Shapley Value of Uncertain Coalitional Game based on Hurwicz Criterion with Application to Water Resource Allocation

Coalitional game studies the situation where the players cooperate. In an actual game, due to a lack of information, the payoffs are generally hard to be precisely estimated. To deal with this problem, researchers of uncertainty theory supposed the transferable payoffs to be uncertain variables and proposed the uncertain coalitional game. Prior scholars have discussed the uncertain core, uncertain Shapley value, and uncertain stable set under the expected value criterion and optimistic value criterion as solution concepts for an uncertain coalitional game. However, the expected value criterion does not consider the players’ attitude to the risk, and the optimistic criterion is too extreme to maximize the maximum uncertain payoff. Therefore, we propose the $$(\alpha ,\rho )$$ ( α , ρ ) -Hurwicz–Shapley value as the solution based on the Hurwicz criterion to overcome severe cases. Besides, several properties of the $$(\alpha ,\rho )$$ ( α , ρ ) -Hurwicz–Shapley value are discussed, and the uniqueness is proved. At last, an example of the cooperation of water resource users is offered to illustrate the validity of the $$(\alpha ,\rho )$$ ( α , ρ ) -Hurwicz–Shapley value.