N-person Cooperative Game Research Articles

Optimal resource allocation in hierarchical organizations under uncertainty: An interval-valued n-person cooperative game approach

In many real production scenarios, departmental organizations often exhibit a hierarchical structure, where departments cooperate with subordinate departments to optimize resource allocation and maximize their respective benefits. However, due to a lack of information or data, many model parameters in the allocation process cannot be precisely defined. In response to this challenge, an interval n-person hierarchical resource allocation model is proposed to achieve maximum economic benefit in uncertain environments. Based on the concepts of satisfactory degrees of comparing intervals and interval-valued cores of interval-valued n-person cooperative games, an auxiliary nonlinear programming model and method are developed to solve the interval-valued cores of such cooperative games. The approach explicitly considers the inclusion and/or overlap relations between intervals, whereas the traditional interval ranking method may not guarantee the existence of interval-valued cores. The proposed method offers cooperative opportunities under uncertain conditions. Finally, the feasibility and applicability of the models and methods are demonstrated through a numerical example and comparison with other methods.

The Choquet integral-based Shapley function for n-person cooperative games with probabilistic hesitant fuzzy coalitions

The use of probabilistic hesitant fuzzy elements (PHFEs) to represent the alliance status in cooperative games can describe the player's participation levels and associated probability. This paper proposes an effective Shapley function for probabilistic hesitant fuzzy cooperative games based on generalized Choquet integral. We first define the probabilistic hesitant fuzzy coalition of cooperative games. After that, a supplementary model of incomplete probability information is established. Without losing any probability information, an adjustment algorithm is developed to derive the consistent probability of coalitions. Then the characteristic function of probabilistic hesitant fuzzy cooperative games is constructed based on generalized Choquet integral. The Choquet integral-based Shapley function is presented according to the proposed characteristic function. Some desirable properties of Shapley function are discussed in detail. The CIG algorithm is established for generating consistent imputation of cooperative games. Finally, the proposed algorithm is applied to the cooperative production of alloy materials.

Retraction

Abstract Original Article Title: Existence of k-People Stable Alliance in n-Player Cooperative Games Authors: Zheng Tao,1 Yuanwei Hu,2 Ziyi Pan,2 Junyi Zhao,2 and Dongao Xing2 This article has been retracted upon request from the authors due to errors they found in the conclusions. ASTM International February 2022

Editorial: Special Section on Human-Centered Artificial Intelligence with Big Data Applications

Editorial: Special Section on Human-Centered Artificial Intelligence with Big Data Applications

Owen-stable coalition partitions in games with vector payoffs

The paper is devoted to the study of multicriteria cooperative games with vector payoffs and coalition partition. The imputation which is based on the concept of the Owen value is proposed. We use it for the definition of stable coalition partition for bicriteria games. In three person cooperative game with 0-1 characteristic function the conditions under which the coalition partition is stable are found.

Matrix expression of Shapley values and its application to distributed resource allocation

The symmetric and weighted Shapley values for cooperative n-person games are studied. Using the semi-tensor product of matrices, it is first shown that a characteristic function can be expressed as a pseudo-Boolean function. Then, two simple matrix formulas are obtained for calculating the symmetric and weighted Shapley values. Finally, using these new formulas, a design technique for the agents’ payoff functions in distributed resource allocation problems is proposed. It is possible to design payoff functions with the weighted Shapley value by the nonsymmetric weights defined on the players, thus ensuring that the optimal allocation is a pure Nash equilibrium. Practical examples are presented to illustrate the theoretical results.

A Game Theoretic Approach to Estimate Fair Cost of VM Placement in Cloud Data Center

Pricing of virtual machines (VMs) with different dimensions is a challenging task. VM pricing involves both capital and operational expenditures. Capital expenditure is fixed in nature, while operational expenditure is variable. A fraction of capital expenditure is included for VM pricing. An individual pays the cloud service provider (CSP) for his requested VM. If the users cooperate among themselves they may end up paying less to CSP for their requested VMs vis-a-vis they would have paid had they requested individually. In this paper, an n -person cooperative game is adopted to determine the price that users would pay for their requested VMs under a cooperative environment. Shapley value is used to estimate the fraction of capital expenditure that would be included in the VM price. An integer linear programming is proposed for energy-efficient placement. For evaluation, VM configurations and pricing of popular CSPs—Microsoft Azure and Amazon EC2—are considered. Results show that users would pay less for their requested VMs if they cooperate. The energy consumed by the proposed VM placement technique is compared with first fit decreasing (FFD) and enhanced first fit decreasing (EFFD). It is observed that the proposed technique consumes lesser energy compared to FFD and EFFD.

Cooperative game-theoretic perspectives on global climate action: Evaluating international carbon reduction agreements

ABSTRACTThe theory of cooperative n-person games offers a rigorous approach for analysing multilateral real-world agreements, but its practical application is hindered by the exacting data requirements demanded by the fully specified theoretical models. In this article, we demonstrate how the formal analytic can be made more amenable to application. We utilize our approach to model international climate negotiations as an n-person cooperative game, the solution of which allocates carbon reductions across the grand coalition of nations. Using a simplified game to represent the carbon reduction allocation problem, we obtain theoretical solutions using a game-theoretic concept known as the proportional nucleolus. The solution to the game allows us to ideally determine countries’ relative percentage carbon reductions. These theoretical results are compared against actual commitments established in the Paris Agreement of 2015. The paper discusses the implications of the game-theoretic results, including the significant under-commitment of nations such as the United States. More generally, the approach developed herein provides an illustration of how rigorous game-theoretic methods can be adapted to the practical considerations of policy analysis.

Interval-valued n-person cooperative games with satisfactory degree constraints

The aim of this study is to develop several nonlinear programming models for interval-valued cooperative games in which taking into account the decision makers’ risk attitudes. First, we investigate several existing used satisfactory degree comparison methods for ranking interval-valued fuzzy numbers, and point out by an example that the method proposed by Liu et al. (Soft Comput 22:2557–2565, 2018) is more efficient than the method proposed by Hong and Li (Oper Res 17:1–19, 2016). Second, by taking into account decision makers’ risk attitudes, several corresponding nonlinear programming models are constructed based on satisfactory degree formulas that were proposed by Liu et al. (2018). Third, an illustrative example in conjunction with comparative analyses are employed to demonstrate the validity and applicability of the proposed models. Finally, to further highlight the validity of the proposed method, we discuss the relationship of the satisfactory degree formulas between Hong and Li (2016)’s method and Xu and Da (J Syst Eng 18:67–70, 2003)’s method.

An approach to solve multi-objective linear production planning games with fuzzy parameters

In this research, n-person cooperative games, arising from multi-objective linear production planning problem with fuzzy parameters, are considered. It is assumed that the fuzzy parameters are fuzzy numbers. The fuzzy multi-objective game problem is transformed to a single-objective game problem by group AHP method. The obtained problem is converted to a problem with interval parameters by considering the nearest interval approximation of the fuzzy numbers. Then, optimistic and pessimistic core concepts are introduced. The payoff vectors of the players are obtained by the duality theorem of linear programming. Finally, validity and applicability of the method are illustrated by a practical example.

The general prenucleolus of n-person cooperative fuzzy games

In this paper, we define the concept of the general prenucleolus minimizing the player complaint vector in the lexicographic order over the preimputation set for the cooperative games with fuzzy coalitions. For the general prenucleolus, we get a sufficient condition that the complaints of all players are in equal amount under the case of linear complaint functions concluding the proposed player excess (Sakawa and Nishizaki (1994) [15]). As a result, we can obtain the proposed fuzzy solutions with the corresponding linear complaint functions, such as the equalizer solution (Molina and Tejada (2002) [5]) and Shapley function etc. We prove that the least square general prenucleolus minimizing the variance of the resulting player complaints, is the general prenucleolus. Thus, an optimal solution considered from two aspects of the lexicographic order and the least square criterion, is obtained. In addition, several general prenucleoli are proposed in terms of specific situations.

Why macroeconomic coordination may not be possible in a monetary union: A game theoretic approach

In order to explore the possibility of macroeconomic policy coordination in monetary unions, we model the monetary union as an n-person cooperative game. A key equilibrium concept of this game is the core, which is defined as the set of outcomes that can be blocked by no coalition. It follows that in a monetary union, coordination is possible if the monetary game possesses a core, i.e., when the joint outcome, obtained if all member countries coordinate their activities, cannot be challenged by anyone. Thus, coordination is possible in all cases, in which the existing economic conditions eliminate all outcomes that any subset of countries could improve upon. And since these economic conditions are summarized by the characteristic function of the game, coordination (or the failure of coordination) of economic policies in a monetary union is determined by its properties.

Nonlinear programming method for interval-valued n-person cooperative games

The aim of this paper is to develop a nonlinear programming method for computing the elements of the interval-valued cores of n-person cooperative games in which coalitions’ values are expressed with intervals, which are often called interval-valued n-person cooperative games for short. With finding out the maximum satisfactory degree in the situation with the features of inclusion and/or overlap relations between intervals, this paper tries to explore the cooperation chance in this type of cooperative games. Firstly, we define the concept of interval-valued cores of interval-valued n-person cooperative games and satisfactory degrees (or ranking indexes) of comparing intervals with the features of inclusion and/or overlap relations. Hereby, we propose the auxiliary nonlinear programming model and method for solving interval-valued cores of any interval-valued n-person cooperative games. The developed method can provide cooperative chance under the situation of inclusion and/or overlap relations between intervals, while the traditional interval ranking method may not assure that the interval-valued cores exist. This method is a complement to the traditional methods rather than the alternative one. The feasibility and applicability of the model and method proposed in this paper are illustrated with a numerical example.

N-Person cooperative games and geometrical aspects: a brief survey and new perspectives

In this paper we consider cooperative games and various analytical tools in order to study behavioral situations of a set of players and to analyze different classes of games in a cooperative way. Following these approaches we can consider these classes of games as cones characterized by particular sets of generators. Mathematics Subject Classification: 91A12

Approximating Nash Equilibrium Using Genetic Algorithm

The problem of detecting the Nash equilibrium of a non cooperative n-person game is solved by introducing a non-linear optimization model that enables evolutionary search operators to converge towards Nash Equilibrium of a game.

CHOQUET EXTENSION OF COOPERATIVE GAMES

A multilinear extension of the n-person cooperative game was introduced by Owen in 1972, and a new extension method is proposed in this paper. For n-person cooperative games, any coalition can equivalently be represented by its characteristic vectors. By means of the Choquet integral, a new fuzzy extension, called the Choquet extension, is developed. Furthermore, a Shapley function in this class of fuzzy cooperative games with the Choquet extension form is defined. Axioms of the Shapley function are proposed, and an explicit formula for the Shapley function is given. Finally, an equivalent definition of this Shapley function is discussed.

Finding the nucleolus of any n-person cooperative game by a single linear program

In this paper we show a new method for calculating the nucleolus by solving a unique minimization linear program with O(4n) constraints whose coefficients belong to {−1,0,1}. We discuss the need of having all these constraints and empirically prove that they can be reduced to O(kmax2n), where kmax is a positive integer comparable with the number of players. A computational experience shows the applicability of our method over (pseudo)random transferable utility cooperative games with up to 18 players.

A Closed-form Formula for the Fair Allocation of Gains in Cooperative N-person Games

Abstract This paper provides a closed-form optimal solution to the multi-objective model of the fair allocation of gains obtained by cooperation among all players. The optimality of the proposed solution is first proved. Then, the properties of the proposed solution are investigated. At the end, a numerical example in inventory control environment is given to demonstrate the application and to compare the results with the ones of an existing method.

A STRATEGIC COOPERATIVE GAME-THEORETIC MODEL FOR MARKET SEGMENTATION WITH APPLICATION TO BANKING IN EMERGING ECONOMIES

Market segmentation is essential to target efficaciously core-segment customers and to obtain a competitive advantage. Firms when confronted by the range of market segments, have difficulty in deciding the core-segment customers who are the most probable purchasers of their product and services. We propose a novel fuzzy group multi-criteria method for market entry and segment evaluation and selection. This proposed method provides a comprehensive and systematic framework that combines bi-level multi-objective optimization with real option analysis (ROA) and fuzzy cooperative n-person game theory. The contribution of the proposed segment evaluation and selection method is fivefold: (1) it addresses the gaps in the marketing literature on the efficacious and effective assessment of market segments; (2) it provides a comprehensive and systematic framework that combines bi-level multi-objective optimization with ROA and fuzzy cooperative n-person game theory; (3) it considers fuzzy logic and fuzzy sets to represent ambiguous, uncertain or imprecise information; (4) it does not insist on consensus but synthesizes a representative outcome based on qualitative judgments and quantitative data; and (5) it is applicable to national and international market segmentation. The practical application of this proposed framework illustrates the efficacy of the procedures and algorithms.

Note on non-uniform bin packing games

A non-uniform bin packing game is an N-person cooperative game, where the set N is defined by k bins of capacities b1,…,bk and n items of sizes a1,…,an. The objective function v of a coalition is the maximum total value of the items of that coalition which can be packed to the bins of that coalition. We investigate the taxation model of Faigle and Kern (1993) [2] and show that the 1/2-core is always nonempty for such bin packing games. If all items have size strictly larger than 1/3, we show that the 5/12-core is always non-empty. Finally, we investigate the limiting case k→∞, thereby extending the main result in Faigle and Kern (1998) [3] to the non-uniform case.