scholarly journals Example of a Finite Game with No Berge Equilibria at All

Games ◽  
2019 ◽  
Vol 10 (1) ◽  
pp. 7 ◽  
Author(s):  
Jarosław Pykacz ◽  
Paweł Bytner ◽  
Piotr Frąckiewicz
Keyword(s):  
Normal Form ◽  
Mixed Strategies ◽  
Finite Games ◽  
Berge Equilibrium ◽  
Finite Game ◽  
Player Game

The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal-form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure, nor in mixed strategies is given.

scholarly journals Weak Berge Equilibrium in Finite Three-person Games: Conception and Computation

2020 ◽  
Vol 11 (1) ◽  
pp. 127-134
Author(s):  
Konstantin Kudryavtsev ◽  
Ustav Malkov
Keyword(s):  
Cooperative Games ◽  
Hippocratic Oath ◽  
The Other ◽  
Mixed Strategies ◽  
Finite Games ◽  
Moral Basis ◽  
Other Hand ◽  
Berge Equilibrium ◽  
Do No Harm ◽  
Non Cooperative Games

AbstractThe paper proposes the concept of a weak Berge equilibrium. Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm”. On the other hand, any Berge equilibrium is a weak Berge equilibrium. But, there are weak Berge equilibria, which are not the Berge equilibria. The properties of the weak Berge equilibrium have been investigated. The existence of the weak Berge equilibrium in mixed strategies has been established for finite games. The weak Berge equilibria for finite three-person non-cooperative games are computed.

The Computational Complexity of Finding a Mixed Berge Equilibrium for a k-Person Noncooperative Game in Normal Form

2018 ◽  
Vol 20 (04) ◽  
pp. 1850010
Author(s):  
Ahmad Nahhas ◽  
H. W. Corley
Keyword(s):  
Computational Complexity ◽  
Normal Form ◽  
Noncooperative Game ◽  
Equilibrium Strategy ◽  
Mixed Strategies ◽  
Case Scenario ◽  
Worst Case ◽  
Form Game ◽  
Normal Form Game ◽  
Berge Equilibrium

The mixed Berge equilibrium (MBE) is an extension of the Berge equilibrium (BE) to mixed strategies. The MBE models mutually support in a [Formula: see text]-person noncooperative game in normal form. An MBE is a mixed-strategy profile for the [Formula: see text] players in which every [Formula: see text] players have mixed strategies that maximize the expected payoff for the remaining player’s equilibrium strategy. In this paper, we study the computational complexity of determining whether an MBE exists in a [Formula: see text]-person normal-form game. For a two-person game, an MBE always exists and the problem of finding an MBE is PPAD-complete. In contrast to the mixed Nash equilibrium, the MBE is not guaranteed to exist in games with three or more players. Here we prove, when [Formula: see text], that the decision problem of asking whether an MBE exists for a [Formula: see text]-person normal-form game is NP-complete. Therefore, in the worst-case scenario there does not exist a polynomial algorithm that finds an MBE unless P=NP.

BENEFIT FUNCTION AND DUALITY IN FINITE NORMAL FORM GAMES

2007 ◽  
Vol 09 (03) ◽  
pp. 495-513
Author(s):  
WALTER BRIEC
Keyword(s):  
Game Theory ◽  
Normal Form ◽  
Mixed Strategies ◽  
Numerical Function ◽  
Normal Form Games ◽  
Benefit Function ◽  
Pareto Efficient

Luenberger (1992, 1994) introduced a function he terms the benefit function, that converts preferences into a numerical function and has some cardinal meaning. In this paper, we show that the benefit function enjoys many interesting properties in a game theory context. We point out that the benefit function can be adapted to compare the mixed profiles of a game. Along this line, inspired from the Luenberger's approach, we propose a dual framework and establish a characterization of Nash equilibriums in terms of the benefit function. Moreover, some criterions are provided to identify the efficient mixed strategies of a game (which differ from the Pareto efficient strategies). Finally, we go a bit further proposing some issue in comparing profiles and equilibriums of a game. This we do using the so-called Σ-subdifferential of the benefit function.

scholarly journals A Mixed Cooperative Dual to the Nash Equilibrium

Game Theory ◽  
2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
H. W. Corley
Keyword(s):  
Nash Equilibrium ◽  
Mixed Strategy ◽  
The Other ◽  
Mixed Strategies ◽  
Mutual Support ◽  
Equilibrium Models ◽  
Expected Payoff ◽  
Berge Equilibrium

A mixed dual to the Nash equilibrium is defined for n-person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other n-1 players’ strategies. Conversely, in the dual equilibrium every n-1 players have mixed strategies that maximize the remaining player’s expected payoff. Hence this dual equilibrium models mutual support and cooperation to extend the Berge equilibrium from pure to mixed strategies. This dual equilibrium is compared and related to the mixed Nash equilibrium, and both topological and algebraic conditions are given for the existence of the dual. Computational issues are discussed, and it is shown that for each n>2 there exists a game for which no dual equilibrium exists.

scholarly journals A generalization of Nash's theorem with higher-order functionals

2013 ◽  
Vol 469 (2154) ◽  
pp. 20130041 ◽  
Author(s):  
Julian Hedges
Keyword(s):  
Normal Form ◽  
Proof Theory ◽  
Mixed Strategy ◽  
Recent Theory ◽  
Sequential Games ◽  
New Class ◽  
Finite Games ◽  
Minimax Strategies ◽  
Game Theoretic ◽  
Axiom Of Countable Choice

The recent theory of sequential games and selection functions by Escardó & Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is generalized to games defined by selection functions. A normal form construction is given, which generalizes the game-theoretic normal form, and its soundness is proved. Minimax strategies also generalize to the new class of games, and are computed by the Berardi–Bezem–Coquand functional, studied in proof theory as an interpretation of the axiom of countable choice.

scholarly journals Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

2017 ◽  
Author(s):  
Theodore Andronikos ◽  
Alla Sirokofskich ◽  
Kalliopi Kastampolidou ◽  
Magdalini Varvouzou ◽  
Konstantinos Giannakis ◽  
...  
Keyword(s):  
Game Theory ◽  
Quantum Computation ◽  
Finite Automata ◽  
State Machines ◽  
Quantum Game ◽  
Finite Games ◽  
New Concepts ◽  
Finite Game ◽  
Finite State ◽  
Deterministic Automata

The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The PQ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper we investigate all possible finite games that can be played between the two players Q and Picard of the original PQ game. For this purpose we establish a rigorous connection between finite automata and the PQ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the PQ game. What this means is that from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.

scholarly journals Strong and Safe Nash Equilibrium in Some Repeated 3-Player Games

2019 ◽  
Vol 65 (3) ◽  
pp. 271-295 ◽  
Author(s):  
Tadeusz Kufel ◽  
Sławomir Plaskacz ◽  
Joanna Zwierzchowska
Keyword(s):  
Nash Equilibrium ◽  
Normal Form ◽  
Prisoner's Dilemma ◽  
Prisoner’S Dilemma ◽  
Repeated Game ◽  
Equilibrium Payoff ◽  
Equilibrium Strategies ◽  
Strong Nash Equilibrium ◽  
Player Game ◽  
Stage Game

The paper examines an infinitely repeated 3-player extension of the Prisoner’s Dilemma game. We consider a 3-player game in the normal form with incomplete information, in which each player has two actions. We assume that the game is symmetric and repeated infinitely many times. At each stage, players make their choices knowing only the average payoffs from previous stages of all the players. A strategy of a player in the repeated game is a function defined on the convex hull of the set of payoffs. Our aim is to construct a strong Nash equilibrium in the repeated game, i.e. a strategy profile being resistant to deviations by coalitions. Constructed equilibrium strategies are safe, i.e. the non-deviating player payoff is not smaller than the equilibrium payoff in the stage game, and deviating players’ payoffs do not exceed the nondeviating player payoff more than by a positive constant which can be arbitrary small and chosen by the non-deviating player. Our construction is inspired by Smale’s good strategies described in Smale’s paper (1980), where the repeated Prisoner’s Dilemma was considered. In proofs we use arguments based on approachability and strong approachability type results.

scholarly journals Course notes on finite games and rational choice

2019 ◽  
Author(s):  
Riccardo Bruni
Keyword(s):  
Normal Form ◽  
Rational Choice ◽  
History Of Science ◽  
Extensive Form ◽  
Finite Games ◽  
History Of ◽  
Basic Concepts ◽  
University Of Florence ◽  
University Course ◽  
The University

This book collects notes that were prepared for a university course taught in the Spring of 2018, and delivered to an audience of students enrolled in the Master course in Logic, philosophy and history of science of the University of Florence. The goal of the course was to introduce students to some basic concepts from the area of research generally known as decision theory. This is done by focussing on the concept of ‘rational choice’, which is analyzed, methodologically speaking, by the means of the theory of games. To minimize prerequisites it was decided to restrict the attention to the theory of finite games in particular. The topics treated are vary, and belongs to both the theory of games ‘in normal form’ as well as that of games ‘in extensive form’, as they are usually referred to. The classical issues in both fields, such as the theory of ‘equilibria’ and the study of properties such as determinacy, are carefully discussed to make them clear to beginners and are addressed from a novel perspective which makes use of formal methods that are typical of researches connected with the study of logic.

scholarly journals A new tensor approach of computing pure and mixed Unilateral Support Equilibria

2021 ◽  
Author(s):  
Elias Safatly ◽  
Joanna Abdou
Keyword(s):  
High Dimensional ◽  
Tensor Form ◽  
Numerical Examples ◽  
Numerical Example ◽  
Berge Equilibrium ◽  
Mixed Use ◽  
Player Game ◽  
Mathematical Foundations

In this paper, we locate pure unilateral support equilibrium (USE) among pure Nash and pure Berge equilibrium using tensors. The differences between these equilibria are shown using tensor form of a game and are illustrated with numerical examples. Tensors will help to specify the location of each equilibrium using a system of coordinates based on tensors which will bring a solid mathematical foundations of all equilibria and will provide the possibility to solve high dimensional problems as we will see in a numerical example with a $15-$player game. Additionally, we extend the notion of pure USE to mixed USE when the sets of strategies of all players are discrete and finite. We prove a lemma dedicated to inaugurate a method of computing mixed USE profiles. We write corresponding formulas using tensors and their operations, then we illustrate this method by a numerical example of a $7-$player game.

A Nonlinear Programming Approach to Determine a Generalized Equilibrium for N-Person Normal Form Games

2017 ◽  
Vol 19 (03) ◽  
pp. 1750011 ◽  
Author(s):  
Ahmad Nahhas ◽  
H. W. Corley
Keyword(s):  
Nonlinear Programming ◽  
Normal Form ◽  
Objective Function Value ◽  
Programming Approach ◽  
Nonlinear Program ◽  
Change Strategies ◽  
Mixed Strategies ◽  
Expected Payoff ◽  
Normal Form Games ◽  
Generalized Equilibrium

A generalized equilibrium (GE) for finite [Formula: see text]-person normal form games is defined as a collection of mixed strategies with the following property: no player in some subset [Formula: see text] of the players can achieve a better expected payoff if players in an associated set [Formula: see text] change strategies unilaterally. A GE is proved to exist for a game if and only if the maximum objective function value of a certain nonlinear programming problem is zero, in which case the solution to the nonlinear program yields a GE.

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