978-0393919684 Chapter 12 Lecture Note

Unlock access to all the studying documents.

View Full Document

CHAPTER 12

Evolutionary Games

Teaching Suggestions

How much time you want to spend on this material will depend on the focus of your

course. For many social science courses, a general exposure to the ideas, based on a quick run

through the special examples of the Prisoners’ Dilemma and Chicken games will suffice. If

your class is comfortable with a bit of math, you can do the two together using the algebraic

formulation of the Hawk-Dove game in Section 6.

We have found this last method to be a satisfying way to approach the topic of

evolutionary games when you have time to devote only a single lecture to the topic. (If you

want to use this type of presentation but avoid the algebra, you can substi tute numbers for V

and C in the following analysis. Exercise S3 provides one possible story and payoff structure.)

You can set up the framework of the analysis first, comparing it with rational game-theoretic

ideas. (Players here are phenotypes with hardwired strategies; fitness represents payoff to the

players; equilibrium can occur as a monomorphism—pure ESS—or polymorphism—mixed

ESS.) After you present the general hawk-dove payoffs, it is a straightforward task to show

that when V > C, there is a prisoners’ dilemma structure to the payoffs and that when V < C,

there is a chicken structure.

For the prisoners’ dilemma case, the fitness graphs look as follows:

The evolutionary stable proportion of doves in the population is 0 when the game resembles

a prisoners’ dilemma. Hawk is the unique ESS. (You can note that hawk is the dominant

strategy for both players in the Hawk-Dove game if it is analyzed using standard, rational

game-theoretic analysis.)

You may also want to draw a link between the repeated Prisoners’ Dilemma game

analyzed in this chapter and Axelrod’s prisoners’ dilemma tournaments discussed in Chapter

10. One point that could be made is that, since we now know that evolutionary success

sometimes depends on starting conditions, Tit-for-tat may have failed to win Axelrod’s tour-

naments if the starting population of strategies had been sufficiently different.

For the chicken case, rational game-theoretic analysis found two pure-strategy

equilibria as well as a mixed-strategy equilibrium. For the Hawk-Dove game in the text, this

type of analysis finds the equilibrium mixture to put probability V/C on hawk and probability 1

– V/C = (C – V)/2 on dove. To make use of evolutionary analysis in this game, however, we

need to look at the fitness graphs again:

This time, (V – C)/2 < 0, so the fitness lines cross each other. For low values of d,

dove fitness is higher than hawk fitness, while for high values of d, hawk fitness is higher than

dove fitness; the value of d at the intersection is 1 – V/C.

But what is the ESS in this case? Clearly a predominantly hawk population could be

invaded by a mutant dove (since dove fitness is higher for small d) but a predominantly dove

population can also be invaded by a mutant hawk (since hawk fitness is higher for large d).

Thus, neither d = 0 (all hawk) nor d = 1 (all dove) represents a stable equilibrium; neither hawk

nor dove is an ESS. (Another way to think about this is that neither of the pure strategies is

evolutionary stable.) The other two possibilities are (1) a stable polymorphic population (the

population is mixed), and (2) a population in which each individual is hardwired to play a

mixed strategy.

To test for a stable polymorphism, we look at the dynamics of the population

proportion of doves in the fitness graph above. The analysis just above showed that, from either

extreme, d converges to some central value; that value is at the intersection of the two fitness

graphs, d = 1 – V/C. The graph indicates that when the population proportion of doves is 1 –

V/C and the population proportion of hawks is V/C, there is a stable polymorphic ESS.

The final possibility is that there might be mixers—phenotypes hardwired to play hawk

some proportion of the time and dove the rest of the time—in the population and that mixing

could be an ESS. If you have already calculated the equilibrium mixed strategy from the

rational analysis of this game, that is the mixture to suggest as a possible ESS for the mixing

phenotype. If you haven’t calculated it yet, you will want to do so now and then suggest it as

the appropriate way to hardwire the mixing phenotype. The challenge then is to show that

mixing could be an ESS when V < C. You will want to use the payoff table shown here that

includes the third possibility for each phenotype:

Column

Hawk Dove Mixer

Row

Hawk (V – C)/2, (V – C)/2 V, 0

(1 – V/C)V/2,

V/C(V – C)/2

Dove 0, V V/2, V/2

(1 – V/C)V/2,

V/2(V/C + 1)

Mixer

V/C(V – C)/2,

(1 – V/C)V/2

V/2(V/C + 1),

(1 – V/C)V/2

(1 – V/C)V/2,

(1 – V/C)V/2

To see if Mixer is an ESS, you need to see if any mutant doves or any mutant hawks

could invade a predominantly mixer population. Here, you can appeal either to fitness graphs

(as above) or to the algebra alone, since you will have shown several fitness graphs already. If

there is a small proportion d of mutant doves in a predominantly mixer population, then dove

fitness is dV/2 + (1 – d)V/2(1 – V/C) and mixer fitness is dV/2(V/C + 1) + (1 – d)V/2(1 – V/C).

Note that the last parts of the two expressions are identical so doves can invade only if V/2 >

V/2(V/C + 1). But V/C + 1 > 1, so this cannot hold and doves cannot invade. Similarly, hawks

cannot invade. Mixer is an ESS in this version of the game.

If you have time, you could consider a game in which phenotypes are more fit when

more abundant; such a game would have a payoff structure similar to the assurance-type game

in Chapter 4. Then you will get two stable monomorphic equilibria at the extremes. There will

also be a polymorphic equilibrium in such a game, but it will be unstable.

If you have even more time to spare for this topic, and especially if your class has a

significant component of science and biology students, this chapter provides a great

opportunity to engage their interest, have them do some extra reading, and take leadership roles

in class discussions. We referred in passing to several books in the chapter, most notably:

Lee Dugatkin, Cheating Monkeys and Citizen Bees: The Nature of Cooperation in Animals and

Humans (Cambridge, Mass.: Harvard University Press, 1999).

Matt Ridley, The Red Queen: Sex and the Evolution of Human Nature (New York: Penguin,

1993).

________. The Origins of Virtue: Human Instincts and the Evolution of Cooperation (New

York: Penguin, 1996).

Karl Sigmund, Games of Life: Explorations in Ecology, Evolution, and Behavior (New York:

Penguin, 1993).

Robert Wright, Non-Zero: The Logic of Human Destiny (New York: Pantheon, 2000).

Evolutionary biology is a flourishing field and a good topic for semipopular books, so

new ones keep emerging. Research articles are also abundant if your class is truly

sophisticated. You should watch for such new emerging material, perhaps using the reading

lists of colleagues in biology as a source. Many of the semipopular and popular books have

irresistible titles, so your students may actually read them. Many of the books, especially the

ones by Dugatkin and by Ridley cited above, enable you to connect this material with the

discussion of collective games in Chapter 11.

If you want to spend even more time on this material, you can make it come to life

using some simulations of patterns that evolve according to specified rules of birth and death.

Among the best known of such simulations is John Conway’s game of life; the game itself can

be found at www.bitstorm.org/gameoflife. Also, the Santa Fe Institute is active in research on

evolving cellular automata that can mimic the patterns of life cycles of some actual organisms.

You can find this work at their Web site at www.santafe.edu.

For those who want to cover the population dynamics in the Battle of the Two Sexes

game, you may find it easier to explain to a nonmathematical class if you first present it broken

into two pieces. (This is best achieved with a series of overheads.) First draw the graph with a

horizontal line at y = 2/3 and show x decreasing above the line and increasing below it. Then

draw a separate graph with a vertical line at x = 2/3 and show y increasing to the left of it and

decreasing to the right of it. These two diagrams can then be combined (stacked if you are

using overheads) to get the one shown in the text.

For the rare class that is sufficiently mathematically trained to cope with simple

differential equations, the dynamics of Figures 12.12 and 12.15 can also be made more precise.

We cover the mathematics of the Rock–Scissors–Paper game now in the optional Exercise U10

in this chapter. For the Battle of the Two Sexes game, similar calculations proceed as follows:

Suppose that dx/dt = 2/3 – y and dy/dt = 2/3 – x. Then d([2/3 – x]2 – [2/3 – x]2)/dt = –2 (2/3 – x)

dx/dt + 2(2/3 – y)dy/dt = 0. Therefore, the time paths are hyperbolas along which (2/3 – x)2 –

(2/3 – x)2 = constant.