Special Issue "Mathematical Game Theory"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 August 2018)

Special Issue Editor

Guest Editor
Prof. Dr. Vladimir Mazalov

Institute of Applied Mathematical Research of Karelian Research Centre, Russian Academy of Sciences, Petrozavodsk, Karelia 185910, Russia
Website | E-Mail
Interests: game theory; decision analysis; dynamic programming; bargaining models; networking games; behavioral models

Special Issue Information

Dear Colleagues,

Rapid developments in technology, communication, industrial organization, economic integration and international trade have stimulated the appearance of different practical statements in the description of agent interaction, based on the game theory. A strategic approach to decision-making is very useful in many areas, such as bargaining, resource allocation, fishery, competition and cooperation, pollution control, networking, and competitive mobile systems. The main tools in the analysis of game models are mathematical methods. In dynamic games, the Hamilton-Jacobi-Bellman equation and Pontryagin maximum principle are very useful. Dynamic games theory has many applications in many fields, including biology, computer science, ecology, economics and management. In networking games, the result of interactions between agents are defined by a certain network. Networking games are games on graphs; graph-theoretic models are very important in this field.  This direction in game theory has appeared in connection with the emergence of new information technologies. First of all it's the global Internet, mobile communications, distributed and cloud computing and social networks. In routing games, players choose information transfer channels with limited bandwidths.  Equilibrium, here, is a result of the application of the optimization theory.  Social networks appear lead to many new game-theoretic problem formulations. Users of such networks are united in communities, forming networks of different topologies. An analysis of the structure of such a graph is important in of itself, but is also important in being able to evaluate the results of equilibrium game-theoretic interactions in such networks. The spectrum of mathematical approaches in game theory is very wide.

This Special Issue contains papers that cover the wide range of mathematical methods used in game theory, including recent advances in areas of high potential for future works, as well as new developments in classical results. It will be of interest to anyone doing theoretical research in game theory or working on one its numerous applications.

Prof. Dr. Vladimir Mazalov
Guest Editor

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Keywords

  • Competition and cooperation
  • Dynamic games
  • Networking games
  • Behavioral game theory
  • Potential games
  • Bargaining models
  • Hamilton-Jacobi-Bellman equation
  • Pontryagin maximum principle
  • Applications in resource allocation, fishery, pollution control, networking

Published Papers (8 papers)

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Research

Open AccessArticle Price and Treatment Decisions in Epidemics: A Differential Game Approach
Mathematics 2018, 6(10), 190; https://doi.org/10.3390/math6100190
Received: 30 August 2018 / Revised: 26 September 2018 / Accepted: 28 September 2018 / Published: 2 October 2018
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Abstract
We consider a pharmaceutical company that sells a drug that is useful in the treatment of an infectious disease. A public authority buys the drug to heal at least a portion of the infected population. The authority has an overall budget for all [...] Read more.
We consider a pharmaceutical company that sells a drug that is useful in the treatment of an infectious disease. A public authority buys the drug to heal at least a portion of the infected population. The authority has an overall budget for all health care costs in the country and can only allocate a (small) part of the budget to the purchase of the drug. The government chooses the amount of drug to be purchased in order to minimize both the number of infectious people and the perceived cost of the operation along a given time horizon. This cost can be modeled through a linear or quadratic function of the monetary cost (as generally happens in the literature) or through a specific function (blow-up) that makes the budget constraint endogenous. The pharmaceutical company chooses the price of the drug in order to maximize its profit and knowing the budget constraints of the buyer. The resulting differential game is studied by supposing the simplest possible dynamics for the population. Two different games are proposed and their solutions are discussed: a cooperative game in which the two players bargain for the price of the drug and the quantity is purchased with the aim of maximizing the overall payoff and a competitive game in which the seller announces a price strategy to the buyer and binds to it; the buyer reacts by choosing the quantity to be purchased. In the case of linear or quadratic costs, the solution provided (for budget levels is not high enough) that the government spends the entire budget to purchase the drug. This drawback does not occur when the blow-up cost function is used. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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Open AccessArticle Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration
Mathematics 2018, 6(9), 165; https://doi.org/10.3390/math6090165
Received: 25 July 2018 / Revised: 24 August 2018 / Accepted: 31 August 2018 / Published: 11 September 2018
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Abstract
A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with [...] Read more.
A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with respect to the jump. We investigate properties of optimal trajectories and of imputation distribution procedures if the game is played cooperatively. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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Open AccessArticle A Game-Theoretic Loss Allocation Approach in Power Distribution Systems with High Penetration of Distributed Generations
Mathematics 2018, 6(9), 158; https://doi.org/10.3390/math6090158
Received: 31 July 2018 / Revised: 28 August 2018 / Accepted: 3 September 2018 / Published: 6 September 2018
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Abstract
Allocation of the power losses to distributed generators and consumers has been a challenging concern for decades in restructured power systems. This paper proposes a promising approach for loss allocation in power distribution systems based on a cooperative concept of game-theory, named Shapley [...] Read more.
Allocation of the power losses to distributed generators and consumers has been a challenging concern for decades in restructured power systems. This paper proposes a promising approach for loss allocation in power distribution systems based on a cooperative concept of game-theory, named Shapley Value allocation. The proposed solution is a generic approach, applicable to both radial and meshed distribution systems as well as those with high penetration of renewables and DG units. With several different methods for distribution system loss allocation, the suggested method has been shown to be a straight-forward and efficient criterion for performance comparisons. The suggested loss allocation approach is numerically investigated, the results of which are presented for two distribution systems and its performance is compared with those obtained by other methodologies. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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Open AccessArticle Dynamic Multicriteria Games with Finite Horizon
Mathematics 2018, 6(9), 156; https://doi.org/10.3390/math6090156
Received: 31 July 2018 / Revised: 8 August 2018 / Accepted: 3 September 2018 / Published: 5 September 2018
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Abstract
The approaches to construct optimal behavior in dynamic multicriteria games with finite horizon are presented. To obtain a multicriteria Nash equilibrium, the bargaining construction (Nash product) is adopted. To construct a multicriteria cooperative equilibrium, a Nash bargaining scheme is applied. Dynamic multicriteria bioresource [...] Read more.
The approaches to construct optimal behavior in dynamic multicriteria games with finite horizon are presented. To obtain a multicriteria Nash equilibrium, the bargaining construction (Nash product) is adopted. To construct a multicriteria cooperative equilibrium, a Nash bargaining scheme is applied. Dynamic multicriteria bioresource management problem with finite harvesting times is considered. The players’ strategies and the payoffs are obtained under cooperative and noncooperative behavior. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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Open AccessArticle Accounting Games: Using Matrix Algebra in Creating the Accounting Models
Mathematics 2018, 6(9), 152; https://doi.org/10.3390/math6090152
Received: 23 July 2018 / Revised: 13 August 2018 / Accepted: 14 August 2018 / Published: 31 August 2018
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Abstract
The aim of this paper is to show the mathematical basis for a precise treatment of double-entry bookkeeping, which was first developed in the nineteenth century by Sir William Rowan Hamilton. This is done by using basic notions of matrix algebra founded on [...] Read more.
The aim of this paper is to show the mathematical basis for a precise treatment of double-entry bookkeeping, which was first developed in the nineteenth century by Sir William Rowan Hamilton. This is done by using basic notions of matrix algebra founded on the idea of ordered pairs. We also reveal how complex numbers and rationals (fractions) developed in mainstream accounting science and became a leading platform for the ongoing processes within Industry 4.0. The paper concludes with examples of how accounting operations can be represented by matrix equations with the result of generating a final report. The author presents a mathematical model of accounting which is independent of specific existential forms, but which is capable of undertaking the form of any of them and thus which has the potential of being understood and accepted by specialists globally. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
Open AccessArticle Inverse Stackelberg Solutions for Games with Many Followers
Mathematics 2018, 6(9), 151; https://doi.org/10.3390/math6090151
Received: 30 July 2018 / Revised: 23 August 2018 / Accepted: 27 August 2018 / Published: 30 August 2018
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Abstract
The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity [...] Read more.
The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity conditions, establish the existence theorem. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
Open AccessArticle A Markovian Mechanism of Proportional Resource Allocation in the Incentive Model as a Dynamic Stochastic Inverse Stackelberg Game
Mathematics 2018, 6(8), 131; https://doi.org/10.3390/math6080131
Received: 4 July 2018 / Revised: 23 July 2018 / Accepted: 27 July 2018 / Published: 30 July 2018
Cited by 1 | PDF Full-text (237 KB) | HTML Full-text | XML Full-text
Abstract
This paper considers resource allocation among producers (agents) in the case where the Principal knows nothing about their cost functions while the agents have Markovian awareness about his/her strategies. We use a dynamic setup of the stochastic inverse Stackelberg game as the model. [...] Read more.
This paper considers resource allocation among producers (agents) in the case where the Principal knows nothing about their cost functions while the agents have Markovian awareness about his/her strategies. We use a dynamic setup of the stochastic inverse Stackelberg game as the model. We suggest an algorithm for solving this game based on Q-learning. The associated Bellman equations contain functions of one variable for the Principal and also for the agents. The new results are illustrated by numerical examples. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
Open AccessArticle Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game
Mathematics 2018, 6(2), 20; https://doi.org/10.3390/math6020020
Received: 29 October 2017 / Revised: 17 January 2018 / Accepted: 26 January 2018 / Published: 1 February 2018
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Abstract
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 [...] Read more.
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 P Q 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 P Q game. For this purpose, we establish a rigorous connection between finite automata and the P Q 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 P Q 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. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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