scholarly journals On the exact polynomial time algorithm for a special class of bimatrix game

2013 ◽  
Vol 54 ◽  
Author(s):  
Jonas Mockus ◽  
Martynas Sabaliauskas
Keyword(s):  
Game Theory ◽  
Linear Programming ◽  
Nash Equilibrium ◽  
Polynomial Time ◽  
Explicit Solution ◽  
Special Class ◽  
Polynomial Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Bimatrix Game

The Strategy Elimination (SE) algorithm was proposed in [2] and implemented by a sequence of Linear Programming (LP) problems. In this paper an efficient explicit solution is developed and the convergence to the Nash Equilibrium is proven.Keywords: game theory, polynomial algorithm, Nash equilibrium.

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.

scholarly journals Discrepancy Games

10.37236/1948 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Noga Alon ◽  
Michael Krivelevich ◽  
Joel Spencer ◽  
Tibor Szabó
Keyword(s):  
Game Theory ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Vertex Set

We investigate a game played on a hypergraph $H=(V,E)$ by two players, Balancer and Unbalancer. They select one element of the vertex set $V$ alternately until all vertices are selected. Balancer wins if at the end of the game all edges $e\in E$ are roughly equally distributed between the two players. We give a polynomial time algorithm for Balancer to win provided the allowed deviation is large enough. In particular, it follows from our result that if $H$ is $n$-uniform and has $m$ edges, then Balancer can achieve having between $n/2-\sqrt{\ln(2m)n/2}$ and $n/2+\sqrt{\ln(2m)n/2}$ of his vertices on every edge $e$ of $H$. We also discuss applications in positional game theory.

scholarly journals On the Nash equilibrium in the inspector problem

2014 ◽  
Vol 55 ◽  
Author(s):  
Martynas Sabaliauskas ◽  
Jonas Mockus
Keyword(s):  
Nash Equilibrium ◽  
Polynomial Time ◽  
Explicit Solution ◽  
Security Systems ◽  
Bimatrix Game ◽  
Complete Problem ◽  
Elimination Algorithm ◽  
Np Complete ◽  
Special Case

Inspector problem represents an economic duel of inspector and law violator and is formulated as a bimatrix game. In general, bimatrix game is NP-complete problem. The inspector problem is a special case where the equilibrium can be found in polynomial time. In this paper, a generalized version of the Inspector Problem is described with the aim to represent broader family of applied problems, including the optimization of security systems. The explicit solution is provided and the Modified Strategy Elimination algorithm is introduced.

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