scholarly journals On Sparse Discretization for Graphical Games

2020 ◽  
Vol 69 ◽  
pp. 67-84
Author(s):  
Luis Ortiz
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Sufficient Conditions ◽  
Research Note ◽  
Graphical Games ◽  
Approximate Nash Equilibrium ◽  
Approximate Nash Equilibria ◽  
Polymatrix Games ◽  
The Impact ◽  
Game Representation

Graphical games are one of the earliest examples of the impact that the general field of graphical models have had in other areas, and in this particular case, in classical mathematical models in game theory. Graphical multi-hypermatrix games, a concept formally introduced in this research note, generalize graphical games while allowing the possibility of further space savings in model representation to that of standard graphical games. The main focus of this research note is discretization schemes for computing approximate Nash equilibria, with emphasis on graphical games, but also briefly touching on normal-form and polymatrix games. The main technical contribution is a theorem that establishes sufficient conditions for a discretization of the players’ space of mixed strategies to contain an approximate Nash equilibrium. The result is actually stronger because every exact Nash equilibrium has a nearby approximate Nash equilibrium on the grid induced by the discretization. The sufficient conditions are weaker than those of previous results. In particular, a uniform discretization of size linear in the inverse of the approximation error and in the natural game-representation parameters suffices. The theorem holds for a generalization of graphical games, introduced here. The result has already been useful in the design and analysis of tractable algorithms for graphical games with parametric payoff functions and certain game-graph structures. For standard graphical games, under natural conditions, the discretization is logarithmic in the game-representation size, a substantial improvement over the linear dependency previously required. Combining the improved discretization result with old results on constraint networks in AI simplifies the derivation and analysis of algorithms for computing approximate Nash equilibria in graphical games.

scholarly journals Semidefinite Programming and Nash Equilibria in Bimatrix Games

2020 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Jeffrey Zhang
Keyword(s):  
Nash Equilibrium ◽  
Semidefinite Programming ◽  
Nash Equilibria ◽  
Valid Inequalities ◽  
Bimatrix Games ◽  
Competitive Game ◽  
Approximate Nash Equilibria ◽  
Hard Problems ◽  
Rank 2

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.

scholarly journals Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

2021 ◽  
Author(s):  
Matteo Basei ◽  
Haoyang Cao ◽  
Xin Guo
Keyword(s):  
Nash Equilibria ◽  
Stochastic Games ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Mean Field Games ◽  
Model Parameters ◽  
Management Problem ◽  
Player Game ◽  
The Impact ◽  
Impulse Controls

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to infinity, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with [Formula: see text]. As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.

scholarly journals Constructive Proof of the Existence of Nash Equilibrium in a Finite Strategic Game with Sequentially Locally Nonconstant Payoff Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yasuhito Tanaka
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Constructive Proof ◽  
Constructive Mathematics ◽  
Strategic Game ◽  
Sperner’S Lemma ◽  
Approximate Nash Equilibria ◽  
Payoff Functions

We will constructively prove the existence of a Nash equilibrium in a finite strategic game with sequentially locally nonconstant payoff functions. The proof is based on the existence of approximate Nash equilibria which is proved by Sperner's lemma. We follow the Bishop-style constructive mathematics.

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

2019 ◽  
Vol 33 ◽  
pp. 1926-1932
Author(s):  
Michail Fasoulakis ◽  
Evangelos Markakis
Keyword(s):  
Nash Equilibrium ◽  
Polynomial Time ◽  
Nash Equilibria ◽  
Polynomial Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Bimatrix Games ◽  
Approximate Nash Equilibria ◽  
Pure Strategies ◽  
Approximation Parameter

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.

Computing Approximate Nash Equilibria in Polymatrix Games

Algorithmica ◽  
2015 ◽  
Vol 77 (2) ◽  
pp. 487-514 ◽  
Author(s):  
Argyrios Deligkas ◽  
John Fearnley ◽  
Rahul Savani ◽  
Paul Spirakis
Keyword(s):  
Nash Equilibria ◽  
Approximate Nash Equilibria ◽  
Polymatrix Games

scholarly journals On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games

2018 ◽  
Vol 20 (03) ◽  
pp. 1840001
Author(s):  
Stefanos Leonardos ◽  
Costis Melolidakis
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Sufficient Conditions ◽  
Solution Concept ◽  
Optimal Strategies ◽  
Payoff Matrix ◽  
Matrix Game ◽  
Bimatrix Games ◽  
Equilibrium Payoff ◽  
The Relationship

Given a bimatrix game, the associated leadership or commitment games are defined as the games at which one player, the leader, commits to a (possibly mixed) strategy and the other player, the follower, chooses his strategy after being informed of the irrevocable commitment of the leader (but not of its realization in case it is mixed). Based on a result by Von Stengel and Zamir [2010], the notions of commitment value and commitment optimal strategies for each player are discussed as a possible solution concept. It is shown that in nondegenerate bimatrix games (a) pure commitment optimal strategies together with the follower’s best response constitute Nash equilibria, and (b) strategies that participate in a completely mixed Nash equilibrium are strictly worse than commitment optimal strategies, provided they are not matrix game optimal. For various classes of bimatrix games that generalize zero-sum games, the relationship between the maximin value of the leader’s payoff matrix, the Nash equilibrium payoff and the commitment optimal value are discussed. For the Traveler’s Dilemma, the commitment optimal strategy and commitment value for the leader are evaluated and seem more acceptable as a solution than the unique Nash equilibrium. Finally, the relationship between commitment optimal strategies and Nash equilibria in [Formula: see text] bimatrix games is thoroughly examined and in addition, necessary and sufficient conditions for the follower to be worse off at the equilibrium of the leadership game than at any Nash equilibrium of the simultaneous move game are provided.

scholarly journals Sampling Individually Fundamental Simplexes as Sets of Players’ Mixed Strategies in Finite Noncooperative Game for Applicable Approximate Nash Equilibrium Situations with Possible Concessions

2016 ◽  
Vol 40 (1) ◽  
pp. 105-143
Author(s):  
Vadim Romanuke
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Finite Lattice ◽  
Noncooperative Game ◽  
Mixed Strategies ◽  
Approximate Equilibrium ◽  
Approximate Nash Equilibrium ◽  
Universal Approach ◽  
Finite Lattices

In finite noncooperative game, a method for finding approximate Nash equilibrium situations is developed. The method is prior-based on sampling fundamental simplexes being the sets of players’ mixed strategies. Whereas the sampling is exercised, the sets of players’ mixed strategies are mapped into finite lattices. Sampling steps are envisaged dissimilar. Thus, each player within every dimension of its simplex selects and controls one’s sampling individually. For preventing approximation low quality, however, sampling steps are restricted. According to the restricted sampling steps, a player acting singly with minimal spacing over its lattice cannot change payoff of any player more than by some predetermined magnitude, being specific for each player. The finite lattice is explicitly built by the represented routine, where the player’s mixed strategies are calculated and arranged. The product of all the players’ finite lattices approximates the product of continuous fundamental simplexes. This re-defines the finite noncooperative game in its finite mixed extension on the finite lattices’ product. In such a finite-mixed-extension-defined game, the set of Nash equilibrium situations may be empty. Therefore, approximate Nash equilibrium situations are defined by the introduced possible payoff concessions. A routine for finding approximate equilibrium situations is represented. Approximate strong Nash equilibria with possible concessions are defined, and a routine for finding them is represented as well. Acceleration of finding approximate equilibria is argued also. Finally, the developed method is discussed to be a basis in stating a universal approach for the finite noncooperative game solution approximation implying unification of the game solvability, applicability, realizability, and adaptability.

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

2012 ◽  
Vol 2 (2) ◽  
Author(s):  
Urszula Boryczka ◽  
Przemyslaw Juszczuk
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Probability Distributions ◽  
Pure Strategy ◽  
Global Optimum ◽  
Adaptive Mutation ◽  
De Algorithm ◽  
Approximate Nash Equilibria ◽  
Zero Sum ◽  
Continuous Problem

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.

A Theory of Patronized Goods: Inefficient and Efficient Equilibria

2011 ◽  
pp. 65-87 ◽  
Author(s):  
A. Rubinstein
Keyword(s):  
Nash Equilibrium ◽  
Social Policy ◽  
Nash Equilibria ◽  
Institutional Environment ◽  
Social Motivation ◽  
Pareto Optimal ◽  
Market Failures ◽  
Equilibrium Conditions ◽  
Policy Issues

The article considers some aspects of the patronized goods theory with respect to efficient and inefficient equilibria. The author analyzes specific features of patronized goods as well as their connection with market failures, and conjectures that they are related to the emergence of Pareto-inefficient Nash equilibria. The key problem is the analysis of the opportunities for transforming inefficient Nash equilibrium into Pareto-optimal Nash equilibrium for patronized goods by modifying the institutional environment. The paper analyzes social motivation for institutional modernization and equilibrium conditions in the generalized Wicksell-Lindahl model for patronized goods. The author also considers some applications of patronized goods theory to social policy issues.

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