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Mathematical Game Theory 2019
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Mathematical Game Theory 2019

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 December 2019) | Viewed by 25533

Special Issue Editor

Prof. Dr. Vladimir Mazalov

E-Mail Website
Guest Editor
Institute of Applied Mathematical Research of Karelian Research Centre, Russian Academy of Sciences, Petrozavodsk, 185910 Karelia, Russia
Interests: game theory; decision analysis; dynamic programming; bargaining models; networking games; behavioral models
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This issue is a continuation of the previous successful Special Issue "Mathematical Game Theory" in Mathematics.

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. The main tools in the analysis of game models are mathematical methods. The spectrum of mathematical approaches in game theory is very wide.  In dynamic games, the Hamilton-Jacobi-Bellman equation and Pontryagin maximum principle are very useful. The mean-field approach studies the situations that involve a very large number of “rational players” where each player chooses his optimal strategy in view of the global information that is available to him and that results from the actions of all players. Dynamic games theory has various applications in many fields, including resource allocation, pollution control, fishery, and energy-efficient power control.  Networking games are games on graphs. This direction in game theory has appeared in connection with the emergence of new information technologies, in particular, global Internet, mobile communications, distributed and cloud computing, and social networks. The online social networks have given impulse to the development of new graph-theoretical methods for network analysis. Users of such networks are united in communities, forming networks of different topologies. An analysis of the structure of such graphs is important not only in itself but also for being able to evaluate the results of equilibrium game-theoretic interactions in such networks. Social network analysis methods are applied in many fields, such as economics, physics, biology, and information technologies. In routing games, players choose information transfer channels with limited bandwidths. Here, equilibrium is a result of the application of the optimization theory. 

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

Prof. Dr. Vladimir Mazalov
Guest Editor

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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

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Published Papers (9 papers)

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Research

20 pages, 871 KiB  
Article
Drivers’ Skills and Behavior vs. Traffic at Intersections
by Krzysztof J. Szajowski and Kinga Włodarczyk
Mathematics 2020, 8(3), 433; https://doi.org/10.3390/math8030433 - 16 Mar 2020
Cited by 4 | Viewed by 2807
Abstract
The aim of the work is to connect individual behavior of drivers with traffic intensity. By diversifying the populations of drivers into two categories, often considered in this type of an analysis, CO (cooperative) and DE (defective), the tendency of drivers from each [...] Read more.
The aim of the work is to connect individual behavior of drivers with traffic intensity. By diversifying the populations of drivers into two categories, often considered in this type of an analysis, CO (cooperative) and DE (defective), the tendency of drivers from each of these groups to deviate from compliance with traffic rules is established. The effective driver behavior translates into disrupting traffic by slowing it down. Participant interactions are described using game theories that provide information for simulations algorithms based on cellular automata. Three different ways of using this combination of descriptions of traffic participants to examine the impact of their behavior on the traffic dynamics are shown. Directions of the further, detailed analysis are indicated, which requires basic research in the field of game theory models. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
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16 pages, 433 KiB  
Article
The Euler-Equation Approach in Average-Oriented Opinion Dynamics
by Vladimir Mazalov and Elena Parilina
Mathematics 2020, 8(3), 355; https://doi.org/10.3390/math8030355 - 5 Mar 2020
Cited by 11 | Viewed by 3046
Abstract
We consider the models of average-oriented opinion dynamics. An opinion about an event is distributed among the agents of a social network. There are an optimization problem and two game-theoretical models when players as centers of influence aim to make the opinions of [...] Read more.
We consider the models of average-oriented opinion dynamics. An opinion about an event is distributed among the agents of a social network. There are an optimization problem and two game-theoretical models when players as centers of influence aim to make the opinions of the agents closer to the target ones in a finite time horizon minimizing their costs. The optimization problem and the games of competition for the agents’ opinion are linear-quadratic and solved using the Euler-equation approach. The optimal strategies for optimization problem and the Nash equilibria in the open-loop strategies for the games are found. Numerical simulations demonstrate theoretical results. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
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12 pages, 246 KiB  
Article
Multiobjective Games for Detecting Abnormally Expressed Genes
by Vito Fragnelli and Lucia Pusillo
Mathematics 2020, 8(3), 350; https://doi.org/10.3390/math8030350 - 5 Mar 2020
Cited by 3 | Viewed by 1765
Abstract
A class of multiobjective games with applications to a medicine setting is studied. We consider the vector Shapley value and the vector Banzhaf value for a multicriteria game and we apply them to a microarray game. We give an axiomatic characterization too. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
20 pages, 488 KiB  
Article
A Game Theoretic Model of Choosing a Valuable Good via a Short List Heuristic
by David M. Ramsey
Mathematics 2020, 8(2), 199; https://doi.org/10.3390/math8020199 - 6 Feb 2020
Cited by 5 | Viewed by 2098
Abstract
The Internet gives access to a huge amount of data at the click of a mouse. This is very helpful when consumers are making decisions about which product to buy. However, the final decision to purchase is still generally made by humans who [...] Read more.
The Internet gives access to a huge amount of data at the click of a mouse. This is very helpful when consumers are making decisions about which product to buy. However, the final decision to purchase is still generally made by humans who have limited memory and perception. The short list heuristic is often used when there are many offers on the market. Searchers first find information about offers via the Internet and on this basis choose a relatively small number of offers to view in real life. Although such rules are often used in practice, little research has been carried out on determining, for example, what the size of the short list should be depending on the parameters of the problem or modelling how the short list heuristic can be implemented when there are multiple decision makers. This article presents a game theoretic model of such a search procedure with two players. These two players can be interpreted, for example, as a couple searching for a flat or a second-hand car. The model indicates that under such a search procedure the roles of searchers should only be divided when the preferences of the players are coherent or there is a high level of goodwill between them. In other cases, dividing the roles leads to a high level of conflict. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
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9 pages, 261 KiB  
Article
Characteristic Function and Time Consistency for Two-Stage Games with Network Externalities
by Artem Sedakov
Mathematics 2020, 8(1), 38; https://doi.org/10.3390/math8010038 - 1 Jan 2020
Cited by 3 | Viewed by 1952
Abstract
Time consistency is a property of the solution to a cooperative dynamic game which guarantees that this solution remains stable with respect to its revision by players over time. The fulfillment of this property is directly related to the characteristic function and its [...] Read more.
Time consistency is a property of the solution to a cooperative dynamic game which guarantees that this solution remains stable with respect to its revision by players over time. The fulfillment of this property is directly related to the characteristic function and its behavior with the course of the game as any solution is based on this function. In this paper, we will examine the characteristic functions for two economic models with network externalities represented by a two-stage network game using the theory developed for this class of games. For a network game with positive externalities represented by a public goods provision model, we demonstrate a sufficient condition for time consistency. For a network game with negative externalities represented by a market competition model, we show that the cooperative solution is always time consistent. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
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19 pages, 845 KiB  
Article
A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options
by Sergey Smirnov
Mathematics 2019, 7(12), 1246; https://doi.org/10.3390/math7121246 - 17 Dec 2019
Cited by 6 | Viewed by 2771 | Correction
Abstract
This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of [...] Read more.
This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
31 pages, 1249 KiB  
Article
Moving Information Horizon Approach for Dynamic Game Models
by Ovanes Petrosian, Lihong Shi, Yin Li and Hongwei Gao
Mathematics 2019, 7(12), 1239; https://doi.org/10.3390/math7121239 - 14 Dec 2019
Cited by 7 | Viewed by 3088
Abstract
In the paper, a new class of dynamic game models with a moving information horizon or dynamic updating is studied. In this class of games, players do not have full information about the game structure (motion equations, payoff functions) on the interval on [...] Read more.
In the paper, a new class of dynamic game models with a moving information horizon or dynamic updating is studied. In this class of games, players do not have full information about the game structure (motion equations, payoff functions) on the interval on which the game is defined. It is supposed that the players at each stage of the dynamic game have only truncated information about the game structure defined by the information horizon. Cooperative and noncooperative settings are considered in the paper. Results are illustrated using the oligopoly advertising game model, and comparison between the solution in the initial game model and in the game model with moving information horizon is presented. Simulation results are presented. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
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16 pages, 318 KiB  
Article
Robust Portfolio Optimization in an Illiquid Market in Discrete-Time
by Nikolay Andreev
Mathematics 2019, 7(12), 1147; https://doi.org/10.3390/math7121147 - 24 Nov 2019
Cited by 2 | Viewed by 2604
Abstract
We present a robust dynamic programming approach to the general portfolio selection problem in the presence of transaction costs and trading limits. We formulate the problem as a dynamic infinite game against nature and obtain the corresponding Bellman-Isaacs equation. Under several additional assumptions, [...] Read more.
We present a robust dynamic programming approach to the general portfolio selection problem in the presence of transaction costs and trading limits. We formulate the problem as a dynamic infinite game against nature and obtain the corresponding Bellman-Isaacs equation. Under several additional assumptions, we get an alternative form of the equation, which is more feasible for a numerical solution. The framework covers a wide range of control problems, such as the estimation of the portfolio liquidation value, or portfolio selection in an adverse market. The results can be used in the presence of model errors, non-linear transaction costs and a price impact. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
15 pages, 34174 KiB  
Article
Cournot Duopoly Games: Models and Investigations
by S. S. Askar and A. Al-khedhairi
Mathematics 2019, 7(11), 1079; https://doi.org/10.3390/math7111079 - 8 Nov 2019
Cited by 6 | Viewed by 4701
Abstract
This paper analyzes Cournot duopoly games that are constructed based on Cobb–Douglas preferences. We introduce here two models whose dynamic adjustments depend on bounded rationality, dynamic adjustment, and tit-for-tat mechanism. In the first model, we have two firms with limited information and due [...] Read more.
This paper analyzes Cournot duopoly games that are constructed based on Cobb–Douglas preferences. We introduce here two models whose dynamic adjustments depend on bounded rationality, dynamic adjustment, and tit-for-tat mechanism. In the first model, we have two firms with limited information and due to that they adopt the bounded rationality mechanism. They update their productions based on the changing occurred in the marginal profit. For this model, its fixed point is obtained and its stability condition is calculated. In addition, we provide conditions by which this fixed point loses its stability due to flip and Neimark–Sacker bifurcations. Furthermore, numerical simulation shows that this model possesses some chaotic behaviors which are recovered due to corridor stability. In the second model, we handle two different mechanisms of cooperation. These mechanisms are dynamic adjustment process and tit-for-tat strategy. The players who use the dynamic adjustment increase their productions based on the cooperative output while, in tit-for-tat mechanism, they increase the productions based on the cooperative profit. The local stability analysis shows that adopting tit-for-tat makes the model unstable and then the system becomes chaotic for any values of the system’s parameters. The obtained results show that the dynamic adjustment makes the system’s fixed point stable for a certain interval of the adjustment parameter. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2019)
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