‎The main aim of this paper is to introduce the core and nucleolus notions of cooperative games with multiple scenarios in uncertain environment‎. ‎Taking imprecision of information into account‎, ‎we incorporate fuzzy coalition values‎, ‎which are represented by intuitionistic fuzzy numbers‎. ‎They can be applied as an appropriate approach to define a fuzzy set in the case that available information is not sufficient for defining an imprecise concept by means of a conventional fuzzy set‎. ‎The characteristic function of such games associates a coalition with a vector containing the intuitionistic fuzzy components‎. ‎The notion of expected interval is defined and computed for the intuitionistic fuzzy numbers‎. ‎Then‎, ‎an approach is proposed to transform the problem into a single-objective cooperative game with interval-valued payoffs‎. ‎The concepts of core and nucleolus are considered‎. ‎It is shown that the core is nonempty in these games‎. ‎A method is proposed to compute the nucleolus of such the problems‎. ‎Finally‎, ‎the validity and applicability of the approach are illustrated by a numerical example.

This paper shows that cooperative grey game theory can help us to establish a fair cost share between private organizations for supporting the temporary housing problem by using facility location games under uncertainty. Temporary accommodation might be a method that ought to get started before a tragedy happens, as a preventative pre-planning. In spite of being temporary constructions, the housing buildings are one of the most essential parts to produce in emergency situations, to contribute to the reconstruction and to recover better. Our study is based on a default earthquake in Izmir of western Turkey. A number of tents are being built in three cities near Izmir. Two companies are selected to distribute the tents in a fair way between the three cities. For this purpose, we use cooperative grey game theory to help us define a fair cost allocation between private organizations for supporting the housing problem by using facility location games under uncertainty.

Evaluation of decision-making units is very importantin economic and management systems. Data envelopment analysis is one of the scientific and practical techniques for evaluating decision making units. In the conventional models of data envelopment analysis, the units are divided into efficient and inefficient categories, where the efficiency of each efficient unit is one and there is no distinction between efficient units. This study intends to propose a new way of ranking efficient units based on the concepts of cooperative game. The proposed method is that efficient units are considered as players of a cooperative game. A subset of these players is defined as the coalitionS. Then the sum of the efficiency of the inefficient units according to the production possibility set that is created by the inefficient units and the members of the coalitionS is defined as the characteristic function of S, which is used to determine the marginal effect of the efficient units in the various coalitions. Finally, the Shapley value is used to determine the cooperative game solution and rank the efficient units. In the same way that it is able to rank non-extreme units, inefficient units are also effective in ranking, alsois feasiblein any circumstances of return to scale.

This article by modeling a non-cooperative dynamic game tries to improve profitability and competition. This paper has considered how the manufacturer interacts with multiple competitor distributors. Each distributor also determines the optimal distribution price and inventory replenishment policies to maximize their profits. The issue form a non-cooperative dynamic game. Distributors formulate sub-games and finally, have formed the main game with the manufacturer. After designing the game, we determined the Nash equilibrium. We use the concept of "Nash equilibrium" to analyze supplier, manufacturer and distributor strategies in the overall game with the manufacturer. In order to achieve the Nash equilibrium, we use decisions as input parameters. In this case, each player, in addition to making the right decision, can make decisions in order to prepare for possible changes in the decisions of other actors and thereby maximize their profit. As long as actors are reluctant to change their decisions, the process continues. For this purpose, we used analytical method and solution procedure. The results indicate that by increasing the market scale, increasing price sensitivity, increasing the degree of replacement of products, as well as increasing production costs, distributor's profit increases. In this paper, Lingo software was used for calculations.

The legislature is a manifestation of national sovereignty and one of the most important issues of policy and decision making in the country. Parliamentarians not only express their views on the status of the law, opinions but rather the selection of the Cabinet of Ministers and their removal, the comment declares themselves. Therefore the power of a parliamentary member is to apply his vote through voting. But how can this voting power be changed? The purpose of this paper is to measure the voting power of the members related to major political coalitions in the Islamic parliament using the theory of games. Using Shaply-Shubik's index, voting power of the three coalitions, reformist, and independent were measured in three election periods of 2007, 2011, and 2015. The results show that in the eighth period (2007-2011), all of voting power was for the principlists and the other two coalitions were dummy coalitions. In the ninth period (2011-2015), the voting power was again restored by the principlists and the other two coalitions were dummy coalitions. In the tenth period (2015-2019), despite the difference in the seats of the principlists and reformist and independent coalitions, the voting power is equally divided among all three coalitions. This is a very interesting conclusion that the number of seats in a coalition does not always reflect the strength of that coalition. It is suggested that the Islamic parliament Research Center use this approach in measuring the voting power of the fractions in the specialized parliamentary committees.

Data Envelopment Analysis is a non-parametric method for evaluating the e, ciency of those Decision Making Units (DMUs) that have the same functionality and use multiple inputs to generate multiple outputs. DMUs may sometimes be divided into several groups according to a series of criteria, and it is intended to evaluate a group of similar DMUs. In this paper groups are fairly evaluated under a com-mon platform, each group was considered as a player in a cooperative game and a subset of groups was considered as a coalition. Assuming the Production Possibility Set(PPS) is made up of DMUs belonging to the groups that are a coalition's member, a characteristic function was de, ned in terms of the sum of the e, ciency of all units to determine the marginal e, ect of each group in di, erent coalitions. The groups were then evaluated using the Shapley Value as a unique solution of the cooperative game. Some Examples were provided to describe and apply the method.

The Smart Grid is a large-scale electric power grid by using digital technology that integrates the components of power generations (producers and consumers) in order to enhance the reliability, security, and efficiency by utilizing sensing, processing, measurement, and control. Now, the technologies should lie on the abstract mathematics firmly to model and control the operations. Game theory is a mathematical tool to model systems with Multi-agents (rational decision-makers). Game theory models the behavior of independent and rational agents which in they want to maximize their profits. To this end, we model the behavior of agents and the possible behaviors profit. In the new system, users are able to generate the required energy from renewable energy sources such as solar panels, electric generators, and so forth. Hence, in these complex systems, we would make the use of the mathematical tools to model the interactions and features between agents. In this paper, three basic questions will be answered which I) Why cooperation between agents is valuable. II) Which forms of cooperation are forming and III) Why and how game theory can model the cooperation. In the following, we survey a number of Game theory-based applications especially cooperative game theory to solve relevant problems in Smart Grid. The applications of the Game theory have been presented in three scopes: Micro-grid, Demand Side Management (DSM), and Communications that we will describe them in detail in the following.

In the game theory, the well-known solution to obtain the best profit in non-repeated games as much as possible is the Nash equilibrium. However, in some repeated non-cooperative games, agents can achieve more profit than the Nash equilibrium by tacit collusion. One of the methods to achieve profit more than Nash equilibriums in tacit collusion is reinforcement learning. However, reinforcement learning-based methods consider only one step in the learning process. To achieve and improve profit in these games, more than one step can be used. In this regard, a learning-based forwarding N-steps algorithm called Forwarding Steps (ForSts) is proposed in this paper. The main idea behind ForSts is to improve the performance of agents in noncooperative games by observing the last N-step rewards. As ForSts is used in the game theory to learn tacit collusion, it is evaluated by the iterated prisoner’ s dilemma and the Cournot market. Prisoner’ s Dilemma is an example of a traditional game. The results show that in the iterated prisoner’ s dilemma, the agents using ForSts achieve better profit than the agents playing in the Nash equilibrium. Also, in the Cournot electricity market, sum of the profit of agents using ForSts is 3. 614% more than the sum of profit of agents` playing in the Nash equilibrium.

In this paper, an interactive approach for solving Multi-Objective Nonlinear Programming (MONLP) problem has introduced. This approach combines with the Reference Direction (RD) introduced by Narula et al. [10] and the Attainable Reference Point (ARP) method introduced by Wang et al. [17]. In the interactive approach, we still start with a weak efficient solution as the first step and use the corresponding objective values to improve the weighting coefficients of the augmented Lexicographic Weighted Tchebycheff Programming (LWTP) problem; hence we modify the reference point in the case of an unsatisfactory solution for the Decision Maker (DM) he (she) wishes. The cooperative continuous static game is introduced as an application and hence the stability set of the first kind is determined corresponding to its solution through the interactive approach. Finally, a numerical example has given to the utility of our interactive approach.

Pricing and advertising are two important marketing strategies in the supply chain management which lead to increase customer demand, therefore higher profit for members of supply chains. This paper considers advertising, and pricing decisions simultaneously for a three-level supply chain with one supplier, one manufacturer and one retailer. The amount of market demand is influenced by pricing and advertising. In this paper, three well-known approaches in the game theory including the Nash, Stackelberg and Cooperative games are exploited to study the effects of pricing and advertising decisions on the supply chain. Using these approaches, we identify optimal decisions in each case for the supplier, the manufacturer and the retailer. Also, we compare the outcomes decisions among the mentioned games. The results show that, the Cooperative and the Nash games have the highest and lowest advertising expenditure, respectively. The price level in the Nash game is more than the Stackelberg game for all three levels, and the retailer price in the Stackelberg and Cooperative games are equal. The system has the highest profit in the Cooperative game. Finally, the Nash bargaining model will be presented and explored to investigate the possibilities for profit sharing.