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On the Behavior of Strategies in Hawk-Dove Game Journal of Game Theory 2016, 5(1): 9-15 DOI: 10.5923/j.jgt.20160501.02 On the Behavior of Strategies in Hawk-Dove Game Essam EL-Seidy Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt Abstract The emergence of cooperative behavior in human and animal societies is one of the fundamental problems in biology and social sciences. In this article, we study the evolution of cooperative behavior in the hawk dove game. There are some mechanisms like kin selection, group selection, direct and indirect reciprocity can evolve the cooperation when it works alone. Here we combine two mechanisms together in one population. The transformed matrices for each combination are determined. Some properties of cooperation like risk-dominant (RD) and advantageous (AD) are studied. The property of evolutionary stable (ESS) for strategies used in this article is discussed. Keywords Hawk-dove game, Evolutionary game dynamics, of evolutionary stable strategies (ESS), Kin selection, Direct and Indirect reciprocity, Group selection difference between both games has a rather profound effect 1. Introduction on the success of both strategies. In particular, whilst by the Prisoner’s Dilemma spatial structure often facilitates Game theory provides a quantitative framework for cooperation this is often not the case in the hawk–dove game analyzing the behavior of rational players. The theory of ([14]). iterated games in particular provides a systematic framework The Hawk-Dove game ([18]), also known as the snowdrift to explore the players' relationship in a long-term. It has been game or the chicken game, is used to study a variety of topics, an important tool in the behavioral and biological sciences from the evolution of cooperation to nuclear brinkmanship and it has been often invoked by economists, political ([8], [22]). The basic idea of the Hawk-Dove (or chicken) scientists, anthropologists and other scientists who were game is that two opponents compete for a resource. The interested in human cooperation ([1], [2], [11], [12]). resource brings a benefit to the one who wins it. In this The evolutionary game theory has proven to be excellent game, opponents have the opportunity to play Hawk and for studying the evolution and success of different behavioral fight (i.e., defect), or play퐵 Dove and give way (i.e., patterns in human as well as animal societies ([18]). The two cooperate). The payoffs are maximized when both players games receiving the most attention are the hawk–dove and give way and play Dove (cooperate). Unfortunately, in a the Prisoner’s Dilemma game ([3]). In both games, the world of doves, it pays to defect and play Hawk (defect). cooperative strategy, i.e. dove strategy in the hawk– dove This game’s payoffs can be couched in terms of costs and game, warrants the highest collective payoff that is equally benefits and modified to consider the joint consequences of kin selection and reciprocal altruism. Fighting is, however, shared among the players. Mutual cooperation is, however, dangerous and the looser of a fight has to bear a cost . If a challenged by the defecting strategy, i.e. hawk strategy in the Hawk meets a Hawk, they will fight and one of them will win Hawk–Dove game, that promises the defector a higher the resource. Thus, the average payoff of a Hawk meeting퐶 a income at the expense of the neighboring cooperator. The Hawk is ( )/2. If a Hawk meets a Dove the Dove crucial difference that distinguishes both games is the way immediately withdraws, so its payoff is zero, while the defectors are punished when facing each other. In the payoff of the퐵 −Hawk퐶 is B. If two Doves meet, the one who first Prisoner’s Dilemma game, a defector encountering another gets hold of the resource keeps it while the other does not defector still earns more than a cooperator facing a defector, fight for it. Thus, the average payoff for a Dove meeting a whilst in the hawk–dove game the ranking of these two Dove is /2. The strategic form of the game is given by the payoffs is switched. Thus, in the hawk–dove game a payoff matrix: cooperator facing a defector earns more than a defector 퐵 playing with another defector. This seemingly minute 퐻 퐷 퐵−퐶 (1) * Corresponding author: 0 퐵 [email protected] (Essam EL-Seidy) 퐻 2 퐻 In Nowak and Taylor� ([23]), they퐵� studied five mechanisms Published online at http://journal.sapub.org/jgt 퐷 퐷2 Copyright © 2016 Scientific & Academic Publishing. All Rights Reserved for the evolution of the cooperative behavior in the 10 Essam EL-Seidy: On the Behavior of Strategies in Hawk-Dove Game prisoner’s dilemma game, direct and indirect reciprocity, kin strategy. The strategy ought to additionally have the selection, group selection, and network reciprocity. These advantage of doing great when set against adversaries mechanisms have been proposed to explain the evolution of utilizing the same strategy. This is important, because a cooperative behavior. The kin selection focuses on successful strategy is likely to be common and the player will cooperation among individuals who are closely related probably have to compete with others who are employing it. genetically (Hamilton, 1964), whereas direct reciprocity It does not have to be a single evolutionary strategy. It can be focus on the selfish incentives for cooperation in repeated a combination of strategies, or a combination of players interactions ([4], [25]). The indirect reciprocity show how where every utilize only one strategy. Maynard Smith and cooperation in larger groups can emerge when the Price specify two conditions for a strategy to be an . cooperators can build a reputation. Network reciprocity Either 푋 퐸푆푆 operates in structured populations, where cooperators can I. ( , ) > ( , ), that is, the payoff for playing prevail over defectors by forming clusters ([1], [19]). against (another playing) is greater than that for Elseidy and Almuntaser ([27]) studied the evolution of playing휋 푋 푋 any휋 other푌 푋 strategy against for all 푋 cooperative behavior in Prisoner’s Dilemma game. They 푋 Or used a combination of mechanisms which allow the 푌 푋 푋 ≠ 푌 cooperation between the players to emerge and evolve, also II. ( , ) = ( , ) and ( , ) > ( , ) , that is, the they derived the fundamental conditions that makes the payoff of playing against itself is equal to that of 휋 푋 푋 휋 푌 푋 휋 푋 푌 휋 푌 푌 cooperative behavior evolutionary stable strategy. In this playing against but the payoff of playing 푋 article, we use some combination of mechanisms to study the against is less than that of playing against for all 푌. 푋 푌 behavior of strategies in the Hawk-Dove game after we find Note that either푌 (I) or (II) will do and that 푋the previous푌 is a the corresponding transformed matrix for each combination. 푋 ≠ 푌 We derive the necessary condition for evolution of fighting more grounded condition than the recent. Clearly, if (I) gets, (play Hawk)or cooperation (play Dove) between players and the Y invader commonly loses against X, and along these evolutionary stability property (ESS) of the cooperative lines it can't even start to increase with any achievement. If behavior and knowing that we intended to cooperate here (II) gets, the Y invader does as well against X as X itself, but both players will play Dove (give way). We derive the it loses to X against other Y invaders, and therefore it cannot multiply. In short, players cannot successfully invade a necessary conditions that make the hawks’ players risk- population of X players. It is conceivable to present a dominant (RD) and advantageous (AD) in a population in the strategy that is stronger푌 than an , namely, an unbeatable context of the hawk-dove game. strategy. Strategy is unbeatable if, given whatever other strategy : 퐸푆푆 푋 2. Evolutionary Game Theory Π( , ) > Π( , ) Π( , ) > Π( , ). 푌 An unbeatable strategy is the most powerful strategy, The first ideas of evolutionary game theory showed up in 푋 푋 푌 푋 푎푛푑 푋 푌 푌 푌 the papers by Hamilton [13], Trivers [25], and Maynard because it strictly dominates any other strategy; however it is Smith and Price [18]. Evolutionary game theory studies the additionally uncommon, and subsequently in exceptionally behavior of large populations of agents who repeatedly constrained use. engage in strategic interactions. Evolutionary game theory varies from classical game theory by concentrating more on 2.2. Evolutionary Game Dynamics the dynamics of strategy change as impacted not singularly Evolutionary game dynamics is the application of by the nature of the various competing strategies, but by the population dynamical methods to game theory. It has been impact of the frequency with which those various competing introduced by evolutionary biologists (such as William D. strategies are found in the population. Evolutionary game Hamilton and John Maynard Smith), anticipated in part by theory has turned out to be precious in clarifying numerous classical game theorists. Consider a game between two mind boggling and testing parts of science. It has been strategies, and , given by the payoff matrix: especially useful in building up the premise of altruistic behaviors inside of the connection of Darwinian procedure. 푋 푌 푋 푌 (2) In spite of its cause and unique reason, evolutionary game 푋 푎 푏 theory has become of increasing interest to economists, The entries denote the payoffs� � for the row player. Thus, sociologists, anthropologists, and philosophers ([1], [7], strategy obtains payoff푌 푐 푑 when playing another [17]). player, but payoff when playing a player. Likewise, strategy 푋 obtains payoff when푎 playing an player and푋 2.1. Evolutionarily Stable Strategies ( ) payoff when playing푏 a player. 푌 푌 > > 푐 푋 . The idea of evolutionary stable strategies푬푺푺 was defined and 1.
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