A generalization of the Nash equilibrium theorem on bimatrix games

1996 ◽  
Vol 25 (1) ◽  
pp. 1-12 ◽  
Author(s):  
M. Seetharama Gowda ◽  
Roman Sznajder
Keyword(s):  
Nash Equilibrium ◽  
Bimatrix Games ◽  
Equilibrium Theorem

scholarly journals Fixed point theorems in minimal generalized convex spaces

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 165-176 ◽  
Author(s):  
Rahmat Darzi ◽  
Rostamian Delavar ◽  
Mehdi Roohi
Keyword(s):  
Fixed Point ◽  
Nash Equilibrium ◽  
Fixed Point Theorem ◽  
Fixed Point Theorems ◽  
Coincidence Point ◽  
Kkm Principle ◽  
Generalized Convex Space ◽  
Ky Fan ◽  
Equilibrium Theorem ◽  
Convex Spaces

This paper deals with coincidence and fixed point theorems in minimal generalized convex spaces. By establishing a kind of KKM Principle in minimal generalized convex space, we obtain some results on coincidence point and fixed point theorems. Generalized versions of Ky Fan?s lemma, Fan-Browder fixed point theorem, Nash equilibrium theorem and some Urai?s type fixed point theorems in minimal generalized convex spaces are given.

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.

Analysis on Cournot Oligopoly Game Models With Two Firms and More Than Two Firms

2008 ◽  
Author(s):  
Junyan Wang
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Cooperative Game ◽  
Pure Strategy ◽  
Cournot Oligopoly ◽  
Cournot Equilibrium ◽  
Cournot Game ◽  
Existence Conditions ◽  
Equilibrium Theorem

This paper reviews the standard game models and its Nash equilibrium and then analyses Cournot oligopoly game from two firms to the case with more than two firms. Due to Cournot equilibrium point, the concept of Cournot equilibrium point is the same as the concept as the non-cooperative game with pure strategy but the strategy can be chosen in Cournot game is infinity and it can not be obtained base on Nash equilibrium theorem. Finally, the existence conditions of Cournot equilibrium point are given and the theorem and its proof of the existence Cournot equilibrium point are given too.

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

2013 ◽  
Vol 2 (2) ◽  
pp. 54
Author(s):  
Anggi Mutia Sani
Keyword(s):  
Nash Equilibrium ◽  
Saddle Point ◽  
Algebraic Approach ◽  
Column Vector ◽  
Bimatrix Games ◽  
Saddle Point Matrices

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.

scholarly journals A Fast Approach to Bimatrix Games with Intuitionistic Fuzzy Payoffs

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Min Fan ◽  
Ping Zou ◽  
Shao-Rong Li ◽  
Chin-Chia Wu
Keyword(s):  
Nash Equilibrium ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Bimatrix Games ◽  
Bimatrix Game ◽  
Intuitionistic Fuzzy ◽  
Fast Approach ◽  
Linear Programming Methods ◽  
Programming Algorithms ◽  
Intuitionistic Fuzzy Value

The aim of this paper is to develop an effective method for solving bimatrix games with payoffs of intuitionistic fuzzy value. Firstly, bimatrix game model with intuitionistic fuzzy payoffs (IFPBiG) was put forward. Secondly, two kinds of nonlinear programming algorithms were discussed with the Nash equilibrium of IFPBiG. Thirdly, Nash equilibrium of the algorithm was proved by the fixed point theory and the algorithm was simplified by linear programming methods. Finally, an example was solved through Matlab; it showed the validity, applicability, and superiority.

scholarly journals Justifying optimal play via consistency

2019 ◽  
Vol 14 (4) ◽  
pp. 1185-1201
Author(s):  
Florian Brandl ◽  
Felix Brandt
Keyword(s):  
Nash Equilibrium ◽  
Minimax Theorem ◽  
Bimatrix Games ◽  
Zero Sum Games ◽  
Zero Sum ◽  
Normative Foundations ◽  
Coherent Behavior

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.

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