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20 pages, 829 KiB

Open AccessArticle

Characterizations, Potential, and an Implementation of the Shapley-Solidarity Value

by Jun Su, Yuan Liang, Guangmin Wang and Genjiu Xu

Mathematics 2020, 8(11), 1965; https://doi.org/10.3390/math8111965 - 05 Nov 2020

Viewed by 1402

In this paper, we provide cooperative and non-cooperative interpretations of the Shapley–Solidarity value for cooperative games with coalition structure. Firstly, we present two new characterizations of this value based on intracoalitional quasi-balanced contributions property. Secondly, we study a potential function of the Shapley–Solidarity [...] Read more.

In this paper, we provide cooperative and non-cooperative interpretations of the Shapley–Solidarity value for cooperative games with coalition structure. Firstly, we present two new characterizations of this value based on intracoalitional quasi-balanced contributions property. Secondly, we study a potential function of the Shapley–Solidarity value. Finally, we propose a new bidding mechanism for the Solidarity value and then apply the result to the Shapley–Solidarity value. Full article

(This article belongs to the Special Issue Game Theory)

16 pages, 334 KiB

Open AccessArticle

Rational Behavior in Dynamic Multicriteria Games

by Anna Rettieva

Mathematics 2020, 8(9), 1485; https://doi.org/10.3390/math8091485 - 02 Sep 2020

Cited by 4 | Viewed by 1227

Abstract

We consider a dynamic, discrete-time, game model where n players use a common resource and have different criteria to optimize. To construct a multicriteria Nash equilibrium the bargaining solution is adopted. To design a multicriteria cooperative equilibrium, a modified bargaining scheme that guarantees [...] Read more.

We consider a dynamic, discrete-time, game model where n players use a common resource and have different criteria to optimize. To construct a multicriteria Nash equilibrium the bargaining solution is adopted. To design a multicriteria cooperative equilibrium, a modified bargaining scheme that guarantees the fulfillment of rationality conditions is applied. The concept of dynamic stability is adopted for dynamic multicriteria games. To stabilize the multicriteria cooperative solution a time-consistent payoff distribution procedure is constructed. The conditions for rational behavior, namely irrational-behavior-proofness condition and each step rational behavior condition are defined for dynamic multicriteria games. To illustrate the presented approaches, a dynamic bi-criteria bioresource management problem with many players is investigated. Full article

(This article belongs to the Special Issue Game Theory)

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14 pages, 361 KiB

Open AccessFeature PaperArticle

On a Simplified Method of Defining Characteristic Function in Stochastic Games

by Elena Parilina and Leon Petrosyan

Mathematics 2020, 8(7), 1135; https://doi.org/10.3390/math8071135 - 11 Jul 2020

Cited by 2 | Viewed by 2114

Abstract

In the paper, we propose a new method of constructing cooperative stochastic game in the form of characteristic function when initially non-cooperative stochastic game is given. The set of states and the set of actions for any player is finite. The construction of [...] Read more.

In the paper, we propose a new method of constructing cooperative stochastic game in the form of characteristic function when initially non-cooperative stochastic game is given. The set of states and the set of actions for any player is finite. The construction of the characteristic function is based on a calculation of the maximin values of zero-sum games between a coalition and its anti-coalition for each state of the game. The proposed characteristic function has some advantages in comparison with previously defined characteristic functions for stochastic games. In particular, the advantages include computation simplicity and strong subgame consistency of the core calculated with the values of the new characteristic function. Full article

(This article belongs to the Special Issue Game Theory)

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17 pages, 800 KiB

Open AccessArticle

Maximizing the Minimal Satisfaction—Characterizations of Two Proportional Values

by Wenzhong Li, Genjiu Xu and Hao Sun

Mathematics 2020, 8(7), 1129; https://doi.org/10.3390/math8071129 - 10 Jul 2020

Cited by 2 | Viewed by 1451

Abstract

A class of solutions are introduced by lexicographically minimizing the complaint of coalitions for cooperative games with transferable utility. Among them, the nucleolus is an important representative. From the perspective of measuring the satisfaction of coalitions with respect to a payoff vector, we [...] Read more.

A class of solutions are introduced by lexicographically minimizing the complaint of coalitions for cooperative games with transferable utility. Among them, the nucleolus is an important representative. From the perspective of measuring the satisfaction of coalitions with respect to a payoff vector, we define a family of optimal satisfaction values in this paper. The proportional division value and the proportional allocation of non-separable contribution value are then obtained by lexicographically maximizing two types of satisfaction criteria, respectively, which are defined by the lower bound and the upper bound of the core from the viewpoint of optimism and pessimism respectively. Correspondingly, we characterize these two proportional values by introducing the equal minimal satisfaction property and the associated consistency property. Furthermore, we analyze the duality of these axioms and propose more approaches to characterize these two values on basis of the dual axioms. Full article

(This article belongs to the Special Issue Game Theory)

29 pages, 1169 KiB

Open AccessArticle

Resource Exploitation in a Stochastic Horizon under Two Parametric Interpretations

by José Daniel López-Barrientos, Ekaterina Viktorovna Gromova and Ekaterina Sergeevna Miroshnichenko

Mathematics 2020, 8(7), 1081; https://doi.org/10.3390/math8071081 - 03 Jul 2020

Cited by 5 | Viewed by 2299

Abstract

This work presents a two-player extraction game where the random terminal times follow (different) heavy-tailed distributions which are not necessarily compactly supported. Besides, we delve into the implications of working with logarithmic utility/terminal payoff functions. To this end, we use standard actuarial results [...] Read more.

This work presents a two-player extraction game where the random terminal times follow (different) heavy-tailed distributions which are not necessarily compactly supported. Besides, we delve into the implications of working with logarithmic utility/terminal payoff functions. To this end, we use standard actuarial results and notation, and state a connection between the so-called actuarial equivalence principle, and the feedback controllers found by means of the Dynamic Programming technique. Our conclusions include a conjecture on the form of the optimal premia for insuring the extraction tasks; and a comparison for the intensities of the extraction for each player under different phases of the lifetimes of their respective machineries. Full article

(This article belongs to the Special Issue Game Theory)

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20 pages, 403 KiB

Open AccessArticle

Subgame Consistent Cooperative Behavior in an Extensive form Game with Chance Moves

by Denis Kuzyutin and Nadezhda Smirnova

Mathematics 2020, 8(7), 1061; https://doi.org/10.3390/math8071061 - 01 Jul 2020

Cited by 4 | Viewed by 1772

Abstract

We design a mechanism of the players’ sustainable cooperation in multistage n-person game in the extensive form with chance moves. When the players agreed to cooperate in a dynamic game they have to ensure time consistency of the long-term cooperative agreement. We [...] Read more.

We design a mechanism of the players’ sustainable cooperation in multistage n-person game in the extensive form with chance moves. When the players agreed to cooperate in a dynamic game they have to ensure time consistency of the long-term cooperative agreement. We provide the players’ rank based (PRB) algorithm for choosing a unique cooperative strategy profile and prove that corresponding optimal bundle of cooperative strategies satisfies time consistency, that is, at every subgame along the optimal game evolution a part of each original cooperative trajectory belongs to the subgame optimal bundle. We propose a refinement of the backwards induction procedure based on the players’ attitude vectors to find a unique subgame perfect equilibrium and use this algorithm to calculate a characteristic function. Finally, to ensure the sustainability of the cooperative agreement in a multistage game we employ the imputation distribution procedure (IDP) based approach, that is, we design an appropriate payment schedule to redistribute each player’s optimal payoff along the optimal bundle of cooperative trajectories. We extend the subgame consistency notion to extensive-form games with chance moves and prove that incremental IDP satisfies subgame consistency, subgame efficiency and balance condition. An example of a 3-person multistage game is provided to illustrate the proposed cooperation mechanism. Full article

(This article belongs to the Special Issue Game Theory)

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13 pages, 279 KiB

Open AccessArticle

The Relationship between the Core and the Modified Cores of a Dynamic Game

by Artem Sedakov and Hao Sun

Mathematics 2020, 8(6), 1023; https://doi.org/10.3390/math8061023 - 23 Jun 2020

Viewed by 1735

Abstract

The core as a solution to a cooperative game has the advantage that any imputation from it is undominated. In cooperative dynamic games, there is a known transformation that demonstrates another advantage of the core—time consistency—keeping players adhering to it during the course [...] Read more.

The core as a solution to a cooperative game has the advantage that any imputation from it is undominated. In cooperative dynamic games, there is a known transformation that demonstrates another advantage of the core—time consistency—keeping players adhering to it during the course of the game. Such a transformation may change the solution, so it is essential that the new core share common imputations with the original one. In this paper, we will establish the relationship between the original core of a dynamic game and the core after the transformation, and demonstrate that the latter can be a subset of the former. Full article

(This article belongs to the Special Issue Game Theory)

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24 pages, 1364 KiB

Open AccessArticle

On the Effect of the Absorption Coefficient in a Differential Game of Pollution Control

by Ekaterina Marova, Ekaterina Gromova, Polina Barsuk and Anastasia Shagushina

Mathematics 2020, 8(6), 961; https://doi.org/10.3390/math8060961 - 12 Jun 2020

Cited by 2 | Viewed by 1519

Abstract

We consider various approaches for a characteristic function construction on the example of an n players differential game of pollution control with a prescribed duration. We explore the effect of the presence of an absorption coefficient in the game on characteristic functions. As [...] Read more.

We consider various approaches for a characteristic function construction on the example of an n players differential game of pollution control with a prescribed duration. We explore the effect of the presence of an absorption coefficient in the game on characteristic functions. As an illustration, we consider a game in which the parameters are calculated based on the real ecological situation of the Irkutsk region. For this game, we compute a number of characteristic functions and compare their properties. Full article

(This article belongs to the Special Issue Game Theory)

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16 pages, 280 KiB

Open AccessArticle

Hotelling’s Duopoly in a Two-Sided Platform Market on the Plane

by Vladimir Mazalov and Elena Konovalchikova

Mathematics 2020, 8(6), 865; https://doi.org/10.3390/math8060865 - 27 May 2020

Cited by 2 | Viewed by 2956

Abstract

Equilibrium in a two-sided market represented by network platforms on the plane and heterogeneous agents is investigated. The advocated approach is based on the duopoly model which implies a continuum of agents of limited size on each side of the market and examines [...] Read more.

Equilibrium in a two-sided market represented by network platforms on the plane and heterogeneous agents is investigated. The advocated approach is based on the duopoly model which implies a continuum of agents of limited size on each side of the market and examines the agents’ heterogeneous utility with the Hotelling specification. The exact values were found for the equilibrium in the case of duopoly in a two-sided market with two platforms on the plane. The dependence of the platforms’ benefits on network externalities was investigated. The problem of the optimal location of platforms in the market was considered. Full article

(This article belongs to the Special Issue Game Theory)

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20 pages, 473 KiB

Open AccessArticle

IDP-Core: Novel Cooperative Solution for Differential Games

by Ovanes Petrosian and Victor Zakharov

Mathematics 2020, 8(5), 721; https://doi.org/10.3390/math8050721 - 04 May 2020

Cited by 5 | Viewed by 1943

Abstract

IDP-core is a new cooperative solution for dynamic and differential games. A novel approach of constructing solutions for dynamic and differential games was employed in which the time consistency property was used as the main axiom property for the cooperative solution. Another new [...] Read more.

IDP-core is a new cooperative solution for dynamic and differential games. A novel approach of constructing solutions for dynamic and differential games was employed in which the time consistency property was used as the main axiom property for the cooperative solution. Another new and important approach used for constructing IDP-core is the IDP dominance, which allows to select undominated imputation distribution procedures and construct the cooperative solution or imputation set. This approach shows the potential of using the time consistency property as the main axiom for solutions in various fields such as Social Choice and Mechanism Design. The overall procedure for defining the cooperative solution is also new since IDP-core was constructed using imputation distribution procedures but not by using imputations directly. Full article

(This article belongs to the Special Issue Game Theory)

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11 pages, 245 KiB

Open AccessArticle

An Optimal Pursuit Differential Game Problem with One Evader and Many Pursuers

by Idris Ahmed, Poom Kumam, Gafurjan Ibragimov, Jewaidu Rilwan and Wiyada Kumam

Mathematics 2019, 7(9), 842; https://doi.org/10.3390/math7090842 - 11 Sep 2019

Cited by 6 | Viewed by 3038

Abstract

The objective of this paper is to study a pursuit differential game with finite or countably number of pursuers and one evader. The game is described by differential equations in

l_{2}

-space, and integral constraints are imposed on the control function of [...] Read more.

The objective of this paper is to study a pursuit differential game with finite or countably number of pursuers and one evader. The game is described by differential equations in

l_{2}

-space, and integral constraints are imposed on the control function of the players. The duration of the game is fixed and the payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. However, we discuss the condition for finding the value of the game and construct the optimal strategies of the players which ensure the completion of the game. An important fact to note is that we relaxed the usual conditions on the energy resources of the players. Finally, some examples are provided to illustrate our result. Full article

(This article belongs to the Special Issue Game Theory)

Review

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31 pages, 2071 KiB

Open AccessReview

Applications of Game Theory in Project Management: A Structured Review and Analysis

by Mahendra Piraveenan

Mathematics 2019, 7(9), 858; https://doi.org/10.3390/math7090858 - 17 Sep 2019

Cited by 18 | Viewed by 13512

Abstract

This paper provides a structured literature review and analysis of using game theory to model project management scenarios. We select and review thirty-two papers from Scopus, present a complex three-dimensional classification of the selected papers, and analyse the resultant citation network. According to [...] Read more.

This paper provides a structured literature review and analysis of using game theory to model project management scenarios. We select and review thirty-two papers from Scopus, present a complex three-dimensional classification of the selected papers, and analyse the resultant citation network. According to the industry-based classification, the surveyed literature can be classified in terms of construction industry, ICT industry or unspecified industry. Based on the types of players, the literature can be classified into papers that use government-contractor games, contractor–contractor games, contractor-subcontractor games, subcontractor–subcontractor games or games involving other types of players. Based on the type of games used, papers using normal-form non-cooperative games, normal-form cooperative games, extensive-form non-cooperative games or extensive-form cooperative games are present. Also, we show that each of the above classifications plays a role in influencing which papers are likely to cite a particular paper, though the strongest influence is exerted by the type-of-game classification. Overall, the citation network in this field is sparse, implying that the awareness of authors in this field about studies by other academics is suboptimal. Our review suggests that game theory is a very useful tool for modelling project management scenarios, and that more work needs to be done focusing on project management in ICT domain, as well as by using extensive-form cooperative games where relevant. Full article

(This article belongs to the Special Issue Game Theory)

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