Multicriteria Games

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

Deadline for manuscript submissions: closed (28 February 2023) | Viewed by 4928

Special Issue Editor

Institute of Applied Mathematical Research, Karelian Research Centre of RAS, 11, Pushkinskaya str., 185910 Petrozavodsk, Russia
Interests: game theory; dynamic games; optimal control; multicriteria games

Special Issue Information

Dear Colleagues,

Mathematical models that take into account the presence of several objective functions of participants in conflict-controlled processes are closer to reality. Players often seek to achieve several goals simultaneously, which can be incomparable. Thus, conflicting interests appear not only between decision-makers but also within each individual due to the different criteria. The multicriteria approach allows determining an optimal behavior in such game-theoretic models of real-life processes.

The theory of multicriteria games is a confluence of game theory and multicriteria optimization. In 1959, Shapley introduced the definition of a game with vector payoffs. Since then, much research has been dedicated to the multicriteria games and the concepts of their solutions. Similar to multi-objective optimization, in static games with vector payoffs the solution concepts are usually based on the Pareto set or some convolutions of the criteria. In differential multicriteria games the ideas of Pontryagin and Krasovskii are adopted. The applications of games with vector payoffs are appeared in journals of engineering science, economics, optimization, finance, logistic, medicine, ecology and social science. Renewed interest in recent years has been caused by the realization that multicriteria game-theoretic models can explain a decision makers' real-life behavior.

The purpose of this Special Issue is to gather a collection of the recent advances in the theory of multicriteria games, including static and dynamic, noncooperative and cooperative settings and numerous applications.

Prof. Dr. Anna Rettieva
Guest Editor

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Keywords

  • Dynamic multicriteria games
  • Static multicriteria games
  • Games with vector payoffs
  • Multistage multicriteria games
  • Multicriteria matrix games
  • Differential games with vector payoffs

Published Papers (4 papers)

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Research

15 pages, 370 KiB  
Article
Sustainable Cooperation in a Bicriteria Game of Renewable Resource Extraction
by Denis Kuzyutin and Nadezhda Smirnova
Mathematics 2023, 11(6), 1497; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061497 - 19 Mar 2023
Cited by 1 | Viewed by 756
Abstract
We study a multi-objective finite-horizon game model of renewable, common resource extraction where the players have two separate objectives (one is economic success; the other describes the players’ environmental concern). We derive the cooperative strategy and the subgame-perfect Pareto equilibrium in linear-state non-stationary [...] Read more.
We study a multi-objective finite-horizon game model of renewable, common resource extraction where the players have two separate objectives (one is economic success; the other describes the players’ environmental concern). We derive the cooperative strategy and the subgame-perfect Pareto equilibrium in linear-state non-stationary feedback strategies by employing the dynamic programming approach. Since the utility is transferable only based on the economic criterion, we need to revise the concept of time consistency and the payoff-distribution procedure to provide a mechanism for sustainable long-term cooperation. All the results are illustrated with a numerical example. Full article
(This article belongs to the Special Issue Multicriteria Games)
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28 pages, 1057 KiB  
Article
Germeier’s Scalarization for Approximating Solution of Multicriteria Matrix Games
by Natalia Novikova and Irina Pospelova
Mathematics 2023, 11(1), 133; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010133 - 27 Dec 2022
Cited by 1 | Viewed by 801
Abstract
In this paper, we study the properties of Germeier’s scalarization applied for solving multicriteria games. The equilibria and the equilibrium values of such games, as a rule, make sets, and the problems of parametrizing and approximating these sets arise. Shapley proved that Nash [...] Read more.
In this paper, we study the properties of Germeier’s scalarization applied for solving multicriteria games. The equilibria and the equilibrium values of such games, as a rule, make sets, and the problems of parametrizing and approximating these sets arise. Shapley proved that Nash equilibrium of multicriteria matrix game can be found by solving a two-parametric family of scalar games obtained with the help of linear scalarization of the criteria vector. We show that Germeier’s scalarization parametrizes the equilibria of the multicriteria game by using one-parametric family of scalar games. Germeier’s scalarization has certain advantages over the linear one, and we suggest it for approximating the multicriteria game equilibria with a finite set. For two-criteria games with 2×2 matrices, we show by examples that there is no continuity of the values of scalar games in the scalarizing parameters. We prove one-sided (from the left or from the right) continuity for the game values. As a result, we come to convergence in Hausdorff metric for the set of equilibrium values obtained for ϵ-net on the simplex of scalarizing parameters to the value of the multicriteria game as ϵ0. The constructed finite approximation may be helpful in practical applications, where players try to find a compromise in an iterative negotiating procedure under multiple criteria. Full article
(This article belongs to the Special Issue Multicriteria Games)
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15 pages, 320 KiB  
Article
Dynamic Multicriteria Game with Pollution Externalities
by Anna Rettieva
Mathematics 2022, 10(22), 4238; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224238 - 13 Nov 2022
Cited by 2 | Viewed by 935
Abstract
The multicriteria approach deals with real-life applications of game theory. However, the existing game-theoretic statements with the joint analysis of resource extraction and pollution dynamics have not considered the multiple objectives of the players. To address this issue, a dynamic multicriteria game is [...] Read more.
The multicriteria approach deals with real-life applications of game theory. However, the existing game-theoretic statements with the joint analysis of resource extraction and pollution dynamics have not considered the multiple objectives of the players. To address this issue, a dynamic multicriteria game is proposed: many players exploit a common resource and seek to optimize different criteria under pollution externalities. Two interconnected state variables (resource stock and pollution level) are introduced and studied. The pollution level depends on exploitation strategies, and the players have an environmental objective to reduce the accumulated pollution. The noncooperative and cooperative behavioral strategies of the players are analyzed. A linear dynamic multicriteria bioresource management problem with pollution externalities is investigated to illustrate the solution concepts proposed. The differences between the noncooperative and cooperative cases, as well as between the models with and without environmentally concerned players, are treated analytically and numerically. As shown by the results, the cooperative behavior reduces pollution in both statements, bringing to sparing bioresource exploitation. Full article
(This article belongs to the Special Issue Multicriteria Games)
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34 pages, 534 KiB  
Article
A Dynamic Multi-Objective Duopoly Game with Capital Accumulation and Pollution
by Bertrand Crettez, Naila Hayek and Peter M. Kort
Mathematics 2021, 9(16), 1983; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161983 - 19 Aug 2021
Cited by 6 | Viewed by 1574
Abstract
This paper studies a discrete-time dynamic duopoly game with homogenous goods. Both firms have to decide on investment where investment increases production capacity so that they are able to put a larger quantity on the market. The downside, however, is that a larger [...] Read more.
This paper studies a discrete-time dynamic duopoly game with homogenous goods. Both firms have to decide on investment where investment increases production capacity so that they are able to put a larger quantity on the market. The downside, however, is that a larger quantity raises pollution. The firms have multiple objectives in the sense that each one maximizes the discounted profit stream and appreciates a clean environment as well. We obtain some surprising results. First, where it is known from the continuous-time differential game literature that firms invest more under a feedback information structure compared to an open-loop one, we detect scenarios where the opposite holds. Second, in a feedback Nash equilibrium, capital stock is more sensitive to environmental appreciation than in the open-loop case. Full article
(This article belongs to the Special Issue Multicriteria Games)
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