The Shapley value based on hesitant fuzzy linguistic comprehensive entropy and its application in noncooperative-cooperative biform game

•New construction approach of exponential entropy measures for HFLTSs is derived.•Confidence level based on hesitant fuzzy linguistic comprehensive entropy is given.•Biform game with hesitant fuzzy linguistic Shapley value is defined. Profit allocation is a decision-making process, and noncooperative-cooperative biform game can effectively depict the real competitive-cooperative relationship. This paper aims to construct an effective Shapley function of biform game in hesitant fuzzy linguistic environment, which can reasonably analyze the formation and the distribution of profits by taking into account the individual hesitation and information fuzziness. Firstly, we propose a new construction method of an entropy for hesitant fuzzy linguistic term sets (HFLTSs). This entropy measure can describe the linguistic information from the aspects of fuzziness and hesitation, which improves the flaws of the existing entropy for HFLTSs. After that, a function for calculating confidence level is defined based on comprehensive entropy. Then in order to study the profit allocation in a competitive-cooperative environment, we define hesitant fuzzy linguistic biform game and its characteristic function, and the effectiveness of the proposed Shapley value function according to an axiomatic system is proved. Finally, the feasibility of this approach is indicated by an example of profit allocation of cooperation in watershed management considering information asymmetry.