Game Theory Problem Research Articles

A Game Theory Approach Using Excel

Understanding the behavior of companies in market conditions has been an important topic of studies for the economists and mathematicians. Many companies have treated their relationship of service providers as a game. Thus, the attitudes of the players can be endowed with rational thinking, which leads to the conclusion that the use of game theory as a tool to understand such behavior is of great importance. Based on the previous studies, this paper presents a study on the cooperative game theory, discussing the Nash equilibrium in pure and mixed strategies, treating solutions using the minimax John Von Neumann theorem and illustrating a mathematical modeling of a game theory problem. Furthermore, a solution of linear programming using Microsoft Excel is also proposed and presented. The methodology adopted to model the problem may help students to familiarize themselves to game theory.

An overview of ORP $$^3$$ 3 -OR for young researchers and practitioners

This volume of the Annals of Operations Research is devoted to the ORP3 or “the OR Peripatetic Post-Graduate Programme” Conference, held at the facilities of the University of Cadiz (Spain) from September 13 to 17, 2011. ORP3 is a sequence of conferences sponsored by EURO – The Association of European Operational Research Societies. Its objective is to promote collaboration and scientific and social exchanges between young OR researchers and practitioners. Before the 2011 meeting in Cadiz, four conferences were held: Paris (2001), Lambrecht (2003), Valencia (2005), and Guimaraes (2007). The ORP3 events are not restricted to specific fields and they accept any topic within the general scope of operations research. Nevertheless, in Cadiz the following topics were especially welcome: integer programming and combinatorial optimization, discrete optimization, graphs and networks, location theory, transportation and logistics, game theory, network optimization,multicriteria optimization, productionmanagement, supply chainmanagement, and routing problems. Due to capacity constraints, participation in the Cadiz meeting was limitted to a group of 36. Applicants were considered only if they were either young PhD students, young postdocs, or young faculty in OR (having completed the doctoral thesis a maximum of two years before the deadline for submission), or young OR analysts (having a maximum of two years of professional experience at the deadline for submission). The selection of participants by the Scientific Committee was based only on the quality of their full paper submissions. Papers were expected to contain high quality material that would be publishable in an international

An exact double-oracle algorithm for zero-sum extensive-form games with imperfect information

Developing scalable solution algorithms is one of the central problems in computational game theory. We present an iterative algorithm for computing an exact Nash equilibrium for two-player zero-su...

Game-theory Model based on Carrier Sense Multiple Access Protocol in Wireless Network

In order to solve game-theory model problem based on carrier sense multiple access protocol with collision avoidance in wireless network, a game-theory-based general and compatible modeling process is proposed, where the transmission probability of a node in a slot time is regarded as mixed game and network performance parameter M is taken as revenue function of game so as to build M rounds non-cooperative mixed strategy game with N gamers. After performing M rounds game, it will acquire M Nash-equilibria which is not equal to each other, and then these Nash-equilibria are taken as the input of single objective programming model with many constraints so as to obtain compromise equilibria. Finally, a minimum contention window based on compromise equilibria is designed. The throughput and successful transmission rate are analyzed through the simulation, and the simulation experiment results show our proposed modeling process has the more throughputs and the lower drop performance than the existing methods.

New Challenges in the Degree in Mathematics: Applications at Work

The new “Degree in Mathematics” of the University of Murcia implements in 2011/12 its third year, which includes the compulsory subject “Modeling laboratory”. Its syllabus says “the main aim is that the student be able to afford a problem from other sciences, model it using mathematical techniques, give a solution (even an approximate one) and check and interpret the obtained solution.” For that, “a set of projects will be offered to the students, to work on individually and in groups”. In this paper we show the decisions made by the teachers to put the subject at work, as well as the first results obtained. In particular, we describe the topics considered in the subject (which include discrete models of numerical analysis, differential equations models, (linear, integer and nonlinear) optimization problems, heuristic optimization, multiobjective optimization, routing problems, location problems, game theory problems and project management), the way they are explained, and the activities that the students have to do (both individually and in small groups) to pass.

PPAD-completeness of polyhedral versions of Sperner’s Lemma

We show that certain polyhedral versions of Sperner’s Lemma, where the colouring is given explicitly as part of the input, are PPAD-complete. The proofs are based on two recent results on the complexity of computational problems in game theory: the PPAD-completeness of 2-player Nash, proved by Chen and Deng, and of Scarf’s Lemma, proved by Kintali. We give a strengthening of the latter result, show how colourings of polyhedra provide a link between the two, and discuss a special case related to vertex covers.

Game Optimization Theory and Application in Distribution System Expansion Planning, Including Distributed Generation

Based on Game Theory and Multi-objective optimization problems (MOP), Game Optimization Theory (GOT) is discussed in this paper. Optimization Stability Analysis (OSA), Distance Entropy Multi-Objective Particle Swarm Optimization (DEMPSO) and Fuzzy Multi-weights Decision-making Method (FMW) are proposed as well. Game Optimization Theory, which is a comprehensive system, could not only handle multi-objective optimization problems effectively, but also could offset the disadvantages of traditional optimization theories, such as lack of framework and the insufficient consideration of relevant elements. In this paper GOT is used for the first time in solving the distribution systems planning (DSP) issue by implementing distributed generation. The proposed model integrates costs, losses, and voltage index to achieve optimal size and site of distributed generation. The model allows minimizing total system costs, system power losses and maximizing voltage improvement. A detailed DSP example is used for verifying the effectiveness and reasonableness of GOT in this context.

Dynamic decision-making in uncertain environments II. Allais paradox in human behavior

In both animal and human behavioral repertoires, classical expected utility theory is considered a fundamental element of decision making under conditions of uncertainty. This theory has been widely applied to problems of animal behavior and evolutionary game theory, as well as to human economic behavior. The Allais paradox hinges on the expression of avoidance of bankruptcy by humans, or death by starvation in animals. This paradox reveals that human behavioral patterns are often inconsistent with predictions under the classical expected utility theory as formulated by von Neumann and Morgenstern. None of the many attempts to reformulate utility theory has been entirely successful in resolving this paradox with rigorous logic. We present a simple, but novel approach to the theory of decision making, in which utility is dependent on current wealth, and in which losses are more heavily weighted than gains. Our approach resolves the Allais paradox in a manner that is consistent with how humans formulate decisions under uncertainty. Our results indicate that animals, including humans, are in principle risk-averse. Our restructuring of dynamic utility theory presents a basic optimization scheme for sequential or dynamic decisions in both animals and humans.

The Twentieth International Conference on Automatic Control "AUTOMATICS-2013"

International conferences on automatic control are initiated by Ukrainian Association on Automatic Control (UAAC) and held annually since 1994. In September 25–27 on the basis of Admiral Makarov National Shipbuilding University the town of Nikolaev held XX International Conference on Automatic Control “AUTOMATICS–2013”. The conference was devoted to 100birthday anniversary of the outstanding Ukrainian scientist in the field of technical cybernetics, Academician of National Academy of Sciences of Ukraine A.G. Ivakhnenko. The goal of the Conference is the discussion of research and development results in the field of control of complex objects of different nature, discussion of development tendencies in the control science, establishment of scientific contacts, involvement of talented young people in scientific investigations. The organizers of Conference were National Academy of Sciences (NAS) of Ukraine, Ministry of Education and Science (MES) of Ukraine, Ukrainian Association on Automatic Control, National Committee of Russia on Automatic Control, V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, Institute of Space Research of NAS and SSA of Ukraine, International Scientific-Educational Center of Information Technologies and Systems of NAS and MES of Ukraine, Moscow M.V. Lomonosov State University, Admiral Makarov National Shipbuilding University, Academy of Sciences of Shipbuilding of Ukraine. Subject matters of the Conference were presented in six sections. • Mathematical problems of control, optimization and game theory. • Control and identification under uncertainty conditions. • Automatic control in technical systems. • Control of aerospace, marine and other moving objects.

A Comparative Study between Genetic Algorithm and LINGO

The research is a comparison of solving some game theory problems by using the genetic algorithm and LINGO software package to define the best one in solving such problems. The research concluded that the least value of the match was accomplished by using genetic algorithm in contrast with solving by LINGO . The time ratio by which every player can use the available strategy to achieve the optimum value of the match was also defined .

Scheduling the usage of radar and jammer during peace and war time

The challenge between one's own radar and an enemy's jammer is modelled as a game theory problem. Using this model, the equilibrium point of the game is studied under different conditions. Also regarding the costs of both war and peace situations, the best choice of each part of the game during peace and war time is illustrated and studied.

Computing tournament solutions using relation algebra and RelView

We describe a simple computing technique for the tournament choice problem. It rests upon relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView’s labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.

On the practical usage of genetic algorithms in ecology and evolution

Summary Genetic algorithms are a heuristic global optimisation technique mimicking the action of natural selection to solve hard optimisation problems, which has enjoyed growing usage in evolution and ecology. However, there is little standardisation in how genetic algorithms are implemented in these fields and even less practical advice for those hoping to implement genetic algorithms for their own models. This article aims to demystify genetic algorithms and provide assistance to researchers; basic programming knowledge is important for working with genetic algorithms, but none is required to read this article. I review the basics of genetic algorithm methodology and provide suggestions on problems that may or may not benefit from genetic algorithm methods. The genetic operators (selection, replacement, mutation, crossover) and their rate parameters (mutation rate, etc.) are a source of potential confusion and to ease their selection, I present recommendations informed by historical usage and best practices, concluding with potential pitfalls. Good reasons for employing genetic algorithms include: solving optimisation problems beyond the reach of analytical techniques, relaxing model assumptions, evolving behaviour in individual‐based models, and simulating co‐evolutionary processes. However, genetic algorithms are not always the correct technique to use. Simpler methods may sometimes outperform genetic algorithms, and certain problem features may cause trouble. Genetic algorithms strike a balance between exploration of the search space and exploitation of good solutions, driven by the choice of genetic operators and rate parameters. I outline a basic set of parameter values and operator choices for genetic algorithm models as a starting point and provide advice on how to modify them to explore the exploitation/exploration trade‐off. Finally, I provide general advice on analysis and reporting of these models. There are a number of potential issues that can surprise unwary users, such as rate parameter values that lead to pathological outcomes; I suggest ways to detect and correct these problems. Also, contrary to popular usage, genetic algorithms can find solutions to game theory problems but cannot guarantee their stability. Finally, while genetic algorithms offer great power and flexibility by drawing inspiration from evolutionary processes, they are (usually) not a faithful model of genetics or evolution.

Quantum Mechanism Helps Agents Combat with 'Bad' Social Choice Rules

Quantum strategies have been successfully applied in game theory for years. However, as a reverse problem of game theory, the theory of mechanism design is ignored by physicists. In this paper, we generalize the classical theory of mechanism design to a quantum domain. The main result is that by virtue of quantum mechanism, agents who satisfy a certain condition can combat with 'bad' social choice rules instead of being restricted by the traditional mechanism design theory.

Min–max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω ⊂ R n , n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min–max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.

Game Theory

Designing suitable robotic controllers for automating movement-based rehabilitation therapy requires an understanding of the interaction between patient and therapist. Current approaches do not take into account the highly dynamic and interdependent nature of this relationship. A better understanding can be accomplished through framing the interaction as a problem in game theory. The main strength behind this approach is the potential to develop robotic control systems which automatically adapt to patient interaction behavior. Agents learn from experiences, and adapt their behaviors so they are better suited to their environment. As the models evolve, structures, patterns and behaviors emerge that were not explicitly programmed into the original models, but which instead surface through the agent interactions with each other and their environment. This paper advocates the use of such agent based models for analysing patient-therapist interactions with a view to designing more efficient and effective robotic controllers for automated therapeutic intervention in motor rehabilitation. The authors demonstrate in a simplified implementation the effectiveness of this approach through simulating known behavioral patterns observed in real patient-therapist interactions, such as learned dependency.

The fluid-structure interaction model with both control and disturbance at the interface: a game theory problem via an abstract approach

We consider an established hyperbolic–parabolic fluid-structure interaction model with control and disturbance acting at the interface between the two media. Here, the structure (modelled by the system of dynamic elasticity) is immersed in a fluid (modelled by the linearized Navier–Stokes equations). A game theory problem between control and disturbance is studied, when both act at the interface. To this end, the main mathematical difficulty that one encounters is the fact that such model fails to satisfy the ‘singular estimate’ from, say, control to state. This is a critical obstacle, as this is precisely the foundational property used in the development by Triggiani and Zhang [Min–max game theory and non-standard differential Riccati equations under singular estimates for e At B and e At G in the absence of analyticity, Set-Valued Var. Anal., 17 (2009), pp. 245–283.] for a full theory to include the solvability of the Differential Riccati equation in the study of the associated min–max game theory problem. Failure of the ‘singular estimate’ property is due to a mismatch between the parabolic and hyperbolic component of the overall coupled dynamics. Next, by introducing suitable observation or output operators, with incremental smoothness on the trajectory, it is shown that the resulting system satisfies a modified singular estimate, this time from the control to the observation space. This then allows one to adapt (and generalize) the complete min–max theory by Triggiani and Zhang to the present fluid-structure interaction model. More precisely, the approach followed is based on an abstract setting, of which the fluid-structure interaction model is a canonical illustration, which enjoys a desirable property not assumed for the abstract model.

Efficient social contracts and group selection

We consider the Stag Hunt in terms of Maynard Smith’s famous Haystack model. In the Stag Hunt, contrary to the Prisoner’s Dilemma, there is a cooperative equilibrium besides the equilibrium where every player defects. This implies that in the Haystack model, where a population is partitioned into groups, groups playing the cooperative equilibrium tend to grow faster than those at the non-cooperative equilibrium. We determine under what conditions this leads to the takeover of the population by cooperators. Moreover, we compare our results to the case of an unstructured population and to the case of the Prisoner’s Dilemma. Finally, we point to some implications our findings have for three distinct ideas: Ken Binmore’s group selection argument in favor of the evolution of efficient social contracts, Sewall Wright’s Shifting Balance theory, and the equilibrium selection problem of game theory.

Algorithms of optimal control methods for solving game theory problems

PurposeThe aim of the paper is to present the theory and algorithms based on the methods of systems optimal control for a numerical solution of a defined mathematical model of a system as well as that of a mathematical model of game theory.Design/methodology/approachThe paper brings a formulation of the mathematical model of a problem of systems optimal control with distributed parameters in Hilbert space. The mathematical model of the optimal control problem includes equations that also occur in the defined mathematical model of the theory of a two player zero‐sum game. Optimization problems of game theory have been defined for the purpose of finding a saddle point of a functional satisfying task constraints ε>0.FindingsIn order to find a saddle point of a functional and that one of a functional with a limitation, a designed algorithm of an iterative gradient method is presented. Furthermore, the paper contains a concept of algorithms designing that can be applied to a numerical solution of the defined problem of game theory. These algorithms can be realized on the basis of the methods of systems optimal control. After an adjoint state of the system is defined, a saddle point of a functional will be characterized by equations and inequalities.Originality/valueThe contribution of the paper lies in the formulation of the theorems which express the necessary and sufficient conditions of optimality for saddle points of a functional. Furthermore, it has been proved that algorithms of methods of systems optimal control with distributed parameters can be used for the solution of a mathematical model of game theory. The paper contains original results achieved by the authors within scientific projects.

Foreword: special issue on games and decisions in risk and reliability analysis

The objective of this special issue of Applied Stochastic Models in Business and Industry is to introduce a new theme, the use of game theory and decision theory in reliability and risk analysis. In so doing, the special issue brings together novel research from disciplines that have a lot to contribute to this theme, including economics, engineering, finance, medical sciences, probability, and statistics. The issue contains six papers. Four of these papers were presented at the First Symposium on Games and Decisions in Reliability and Risk which was held at The George Washington University on May 27–28, 2009. The discussion paper by David Banks, Francesca Petralia, and Shouqiang Wang introduces adversarial risk analysis as an alternative solution approach to game theory problems. The approach is implemented to gambling problems based on Borel games. The paper is discussed by Joseph B. Kadane and Nicholas G. Polson. The paper by Ethem Çanakoglu and S üleyman Özekici considers optimal portfolio selection, where distributions of asset returns change according to a latent Markovian process. A dynamic programming approach is developed to obtain a characterization of the optimal portfolio policy under the resulting hidden Markov model. Nader Ebrahimi, Ehsan S. Soofi, and Shaoqiong Zhao present properties of information measures for the Dirichlet family and related distributions. A new information characterization of the Dirichlet distribution in terms of survival and hazard gradient constraints is introduced. The paper by Rabin E. J. Neslo and Roger M. Cooke introduces probabilistic inversion methods to characterize a population of stakeholders' preferences via a probability distribution of utilities. The method is suitable for multicriteria evaluation problems, where utilities are inferred using data from discrete choices. The methodology is applied for the evaluation of actual health states data. Nicholas G. Polson and Morten Sorensen develop a simulation-based approach to stochastic dynamic programming problems. The presented approach is an alternative to the Q-learning algorithm of Watkins and Dayan (1992, Machine Learning, 8:279–292). The approach is illustrated using a dynamic investment problem. Nozer D. Singpurwalla gives an expository perspective on mathematical issues associated with the failure rate. More specifically, the circumstances, where the exponentiation formula overestimates the probability of survival, are discussed. The point is illustrated using Volterra's product integrals.