Abstract

We analyze, 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 give a full account of the equilibrium points and their stability and derive the Nash equilibria. We also give a detailed account of the local bifurcations that the system exhibits based on choices of the typical Hawk-Dove parameters and . We also give details on the connections between the results found in this work and those of the standard two-strategy Hawk-Dove game. We conclude the paper with some examples of numerical simulations that further illustrate some global behaviours of the system.

1. Introduction

The Hawk-Dove game is one of the first examples of a pairwise game that was used to model the conflict between animals [1]. The basic idea is that “Hawks” and “Doves” represent two types of behaviours (actions or pure strategies) that could be exhibited by animals of the same species [2]. In the standard Hawk-Dove game, individuals can use one of two possible pure strategies. In one case, they can be aggressive/a “Hawk,” which is typically denoted by , or be nonaggressive/a “Dove,” which is typically denoted by . Then, at various times, individuals in this population can have a conflict over a resource which has value , where the winner of the conflict gets the resource and the loser pays a cost .

The Hawk-Dove game has been studied in the context of replicator dynamics a number of times over the past several years. Some examples of these studies include [3–17].

In replicator dynamics, it is assumed that individuals are programmed to use only pure strategies from a finite set . It can be shown [2] that the dynamical evolution of the proportion of individuals using strategy , , is given by where is the payoff to individuals using strategy , while is known as the average payoff and is defined as Further to (1), one also has the constraint

In this paper, we wish to consider an asymmetric pairwise Dove-Hawk game. Following [2], specifically, this is where two individuals are contesting ownership of a territory that one of them controls. One assumes that the value of the territory and costs of contest are the same for both players. Unlike the standard Hawk-Dove game described above, players can now condition their behaviour on the role that they occupy, which is typically denoted as owner or intruder. So, the pure strategies now take the forms play Hawk if owner and play Dove if intruder, which we will be denoted by . Therefore, there is a set of four pure strategies:

From these arguments, it can be shown [2] that the payoff matrix is given by Table 1.

HH HD DH DD HH , , , , HD , , , , DH , , , , DD , , , ,

We note that the replicator dynamics of this four-strategy asymmetric Hawk-Dove game have not been analyzed from a dynamical systems perspective in the literature to the best of the authors’ knowledge. However, some examples of related asymmetric games can be found in [18–26].

2. The Dynamical Equations

Let us denote by the proportion of individuals who use strategies , , , and , respectively. Then, from the payoff matrix in Table 1 and (1)-(2), the replicator dynamics are given by the following dynamical system: where and, from (3),

This four-dimensional dynamical system can be reduced to three dimensions if we set, via (7), . Therefore, in what follows, we will study the following unconstrained three-dimensional system:

3. A Local Stability Analysis

From (8), we now present the equilibrium points along with their eigenvalues and local stability. The Jacobian matrix, denoted by , corresponding to this dynamical system is a matrix, whose entries are listed as follows:

3.1. Equilibrium Point 1

The first 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 if

3.2. Equilibrium Point 2

The second equilibrium point was found to be The corresponding eigenvalues of were found to be This point is neither a stable nor an unstable node. However, it is a saddle point under the following conditions:

3.3. Equilibrium Point 3

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:

3.4. Equilibrium Point 4

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:

3.5. Equilibrium Point 5

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.

3.6. Equilibrium Point 6

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.

3.7. Equilibrium Point 7

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 (35).

4. Local Bifurcations

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 mechanism for these bifurcations can be seen as follows.

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. Whenever , , as one goes from to , goes from being an unstable node to a stable node, 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. Finally, whenever , , as we go from to , goes from being a stable node to an unstable one, while and go from being unstable nodes to stable ones.

5. Nash Equilibria

Determining the future asymptotic behaviour of the replicator dynamics is of importance since, by Theorem 9.15 in [2], if is an asymptotically stable fixed point of the dynamical system, then the symmetric strategy pair is a Nash equilibrium.

Following [28], we note that, first, by Lyapunov’s theorem, if all eigenvalues of the linear part of a vector field at a singular point have a negative real part, the singular point is asymptotically stable.

From our stability analysis of the various equilibrium points in the preceding sections, we therefore observe the following Nash equilibria of the replicator dynamics depending on the choices of and :(1) , is asymptotically stable is a Nash equilibrium.(2) , is asymptotically stable is a Nash equilibrium.(3) , is asymptotically stable is a Nash equilibrium.(4) , is asymptotically stable is a Nash equilibrium.

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].

6. Connections with the Two-Strategy Hawk-Dove Game

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 (45). 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.

7. Some Numerical Simulations

In this section, we present some numerical simulations of the work above. These simulations were completed in MATLAB using the ODE23s solver with a variety of initial conditions which are denoted with asterisks in Figures 1, 2, 3, 4, and 5.





















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, , .

8. Conclusions

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 . We also gave details on the connections between the results we found and those of the standard two-strategy Hawk-Dove game. We concluded the paper with some examples of numerical simulations that further illustrate some global behaviours of the system.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was partially supported by a grant given to Michael C. Haslam from the Natural Sciences and Engineering Research Council of Canada.