To find the Nash equilibrium point of this game, Let be the probability of playing hawk if you are player 1 and let be the probability of playing hawk if you are player 2.
The payoffs to the two players are: Which simplifies to Thus So the optimal is given by Similarly, So the optimal is given by This gives the diagram depicted in Figure 1.
The best response functions intersect in three places, each of which is a Nash equilibrium. However, the only symmetric Nash equilibrium, in which the players cannot condition their moves on whether they are player 1 or player 2, is the mixed-strategy Nash equilibrium. Figure 1. Nash equilibria in the Hawk-Dove Game 3. Hawk-Dove Game among Kin Selection Kinship theory is based on the commonly observed cooperative behaviors such as altruism exhibited by parents toward their children, nepotism in human societies, etc.
This rule states that altruism or less aggression is favored when the following inequality holds: where r is the genetic relatedness of two interacting agents, b is the fitness benefit to the beneficiary, and c is the fitness cost to the altruist. This rule suggests that agents should show more altruism and less aggression toward closer kin [21]. A simple way to study games between relatives was proposed by Maynard Smith for the Hawk- Dove game.
In this section, we will study the Hawk-Dove game in which there is e relationship between the players. Consider a population where the average relatedness between players is given by r , which is a number between 0 and 1. There are two possible methods to study the games between relatives.
The "inclusive fitness " method adds to the payoff of a player r times the payoff to his co-player. The personal fitness method, proposed by Grafen [28] modifies the fitness of the player by allowing for the fact that a player is more likely than other players of the population to meet co-player adopting the same strategy as himself. We regard the inclusive fitness method to study the Hawk-Dove game [16] , [10] , [26]. Hawk-dove Game among Direct Reciprocity Direct reciprocity is considered to be a powerful mechanism for the evolution of cooperation, and it is generally assumed that it can lead to high levels of cooperation.
Direct reciprocity has been studied by many authors [25], [6]. In every round the two players must choose between cooperation and defection fight of give way. With probability w there is another round. With probability the game is over. Consequently, the average number of interactions between two individuals is. TFT starts with cooperation and then does whatever the opponent has done in the past move. On the off chance that two hawks i.
If there are adequately numerous rounds, then direct reciprocity can prompt this behavior. Direct Reciprocity with Kin Selection in Hawk-Dove Game We will now consider that individuals use direct reciprocity with their relatives. Group Selection among the Hawk-Dove Game Selection does not only act on individuals, but also on groups. A group of cooperators might be more successful than a group of defectors.
A simple model of group selection works as follows: A population is subdivided into m groups. The maximum size of a group is n. Individuals interact with others in the same group according to a Hawk-Dove game. The payoff matrix that describes the interactions between individuals of the same group is given by: 6 Between groups there is no game dynamical interaction in our model, but groups divide at rates that are proportional to the average fitness of individuals in that group.
The multi-level selection is an emerging property of the population structure. Therefore, one can say that fighting groups have a constant payoff , while the groups which give away have a constant payoff.
Hence, in a sense between groups the game can take the form as follows: 7 For a large n and m , the essence of the overall selection dynamics on two levels can be described by a single payoff matrix, which is the sum of the matrix 6 multiplied by the group size, n , and matrix 7 multiplied by the number of groups, m [21]. The result is: 8 The fighting will be stable if and hawks will invade doves if. If and respectively. Hawks are RD if and will be AD if.
Conclusions Direct reciprocity can lead to the evolution of cooperative behavior give way but if it works together with kin selection it can lead to a strong cooperation between players.
We found that, the necessary condition for evolution of cooperative behavior is: , the population of cooperators will be RD if:. Where r is the average relatedness between individuals , which is a number between 0 and 1, and w is the probability of next round. When the group selection works with kin selection, then our fundamental conditions, that we derived , showed that fighting can be maintained in the population, even when the average relatedness is low, groups are large and even if the benefits of fighting are low, if , fighting between players can emerge, otherwise the cooperation give way behavior will be maintain in this situation.
Hawks strategy will be risk-dominant RD whenever , and will be advantageous AD if. For cooperation give way to prove stable, the future must have a sufficiently large shadow.
An indefinite number of interactions, therefore, is a condition under which cooperation give way can emerge. References [1] Alexander, RD. Aldine de Gruyter, New York. Essays in game theory and mathematical economics in honor of Oskar Morgenstern, 4: Science, — Science — Science, , Cybernetics and Systems: An International Journal, — Trends in Ecology and Evolution. The American Economic Review, Proc R Soc B; — Nature — Cambridge Univ.
However, both incur a display cost because of the energy needed to perform the display. Some game theorists do not factor in the display cost when analyzing this game Cressman , Weibull The resource can be any arbitrary value.
In this case, we will assign it a value of Many assumptions are made when we create the matrix game for these situations. Assumptions: 1 Resources and fitness points are the same at every encounter. The following section will list the different payoffs when a hawks and doves meet.
Hawk vs. The winning hawk will win the 50 fitness points and the other will incur a cost of points due to damage from the battle. Since both hawks have an equal chance of winning, we calculate the expected value of each encounter. Dove - When a hawk meets a dove, the hawk will begin to attack and the dove will retreat.
The hawk will be left with the resource every time, winning 50 fitness points. The dove will end up uninjured with no gain and no loss. Thus, one will end up with 40 points, the the other with points. Dove vs. Dove - When a dove meets a dove, both will display until one of them leaves. There will be no conflict, and one of the doves will end up with the 50 points. However, both doves incur a cost of 10 points because of the time and energy wasted displaying. Thus, if we construct a payoff matrix, it will look like this: Hawk Dove Hawk , 50,0 Dove 0,50 15,15 This is the basic hawk-dove game with the two basic strategies.
There are also other possible strategies that can be introduced into the matrix. One of these strategies is called the Bully strategy. In this situation, the animal will attempt to attack, and if it sees it is facing a hawk, it will retreat unharmed. If it faces a dove, the dove will retreat and it will be left with the prize. When a bully meets a bully, one will run away faster than the other, and one will be left unexpectedly with the resource. Another strategies is the Retaliator strategy.
The retaliator will remain passive like a dove unless the opponent attacks. When an opponent attacks, it will attack with its full force until it wins or loses. The following are the additional payoffs with these added encounters. Bully vs. Hawk - When a bully faces a hawk, it will attempt to attack, but when it sees its opponent is a hawk, it will flee. The hawk will be left with 50 fitness points every time, and the bully with 0.
Dove - When a bully faces a dove, it will attempt to attack and this time the dove will flee. The bully will be left with 50 fitness points and the dove will lose nothing. Bully - When a bully faces a bully, both attempt to attack, but then when they each see the other attack, both will attempt to flee.
One will escape faster than the other and the reward will be left unexpectedly to one. Since each one is left with the resource half of the time, we can compute the expected value. Retaliator vs. Hawk - When a retaliator faces a hawk, the hawk will attack first and then the retaliator will behave as a hawk and fight back. The payoff is the same as a hawk vs.
Dove - When a retaliator faces a dove, both will remain passive. Cost of display -- displays generally have costs, although howhigh they are varies -- clearly they have variable costs in terms of energyand time and they may also increase risk of being preyed upon. All of thesetype of measurements, in theory at least, can be translated into fitnessterms.
Important Note: All of these separate payoffs are in unitsof fitness whatever they are! You will see shortly that the values thatare assigned to each payoff is crucial to outcome of the game -- thus accurateestimates are vital in usefulness of any ESS game in understanding a behavior.
Calculation of the payoff to Hawk in Hawk vs. H contests: Relevant variables from eq. Does it seem reasonable that hawks pay no cost in winning? Also, does it seem reasonable that the loser only pays an injury cost? Thinkabout what animals do and about simplifications of models. Forsome discussion of this question, press here but think about it first.
Note ; the costs of losing are added in our model sincewe gave the costs anegative sign to emphasize that they lowered the fitness of theloser. Calculation of the payoff to Hawk when vs. Dove: Relevant variables from eq. Calculation of the payoff to Dove when vs. Hawk: Relevant variables from eq.
So for this particular version of the Hawk vs. Dovegame defined by these payoffs , the pay-off matrix is:. Should you get the same results eachtime?
ANS -- as usual,please try to reason through this one before going to the answer. If you did the problem above, you will realize that neither Hawk norDove are pure ESSs given the payoffs calculated from the equations and valuesfor benefits and costs presented above. When you use the simulation, youwill see that certain benefits and costs can be used to make either of thestrategies pure ESSs, although these might seem to involve unreasonableassumptions.
It is good to keep in mind the fact that the rules you used to determinethat neither strategy was a pure ESS require some reasonable assumptions to review, presshere. If we have no pure ESS, we know that in a two strategy game there willbe a mixed ESS which is defined as the frequencies of the strategies whereboth have equal fitness.
Recall that the fitness of a strategy is the sumof the payoffs times the frequency of their occurrence. Notice that each of the equations for strategy fitness yield a straightline when solved for a series of frequencies. We can understand the solution more clearly if we graph eqs. The intersection of the Hawk and Dove plotsrepresents the frequency of one strategy in this case Hawk where the fitnessesof both strategies are equally fit in terms of payoff units.
Remarks About the Graphical Results ofthe Hawk vs. Dove Game:. At this point you know how to set up and solve a simple game. And youhave a basic familiarity with the Hawks and Doves game. So, you are now ready to explore the Hawks and Doves game in detail usinga simulation that will allow you to alter payoffs by changing benefits andcosts.
The simulation will provide you with a visual representation of thesolution, using the same techniques you have just learned except the computerwill now do the computational work for you! And, you'll get to see somethingnew -- you'll be able to set frequencies of the two strategies and thensee how a population with a given payoff matrix will evolve over time.
Press here to go to apage that explains how to use the simulation and then launches it. Why can't hawks die or get permanentlyknocked out of action? Why must they be miraculously restoredto health?
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