Dove and hawk game

The Brown-Nash-von Neumann dynamics. One key assumption made by the learning rule which yields the replicator dynamics is that imitation is a reliable guide to future payoffs. This can be problematic for two reasons. First, the fact that a strategy has an expected payoff greater than the average payoff of the population may simply be indicative of peculiarities of the current population composition, and not of any particular strategic merit the strategy possesses.

Such a learning rule may end up with much of the population shifting to adopt a strategy, only for its transient fitness benefits to disappear.

Second, if a strategy is not present in the population at all, it has no chance of being adopted by imitation. Notice that such a learning rule attributes a higher degree of rationality to the individual players than the learning rule which generates the replicator dynamics.

This learning rule requires people to know the entire set of possible strategies, as well as the associated payoff matrix, so that they can determine if a strategy presently absent from the population would be worth adopting. Unlike the replicator dynamics, the BNN dynamic can introduce new strategies into the population which are not represented. When the population is started in the states where everyone plays Rock, Paper or Scissors, the BNN dynamic will eventually end up in a state where all strategies are represented.

The Smith Dynamics. One unusual feature of the learning rule which generates the BNN dynamic is that it compares the expected payoff of alternative possible strategies with the average payoff of the population.

One might wonder why performing better than the population average is a sensible point of comparison, since the average payoff of the population is often not actually achievable by any particular strategy available to the players. When plugged into the Sandholm framework above, this learning rule generates an evolutionary dynamic first studied by Smith and is thus known as the Smith dynamic. Given the number of different types of evolutionary dynamics, as seen in section 2.

Answers to these questions turn out to be more subtle and complex than one might first anticipate. One complication, at the outset, is that an ESS is a strategy , possibly mixed, which satisfies certain properties. In contrast, all of the evolutionary dynamics described above model populations where individuals employ only pure strategies. How, then, are we to relate the two concepts? One natural suggestion is to interpret the probabilities which appear in an evolutionarily stable strategy as population frequencies.

When the probabilities are understood in this way, one speaks of an evolutionarily stable state , in order to stress the difference in interpretation. One can then ask under what conditions a particular evolutionary dynamic will converge to an evolutionarily stable state.

In the case of the replicator dynamics, it is immediately apparent that the replicator dynamics need not converge to an evolutionarily stable state. This is because, as noted previously, the replicator dynamics cannot introduce strategies into the population if they are initially absent. Hence, if an evolutionarily stable state requires certain pure strategies to be present, and those pure strategies do not appear in the initial population state, then the replicator dynamics will not converge to the evolutionarily stable state.

This can be seen in a particularly stark form in figure 6, above. This shows that, under the replicator dynamics, there can be instances where even strictly dominated strategies will persist. That said, it can also be seen in figure 6 that whenever there is a nonzero proportion of Defectors present in the population, that they will increase in number and eventually drive Cooperate to extinction. In the limit, because another property of the replicator dynamics is that no strategy which appears can ever go extinct in a finite amount of time.

This motivates the following result:. What the above theorem shows is that, although there are cases where strictly dominated strategies may persist under the replicator dynamics, such cases are rare.

As long as all strategies are initially present, no matter to how small an extent, the replicator dynamics eliminates strictly dominated strategies. However, the same does not hold for weakly dominated strategies. Weakly dominated strategies can can appear in a Nash equilibrium, as figure 7 shows below. Figure 7.

It can be shown see Weibull, that a weakly dominated strategy can never be an ESS. In the case of the game shown in figure 7, this is surprising because the equilibrium generated by the weakly dominated strategy is Pareto optimal and has a much higher expected payoff than any other Nash equilibrium.

However, it is also the case see Skyrms, that the replicator dynamics need not eliminate weakly dominated strategies. In fact, in chapter 2 of Evolution of the Social Contract , Brian Skyrms shows that there are some games in which the replicator dynamics almost always yields outcomes containing a weakly dominated strategy! This shows that there can be considerable disagreement between the evolutionary outcomes of the replicator dynamics and what the static approach identifies as an evolutionarily stable strategy.

For example, consider the BNN dynamic and the Smith dynamic, described in section 2. In both cases, the underlying learning rule which generates those dynamics has some intuitive plausibility. In particular, each of those learning rules can be seen as employing slightly more rational approaches to the problem of strategy revision than the imitative learning rule which generated the replicator dynamics. Yet Hofbauer and Sandholm show that both the BNN dynamic and the Smith dynamic are not guaranteed to eliminate strictly dominated strategies!

Consider the game of figure 8 below. This means that the Twin strategy is strictly dominated by Paper, since there are absolutely no instances in which it is rationally preferable to play Twin instead of Paper. Yet, under the Smith dynamic, there are a nontrivial number of initial conditions which end up trapped in cycles where the Twin strategy is played by a nontrivial portion of the population. The connection between ESSs and stable states under an evolutionary dynamical model is weakened further if we do not model the dynamics using a continuous population model.

The figure below illustrates how rapidly one such population converges to a state where everyone defects. Notice that with these particular settings of payoff values, the evolutionary dynamics of the local interaction model differ significantly from those of the replicator dynamics. Under these payoffs, the stable states have no corresponding analogue in either the replicator dynamics nor in the analysis of evolutionarily stable strategies.

Here, the dynamics of local interaction lead to a world constantly in flux: under these values regions occupied predominantly by Cooperators may be successfully invaded by Defectors, and regions occupied predominantly by Defectors may be successfully invaded by Cooperators. These results demonstrate that, although there are cases where both the static and dynamic approaches to evolutionary game theory agree about the expected outcome of an evolutionary game, there are enough differences in the outcomes of the two modes of analysis to justify the development of each program independently.

The concept of a Nash equilibrium see the entry on game theory has been the most used solution concept in game theory since its introduction by John Nash By best-response, we mean that no individual can improve her payoff by switching strategies unless at least one other individual switches strategies as well.

Yet a difficulty arises with the use of Nash equilibrium as a solution concept for games: if we restrict players to using pure strategies, not every game has a Nash equilbrium.

Figure Payoff matrix for the game of Matching Pennies. Row wins if the two coins do not match, whereas Column wins if the two coins match. While it is true that every noncooperative game in which players may use mixed strategies has a Nash equilibrium, some have questioned the significance of this for real agents.

If it seems appropriate to require rational agents to adopt only pure strategies perhaps because the cost of implementing a mixed strategy runs too high , then the game theorist must admit that certain games lack solutions.

A more significant problem with invoking the Nash equilibrium as the appropriate solution concept arises because some games have multiple Nash equilibria see the section on Solution Concepts and Equilibria , in the entry on game theory. Unfortunately, so many refinements of the notion of a Nash equilibrium have been developed that, in many games which have multiple Nash equilibria, each equilibrium could be justified by some refinement present in the literature.

The problem has thus shifted from choosing among multiple Nash equilibria to choosing among the various refinements. Samuelson , in his work Evolutionary Games and Equilibrium Selection expressed hope that further development of evolutionary game theory could be of service in addressing the equilibrium selection problem.

At present, this hope does not seem to have been realised. As section 2. Furthermore, as section 3 showed, there is an imperfect agreement between what is evolutionary stable, in the dynamic setting, and what is evolutionary stable, in the static setting.

The traditional theory of games imposes a very high rationality requirement upon agents. Since the number of different lotteries over outcomes is uncountably infinite, this requires each agent to have a well-defined, consistent set of uncountably infinitely many preferences. Numerous results from experimental economics have shown that these strong rationality assumptions do not describe the behavior of real human subjects. Humans are rarely if ever the hyperrational agents described by traditional game theory.

The hope, then, is that evolutionary game theory may meet with greater success in describing and predicting the choices of human subjects, since it is better equipped to handle the appropriate weaker rationality assumptions. Indeed, one of the great strengths of the framework introduced by Sandholm is that it provides a general method for linking the learning rules used by individuals, at the micro level, with the dynamics describing changes in the population, at the macro level.

The theory of evolution is a dynamical theory, and the second approach to evolutionary game theory sketched above explicitly models the dynamics present in interactions among individuals in the population. Since the traditional theory of games lacks an explicit treatment of the dynamics of rational deliberation, evolutionary game theory can be seen, in part, as filling an important lacuna of traditional game theory. One may seek to capture some of the dynamics of the decision-making process in traditional game theory by modeling the game in its extensive form, rather than its normal form.

However, for most games of reasonable complexity and hence interest , the extensive form of the game quickly becomes unmanageable. A selection of strategy, then, corresponds to a selection, prior to game play, of what that individual will do at any possible stage of the game. The inability to model the dynamical element of game play in traditional game theory, and the extent to which evolutionary game theory naturally incorporates dynamical considerations, reveals an important virtue of evolutionary game theory.

Evolutionary game theory has been used to explain a number of aspects of human behavior. A small sampling of topics which have been analysed from the evolutionary perspective include: altruism Fletcher and Zwick, ; Gintis et al. The following subsections provide a brief illustration of the use of evolutionary game theoretic models to explain two areas of human behavior. The first concerns the tendency of people to share equally in perfectly symmetric situations.

The second shows how populations of pre-linguistic individuals may coordinate on the use of a simple signaling system even though they lack the ability to communicate.

These two models have been pointed to as preliminary explanations of our sense of fairness and language, respectively. They were selected for inclusion here for three reasons: 1 the relative simplicity of the model, 2 the apparent success at explaining the phenomenon in question, and 3 the importance of the phenomenon to be explained.

One natural game to use for investigating the evolution of fairness is divide-the-cake this is the simplest version of the Nash bargaining game. In chapter 1 of Evolution of the Social Contract , Skyrms presents the problem as follows:. A strategy for a player, in this game, consists of an amount of cake that he would like. Figure 13 illustrates the feasible set for this game. Figure The feasible set for the game of Divide-the-Cake.

Even in the perfectly symmetric situation, answering this question is more difficult than it first appears. To see this, first notice that there are an infinite number of Nash equilibria for this game. Thus the equal split is only one of infinitely many Nash equilibria. One might propose that both players should choose that strategy which maximizes their expected payoff on the assumption they are uncertain as to whether they will be assigned the role of Player 1 or Player 2.

This proposal, Skyrms notes, is essentially that of Harsanyi The problem with this is that if players only care about their expected payoff, and they think that it is equally likely that they will be assigned the role of Player 1 or Player 2, then this, too, fails to select uniquely the equal split. Now consider the following evolutionary model: suppose we have a population of individuals who pair up and repeatedly play the game of divide-the-cake, modifying their strategies over time in a way which is described by the replicator dynamics.

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Oxford Reference. Publications Pages Publications Pages. Recall that the matrix lists the payoffsto both strategies in all possible contests:. We now need to make explicit how we arrive at each payoff. Recall thatthe general form of an equation used to calculate payoffs press here to review is:.

Payoff to Strat. We will use the descriptions of the strategies given previously to write the equationsfor each payoff. But first, let's assign some benefits and costs we coulddo this later, but let's do it now so that we can calculate each payoffas soon as we write its equation :.

Injury to self -- if the injury cuts into the animal's abilityto gain the resource in the future, then the cost of an injury is assesedas a large negative. That is, injury now tends to preclude gain in the future. On the other hand, if there is one and only one chance to gain the resource,should severe injury or death be given a large negative value?

Think aboutthis, we'll revisit this situtation when we run the Hawk and Dove simulation. In the list of cost and benefits above, it is assumed thatinjury costs are large compared to the payoff for gaining the resource.

Give a situation where this relative weighing might accurately reflect theforces acting on an animal. 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. All the other students are doves.

Between the nests, somewhere in the middle of the court is a bench — that is the cage. The doves start from one of the nest, the hawks from the other. Once the hawks leave their nest, the game starts. Once the hawks leave the nest they cannot enter either nest anymore. The nests become safe areas for the doves. The Hawks tag as many doves as possible.