Hawk-dove game nash equilibrium

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.

The replicator dynamics allows us to model how the distribution of strategies in the population changes over time, beginning from a particular initial condition. Figure 14 below shows two evolutionary outcomes under the continuous replicator dynamics. Notice that although fair division can evolve, as in Figure 14 a , it is not the only evolutionary stable outcome, as Figure 14 b illustrates.

Figure Two evolutionary outcomes under the continuous replicator dynamics for the game of divide-the-cake. Of the eleven strategies present, only three are colour-coded so as to be identifiable in the plot, as noted in the legend. What the above shows is that, in a population of boundedly rational agents who modify their behaviours in a manner described by the replicator dynamics, fair division is one, although not the only, evolutionary outcome.

The tendency of fair division to emerge, assuming that any initial condition is equally likely, can be measured by determining the size of the basin of attraction of the state where everyone in the population uses the strategy Demand 5 slices.

However, it is important to realise that the replicator dynamics assumes any pairwise interaction between individuals is equally likely. In reality, quite often interactions between individuals are correlated to some extent. When correlation is introduced, the frequency with which fair division emerges changes drastically. Note that this does not depend on there only being three strategies present: allowing for some correlation between interactions increases the probability of fair division evolving even if the initial conditions contain individuals using any of the eleven possible strategies.

Figure Three diagrams showing how, as the amount of correlation among interactions increases, fair division is more likely to evolve. In figures 15 a and 15 b , there is an unstable fixed point in the interior of space where all three strategies are present in the population. This is the point where the evolutionary trajectories appear to intersect. This fixed point is what is known as a saddle point in dynamical systems theory: the smallest perturbation will cause the population to evolve away from that point to one of the other two attractors.

What, then, can we conclude from this model regarding the evolution of fair division? It all depends, of course, on how accurately the replicator dynamics models the primary evolutionary forces cultural or biological acting on human populations. As Skyrms notes:. This claim, of course, has not gone without comment. In his seminal work Convention , David Lewis developed the idea of sender-receiver games.

Such games have been used to explain how language, and semantic content, can emerge in a community which originally did not possess any language whatsoever. Lewis, , pp. Since the publication of Convention , it is more common to refer to the communicator as the sender and the members of the audience as receivers.

The basic idea behind sender-receiver games is the following: Nature selects which state of the world obtains. The person in the role of Sender observes this state of the world correctly identifying it , and sends a signal to the person in the role of Receiver. The Receiver, upon receipt of this signal, performs a response.

If what the Receiver does is the correct response, given the state of the world, then both players receive a payoff of 1; if the Receiver performed an incorrect response, then both players receive a payoff of 0. Notice that, in this simplified model, no chance of error exists at any stage. The Sender always observes the true state of the world and always sends the signal he intended to send.

Likewise, the Receiver always receives the signal sent by the Sender i. We shall see later why larger sender-receiver games are increasingly difficult to analyse. Since we are considering the case where there is only a single responder, the second requirement is otiose. It is, as Lewis notes, a function from the set of states of the world into the set of signals. This means that it is possible that a sender may send the same signal in two different states of the world.

Such a strategy makes no sense, from a rational point of view, because the receiver would not get enough information to be able to identify the correct response for the state of the world. However, we do not exclude these strategies from consideration because they are logically possible strategies. How many sender strategies are there? This means there are four possible sender strategies. What is a strategy for a receiver?

As in the case of the sender, we allow the receiver to perform the same response for more than one signal. These receiver strategies are:. All other possible combinations of strategies result in the players failing to coordinate. What if the roles of Sender and Receiver are not permanently assigned to individuals?

That is, what if nature flips a coin and assigns one player to the role of Sender and the other player to the role of Receiver, and then has them play the game? It makes a difference whether one considers the roles of Sender and Receiver to be permanently assigned or not. If the roles are assigned at random, there are four signaling systems amongst two players [ 12 ] :.

Signaling systems 3 and 4 are curious. System 3 is a case where, for example, I speak in French but listen in German, and you speak German but listen in French. System 4 swaps French and German for both you and me. Notice that in systems 3 and 4 the players are able to correctly coordinate the response with the state of the world regardless of who gets assigned the role of Sender or Receiver.

The problem, of course, with signaling systems 3 and 4 is that neither Player 1 nor Player 2 would do well when pitted against a clone of himself.

They are cases where the signaling system would not work in a population of players who are pairwise randomly assigned to play the sender-receiver game. In fact, it is straightforward to show that the strategies Sender 2, Receiver 2 and Sender 3, Receiver 3 are the only evolutionarily stable strategies see Skyrms , 89— As a first approach to the dynamics of sender-receiver games, let us restrict attention to the four strategies Sender 1, Receiver 1 , Sender 2, Receiver 2 , Sender 3, Receiver 3 , and Sender 4, Receiver 4.

Figure 16 illustrates the state space under the continuous replicator dynamics for the sender-receiver game consisting of two states of the world, two signals, and two responses, where players are restricted to using one of the previous four strategies. One can see that evolution leads the population in almost all cases [ 13 ] to converge to one of the two signaling systems.

Figure 17 illustrates the outcome of one run of the replicator dynamics for a single population model where all sixteen possible strategies are represented. We see that eventually the population, for this particular set of initial conditions, converges to one of the pure Lewisian signalling systems identified above.

Figure The evolution of a signalling system under the replicator dynamics. When the number of states of the world, the number of signals, and the number of actions increase from 2, the situation rapidly becomes much more complex. Given this, one might think that it would prove difficult for evolution to settle upon an optimal signalling system. The third equilibrium point was found to be The corresponding eigenvalues of were found to be The single zero eigenvalue indicates that this equilibrium point is normally hyperbolic, and the local stability can be determined through the nonzero eigenvalues by the invariant manifold theorem [ 27 ].

In particular, this point is a stable node if It is an unstable node if It is a saddle point under the following conditions:.

The fourth equilibrium point was found to be The corresponding eigenvalues of were found to be This point is a stable node if It is an unstable node if It is a saddle point under the following conditions:.

The fifth equilibrium point was found to be The corresponding eigenvalues of were found to be This point is a stable node if It is an unstable node if From 29 , it can be seen that is in fact never a saddle point of the dynamical system. The sixth equilibrium point was found to be The corresponding eigenvalues of were found to be One sees that since , this point is manifestly nonhyperbolic.

As such, its stability properties cannot be determined through the Jacobian matrix. The final equilibrium point was found to be The corresponding eigenvalues of were found to be This point is a stable node if It is an unstable node if Further, this point is never a saddle point as can be seen from With knowledge of the equilibrium points and their local stability as given in the previous sections, we now attempt to describe bifurcation behaviour exhibited by this dynamical system.

Analyzing bifurcation behaviour is important as this determines the local changes in stability of the equilibrium points of the system. The linearized system in a neighbourhood of takes the form We see that destabilizes when , destabilizes when , and destabilizes when.

The linearized system in a neighbourhood of takes the form We see that destabilizes along the line , while and destabilize when. The linearized system in a neighbourhood of takes the form Therefore, is destabilized by and. The linearized system in a neighbourhood of takes the form Therefore, destabilizes along the line. Further, destabilizes when. Finally, destabilizes when for. The linearized system in a neighbourhood of takes the form Therefore, is destabilized by , , and along the line.

The linearized system in a neighbourhood of takes the form Therefore, destabilizes whenever , or whenever for. The linearized system in a neighbourhood of takes the form We see that, therefore, is destabilized by , , and whenever , for. From these calculations, we can therefore see that, along , as one goes from to , and go from being unstable nodes to stable ones, and vice versa, while goes from being a stable node to an unstable one.

Along the line , as we go from to , and go from being unstable nodes to stable nodes, while and go from being stable nodes to unstable nodes. The existence of these Nash equilibria shows that this asymmetric Hawk-Dove game produces rational behaviour in a population composed of players that are not required to make consciously rational decisions.

In other words, the population is stable when, given what everyone else is doing, no individual would get a better result by adopting a different strategy. This is the so-called population view of a Nash equilibrium, which Nash himself described as the mass action interpretation [ 2 , 29 ]. It is perhaps of interest to discuss our results found above in connection with the standard two-strategy Hawk-Dove game. Following [ 2 ], we note that the payoff matrix for such a game is given by Table 2.

In this case, the replicator dynamics are a simple consequence of 1 - 2. Namely, let denote the proportion of individuals in the population that use strategy in Table 2. Then, the replicator dynamics are governed by the single ordinary differential equation Clearly, 45 has equilibrium points , , and.

Let us denote by the right-hand side of Then, Clearly, when , , which is negative when and positive when. Therefore, the point is a stable node when and an unstable node when. Further, when ,. In this case, the point is a stable node for and. Further, it is an unstable node for and. Finally, when , we have that. This point is a stable node when and , or when and or.

It is an unstable node when and or , or when and. Comparing these cases to the Nash equilibria we found in the full asymmetric game, we see that the case when corresponds to the case of Equilibrium Point 7, where was a Nash equilibrium.

The case in this example corresponds to Equilibrium Points 1 and 4, where and were both found to be Nash equilibria of the full asymmetric replicator dynamics. Certainly, this shows that for any initial population that is not at an equilibrium point. In this section, we present some numerical simulations of the work above.

In Figure 1 , we assume that , ; in Figure 2 , we assume that , ; in Figure 3 , we assume that , ; in Figure 4 , we assume that , ; and in Figure 5 , ,. In this paper, we analyzed, using a dynamical systems approach, the replicator dynamics for the asymmetric Hawk-Dove game in which there is a set of four pure strategies with arbitrary payoffs. We gave a full account of the equilibrium points and their stability and derived the Nash equilibria. In particular, we found that if , , then the strategy pairs and are Nash equilibria.

If , , then the strategy pair is a Nash equilibrium. Finally, if , , then the strategy pair is a Nash equilibrium. We also gave a detailed account of the local bifurcations that the system exhibits based on choices of the typical Hawk-Dove parameters and. So, let's do the analysis again, this time starting with a populationmade entirely of hawks. This would be a nasty place, an asphalt junglewhere you would not want to live.

Lots of injurious fights. Although thesefights don't kill you, they tend to lower everyone's fitness. Yet, justlike with the dove population, no hawk is doing better than any other andthe resources are getting divided equally. Could a DOVE possibly invade this rough place? It might not seem so sincethey always lose fights with hawks.

Yet think about it:. Thus, if a mutant appears in the form of a dove or one wanders in fromelsewhere, it will do quite well relative to hawk and increase in frequency. Thus, Hawk is also not a pure ESS. Notice that in all of the arguments above, we made implicit assumptionsabout the relative values of the resource and the costs of injury and displaythat are consistent with the behavioral descriptions. You probably realizethat if we changed some of these assumptions of relative value, the gamemight turn out differently -- perhaps Hawk or Dove could become an ESS.

Moreover, even if we stick to the qualitative values and to our conclusionthat there is no pure ESS, the technique we have just used will not allowus to predict the frequencies of Dove and Hawk at the mixed ESS. As wasstated earlier, the best models make quantitative predictions since theseare often most easily tested to review testing of models, press here. Thus, in the next section we will use the rules and techniques we previouslylearned to quantitatively analyze the Hawks and Doves game.

The first step of our analysis is to set-up a payoff matrix. 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.