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AbstractMany coalitional values have been introduced in the literature on cooperative games over the last decades, and especially since 2000. The multiplicity of options suggests the convenience of testing the existence of stable coalition structures, in the sense of Hart and Kurz (1983, Econometrica), when payments are made using some of these values. We recall their concept of $$\gamma$$
γ
–stability and give results for the proportional partitional Shapley value, introduced by Alonso–Meijide et al (2015, Discrete Appl. Math.), which shares the utility of any coalition proportionally to the Shapley value of the involved players in the original game.

In recent years, China&rsquo ; s government has encouraged the adoption of the TOT (Transfer-Operate-Transfer) model to realize the marketization of China&rsquo ; s public service stock projects. The TOT model is a cooperation mechanism through sharing investment, revenue and risks between the government and private partner. Therefore, a fair and reasonable revenue sharing method (RSM) is the key to the success of the TOT project. This paper aims to provide a fair and reasonable RSM based on a modified Shapley value with a triangular symmetric fuzzy structure element, which has better motivation, flexibility, forecasting function and dynamic precise distribution function. According to the factors that affect revenue sharing, the Shapley value is improved with initial correction coefficient composed of investment ratio, risk-sharing ratio, execution degree, and fuzzy payment to achieve fairness and reasonableness. The methodology is illustrated by a case study of a TOT project selected from Laohekou city of Hubei province, China. The results testify that the revenue-sharing ratios of participants is positively correlated with the initial correction coefficient, which make the RSM more motivating ; and the Shapley value with fuzzy payment by using triangular symmetric fuzzy element function make the RSM more flexible, and it has both forecasting function and precise dynamic distribution function under project revenue uncertainty.

The interactions of a group of non-identical voting units may be studied by applying the concept of the Shapley value from n-person co-operative game theory. In this theory one assumes that voting units, or players, may form coalitions of various kinds in order to achieve success in the game, and one may assign a "value" to each such coalition based on what it can accomplish against arbitrary coalitions of the remaining players. The relative value of an individual player is calculated by considering how much he brings to each coalition he might join, that is, by how much the value of that coalition increases because of his membership, and summing this figure over all coalitions of which he could be a part.

The number of electric vehicles (EV) in the world has been increasing and is gaining momentum. The large-scale use of EVs in public life has initiated the need to establish EV battery charging services within the power system. Currently, EVs serve as a transportation tool and also as a flexible load. This publication examines the possibility of the owner of an electric vehicle choosing a battery recharging point, as well as of the involvement of several decision makers in the selection of a charging schedule. This problem is important because we assume that a significant proportion of EVs mainly use two parking spaces, one located close to the place of residence and another close to the workplace. We accept and prove that a car charging station can be created by the employer (company) and implemented in the best interests of the employer and the employee (EV owner). For that, a coalition between the company and the EV owner has to be formed. To support rational decisions, this study solves the problem using the cooperative game theory and designs a payment distribution mechanism based on the Shapley value. The results obtained prove that the coalition is beneficial under different conditions, which depend on the capacity of the EV, the distance between the workplace and the place of residence, the difference in the electricity prices of the day, as well as the consumption of the company. In order to estimate the coalition&rsquo ; s gain, it is necessary to take into account the structure of the power tariff system for both the company and the EV owner. Furthermore, we prove that the presence of a coalition allows the company and the EV owner to reduce the annual fee for consumed power. The results of this analysis could be adopted by decision makers such as government agencies, companies, EV owners, and they are recommended for potential investors for the development of transport electrification and smart energy.