A Simple Game: Hawks and Doves

Synopsis: Here you will have a chance to apply what you have learned about games and their solution to a classic two strategy game -- Hawks and Doves. You will be introduced to these strategies which have utility in understanding how fighting and display strategies could co-exist in a population. After this introduction, you will be guided through the construction of a payoff matrix which you will use to determine whether or not Hawk or Dove are pure ESSs. You will also be introduced to a graphical depiction of evolutionary games. This page marks the end of your "basic training" in game theory and is the gateway to using the simulations provided at this website.

Contents:

• Introduction
• Description of the Hawk and Dove Strategies
• A Qualitative Solution to the Hawk vs. Dove Game
• Construction of the Hawk vs. Dove Game
• Listing of Benefits and Costs
• Calculation of the Fitness of Each Strategy
• Graphical Results of a Simple Game
• Direct Mathematical Solution of an Equilibrium
• An Overview and Gateway to the Hawks Doves Java Simulation
• Bottom of the Page -- Links to other pages, end notes, and copyright notice

Introduction

In the last section, we learned the basics of setting up and solving a two strategy game. However, we did not actually construct and solve a game.

In this section, we will construct a classic but very simple game known as Hawks and Doves. These two simplified behavioral strategies employ very different means to obtain resources -- fighting in Hawks and display in Dove. These differences in behavior have marked consequences on the chance of winning and of paying certain types of costs. This leads to very different payoffs.

Goals and How to Use This Frame: Use the Hawks and Doves example to solidify your understanding of basic game theory. As you go through this page, links will be provided to get you quickly to various review topics. Your fundamental goal should be to feel thoroughly comfortable with the basic concepts of evolutionary game theory and with solutions to two strategy games. Gaining this understanding will allow you to get far more out the simulations available at this website.

In addition, as you study the material on this frame:

• Think hard about what species may use strategies like Dove, Hawk or a mix. Try to move beyond the game to application to real animals.
• One of the most important things you should do is to think about how the relative values of the factors that determine the payoffs ultimately affect the equilibrial point of the Hawks and Doves game. An important part of your use of the simulations will be to check out your ideas about how setting different benefits and costs will change the equilibrial point of a two strategy game.
• Think about the limitations of simple models like Hawks and Doves, yet at the same time be sure to think about the insights it has given you.

Return to this frame's contents and top

Game Theory and Evolutionary Stable Strategies -- An Example Using Alternative Strategies in a Symmetrical Contest

As with any game model, our central question is whether or not DOVE and HAWK can coexist and if so, at what frequencies.

Here is a description of the two alternative behaviors:

HAWK: very aggressive, always fights for some resource.

• Fights between hawks are brutal affairs with the loser being the one who first sustains injury. The winner takes sole possession of the resource.
• Although hawks that lose a contest are injured, the mathematics of the game requires that they not die and in fact are fully mended before their next expected contest (see note)
• For simplicity, we will assume that all hawks are equal in fighting ability; that is, each hawk has a 50% chance of winning a Hawk - Hawk conflict. Another way of saying this is that Hawk vs. Hawk contests are symmetrical.

DOVE: never fights for a resource -- it displays in any conflict and if it is attacked it immediately withdraws before it gets injured.

• Thus, in any conflict situation, dove will always lose the resource to a hawk, but it never gets hurt (never sustains a decrease in fitness) when confronting a hawk, and therefore the interactions are neutral with respect to the dove's fitness (see note).
• A corollary to this rule is that doves do not display for very long against hawks. After starting their displays, they immediately recognize that their opponent is a hawk and they withdraw without paying a meaningful display cost. (press here to review the concept of display costs)
• On the other hand, if a dove meets a dove there will be a period of displaying with some cost (time, energy for display) but no injury. We assume that all doves are equally good at displaying and they adapt a strategy of waiting for a random time period (see "War of Attrition Game"). Therefore when two doves face off, each has a 50% chance of winning.
• Notice that both doves will pay essentially the same display cost in any contest. The winner is the individual willing to pay more. However, note that the winner quits displaying essentially at the same time as the loser withdraws (see war of attrition game)

! Notice that we have assumed there are no asymmetries within a strategy -- all hawks are equally good at fighting and all doves are equally good at displays. An animal that wins one contest is just as likely to win or to lose the next. Thus in any contest between members of the same strategy, either contestant has an equal chance of winning -- there is no correlation with past success, condition, whatever. This is clearly not a very reasonable assumption, but we're just starting out so let's keep things simple.

TWO OTHER IMPORTANT ASSUMPTIONS:

• Assume that the attacking animal (the one that either starts first to physically attack or to display) has no knowledge of the strategy that its opponent will play.
• Assume that these interactions increase or decrease the animal's fitness from some baseline fitness. In other words, these interactions simply modify up or down an animal's fitness -- wining a contest does create fitness. This assumption is associated with our convention that injuries and display costs will be assigned negative scores -- losing animals do not have negative fitnesses! (to review discussion of this idea with regards to modeling in general, press here).

Return to this frame's contents and top

Preliminary (Qualitative) Exploration

First, we'll make a qualitative analysis of the game, then we'll use game theory to make a much more quantitative prediction (as was discussed in the introductory material dealing with games) . Let's start with the following question: Are either of the two strategies by themselves impervious to invasion? -- that is, does either represent a pure ESS?

To most people it immediately appears that DOVE is not a pure ESS. Imagine a population entirely of doves. It is probably a very nice place to live and everyone is doing reasonably well without injuries when it comes to conflicts over resources - the worst thing that happens to you is that you waste time and energy displaying. But that is OK, because on the average you win 50% of the encounters. Therefore on the average, you will come out ahead provided the display costs are not large compared to the resource value.

Now, imagine what happens if a HAWK appears by mutation or immigration. The Hawk will do extremely well relative to any dove -- winning every encounter and initially at least suffering no injuries. Thus, its frequency will increase at the expense of dove. Thus, Dove is not a pure ESS. If dove is not an ESS, what about hawk?

So, let's do the analysis again, this time starting with a population made entirely of hawks. This would be a nasty place, an asphalt jungle where you would not want to live. Lots of injurious fights. Although these fights don't kill you, they tend to lower everyone's fitness. Yet, just like with the dove population, no hawk is doing better than any other and the resources are getting divided equally.

Could a DOVE possibly invade this rough place? It might not seem so since they always lose fights with hawks. Yet think about it:

• Doves don't get hurt -- the best way to think about this is that they do not pay high costs in fitness for losing a contest. While they never beat a hawk, they don't get hurt because they flee the moment they realize they are in a contest with a hawk. The result is that, unlike hawk-hawk contests, a dove's fitness is not lowered by a conflict with a hawk (i.e., it is the same as if no contest at all had occurred -- remember that we assume a certain baseline of fitness independent of the outcome of contests - contests only increase or decrease this amount).
• Doves are winners 50% of the time against other doves and they probably lose little in such contests.

Thus, if a mutant appears in the form of a dove or one wanders in from elsewhere, 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 assumptions about the relative values of the resource and the costs of injury and display that are consistent with the behavioral descriptions . You probably realize that if we changed some of these assumptions of relative value, the game might turn out differently -- perhaps Hawk or Dove could become an ESS. Moreover, even if we stick to the qualitative values and to our conclusion that there is no pure ESS, the technique we have just used will not allow us to predict the frequencies of Dove and Hawk at the mixed ESS. As was stated earlier, the best models make quantitative predictions since these are often most easily tested (to review testing of models, press here).

Thus, in the next section we will use the rules and techniques we previously learned to quantitatively analyze the Hawks and Doves game.

Return to this frame's contents and top

Formal Analysis of the Hawk-Dove ESS

The first step of our analysis is to set-up a payoff matrix. Recall that the matrix lists the payoffs to both strategies in all possible contests:

	Opponent
Focal Strategy	Hawk	Dove
Hawk	E(H,H)	E(H,D)
Dove	E(D,H)	E(D,D)

We now need to make explicit how we arrive at each payoff. Recall that the general form of an equation used to calculate payoffs (press here to review) is:

Eq. 1. Payoff(to Strat., when vs. a Strat.) = [(chance of win) * (resource value - cost of win)] + [(chance of loss) * cost of loss]

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

Action	Benefit or Cost (arbitrary units)
Gain Resource	+ 50
Lose Resource	0
Injury to Self	- 100
Cost of Display to Self	- 10

Rationale for these values:

Gain resource -- self-explanatory.

Lose resource -- self-explanatory (nothing gained).

Injury to self -- if the injury cuts into the animal's ability to gain the resource in the future, then the cost of an injury is assesed as 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 about this, we'll revisit this situtation when we run the Hawk and Dove simulation.

? In the list of cost and benefits above, it is assumed that injury costs are large compared to the payoff for gaining the resource. Give a situation where this relative weighing might accurately reflect the forces acting on an animal. ANS

Cost of display -- displays generally have costs, although how high they are varies -- clearly they have variable costs in terms of energy and time and they may also increase risk of being preyed upon. All of these type of measurements, in theory at least, can be translated into fitness terms.

! Important Note: All of these separate payoffs are in units of fitness (whatever they are!). You will see shortly that the values that are assigned to each payoff is crucial to outcome of the game -- thus accurate estimates are vital in usefulness of any ESS game in understanding a behavior.

a. Calculation of the payoff to Hawk in Hawk vs. H contests:
Relevant variables (from eq. 1)

• chance of winning (50% -- i.e. the contests are symmetrical)
• resource value (see table)
• chance of losing (50%)
• costs of losing -- in this case, the cost is an injury cost (see table)
• notice that no costs are paid in winning

? Does it seem reasonable that hawks pay no cost in winning? Also, does it seem reasonable that the loser only pays an injury cost? Think about what animals do and about simplifications of models. For some discussion of this question, press here (but think about it first)

Thus:

Eq. 2: E(H,H) = (0.5 * 50) + (0.5 * -100) = 25 - 50 = -25

Note; the costs of losing are added in our model since we gave the costs a negative sign to emphasize that they lowered the fitness of the loser.

b. Calculation of the payoff to Hawk when vs. Dove:
Relevant variables (from eq. 1)

• chance of winning 100% -- i.e. the contests are asymmetrical
• resource value (see table)
• no costs to the hawk since (a) they never lose and (b) since the dove immediately retreats once it recognizes the hawk

Eq. 3: E(H,D) = 1.0 * 50 - 0 = +50

c. Calculation of the payoff to Dove when vs. Hawk:
Relevant variables (from eq. 1)

• chance of winning o% -- i.e. the contests are asymmetrical and doves always lose
• no costs to the dove since it immediately retreats once it recognizes the hawk

Eq. 4: E(D,H) = 0 * 50 + 1.0 * 0 = 0

d. Calculation of the payoff to Dove when vs. Dove:
Relevant variables (from eq. 1)

• chance of winning (50% -- dove vs. dove contests are symmetrical)
• resource value (see table)
• display cost paid in winning -- both animals will display essentially the same amount of time. The one that wins is the one that is willing to display for a longer period of time at this particular time (see war of attrition).
• chance of losing (50%)
• costs of losing -- as with the winner, the loser also pays a display cost and it is the same as what the winner pays (see table)
• no injury cost -- no violence please, we're doves for heaven's sake!

Eq. 5: E(D,D) = (0.5) * (50 - 10) - (0.5) * (-10) = +15

So for this particular version of the Hawk vs. Dove game (defined by these payoffs), the pay-off matrix is:

	Opponent
Focal Strategy	Hawk	Dove
Hawk	-25	+50
Dove	0	+15

? Using this matrix, see if the Hawk Dove game above meets the criteria for a pure ESS. (hint: review the rules for a pure ESS and then arbitrarily define H as A and test to see if H is a pure ESS with payoffs listed above (Do this for both strategies -- use H and then D as strategy A. Should you get the same results each time?) ANS -- as usual, please try to reason through this one before going to the answer.

Return to this frame's contents and top

Calculations of the Fitness of Each Strategy and Mixed ESSs

If you did the problem above, you will realize that neither Hawk nor Dove are pure ESSs given the payoffs calculated from the equations and values for benefits and costs presented above. (When you use the simulation, you will see that certain benefits and costs can be used to make either of the strategies pure ESSs, although these might seem to involve unreasonable assumptions).

It is good to keep in mind the fact that the rules you used to determine that neither strategy was a pure ESS require some reasonable assumptions (to review, press here).

If we have no pure ESS, we know that in a two strategy game there will be a mixed ESS which is defined as the frequencies of the strategies where both have equal fitness. Recall that the fitness of a strategy is the sum of the payoffs times the frequency of their occurrence

Thus, if we assume that:

Eq. 7a: frequency(Hawk) = h   then: Eq. 7b: frequency(Dove) = (1 - h)

Thus, the fitness of Hawk, W(H), is:

Eq. 8: W(H) = h * E(H,H) + (1-h) * E(H,D)

and the fitness of Dove, W(D) is:

Eq. 9: W(D) = h * E(D,H) + (1-h) * E(D,D)

Notice that each of the equations for strategy fitness yield a straight line when solved for a series of frequencies.

Now since IN A MIXED ESS BOTH STRATEGIES MUST HAVE THE SAME FITNESS, we can determine the equilibrial mix by setting the fitnesses of the two strategies as equal to each other:

Eq. 10: W(H) = W(D) at equilibrium (mixed ESS)

For our game:

Eq. 11: h * E(H,H) + (1 - h) * E(H,D) = h * E(D,H) + (1 - h) * E(D,D)

If we now solve for the frequency of hawk at this equilibrium:

Eq. 12: h / (1 - h) = [E(D,D) - E(H,D)] / [E(H,H) - E(D,H)]

? Calculate the mixed ESS frequencies of Hawk and Dove using the payoff matrix above. ANS.

We can understand the solution more clearly if we graph eqs. 8 and 9 where the solid line is dove (eq. 9) and the dotted line is hawk(eq. 8). The intersection of the Hawk and Dove plots represents the frequency of one strategy (in this case Hawk) where the fitnesses of both strategies are equally fit (in terms of payoff units).

Remarks About the Graphical Results
of the Hawk vs. Dove Game:

The above graph points up a number of interesting things:

• Note that at the equilibrial point, the addition of individuals of either strategy lowers the relative fitness of all members of that strategy.
  • Thus, an increase in Hawks results in moving to the right along the hawk line to a lower fitness than that of dove -- away from equilibrium.
  • Likewise, addition of Doves (movement to the left along the Dove line since their frequency is (1 - h) lowers their fitness relative to Hawk.
  • In both cases, the lowered fitness will eventually result in a reduction of their numbers and a return to the equilibrium frequency.
• Also note that the addition of any Hawk overall lowers everyone's absolute fitness! -- both curves have negative slopes.
• And, the graph provides another way to see that neither Hawk nor Dove are pure ESSs:
  • Hawk does very poorly when in high frequency compared to when it is rare. Thus, it is easily invaded by what might seem to be the most improbable of invaders, the pacific "dove".
  • On the other hand, Dove is not stable since Hawks do extremely well when entering the population
  • Thus, although it is obvious that any Hawks would always be "bad for the species" they cannot be kept out once they appear (so much for group selection being a common phenomenon). For this particular set of payoffs, Dove is not an ESS any more than Hawk is.

Return to this frame's contents and top

Computer Simulation of the Hawks and Dove Game

At this point you know how to set up and solve a simple game. And you have a basic familiarity with the Hawks and Doves game.

So, you are now ready to explore the Hawks and Doves game in detail using a simulation that will allow you to alter payoffs by changing benefits and costs. The simulation will provide you with a visual representation of the solution, using the same techniques you have just learned (except the computer will now do the computational work for you!). And, you'll get to see something new -- you'll be able to set frequencies of the two strategies and then see how a population with a given payoff matrix will evolve over time.

Press here to go to a page that explains how to use the simulation and then launches it.

End Notes

Why can't hawks die or get permanently knocked out of action? Why must they be miraculously restored to health?

The reason is very simple. If this were not the case, then in any population containing more than one hawk, Hawk vs. Hawk contests would cause the frequency of hawk to decrease. The more hawks, the more Hawk vs. Hawk contests and the faster freq(Hawk) will decrease! Notice that the equations we learned earlier for finding the fitness of the strategy all implied a constant freq. of the strategy. Thus, the bad things that happen in Hawk vs. Hawk contests should be seen as changing (in this case lowering -- to see one example of a Hawk vs. Dove payoff matrix press here) the general fitness of hawk individuals in the population without changing their frequencies.

Return to previous place in MAIN text

Return to previous place in end notes

There are a couple of things to notice here. First, no doves get killed. To reiterate the material about freq of Hawk and injury, notice that if injured hawks did drop out, the freq. of Dove would increase. Also notice the difference in the payoff (according to the descriptions of H and D that you have just read or in the same example payoff matrix that we considered with Hawk) -- negative payoffs tend to mean a lowered fitness as a result of the contest but not death; and payoffs of 0 (the payoff to Dove vs. Hawk in this example) mean no effect on fitness -- the dove goes on as before.

Return to previous place in text

The important idea here is that the animal must be able to reproduce even if it loses all of its contests. If not, you might as well count the animal as dead with the same consequences as outlined in the discussion of injuries. Again, the important consequence of the game is that it alters the fitness of the individual but does not kill (or essentially kill) the individual.

Return to previous place in text

Go to:

• Top of the page
• Continue your studies -- a third strategy -- Bourgeois
• Definition Page

Copyright © 1999 by Kenneth N. Prestwich College of the Holy Cross, Worcester, MA USA 01610 email: kprestwi@holycross.edu About Fair Use of these materials Last modified 2 - 22 - 99