A value for communication situations with players having different bargaining abilities
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A value for communication situations with players having different bargaining abilities C. Manuel1
· D. Martín1
Accepted: 3 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The aim of this paper is to extend the Myerson value (Myerson in Math Oper Res 2:225– 229, 1977) to situations in which players in a TU-game, in addition to having cooperation possibilities restricted by a graph, also have different bargaining abilities. Then, we will associate to each player in a communication situation a weight in the interval [0, 1] that measures his bargaining ability. A unitary weight corresponds to a fully cooperative player whereas a null weight corresponds to a player that is not willing to cooperate in any way. Intermediate values modulate the bargaining ability. We modify the original TU-game to a new game which is, in turn, a modification of the Myerson’s graph-restricted game. We will assume that the reduction in the will to cooperate implies that players can not obtain the total dividend of the connected coalitions which must be discounted by an appropriate factor. Then, we propose as a solution for these situations the Shapley value (Shapley, in: Kuhn, Tucker (eds) Annals of mathematics studies, Princeton University Press, Princeton, 1953) of the modified game. This solution extends the Myerson value (and also the Shapley value). Moreover it satisfies monotonicity in the weights. Different characterizations of this rule can be obtained. They are based on properties as bargaining component efficiency, fairness, balanced contributions and balanced bargaining ability contributions, and thus they are parallel to those more prominent existing in the literature for the Myerson value. Keywords Game theory · Communication situation · Weighted game · Bargaining ability · Myerson value
1 Introduction In game theory, a situation in which several actors or players can obtain (transferable) payoffs by cooperating is modeled by means of a TU-game. It is assumed that all possible player’s coalitions are feasible, the value of each coalition depending (in general) on its members.
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C. Manuel [email protected] D. Martín [email protected]
1
Departamento de Estadística y Ciencia de los Datos, Facultad de Estudios Estadísticos, Universidad Complutense de Madrid, Av. Puerta de Hierro s/n, 28040 Madrid, Spain
123
Annals of Operations Research
In his path-breaking work, Myerson (1977) analyzed the situations in which the players in a TU-game are members of a network (graph) that reduces their communications possibilities. He proposed to replace the original game by a new one, the graph-restricted game, in which the outcome of a coalition is the sum of the payoffs obtained in the original game by its maximally connected (in the network) subcoalitions. Given the prominence of the Shapley value (Shapley 1953b) as point solution for TU-games, Myerson proposed the Shapley value of the graph-restricted game (now known as the Myerson value) as a solution for TU-games with cooperation restricted b
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