War of Attrition with Incomplete Information and Fuzzy Players' Types
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NTROL IN SOCIAL ECONOMIC SYSTEMS
War of Attrition with Incomplete Information and Fuzzy Players’ Types A. S. Shvedov National Research University Higher School of Economics, Moscow, Russia e-mail: [email protected] Received October 17, 2019 Revised January 23, 2020 Accepted January 30, 2020
Abstract—The result on existence of a pure-strategy symmetric Bayesian Nash equilibrium in the war of attrition is generalized for fuzzy players’ actions and types. Keywords: games with incomplete information, fuzzy sets, fuzzy random variables, Bayesian Nash equilibria DOI: 10.1134/S0005117920070097
1. INTRODUCTION Over recent decades, games from the “war of attrition” class have received considerable attention. Many results of both theoretical and applied nature have been obtained for these games in application to biological problems, when two animals of the same species enter into a dispute for some prize, such as food or territory, and only one of them can get the prize. However, the animals do not start a battle but rather try to show their opponent their strength and force him/her to retreat. Moreover, the value of the prize for each of the rivals decreases as the time of confrontation increases. In various works, games of this class have been studied with both complete and incomplete information. When considering games with incomplete information, each player knows his own type but does not know the opponent’s type (for example, the type of a player can represent how hungry the animal is). However, each of the players knows the probability distribution for the type of his opponent. The result of the existence of a Bayesian Nash equilibrium in pure strategies for symmetric games with incomplete information of the “war of attrition” class (i.e., for games with identical players, more precisely, with players with the same probability distributions over their types and with the same actions for each type) is presented in [1]. A proof of this result can be found, for example, in [2]. Some subsequent works call this result classical. Various modifications of games in the class “war of attrition” have been considered by many authors. For example, the work [3] gives applications to economic problems. Some surveys of this direction can be found in [4, 5]. For more information on games with incomplete information and on the Bayesian Nash equilibrium see, for example, [6]. The theory of fuzzy sets is used in the consideration of many game theory problems. For games in normal form, we can refer to the survey [7]. Using electronic databases of scientific resources, it is easy to see that the number of works that contain the word “fuzzy” and the word “game” in the title is significant, and many of these works are recent. The theory of fuzzy sets is also used in 1279
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models where there is some control center but the control objects are active, i.e., pursue their own interests [8]. However, as far as we know, games with incomplete information where the types of players are represented by fuzzy numbers (or fuzzy sets) have no
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