Two-person game with hesitant fuzzy payoff: An application in MADM

Hesitant Fuzzy Set (HFS) permits the membership function having a collection of probable values which are more effective for modelling the real-life problems. Multiple Attribute Decision Making (MADM) process apparently assesses multiple conflicting attribute in decision making. In traditional decision making problems, each player is moving independently whereas in reality it is seen that each player aims to maximize personal profit which causes a negative impact on other player. MADM problem treats with candidate to the best alternative corresponding to the several attributes. Here, we present an MADM problem under hesitant fuzzy information and then transforming it into two-person matrix game, referred to herein as MADM game. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is one of the prominent approach for solving the MADM problems. In this work, we develop the TOPSIS based on Ordered Weighted Aggregation (OWA) operator and hybrid hesitant fuzzy normalized Euclidean distance. Then the two approaches, namely Hybrid Hesitant Fuzzy Ordered Weighted Aggregation-TOPSIS (HHFOWA-TOPSIS) and the Linear Programming Problem (LPP) are applied to solve the formulated MADM game. For solving MADM game, we apply LPP by considering the various values of $\alpha, \psi$, and HHFOWA-TOPSIS for finding the optimal alternative according to their scores. An investment selection problem is included to explain the feasibility and superiority of our formulated approaches. A comparison analysis is drawn among the obtained results which are derived from the two approaches. LPP and HHFOWA-TOPSIS provide the best alternative for the proposed problem. Finally, conclusions about our findings and outlooks are described.

Evolutionary Stable Strategies in Games with Fuzzy Payoffs

Evolutionarily stable strategy (ESS) is a key concept in evolutionary game theory. ESS provides an evolutionary stability criterion for biological, social and economical behaviors. In this paper, we develop a new approach to evaluate ESS in symmetric two player games with fuzzy payoffs. Particularly, every strategy is assigned a fuzzy membership that describes to what degree it is an ESS in presence of uncertainty. The fuzzy set of ESS characterize the nature of ESS. The proposed approach avoids loss of any information that happens by the defuzzification method in games and handles uncertainty of payoffs through all steps of finding an ESS. We use the satisfaction function to compare fuzzy payoffs, and adopts the fuzzy decision rule to obtain the membership function of the fuzzy set of ESS. The theorem shows the relation between fuzzy ESS and fuzzy Nash equilibrium. The numerical results illustrate the proposed method is an appropriate generalization of ESS to fuzzy payoff games.

Core for Game with Fuzzy Generalized Triangular Payoff Value

In this paper, we investigate cooperative game with fuzzy payoff value in the generalized triangular fuzzy number directly. Based on the fuzzy max order, we define three kinds of fuzzy cores, i.e., fuzzy strong core, fuzzy non-dominated core and fuzzy weak core. All three kinds of fuzzy cores can be regarded as the generalization of crisp core. Convexity is one of the sufficient conditions for the existence of fuzzy core. By the balanced cooperative game, a necessary and sufficient existence condition of fuzzy strong core is also given. Further, the fuzzy strong core is represented by crisp core, and the relationship between fuzzy strong core and crisp one is shown. For the fuzzy non-dominated core and fuzzy weak core, we show their necessary and sufficient existence condition, and their properties to construct the fuzzy imputations. Hence, the verification of fuzzy non-dominated core and fuzzy weak core become easier. The above three fuzzy cores are all the extensions of crisp core, but their stable conditions are not different. The weak core is least restricted, but is least stable. Hence, we could choose the fuzzy core according the stable neediness of fuzzy cooperative game.

Solving Game Problems Involving Heptagonal and Hendecagonal Fuzzy Payoffs

The main aim of this paper is to deal with a two person zero sum game involving fuzzy payoff matrix comprising of heptagonal and hendecagonal fuzzy numbers. Ranking of fuzzy numbers is a hard task. Many methods have been proposed to rank different fuzzy numbers such as triangular, trapezoidal, hexagonal, octagonal etc. In this paper, a matrix game is considered whose payoffs are heptagonal and hendecagonal fuzzy numbers and ranking method is used to solve the matrix game. By using this proposed approach the fuzzy game problem is converted into crisp problem and then solved by applying the usual game problem techniques. The validity of proposed method is illustrated with the help of two different practical examples; one where the two companies are venturing into online restaurant business and the other where the two political parties with conflicting interests during elections are competing with each other.

Staging option application to residential development: real options approach

Purpose Real option valuation is capable of accounting for uncertainties in residential development projects but still lacks practical adoption due to limited evidence to support application of the theory in practice. The purpose of this paper is to use option valuation to value staging option embedded in residential projects and compare with results from DCF to determine which of the two methods delivers superior results. Design/methodology/approach The fuzzy payoff method (FPOM), a real options model that uses scenario planning approach to generate a range of figures, from which a single-numerical value is computed for decision-making. Findings The results showed that the use of a range of figures was able to represent uncertainties to a higher degree of accuracy than the static DCF. As a result, the FPOM was able to capture about 3 per cent of the value of the project that was missed by the DCF. The staging option offers an opportunity to abandon unprofitable phases of a project, thereby limiting downside losses. Thus, real option models are practically applicable to cases in property sector. Practical implications Residential property developers must consider flexibility in financial feasibility evaluation of development because of the embedded value in uncertain property projects. It is important to account for optionality in financial evaluation of property projects for value maximisation. Originality/value The FPOM has been used for the first time to evaluate a horizontal phasing of a residential development project.