scholarly journals An Algorithm for Computing All Berge Equilibria

Game Theory ◽  
2015 ◽  
Vol 2015 ◽  
pp. 1-2 ◽  
Author(s):  
H. W. Corley ◽  
Phantipa Kwain
Keyword(s):  
Normal Form ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Security Level ◽  
Strategy Profile ◽  
Person Game

An algorithm is presented in this note for determining all Berge equilibria for an n-person game in normal form. This algorithm is based on the notion of disappointment, with the payoff matrix (PM) being transformed into a disappointment matrix (DM). The DM has the property that a pure strategy profile of the PM is a BE if and only if (0,…,0) is the corresponding entry of the DM. Furthermore, any (0,…,0) entry of the DM is also a more restrictive Berge-Vaisman equilibrium if and only if each player’s BE payoff is at least as large as the player’s maximin security level.

Finding the Pure-Strategy Nash Equilibrium Using a Fixed-Point Algorithm

2013 ◽  
Vol 427-429 ◽  
pp. 1803-1806 ◽  
Author(s):  
Zheng Tian Wu ◽  
Chuang Yin Dang ◽  
Chang An Zhu
Keyword(s):  
Fixed Point ◽  
Nash Equilibrium ◽  
Normal Form ◽  
Pure Strategy ◽  
Payoff Function ◽  
Fixed Point Algorithm ◽  
Linear Programming Formulation ◽  
Active Research ◽  
Person Game ◽  
Strategy Nash Equilibrium

It is well known that determining whether a finite game has a pure-strategy Nash equilibrium is an NP-hard problem and it is an active research topic to find a Nash equilibrium recently. In this paper, an implementation of Dang's Fixed-Point iterative method is introduced to find a pure-strategy Nash equilibrium of a finite n-person game in normal form. There are two steps to find one pure-strategy Nash equilibrium in this paper. The first step is converting the problem to a mixed 0-1 linear programming formulation based on the properties of pure strategy and multilinear terms in the payoff function. In the next step, the Dangs method is used to solve the formulation generated in the former step. Numerical results show that this method is effective to find a pure-strategy Nash equilibrium of a finite n-person game in normal form.

scholarly journals A Mixed 0-1 Linear Programming Approach to the Computation of All Pure-Strategy Nash Equilibria of a Finiten-Person Game in Normal Form

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhengtian Wu ◽  
Chuangyin Dang ◽  
Hamid Reza Karimi ◽  
Changan Zhu ◽  
Qing Gao
Keyword(s):  
Linear Programming ◽  
Nash Equilibrium ◽  
Normal Form ◽  
Nash Equilibria ◽  
Selection Process ◽  
Pure Strategy ◽  
Programming Approach ◽  
Main Concern ◽  
Person Game ◽  
Strategy Nash Equilibrium

A main concern in applications of game theory is how to effectively select a Nash equilibrium, especially a pure-strategy Nash equilibrium for a finiten-person game in normal form. This selection process often requires the computation of all Nash equilibria. It is well known that determining whether a finite game has a pure-strategy Nash equilibrium is an NP-hard problem and it is difficult to solve by naive enumeration algorithms. By exploiting the properties of pure strategy and multilinear terms in the payoff functions, this paper formulates a new mixed 0-1 linear program for computing all pure-strategy Nash equilibria. To our knowledge, it is the first method to formulate a mixed 0-1 linear programming for pure-strategy Nash equilibria and it may work well for similar problems. Numerical results show that the approach is effective and this method can be easily distributed in a distributed way.

scholarly journals Representing pure Nash equilibria in argumentation

2021 ◽  
pp. 1-14
Author(s):  
Bruno Yun ◽  
Srdjan Vesic ◽  
Nir Oren
Keyword(s):  
Normal Form ◽  
Nash Equilibria ◽  
Pure Strategy ◽  
Normal Form Games

In this paper we describe an argumentation-based representation of normal form games, and demonstrate how argumentation can be used to compute pure strategy Nash equilibria. Our approach builds on Modgil’s Extended Argumentation Frameworks. We demonstrate its correctness, showprove several theoretical properties it satisfies, and outline how it can be used to explain why certain strategies are Nash equilibria to a non-expert human user.

scholarly journals Distance-Based Equilibria in Normal-Form Games

2020 ◽  
Vol 34 (02) ◽  
pp. 1750-1757
Author(s):  
Erman Acar ◽  
Reshef Meir
Keyword(s):  
Normal Form ◽  
Behavioral Model ◽  
Existence Results ◽  
Solution Concepts ◽  
Worst Case ◽  
Agent Model ◽  
Normal Form Games ◽  
Strategy Profile ◽  
Action Profile

We propose a simple uncertainty modification for the agent model in normal-form games; at any given strategy profile, the agent can access only a set of “possible profiles” that are within a certain distance from the actual action profile. We investigate the various instantiations in which the agent chooses her strategy using well-known rationales e.g., considering the worst case, or trying to minimize the regret, to cope with such uncertainty. Any such modification in the behavioral model naturally induces a corresponding notion of equilibrium; a distance-based equilibrium. We characterize the relationships between the various equilibria, and also their connections to well-known existing solution concepts such as Trembling-hand perfection. Furthermore, we deliver existence results, and show that for some class of games, such solution concepts can actually lead to better outcomes.

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (01) ◽  
pp. 5-7
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I 1, I 2, ···, I n where I j is the strategy ‘play value m j ′ and m 1 < m 2 < ··· < m n . The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[I i , I j ] is the expected payoff for playing I i against I j . We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

COOPERATION IN THE WELL-MIXED TWO-POPULATION SNOWDRIFT GAME WITH PUNISHMENT ENFORCED THROUGH DIFFERENT MECHANISMS

2017 ◽  
Vol 20 (04n05) ◽  
pp. 1750010 ◽  
Author(s):  
ANDRÉ BARREIRA DA SILVA ROCHA
Keyword(s):  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Pure Strategy ◽  
Stable State ◽  
Payoff Matrix ◽  
Globally Asymptotically Stable ◽  
Agent Based ◽  
Evolutionary Pattern ◽  
Snowdrift Game ◽  
Level Of Agreement

I study two mechanisms based on punishment to promote cooperation in the well-mixed two-population snowdrift game (SG). The first mechanism follows a standard approach in the literature and is based on the inclusion of a third additional pure strategy in the payoff matrix of the stage-game. Differently, the second mechanism consists of letting cooperators punish defectors with a given exogenous frequency. In the latter, the pure strategy cooperation is replaced by a mixed strategy in which cooperators randomize between cooperation and punishment against defectors. While both mechanisms share the same result regarding the minimum required level of punishment in order to eliminate defectors in both populations, stability in the mechanism following the second approach is more robust in the sense that extinction of defectors is a globally asymptotically stable state for any interior initial conditions in the phase space. Thus, the second mechanism displays a topologically simpler model but the robustness of the evolutionary equilibrium is improved. Results were obtained analytically through nonlinear differential equations and also using an agent-based simulation. There was a good level of agreement between both approaches with respect to the evolutionary pattern over time and the possible steady-states.

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (1) ◽  
pp. 5-7 ◽  
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I1, I2, ···, In where Ij is the strategy ‘play value mj′ and m1 < m2 < ··· < mn. The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[Ii, Ij] is the expected payoff for playing Ii against Ij. We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

scholarly journals Two-Player Preplay Negotiation Games with Conditional Offers

2016 ◽  
Vol 18 (01) ◽  
pp. 1550017
Author(s):  
Valentin Goranko ◽  
Paolo Turrini
Keyword(s):  
Normal Form ◽  
Bargaining Power ◽  
Payoff Matrix ◽  
Negotiation Process ◽  
Value Of Time ◽  
Bargaining Games ◽  
Bargaining Process ◽  
Normal Form Games ◽  
Normal Form Game ◽  
Made In

We consider an extension of strategic normal form games with a phase before the actual play of the game, where players can make binding offers for transfer of utilities to other players after the play of the game, contingent on the recipient playing the strategy indicated in the offer. Such offers transform the payoff matrix of the original game but preserve its noncooperative nature. The type of offers we focus on here are conditional on a suggested matching offer of the same kind made in return by the receiver. Players can exchange a series of such offers, thus engaging in a bargaining process before a strategic normal form game is played. In this paper, we study and analyze solution concepts for two-player normal form games with such preplay negotiation phase, under several assumptions for the bargaining power of the players, as well as the value of time for the players in such negotiations. We obtain results describing the possible solutions of such bargaining games and analyze the degrees of efficiency and fairness that can be achieved in such negotiation process. We show the similarities and the differences with a variety of frameworks in the literature of bargaining games and games with a preplay phase.

A Necessary and Sufficient Condition for Partial Monotonicity of Noncooperative Games

2020 ◽  
pp. 2050006
Author(s):  
Naoki Matsumoto
Keyword(s):  
Nash Equilibrium ◽  
Pure Strategy ◽  
Noncooperative Games ◽  
Noncooperative Game ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Monotonicity ◽  
Person Game ◽  
Necessary And Sufficient ◽  
Strategy Nash Equilibrium

It is a classical and interesting problem to find a Nash equilibrium of noncooperative games in the strategic form. It is well known that the game always has a mixed-strategy Nash equilibrium, but it does not necessarily have a pure-strategy Nash equilibrium. Takeshita and Kawasaki proved that every noncooperative partially monotone game has a pure-strategy Nash equilibrium, that is, the partial monotonicity is a sufficient condition for a noncooperative game to have a pure-strategy Nash equilibrium. In this paper, we prove the necessary and sufficient condition for a noncooperative [Formula: see text]-person game with [Formula: see text] to be partially monotone. This result is an improvement of Takeshita and Kawasaki’s result.

AN APPLICATION OF A DISCRETE FIXED POINT THEOREM TO A GAME IN EXPANSIVE FORM

2013 ◽  
Vol 30 (03) ◽  
pp. 1340013 ◽  
Author(s):  
HIDEFUMI KAWASAKI ◽  
AKIFUMI KIRA ◽  
SHINPEI KIRA
Keyword(s):  
Fixed Point ◽  
Nash Equilibrium ◽  
Fixed Point Theorem ◽  
Existence Theorem ◽  
Point Theorem ◽  
Pure Strategy ◽  
Contraction Mappings ◽  
Person Game ◽  
Strategy Nash Equilibrium

In this paper, we first present a discrete fixed point theorem for contraction mappings from the product set of integer intervals into itself, which is an extension of Robert's discrete fixed point theorem. Next, we derive an existence theorem of a pure-strategy Nash equilibrium for a noncooperative n-person game from our fixed point theorem. Finally, we show that Kuhn's theorem for a game in expansive form can be explained by our existence theorem.

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