Special Issue "Mathematical Game Theory 2021"

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

Deadline for manuscript submissions: closed (15 September 2021) | Viewed by 7867

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
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Special Issue Information

Dear Colleagues,

This issue is a continuation of the previous successful Special Issue "Mathematical Game Theory 2019" 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 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 their optimal strategy in view of the global information that is available to them 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. Online social networks have driven the development of new graph-theoretical methods for network analysis. The 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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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 (11 papers)

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Research

Article
A Cooperative Network Packing Game with Simple Paths
Mathematics 2021, 9(14), 1683; https://doi.org/10.3390/math9141683 - 17 Jul 2021
Viewed by 587
Abstract
We consider a cooperative packing game in which the characteristic function is defined as the maximum number of independent simple paths of a fixed length included in a given coalition. The conditions under which the core exists in this game are established, and [...] Read more.
We consider a cooperative packing game in which the characteristic function is defined as the maximum number of independent simple paths of a fixed length included in a given coalition. The conditions under which the core exists in this game are established, and its form is obtained. For several particular graphs, the explicit form of the core is presented. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
Inspection—Corruption Game of Illegal Logging and Other Violations: Generalized Evolutionary Approach
Mathematics 2021, 9(14), 1619; https://doi.org/10.3390/math9141619 - 09 Jul 2021
Viewed by 616
Abstract
Games of inspection and corruption are well developed in the game-theoretic literature. However, there are only a few publications that approach these problems from the evolutionary point of view. In previous papers of this author, a generalization of the replicator dynamics of the [...] Read more.
Games of inspection and corruption are well developed in the game-theoretic literature. However, there are only a few publications that approach these problems from the evolutionary point of view. In previous papers of this author, a generalization of the replicator dynamics of the evolutionary game theory was suggested for inspection modeling, namely the pressure and resistance framework, where a large pool of small players plays against a distinguished major player and evolves according to certain myopic rules. In this paper, we develop this approach further in a setting of the two-level hierarchy, where a local inspector can be corrupted and is further controlled by the higher authority (thus combining the modeling of inspection and corruption in a unifying setting). Mathematical novelty arising in this investigation involves the analysis of the generalized replicator dynamics (or kinetic equation) with switching, which occurs on the “efficient frontier of corruption”. We try to avoid parameters that are difficult to observe or measure, leading to some clear practical consequences. We prove a result that can be called the “principle of quadratic fines”: We show that if the fine for violations (both for criminal businesses and corrupted inspectors) is proportional to the level of violations, the stable rest points of the dynamics support the maximal possible level of both corruption and violation. The situation changes if a convex fine is introduced. In particular, starting from the quadratic growth of the fine function, one can effectively control the level of violations. Concrete settings that we have in mind are illegal logging, the sales of products with substandard quality, and tax evasion. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
Differential Games for an Infinite 2-Systems of Differential Equations
Mathematics 2021, 9(13), 1467; https://doi.org/10.3390/math9131467 - 23 Jun 2021
Viewed by 523
Abstract
A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to [...] Read more.
A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
Article
A Monotonic Weighted Banzhaf Value for Voting Games
Mathematics 2021, 9(12), 1343; https://doi.org/10.3390/math9121343 - 10 Jun 2021
Cited by 1 | Viewed by 573
Abstract
The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justified by [...] Read more.
The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justified by the imperfect cooperation. Nevertheless, it is monotonous in the weights. We also obtain three different characterizations of the value. Then we relate it to the Owen multilinear extension. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
Article
Robust Pairwise n-Person Stochastic Duel Game
Mathematics 2021, 9(8), 825; https://doi.org/10.3390/math9080825 - 10 Apr 2021
Cited by 1 | Viewed by 752
Abstract
This paper introduces an extended version of a stochastic game under the antagonistic duel-type setup. The most flexible multiple person duel game is analytically solved. Moreover, the explicit formulas are solved to determine the time-dependent duel game model using the first exceed theory [...] Read more.
This paper introduces an extended version of a stochastic game under the antagonistic duel-type setup. The most flexible multiple person duel game is analytically solved. Moreover, the explicit formulas are solved to determine the time-dependent duel game model using the first exceed theory in multiple game stages. Unlike conventional stochastic duel games, multiple battlefields are firstly introduced and each battlefield becomes a shooting ground of pairwise players in a multiperson game. Each player selects different targets in different game stages. An analogue of this new theory was designed to find the best shooting time within multiple battlefields. This model is fully mathematically explained and is the basis with which to apply a stochastic duel-type game in various practical applications. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
Network Formation with Asymmetric Players and Chance Moves
Mathematics 2021, 9(8), 814; https://doi.org/10.3390/math9080814 - 09 Apr 2021
Viewed by 587
Abstract
We propose a model of network formation as a two-stage game with chance moves and players of various types. First, the leader suggests a connected communication network for the players to join. Second, nature selects a type vector for players based on the [...] Read more.
We propose a model of network formation as a two-stage game with chance moves and players of various types. First, the leader suggests a connected communication network for the players to join. Second, nature selects a type vector for players based on the given probability distribution, and each player decides whether or not to join the network keeping in mind only his own type and the leader’s type. The game is of incomplete information since each player has only a belief over the payoff functions of others. As a result, the network is formed, and each player gets a payoff related to both the network structure and his type. We prove the existence of the Bayesian equilibrium and propose a new definition of the stable partially Bayesian equilibrium defining the network to be formed and prove its existence. The connection between the stable partially Bayesian equilibrium and the Nash equilibrium in the game is examined. Finally, we investigate the characteristics of the network structures under the stable partially Bayesian equilibrium in a three-player game with the major player as well as in the n-player game with a specific characteristic function. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
Strong Time-Consistent Solution for Cooperative Differential Games with Network Structure
Mathematics 2021, 9(7), 755; https://doi.org/10.3390/math9070755 - 01 Apr 2021
Cited by 2 | Viewed by 532
Abstract
One class of cooperative differential games on networks is considered. It is assumed that interaction on the network is possible not only between neighboring players, but also between players connected by paths. Various cooperative optimality principles and their properties for such games are [...] Read more.
One class of cooperative differential games on networks is considered. It is assumed that interaction on the network is possible not only between neighboring players, but also between players connected by paths. Various cooperative optimality principles and their properties for such games are investigated. The construction of the characteristic function is proposed, taking into account the network structure of the game and the ability of players to cut off connections. The conditions under which a strong time-consistent subcore is not empty are studied. The formula for explicit calculation of the Shapley value is derived. The results are illustrated by the example of one differential marketing game. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
A Dynamic Model of Cournot Competition for an Oligopolistic Market
Mathematics 2021, 9(5), 489; https://doi.org/10.3390/math9050489 - 27 Feb 2021
Viewed by 589
Abstract
This paper studies firms’ dynamic interaction in a Cournot market. In each period of the game, the firm decides whether to make a stochastic positioning investment (establishing or maintaining its position in market competition). The market demand is also stochastic (high or low). [...] Read more.
This paper studies firms’ dynamic interaction in a Cournot market. In each period of the game, the firm decides whether to make a stochastic positioning investment (establishing or maintaining its position in market competition). The market demand is also stochastic (high or low). By adopting symmetric Market perfect Nash equilibrium, firms choose strategies to maximize the discounted present value of cash flow. By considering the cases with one, two, and three active firms in the market, respectively, we present the stage game market outcome, show the transition probabilities, find the steady state of the system, and discuss the speed of convergence. Our work allows for two types of uncertainty in firms’ interactions, which contribute to the dynamic oligopoly literature. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
An Intelligent Algorithm for Solving the Efficient Nash Equilibrium of a Single-Leader Multi-Follower Game
Mathematics 2021, 9(5), 454; https://doi.org/10.3390/math9050454 - 24 Feb 2021
Cited by 2 | Viewed by 477
Abstract
This aim of this paper is to provide the immune particle swarm optimization (IPSO) algorithm for solving the single-leader–multi-follower game (SLMFG). Through cooperating with the particle swarm optimization (PSO) algorithm and an immune memory mechanism, the IPSO algorithm is designed. Furthermore, we define [...] Read more.
This aim of this paper is to provide the immune particle swarm optimization (IPSO) algorithm for solving the single-leader–multi-follower game (SLMFG). Through cooperating with the particle swarm optimization (PSO) algorithm and an immune memory mechanism, the IPSO algorithm is designed. Furthermore, we define the efficient Nash equilibrium from the perspective of mathematical economics, which maximizes social welfare and further refines the number of Nash equilibria. In the end, numerical experiments show that the IPSO algorithm has fast convergence speed and high effectiveness. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
Time-Dependent Theme Park Routing Problem by Partheno-Genetic Algorithm
Mathematics 2020, 8(12), 2193; https://doi.org/10.3390/math8122193 - 09 Dec 2020
Cited by 2 | Viewed by 738
Abstract
With the improvement of people’s living standards and entertainment interests, theme parks have become one of the most popular holiday places. Many theme park websites provide a variety of information, according to which tourists can arrange their own schedules. However, most theme park [...] Read more.
With the improvement of people’s living standards and entertainment interests, theme parks have become one of the most popular holiday places. Many theme park websites provide a variety of information, according to which tourists can arrange their own schedules. However, most theme park websites usually have too much information, which makes it difficult for tourists to develop a tourism planning. Therefore, the theme park routing problem has attracted the attention of scholars. Based on the Traveling Salesman Problem (TSP) network, we propose a Time-Dependent Theme Park Routing Problem (TDTPRP), in which walking time is time-dependent, considering the degree of congestion and fatigue. The main goal is to maximize the number of attractions visited and satisfaction and to reduce queues and walking time. To verify the feasibility and the effectiveness of the model, we use the Partheno-Genetic Algorithm (PGA) and an improved Annealing Partheno-Genetic Algorithm (APGA) to solve the model in this paper. Then, in the experimental stage, we conducted two experiments, and the experimental data were divided into real-world problem instances and randomly generated problem instances. The results demonstrate that the parthenogenetic simulated annealing algorithm has better optimization ability than the general parthenogenetic algorithm when the data scale is expanded. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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Article
Exploitation of a Productive Asset in the Presence of Strategic Behavior and Pollution Externalities
Mathematics 2020, 8(10), 1682; https://doi.org/10.3390/math8101682 - 01 Oct 2020
Cited by 2 | Viewed by 715
Abstract
We study the strategic behavior of firms competing in the exploitation of a common-access productive asset, in the presence of pollution externalities. We consider a differential game with two state variables (asset stock and pollution stock), and by using a piecewise-linear approximation of [...] Read more.
We study the strategic behavior of firms competing in the exploitation of a common-access productive asset, in the presence of pollution externalities. We consider a differential game with two state variables (asset stock and pollution stock), and by using a piecewise-linear approximation of the nonlinear asset growth function, we provide a tractable characterization of the symmetric feedback–Nash equilibrium with asymptotically stable steady state(s). The results show that the firm’s strategy takes three forms depending on the pair of state variables and that different options for the model parameters lead to contrasting outcomes in both the short- and long-run equilibria. Full article
(This article belongs to the Special Issue Mathematical Game Theory 2021)
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