Journal of 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 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 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 , 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 ), 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. If and , then dominates In this introduced by the British biologist John Maynard Smith and case,푑 it is always 푌better to use strategy . The George R. Price in a 1973 ([20]). The idea is that one strategy expected푎 푐 payoff푏 of 푑 players푋 is greater than푌 that of in a given contest, on average, will win over any other players for any composition of a well푋-mixed 푋 푌 Journal of Game Theory 2016, 5(1): 9-15 11

population. is an unbeatable strategy in the sense of. Thus a mixed strategy with a probability / of playing If instead < and < , then dominates Hawk and a probability 1 / of playing Dove is and we have푋 exactly the reverse situation . evolutionary stable, i.e. that it cannot be invaded퐵 퐶 by players 2. If > 푎and 푐 < , 푏then 푑both strategies푌 are best푋 playing one of the pure strategies− 퐵 Hawk퐶 or Dove. replies to themselves, which leads to a ‘coordination To find the point of this game, Let game’.푎 In푐 a population푏 푑 where most players use , it is be the probability of playing hawk if you are player 1 and best to use . In a population where most players use let be the probability of playing hawk if you are player훽 , it is best to use . A leads푋 to 2. The payoffs to the two players are: 푋 훾 bi-stability: both strategies are stable against invasion ( ) by푌 the other strategy.푌 Π = + (1 ) + (1 ) (0) + (1 < > 훾훽 퐵−퐶 3. If and , then both strategies are best )1(1 ) /2. replies to each other, 2 훾 − 훽 퐵 − 훼 훾 − Π = ( )/2 + (1 )(0) + (1 ) + 4. If 푎 + 푐 > 푏+ 푑then is risk-dominant ( ). 훾 − 훽 퐵 (1 )(1 ) /2. If both strategies are , then the risk dominant 2 훾훽 퐵 − 퐶 훾 − 훽 − 훾 훽퐵 푎 푏 푐 푑 푋 푅퐷 strategy has the bigger basin of attraction. Which− simplifies훾 − 훽 to퐵 5. If + 2 > + 2 퐸푆푆 then is advantageous ( ) ( ) = ( 1 + ) , . 1 6. If 푎 > 푏 then푐 is푑 a strict푋 Nash equilibrium. 1 Π = 2 (퐵(1 훾 −+ 훽) − 퐶훾훽) , Likewise,퐴퐷 if < then is a strict Nash 1 푎 푐 푋 2 equilibrium. A strategy which is a strict Nash Thus 2 퐵 − 훾 훽 − 퐶훾훽 equilibrium is always푏 푑 an evolutionarily푌 stable strategy > 0 < / ( )([9], [23]). Π = = 0 = / 1 2 푖푓 훽 퐵 퐶 2.3. ESS퐸푆푆 and Nash Equilibrium of the Hawk-Dove Game 휕 퐵 − 퐶훽 < 0 > / � 푖푓 훽 퐵 퐶 휕훾 Recall the Hawk-Dove game, Clearly, Dove is no stable So the optimal is given by 푖푓 훽 퐵 퐶 strategy, since = Π( , ) < Π( , ) = , a population 1 < / 퐵 훼 of doves can be invaded by hawks. Because of = [0,1] = / 2 퐷 퐷 퐻 퐷 퐵 Π( , ) = and Π( , ) = 0 , H is an if 0 푖푓 훽 > 퐵/퐶 퐵−퐶 훼 � 푖푓 훽 퐵 퐶 > . But what if < ? Neither nor is an . 2 Similarly, 푖푓 훽 퐵 퐶 But퐻 we퐻 could ask: What would퐷 퐻 happen to a population퐸푆푆 of > 0 < / individuals퐵 퐶 which are able퐶 to play mixed퐻 strategies?퐷 Maybe퐸푆푆 Π there exists a mixed strategy which is evolutionary stable. = = 0 = / 2 2 푖푓 훾 퐵 퐶 Consider a population consisting of a species, which is 휕 퐵 − 퐶훾 < 1 > / � 푖푓 훾 퐵 퐶 able to play a mixed strategy, i.e. sometimes Hawk and 휕훽 So the optimal is given by 푖푓 훾 퐵 퐶 sometimes Dove with probabilities and 1 1 > / respectively. For a mixed ESS to exist the following must 훽 hold: 훼 − 훼 = [0,1] = / 푆 1 푖푓 훾 < 퐵/퐶 Π( , ) = Π( , ) = Π( , ). 훽 � 푖푓 훾 퐵 퐶 This gives the diagram depicted in Figure 1. The best Suppose that there exists an ESS in which H and D, which 푖푓 훾 퐵 퐶 퐷 푆 퐻 푆 푆 푆 response functions intersect in three places, each of which is are played with positive probability, have different payoffs. a Nash equilibrium. However, the only symmetric Nash Then it is worthwhile for the player to increase the weight equilibrium, in which the players cannot condition their given to the strategy with the highest payoff since this will moves on whether they are player 1 or player 2, is the increase expected utility. But this means that the original mixed-strategy Nash equilibrium ( , ). mixed strategy was not a best response and hence not part of 퐵 퐵 an ESS, which is a contradiction. Therefore, it must be that in 퐶 퐶 an ESS all strategies with positive probability yield the same payoff. Thus: Π( , ) = Π( , ) Π( , ) + (1 )Π( , ) 퐻 푆 퐷 푆 = Π( , ) + (1 )Π( , ) ⇔ 훼 퐻 퐻 − 훼 퐻 퐷 훼 (퐷 퐻 ) + (1− 훼 ) 퐷 =퐷(1 ) 2 2 훼 퐵

⇔ =퐵 − . 퐶 − 훼 퐵 − 훼 Figure 1. Nash equilibria in the Hawk-Dove Game 퐵 ⇔ 훼 퐶 12 Essam EL-Seidy: On the Behavior of Strategies in Hawk-Dove Game

3. Hawk-Dove Game among Kin Implies to < . Selection 퐵−퐶 ● Hawk’player푟 퐵+will퐶 be risk-dominant (RD) whenever Kinship theory is based on the commonly observed Π( , ) + Π( , ) > Π( , ) + Π( , ) cooperative behaviors such as altruism exhibited by parents ( )( ) ( ) toward their children, nepotism in human societies, etc. 퐻 퐻 퐻 퐷 + >퐷 퐻 + 퐷 퐷 Such behaviors toward one’s kin not only decrease the 퐵−퐶 1+ 푟 퐵 1+푟 individual fitness of the donor (while benefiting the fitness of Implies to ⇔ 2 퐵 푟퐵 2 others), they often incur costs – thereby decreasing personal fitness. < . Hamilton’s rule of relatedness provides the foundation of 2퐵−퐶 much of the work on kinship theory. This rule states that ● Hawk’playerwill be푟 advantageous2퐵+퐶 (AD) if: altruism (or less aggression) is favored when the following inequality holds: Π( , ) + 2Π( , ) > Π( , ) + 2Π( , ) – > 0 > / i.e. if: 퐻 퐻 퐻 퐷 퐷 퐻 퐷 퐷 where r is the genetic relatedness of two interacting agents, b ( )(1+ ) 푟푏 푐 표푟 푟 푐 푏 + 2 > + (1 + ) is the fitness benefit to the beneficiary, and c is the fitness 2 퐵 − 퐶 푟 cost to the altruist. This rule suggests that agents should Implies to 퐵 푟퐵 퐵 푟 show more altruism and less aggression toward closer kin ([21]). < . A simple way to study games between relatives was 3퐵−퐶 proposed by Maynard Smith for the Hawk- Dove game . In 푟 3퐵+퐶 this section, we will study the Hawk-Dove game in which there is e relationship between the players. Consider a 4. Hawk-dove Game among Direct population where the average relatedness between players is Reciprocity given by r, which is a number between 0 and 1. There are two Direct reciprocity is considered to be a powerful possible methods to study the games between relatives. The mechanism for the evolution of cooperation, and it is "inclusive fitness " method adds to the payoff of a player r generally assumed that it can lead to high levels of times the payoff to his co-player. The personal fitness cooperation. Direct reciprocity has been studied by many method, proposed by Grafen ([28]) modifies the fitness of authors ([25], [6]). Direct reciprocity is based on the idea ‘I the player by allowing for the fact that a player is more likely help you and you help me’. In every round the two players than other players of the population to meet co-player must choose between cooperation and defection (fight of adopting the same strategy as himself. We regard the give way). With probability w there is another round. With inclusive fitness method to study the Hawk-Dove game ([16], probability 1 – the game is over. Consequently, the [10], [26]). average number of interactions between two individuals is If we assume that there is a relationship between the 1/(1 ). 푤 players, then by using the inclusive fitness method, the In order to determine a fundamental condition for the payoff matrix of the Hawk-Dove game is given by: evolution− 푤 of cooperation in the repeated Hawk-Dove game, we can study the interaction between the dove who play with ( )( ) 퐻 퐷 a strategy ‘always-defect’ ( ) (i.e. Always cooperate and give away ) and the hawk who play with a strategy Tit -For- 퐵−퐶 1+푟 ( ) (3) Tat ( ). starts with퐴푙푙퐷 cooperation and then does 퐻 2 퐵 � 퐵 1+푟 � whatever the opponent has done in the past move. On the off We will analyze 퐷this model푟퐵 for stability2 of fighting chance푇퐹푇 that two푇퐹푇 hawks (i.e. defectors) meet, they defect (hawks’players) ( ), possibility of maintain the the constantly. If two doves meet, they cooperate and give away Doves’players (cooperators) and when the fighting is all the time, so: risk-dominance ( 퐸푆푆) and advantageous strategy ( ). From ( , ) = + + + + this transformed matrix, we get the following outcomes: 2 2 2 2 푅퐷 퐴퐷 퐵 퐵 2 퐵 3 퐵 Π 퐷 퐷 � � 푤 � � 푤 � � 푤 � � ⋯ ● The strategy of playing hawk (i.e defect strategy) will = be ESS (evolutionary stable strategy), if : 2(1 ) 퐵 If a hawk meets a− dove,푤 the TFT’player fighting in the ( )( ) first round and give away a short time later, while the Π( , ) > Π( , ) > 퐵−퐶 1+ 푟 AllD’player gives away in every round, so: 퐻 퐻 퐷 퐻 ⇔ 2 푟퐵

Journal of Game Theory 2016, 5(1): 9-15 13

( , ) = ( ) + + + + 2 2 2 ( )( ) ( ) 2 3 푇퐹푇 +퐴푙푙퐷 퐵 퐵 퐵 ( ) ( ) Π 푇퐹푇 퐷 퐵 푤 � � 푤 � � 푤 � � ⋯ 퐵−퐶 1+푟 푤퐵 1+푟 = + 푇퐹푇 ( ) ( ) (5) +2 1−푤 2 1−푤 2(1 ) ( ) 퐵 ( ) 푤퐵 � 푤퐵 1+푟 퐵 1+푟 � While, the AllD’player퐵 will get 퐴푙푙퐷 − 푤 From this matrix푟퐵 we 2get1− the푤 following2 1−푤 results : ( , ) = (0) + + + + ● will be stable against AllD’players if 2 2 2 ( )( ) ( ) 퐵 2 퐵 3 퐵 > + 푇퐹푇 ( ) ( ) Π 퐷 푇퐹푇 = 푤 � � 푤 � � 푤 � � ⋯ 퐵−퐶 1+푟 푤퐵 1+푟 2(1 ) 푤퐵 2 1−푤(1 ) (12 +1−푤) < 푟퐵 ⇔ And the two TFT’players− 푤 will fight in all the interactions, (1 ) 퐵 − 푟 − 퐶 푟 so: 푤 ● The fighting will be risk퐵-dominant− 푟 ( ) if ( , ) = + + Π( , ) + Π( , ) > Π( , ) + 2 2 2 푅퐷 퐵 − 퐶 퐵 − 퐶 2 퐵 − 퐶 + ( , ) ( ) Π , i.e. when: Π 푇퐹푇 푇퐹푇 +� � 푤+� =� 푤 .� � 푇퐹푇 푇퐹푇 푇퐹푇 퐴푙푙퐷 퐴푙푙퐷 푇퐹푇 ( ) ( )(1 + ) (1 + ) 3 퐵−퐶 퐵−퐶 퐴푙푙퐷 퐴푙푙퐷 + + 2(1 ) 2(1 ) Thus, the payoff matrix푤 �is 2given� by:⋯ 2 1−푤 퐵 − 퐶 푟 푤퐵 푟 (1 + 퐵) (1 + ) > +− 푤 + − 푤 2(1 ) 2(1 ) 푤퐵 푟 퐵 푟 ( ) 푟퐵 ( ) ( ) 푇퐹푇 퐴푙푙퐷+ − 푤 − 푤 ( ) ( ) < 퐵−퐶 푤퐵 ( ) 푇퐹푇 (4) 2퐵 1−푟 −퐶 1+푟 2 1−푤 2 1−푤 ( ) 퐵 ( ) ● TFT’player⇔ has푤 advantageous퐵 1−푟 ( ) if: � 푤퐵 퐵 � 퐴푙푙퐷 Π( , ) + 2Π( , ) ● The TFT’players will2 1− 푤be 2 1if− 푤 퐴퐷 ( ) > ( , ) + +2 ( , ) Π( , ) > Π( , ), i.e. if > Π Π . 퐸푆푆 ( ) ( ) 푇퐹푇 푇퐹푇 푇퐹푇 퐴푙푙퐷 퐵−퐶 푤퐵 i.e. whenever 퐴푙푙퐷 푇퐹푇 퐴푙푙퐷 퐴푙푙퐷 Implies푇퐹푇 to푇퐹푇 퐴퐿퐿퐷 푇퐹푇 2 1−푤 2 1−푤 ( )(1 + ) (1 + ) + 2 + > (4.1) 2(1 ) (1 ) 퐵−퐶 퐵 − 퐶 푟 푤퐵 푟 Therefore, inequality (4.1) represents a base necessity for (1 + 퐵) (1 + ) 퐵 푤 > 2 −+푤 + − 푤 the evolution of fighting between players. If there are (1 ) (1 ) 푤퐵 푟 퐵 푟 adequately numerous rounds, then direct reciprocity can 푟퐵 2 (1 ) (1 + ) < − 푤 −. 푤 prompt this behavior. (1 ) 퐵 − 푟 − 퐶 푟 ● The fighting will be risk-dominant (RD) whenever: ⇔ 푤 퐵 − 푟 Π( , ) + Π( , ) > Π( , ) + Π( , ) ( ) 5. Group Selection among the + + > + 퐻( 퐻 ) 퐻 퐷( ) 퐷( 퐻 ) ( 퐷 퐷) Hawk-Dove Game 퐵−퐶 푤퐵 푤퐵 퐵 ⇔2 1<−푤1 퐵 . 2 1−푤 2 1−푤 2 1−푤 Selection does not only act on individuals, but also on 퐶 groups. A group of cooperators might be more successful ● Fighting⇔푤 will be− 2advantageous퐵 (AD) if: than a group of defectors. There have been many theoretical Π( , ) + 2Π( , ) > Π( , ) + 2Π( , ) and empirical studies of group selection with some ( ) controversy, and most recently there is a Renaissance of such 퐻 퐻 + 2 퐻+퐷 >퐷 퐻 +퐷 퐷 ideas under the heading of ‘multi-level selection’ ([5], [15], 2(1 ) (1 ) 2(1 ) (1 ) 퐵 − 퐶 푤퐵 푤퐵 퐵 [24]). ⇔ 퐵 <− 푤 . − 푤 − 푤 − 푤 A simple model of group selection works as follows: A 3퐵−퐶 population is subdivided into groups. The maximum size 4.1. Direct⇔푤 Reciprocity3퐵 with Kin Selection in Hawk-Dove of a group is . Individuals interact with others in the same 푚 Game group according to a Hawk-Dove game. The payoff matrix that describes푛 the interactions between individuals of the We will now consider that individuals use direct same group is given by: reciprocity with their relatives. One of the simplest strategies of direct reciprocity is Tit-For-Tat ( ). We will consider that all the hawks are using strategy while the doves 퐵−퐶 (6) 푇퐹푇 02 퐵 are using . Then, the payoff matrix is given by: 퐵 푇퐹푇 � � 2 퐴푙푙퐷 14 Essam EL-Seidy: On the Behavior of Strategies in Hawk-Dove Game

Between groups there is no game dynamical interaction in Therefore, group and kin selection together can evolve our model, but groups divide at rates that are proportional to strong fighting than either of them working alone, especially the average fitness of individuals in that group. The when average relatedness is low, groups are large and the multi-level selection is an emerging property of the number of groups is small. population structure. Therefore, one can say that fighting groups have a constant payoff , while the groups which 퐵−퐶 6. Conclusions give away have a constant payoff2 . Hence, in a sense between groups the game can take the퐵 form as follows: Direct reciprocity can lead to the evolution of cooperative 2 behavior (give way) but if it works together with kin selection it can lead to a strong cooperation between players . 퐵−퐶 퐵−퐶 (7) 2 2 We found that, the necessary condition for evolution of � 퐵 퐵 � cooperative behavior is: For a large n and m, the2 essence2 of the overall selection ( ) ( ) < , the population of cooperators will dynamics on two levels can be described by a single payoff ( ) matrix, which is the sum of the matrix (6) multiplied by the 퐵 1−푟 −퐶 1(+푟 ) ( ) be RD if: < . Where r is the average group size, , and matrix (7) multiplied by the number of 푤 퐵 1−푟 ( ) 2퐵 1−푟 −퐶 1+푟 groups, ([21]). The result is: relatedness between individuals , which is a number between 푛 0 and 1, and푤 w is the probability퐵 1−푟 of next round. 푚 ( + ) + When the group selection works with kin selection, then 퐵−퐶 퐵−퐶 (8) ( ) our fundamental conditions, that we derived , showed that 푛 푚 2 푛퐵 푚 2 � 푚퐵 푛+푚 퐵 � fighting can be maintained in the population, even when the The fighting will be2 stable if Π( 2, ) > Π( , ) and average relatedness is low, groups are large and even if the ( ) hawks will invade doves if Π( , ) > Π ( , ) . If benefits of fighting are low, if < , fighting ( ) < and > respectively.퐶 Hawks퐶 are퐷 퐶RD if 푛퐵− 푛+푚 퐶 퐶 퐷 퐷 퐷 between players can emerge, otherwise the cooperation 푚 퐵 푛 퐶 푟 푛 퐵+퐶 푛+푚 >퐵− 퐶and will푚 be AD퐵 if > . (give way) behavior will be maintain in this situation. 2푛 퐶 3푛 퐶 Hawks strategy will be risk-dominant (RD) whenever 푛+2푚 퐵 푛+3푚 퐵 ( ) 5.1. Group Selection with Kin Selection in < , and will be advantageous (AD) if Hawk-Dove Game ( ) 푛 2퐵−(퐶 −2푚퐶) < . We will now consider that there is a relationship between 푟 푛(2퐵+퐶) 3푛퐵− 푛+3푚 퐶 players in the same group, using the Hawk-Dove game, then For cooperation (give way) to prove stable, the future must 푟 푛 3퐵+퐶 the payoff matrix is given by : have a sufficiently large shadow. An indefinite number of [ ( ) ]( ) ( ) interactions, therefore, is a condition under which + 푛 1+푟 +푚 퐵−퐶 푚 퐵−퐶 cooperation (give way) can emerge. [ ( ) ] (9) 2+ 푛퐵 2 � 푚퐵 푛 1+푟 +푚 퐵 � From this transformed푛푟퐵 2 matrix we2 get the following conditions: ● The fighting strategy will be stable if REFERENCES [ (1 + ) + ]( ) [1] Alexander, RD., 1987, The biology of moral systems. Aldine > + 2 2 de Gruyter, New York. 푛 푟 푚 퐵 − 퐶 푚퐵 ( ) < . 푛푟퐵 [2] Aumann, R.J., 1981, Survey of repeated games. Essays in ( ) 푛퐵− 푛+푚 퐶 game theory and mathematical economics in honor of Oskar Morgenstern, 4: 11-42. ● The fighting⇔푟 between푛 퐵 relatives+퐶 in the same group will be ( ) if the inequality: [3] Axelrod, R., & Hamilton, WD., 1981, The evolution of Π( , ) + Π( , ) > Π( , ) + Π( , ) hold, cooperation. Science, 211:1390– 1396. 푅퐷 Implies to [4] Axelrod R., 1984, The Evolution of Cooperation, Basic 퐶 퐶 퐶 퐷 퐷 퐶 퐷 퐷 Books, New York. ( ) < . ( ) [5] Bowles, S., 2006, Group competition, reproductive leveling, 푛 2퐵−퐶 −2푚퐶 and the evolution of human altruism. Science 314: ● Also the fighting푟 between푛 relatives2퐵+퐶 will be ( ) if: 1569–1572. Π( , ) + 2Π( , ) > Π( , ) + 2Π( , ) i.e. when 퐴퐷 [6] Brandt, H., & Sigmund, K., 2006, The good, the bad and the 3 ( + 3 ) discriminator errors in direct and Indirect reciprocity. 퐶 퐶 퐶 퐷 < 퐷 퐶 퐷 퐷 . (3 + ) J.Theor.Biol.239, 183–194. 푛퐵 − 푛 푚 퐶 푟 푛 퐵 퐶

Journal of Game Theory 2016, 5(1): 9-15 15

[7] Darwen, P., & Yao, X., 1995, On evolving robust strategies [17] Maynard Smith, J., 1982, Evolution and the theory of games. for iterated prisoner’s dilemma. In Progress in Evolutionary Cambridge Univ. Press, Cambridge, U.K. Computation, volume 956 of Lecture Notes in Artificial Intelligence, pp 276–292. [18] Maynard Smith, J., & Price, G. R., 1973, The logic of animal conflict. Nature 246(5427): 15–18. [8] Doebeli, M., Hauert, C., & Killingback, T., 2004, The evolutionary origin of cooperators and defectors. Science, [19] Nowak, M., & Sigmund, K., 1998, Evolution of indirect 306, 859862. doi:10.1126/science.1101456. reciprocity by image scoring. Nature 393:573–577. [9] Fogel, D. B., 1991, The evolution of intelligent decision [20] Nowak, M., 2012, Evolving cooperation. Journal of making in gaming. Cybernetics and Systems: An Theoretical Biology 299: 1:8. International Journal, 22:223–236. [21] Nowak , M., 2006, Five rules for the evolution of cooperation. [10] Foster, KR., Wenseleers, T., & Ratnieks, FLW., 2006, Kin Science 314, 1560–1563. selection is the key to altruism. Trends in Ecology and Evolution.21:57–60. [22] Russell, B., 1959, Common sense and nuclear warfare. London: George Allen and Unwin. ISBN 0041720032. [11] Fundenberg, D., & Maskin, E., 2007, Evolution and Repeated Games, mimeo. [23] Taylor, C. & Nowak, M., 2007, Transforming the dilemma. 61(10): 2281–2292. [12] Fundenberg, D., & Maskin, E., 1990, Evolution and cooperation in noisy repeated games. The American [24] Traulsen, A., & Nowak, M., 2006, Evolution of cooperation Economic Review, 274-279. by multilevel selection. PNAS ;103:10952– 10955. [25] Trivers, R., 1971, The evolution of reciprocal altruism. Q. [13] Hamilton , WD., 1964, The genetically evolution of social behavior. J. Theor. Biol. 7:1–16. Rev. Biol. 46:35–37. [26] Hines, W.G.S., & Smith, J.M., 1979, “Games between [14] Hauert, C. & Doebeli, M., 2004, Spatial structure often inhibits the evolution of cooperation in the snowdrift game, relatives”, Journal of Theoretical Biology, 79 - 19-30. Nature 428, 643–646. [27] El Seidy, E., & Almuntaser, A., 2015, On the evolution of cooperative behavior in prisoner’s dilemma, Journal of Game [15] Killingback, T., Bieri, J., & Flatt, T., 2006, Evolution in group-structured populations can resolve the tragedy of the Theory, Vol. 4 No. 1, 2015, pp. 1-5. doi: 10.5923/j.jgt. commons. Proc R Soc B; 273:1477–1481. 20150401.01. [28] Grafen, A., 1979, The hawk-dove game played between [16] Maynard Smith, J., 1964, Group selection and kin selection. Nature 201: 1145–1147. relatives. Animal Behaviour, 27 - 905-907.