Two extensions of the shapley value for cooperative games
Two extensions of the Shapley value for transferable utility cooperative games are provided, both dealing with the costs sharing problem. The first one is based on the following probabilistic model. Let a cooperative (cost) game \((N,c)\) and also a pre-specified cost allocation efficient vector \(x\) be given. Assume that the player set \(N\) splits into two nonempty, complementary coalitions \(S\) and \(N\setminus S\) with some probability. Each such coalition chooses a leader among its members with some probability. Then the overall cost \(c(N)\) is divided among both leaders according to the standard solution for two-player games, that is, in terms of \(c(S)\) and \(c(N\setminus S)\). Next, the leader of each coalition charges to the other members the amounts given by \(x\) and, in turn, is charged with the remainder of \(c(S)\). The pre-specified allocation vector \(x\) is said to be ``consistent'' if the expected amount for each player under the given probability model coincides with the allocation assigned by \(x\). The first main result states that, under three reasonable assumptions on the probabilistic distributions, there exists a unique consistent allocation rule, whose explicit formula is also given. As a corollary, in the case where the probability distributions are uniform -- in a naturally symmetric sense -- it is shown that the unique consistent allocation rule coincides with the Shapley value of the game, as found in [\textit{R. A. Evans}, Games Econ. Behav. 12, 68--80 (1996; Zbl 0846.90134)]. A second result refers to the consistent allocation rules that verify a dummy player property. It offers two conditions equivalent to this: one in terms of additional restrictions to be satisfied by the probability distributions, and the other characterizing the rule as a probabilistic marginalistic allocation in the sense of [\textit{R. J. Weber}, in: The Shapley Value: Essays in Honor of Lloyd S. Shapley, 101--119 (1989; Zbl 0707.90100)]. The second extension concerns \textit{Y. Sprumont}'s [Games Econ. Behav. 2, 378--394 (1990; Zbl 0753.90083)] recursive formula for the Shapley value. Now it is assumed that the player set is partially ordered (poset) and only a class of ``permitted'' coalitions is allowed to work. After introducing the useful notion of ``unrestricted'' player (with regard to a coalition the player belongs to), the first statement is that a probabilistic marginalistic allocation rule can again be recursively determined. Finally, additional restrictions on the sets of unrestricted players lead to state that if the poset game is quasi-concave then the allocation rule lies in the core of the game.
scientific article; zbMATH DE number 1949999