Cooperative grey games and the grey Shapley value

This contribution is located in the common area of operational research and economics, with a close relation and joint future potential with optimization: game theory. We focus on collaborative game theory under uncertainty. This study is on a new class of cooperative games where the set of players is finite and the coalition values are interval grey numbers. An interesting solution concept, the grey Shapley value, is introduced and characterized with the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory. The paper ends with a conclusion and an outlook to future studies

On The Characterizations of The Interval Sequential Equal Contributions Rule

Abstract: This paper deals with the research area of cooperative interval games arising from airport situations with interval data. The major topic of the paper is to present and introduce the interval sequential equal contributions rule. The main result of this study is to give an axiomatic characterization of the interval sequential equal contributions rule. Key words: airport situations, sequential equal contributions rule, interval data. Aralıklı Ardışık Eşit Dağıtım Kuralının Karakterizasyonları Üzerine Özet: Bu makale aralıklı havaalanı durumlarından ortaya çıkan işbirlikçi aralıklı oyunları araştırmak üzerine yazılmıştır. Bu makalenin en büyük katkısı aralıklı ardışık eşit dağıtım kuralından ilk defa bahsetmektir. Bu çalışmanın en önemli sonucu ise aralıklı ardışık eşit dağıtım kuralının aksiyomatik olarak karakterize edilmesidir. Anahtar Kelimeler: havaalanı durumları, ardışık eşit dağıtım kuralı, aralık kavramı

An Axiomatization of the Interval Shapley Value and on Some Interval Solution Concepts

The Shapley value, one of the most common solution concepts in Operations Research applications of cooperative game theory, is defined and axiomatically characterized in different game-theoretical models. In this paper, we focus on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. In this study, we study the properties of the interval Shapley value on the class of size monotonic interval games, and axiomatically characterize its restriction to a special subclass of cooperative interval games by using fairness property, efficiency and the null player property. Further, we introduce the interval Banzhaf value and the interval egalitarian rule. Finally, the paper ends with a conclusion and an outlook to future studies

Cooperative games under bubbly uncertainty

The allocation problem of rewards/costs is a basic question for players, namely, individuals and companies that are planning cooperation under uncertainty. The involvement of uncertainty in cooperative game theory is motivated by the real world in which noise in observation and experimental design, incomplete information and vagueness in preference structures and decision-making play an important role. In this study, a new class of cooperative games, namely, the cooperative bubbly games, where the worth of each coalition is a bubble instead of a real number, is presented. Furthermore, a new solution concept, the bubbly core, is defined. Finally, the properties and the conditions for the non-emptiness of the bubbly core are given. The paper ends with a conclusion and an outlook to related and future studies

Alternative Axiomatic Characterizations of the Grey Shapley Value

The Shapley value, one of the most common solution concepts of cooperative game theory is defined and axiomatically characterized in different game-theoretic models. Certainly, the Shapley value can be used in interesting sharing cost/reward problems in the Operations Research area such as connection, routing, scheduling, production and inventory situations. In this paper, we focus on the Shapley value for cooperative games, where the set of players is finite and the coalition values are interval grey numbers. The central question in this paper is how to characterize the grey Shapley value. In this context, we present two alternative axiomatic characterizations. First, we characterize the grey Shapley value using the properties of efficiency, symmetry and strong monotonicity. Second, we characterize the grey Shapley value by using the grey dividends

On the Grey Equal Surplus Sharing Solutions

The grey uncertainty is a new methodology focusing on the study of problems involving small samples and poor information. It deals with uncertain systems with partially known information through generating, excavating, and extracting useful information from what is available. This paper focuses some division solutions for cooperative games, called the equal surplus sharing solutions. A situation, in which a finite set of players can obtain certain grey payoffs by cooperation can be described by a cooperative grey game. In this paper, we consider some grey division rules, namely the equal surplus sharing grey solutions. Further, we focus on a class of equal surplus sharing grey solutions consisting of all convex combinations of these solutions. An application from Operations Research (OR) situations is also given

An Application of Cooperative Grey Games to Post-Disaster Housing Problem

This paper shows that cooperative grey game theory can help us to establish a fair cost share between private organizations for supporting the temporary housing problem by using facility location games under uncertainty. Temporary accommodation may be a method that ought to get started before the tragedy happens, as a preventative pre-planning. In spite of being temporary constructions, the housing buildings are one of the most essential parts to produce in emergency situations, to contribute to the reconstruction and to recover better. Our study is based on a default earthquake in Izmir of western Turkey. A number of tents are being built in the following three cities, Aydin, Usak, and Balikesir near Izmir as illustrated in Figure 1. Two companies are selected, one is local and another is foreign to distribute the tents in a fair way between the three cities. For this purpose, we use cooperative grey game theory to help us to define a fair cost allocation between private organizations for supporting the housing problem by using facility location games under uncertainty

Transportation interval situations and related games

Basically, uncertainty is present in almost every real-world situation, it is influencing and questioning our decisions. In this paper, we analyze transportation interval games corresponding to transportation interval situations. In those situations, it may affect the optimal amount of goods and consequently whether and how much of a product is transported from a producer to a retailer. Firstly, we introduce the interval Shapley value of a game arising from a transportation situation under uncertainty. Secondly, a one-point solution concept by using a one-stage producere depending on the proportional, the constrained equal awards and the constrained equal losses rule is given. We prove that transportation interval games are interval balanced (I-balanced). Further, the nonemptiness of the interval core for the transportation interval games and some results on the relationship between the interval core and the dual interval optimal solutions of the underlying transportation situations are also provided. Moreover, we characterize the interval core using the square operator and addressing two scenarios such as pessimistic and optimistic

Airport Situations and Games with Grey Uncertainty

In this work, we deal with airport situations where the costs of the pieces of the runway are given by grey numbers. In this context, we expand the Baker-Thompson rule as a solution concept. Some properties regarding an allocation problem of an airport situation under uncertainty is considered and grey solutions are proposed. We introduce grey Baker-Thompson rule. Further, we give the axiomatic characterization of the grey Baker-Thompson rule by using the major and the minor axioms

Cooperative Grey Games: Grey Solutions and an Optimization Algorithm

In this paper, some set-valued solutions using grey payoffs, namely, the grey core, the grey dominance core and the grey stable sets for cooperative grey games, are introduced and studied. Our main results contained are relations between the grey core, the grey dominance core and the grey stable sets of such a game. Moreover, we present a linear programming (LP) problem for the grey core. On the other hand, we suggest a corresponding optimization-basedalgorithm finding the grey core element of a cooperative grey game. Finally, we give an application how cooperative grey game theory can be used to model users' behaviors in various multimedia social networks. The paper ends with aconclusion and an outlook to future investigations