which improves the previous best known guarantee of 2/3.

A linear complementarity based characterization of the weighted independence number and the independent domination number in graphs

The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, characterization of Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation for the weighted independence number of a graph by characterizing it as the maximum weighted ℓ1 norm over the solution set of a linear complementarity problem (LCP). The minimum ℓ1 norm of solutions of this LCP forms a lower bound on the independent domination number of the graph. Unlike the case of the maximum ℓ1 norm, this lower bound is in general weak, but we show it to be tight if the graph is a forest. Using methods from the theory of LCPs, we then obtain a stronger variant of the Lovász theta and a new sufficient condition for a graph to be well-covered, i.e., for all maximal independent sets to also be maximum. This latter condition is also shown to be necessary for well-coveredness of forests. Finally, the reduction of the maximum independent set problem to a linear program with (linear) complementarity constraints (LPCC) shows that LPCCs are hard to approximate.

Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries

We study the deterministic and randomized query complexity of finding approximate equilibria in a k × k bimatrix game. We show that the deterministic query complexity of finding an ϵ-Nash equilibrium when ϵ < ½ is Ω( k 2 ), even in zero-one constant-sum games. In combination with previous results [Fearnley et al. 2013], this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a (3-√5/2 + ϵ)-Nash equilibrium using O ( k .log k /ϵ 2 ) payoff queries, which shows that the ½ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an ϵ-WSNE of a zero-sum bimatrix game using O ( k .log k /ϵ 4 ) payoff queries, and we then use this to obtain a randomized algorithm for finding a (⅔ + ϵ)-WSNE in a general bimatrix game using O ( k .log k /ϵ 4 ) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require Ω( k 2 ) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4 k , even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria.

Unit vector games

AbstractMcLennan and Tourky showed that “imitation games” provide a new view of the computation of Nash equilibria of bimatrix games with the Lemke–Howson algorithm. In an imitation game, the payoff matrix of one of the players is the identity matrix. We study the more general “unit vector games,” which are already known, where the payoff matrix of one player is composed of unit vectors. Our main application is a simplification of the construction by Savani and von Stengel of bimatrix games where two basic equilibrium‐finding algorithms take exponentially many steps: the Lemke–Howson algorithm, and support enumeration.

Sufficient conditions for the existence of Nash equilibria in bimatrix games in terms of forbidden $$2 \times 2$$ 2 × 2 subgames

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its \(2 \times 2\) submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every \(2 \times 2\) subgame has one. Nevertheless, Shapley’s claim can be extended to bimatrix games as follows. We partition all \(2 \times 2\) bimatrix games into fifteen classes \(C = \{c_1, \ldots , c_{15}\}\) depending only on the preferences of two players. A subset \(t \subseteq C\) is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a method to construct all minimal (that is, strongest) NE-theorems based on the procedure of joint generation of transversal hypergraphs given by a special oracle. By this method we obtain all (six) strongest NE-theorems. Let us remark that the suggested approach, which may be called “math-pattern recognition”, is very general: it allows to characterize completely an arbitrary “target” in terms of arbitrary “attributes”.

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.

On the communication complexity of approximate Nash equilibria

We study the problem of computing approximate Nash equilibria of bimatrix games, in a setting where players initially know their own payoffs but not the other player's. In order to find a solution of reasonable quality, some amount of communication is required. We study algorithms where the communication is substantially less than the size of the game. When the communication is polylogarithmic in the number of strategies, we show how to obtain ϵ-approximate Nash equilibrium for ϵ≈0.438, and for well-supported approximate equilibria we obtain ϵ≈0.732. For one-way communication we show that ϵ=12 is the best approximation quality achievable, while for well-supported equilibria, no value of ϵ<1 is achievable. When the players do not communicate at all, ϵ-Nash equilibria can be obtained for ϵ=34; we also provide a corresponding lower bound of slightly more than 12 on the smallest constant ϵ achievable.

SYARAT PERLU DAN SYARAT CUKUP KEBERADAAN DAN KETUNGGALAN KESEIMBANGAN NASH CAMPURAN SEMPURNA PADA BIMATRIX GAMES

In this paper, the necessary and sucient condition for the existence anduniqueness of a completely mixed Nash equilibrium in bimatrix games [A;AT ] are pre-sented. Matrix A 2 Rnn is a payo matrix. The necessary and sucient condition areexpressed with some properties of saddle point matrices, denoted by (A; e), where e isthe column vector with all entries 1. Properties of the saddle point matrices assessedusing the algebraic approach.