Stable strategies analysis based on the utility of Z-number in the evolutionary games
Introduction
Game theory [1] has been applied in lots of applications, such as economics [2], sociology [3], and computer science and logic [4], biology [5], etc. In recent decades of years, after Smith and Price [6] proposed the new fundamental insight from frequency-dependent selection within biology, evolutionary game theory has got great attentions, e.g., comprehensive review on coevolutionary mechanics [7], evolutionary dynamics of group interactions [8], spatial reciprocity influence in evolutionary cooperation game [9], infection control of coupled disease from the perspective of complex networks [10], evolutionary puzzle of cooperation [11], and some other related applications of evolutionary game based, i.e., statistical physics of human cooperation [12], phase transitions in models of human cooperation [13], statistical physics of vaccination [14], etc.
In the real application, for the complexity and uncertainty of the real world, knowledge or opinions are always expressed as a fuzzy linguistic variable, some efforts have been made to predict the evolutionary outcome using fuzzy techniques [15], [16]. Limited work has been done in the area of discussing the role of reliability of uncertain information in such an infrastructure, which is still virgin and may become a fascinating issue. For example, the decision maker wants to obtain the outcome of the next year considering the influence of the competitors, he evaluates an uncertain result, e.g., the outcome of next is about fifty thousand dollars and it is likely. The classical methods including the classical fuzzy method cannot deal with this scenario. Z-number is combined with “restriction” and “reliability”, which is an efficient framework to simulate the thinking of human. It can be formulated as a Z-number, i.e., (about fifty thousand dollars, likely) for the example. In this paper, we extend the crisp payoff matrix into one Z-number based, and a method of analyzing evolutionary stable strategy (ESS) based on the total utility of Z-number is proposed. The proposed frame can degenerate into classical ESS when the information is extremely reliable. At the same time, the proposed method can disclose that the mixed ESS profit decreases with the increasing constraint uncertainty of the information in spite that the ESS is unchangeable. This is consistent with the real situation since the expected profit always decreases with the increasing of the uncertainty. Some simple numerical examples are used to illustrate the effectiveness of the proposed method. In this paper, we employed the Hawk–Dove model to illustrate the basic process of the evolutionary game framework.
The paper is organized as follows. The preliminaries fuzzy sets, Z-number, total utility of Z-number and Hawk–Dove game and evolutionarily stable strategy (ESS) are briefly introduced in Section 2. Section 3 proposed the method of stable strategies analysis based on the utility of Z-number in the evolutionary game. In addition, relation of uncertainty and mixed ESS profit based on total utility of Z-number are discussed in Section 3. Some numerical examples are used to illustrate the effectiveness of the proposed frame in Section 4. Application in avoiding a price war considering the information reliability is discussed in Section 5. At last, this paper is concluded in Section 6.
Section snippets
Preliminaries
In this section, some preliminaries are briefly introduced.
Formulation of the evolutionary game with Z-numbers
In this research, for simplification, we assume there are two players in the Hawk–Dove game, and the payoff matrix is symmetrical. The values of the matrix are all expressed with Z-numbers, which are shown in Table 1.
Convert Z-numbers into crisp numbers using the utility of Z-number
Second, we convert the payoff matrix with Z-numbers into regular crisp payoff matrix using the total utility of Z-number (Eq. (3)). The converted payoff matrix is shown in Table 2.
Solution of the ESS based on the utility of Z-number
Assume that a strategy the probability, p, that a player chooses Hawk (H) action in the contest. If
Numerical examples of ESS using the total utility of Z-number
First, we assume the payoff matrix of two players in a Hawk–Dove game is given as Table 3. The values of the payoff matrix are all crisp numbers. It is easy to see that it is a classical Hawk–Dove game. According to the standard frame of classical HawkDove game, the resource V of a Hawk achieves is 2 when he meets a Dove. At the same time, when two Hawk fight for the resource, the cost C of each Hawk from the damage in a contest is 6.
Since V < C, the ESS of Hawk–Dove game is a mixed strategy.
Application in avoiding a price war considering the information reliability
A possible contribution of the development of Hawk–Dove model can be used to analyze the cost of waging a price war, the strategy-dependent profit, and the managerial security under the influence of the reliability of information.
Assume the possible strategies employed by the two firms are symmetric, where Fight refers to engaging in a price war and Sharing means maintaining price. If the Fight strategy employed, at least a firm will loss. If the Sharing strategy used, the firm will share the
Conclusion
Evolutionary games with the fuzzy set are attracting growing interest. While among previous studies, the role of the reliability of knowledge in such an infrastructure is still virgin and may become a fascinating issue. Z-number is combined with “restriction” and “reliability”, which is an efficient framework to simulate the thinking of human. In this paper, the stable strategies analysis based on the utility of Z-number in the evolutionary games is proposed, which can simulate the procedure of
Acknowledgment
The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61573290, 61503237), and China Scholarship Council.
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2022, Information SciencesCitation Excerpt :Extensive research on Z-numbers has been carried out, some of them include basic methods of processing Z-number, for example, computation methods [2,10,12,35,47], linguistic uncertain Z-numbers [31], linear programming problems [24], the negation of Z-numbers [36], Z-fractional differential equations [34], ranking method [42], Z-network model [29] and Z-clustering [4,5]. Such studies lead to the use of Z-numbers in many fields, for example, business venturing [6], medical diagnosis [46], the appraisal of COVID-19 disease [30], decision making [50], data envelopment analysis and neural network [48], the evolutionary games [32] and growth model [13]. It should be noted that the product of two normal variables is not a normal variable in general.