ch13solutionssolved edit.docx

1、ch13solutionssolved editSolutions to Chapter 13 ExercisesSOLVED EXERCISESS1. (a) The payoff table for the two types of travelers is:HighLowHigh100, 10030, 70Low70, 3050, 50 (b) The graph is: (c) There are three possible equilibria: a stable monomorphic equilibrium of all Low types (h = 0), a stable

2、monomorphic equilibrium of all High types (h = 1), and an unstable polymorphic equilibrium where two-fifths of the population are High types (h = 0.4). S2. Throughout the answers for this exercise, we let x represent the population proportion of the invading type. (a) In a population primarily consi

3、sting of A types with only a small proportion (x) of invading T types, the A-type fitness is F(A) = 864(1 x) + 936x = 864 + 72x and the T-type fitness is F(T) = 792(1 x) + 972x = 792 + 180x. F(A) F(T) as long as 864 + 72x 792 + 180x, or 72 108x, or x F(N) as long as 864 + 216x 648 + 324x, or 216 108

4、x, or x F(A) when 972 324x 1,080 - 216x, or 108 + 108x F(A) when 972 180x 936 72x, or 36 108x, or x 36/108 = 1/3. An all-T population cant be invaded by A types unless the A types are more than one-third of the population, so a small number of mutant As cannot successfully invade. When the mutants a

5、re type N, F(T) = 972(1 x) + 972x = 972, and the mutant Ns also have fitness F(N) = 972(1 x) + 972x = 972. Again, the fitnesses are equal, so Ts and Ns do equally well in the population, and Ts cannot prevent Ns from invading.ColumnATRowA20, 2011, 35T35, 116, 6S3. (a) The payoff table is at right. (

6、b) Let x be the population proportion of T players. Then the As expected years in jail are 20(1 x) + 11x = 20 9x, and the Ts expected years in jail are 35(1 x) + 6x = 35 29x. Then the T type is fitter than the A type if the former spends fewer years in jail: remember that payoffs are years in jail h

7、ere, so smaller numbers are better. Ts are fitter than As when 35 29x 20 9x, or 15 0.75. If more than 75% of the population is already T, then T players are fitter, and their proportions will grow; we have one stable ESS when the population is all T. Similarly, if the population starts with fewer th

8、an 75% T types, then the A types are fitter, and their proportion will grow; we have another stable ESS when the population is all S. Finally, there is another equilibrium in which exactly 75% of the population is type T and 25% of the population is type A; that equilibrium is polymorphic but unstab

9、le.ColumnATNRowA20, 2011, 352, 50T35, 116, 66, 6N50, 26, 66, 6 (c) See the payoff table at right. The strategy N does not do well in this game. In a mixed population that includes all three types, let x and y be the population proportions of T and N, respectively. Then the As expected years in jail

10、equal 20(1 x y) + 11x + 2y, the Ts expected years in jail are 35(1 x y) + 6x + 6y, and the Ns expected years in jail are 50(1 x y) + 6x + 6y. T is strictly fitter than N in this population (remember that smaller numbers are better) and will eventually dominate N. Using the same equations, we see tha

11、t if the population consists initially of only N and A, a mutant T can invade successfully and then eventually dominate N. Even if the population is initially all N, a mutant A will be able to invade. In that case, for a small proportion a of A types, the Ns get 50a + 6(1 a) years in jail, whereas t

12、he As get 20a + 2(1 a); the As are fitter for all values of a, so they can successfully invade. This analysis implies that N cannot be an ESS in this game. The ESS here are the same as those in part (b). S4. The payoff table is shown at right.MaleTheater(prop. y)Movie(prop. 1 y)FemaleTheater(prop. x

13、)1, 10, 0Movie(prop. 1 x)0, 02, 2 Fitness of female T individual = 1y + 0(1 y) = y. Fitness of female M individual = 0y + 2(1 y) = 2(1 y). So for females, the theater type is fitter than the movie type (and the population proportion x of theater types increases) if y 2(1 y) or if y 2/3. Similarly, f

14、or males, the theater type is fitter than the movie type (and the population proportion y of Theater types increases) if x 2(1 x), or if x 2/3. The diagram below shows the dynamics of the game. There are two ESS: (0, 0) and (1,1).S5. (a) The most obvious mixture to play is the mixed-strategy equilib

15、rium of the two-player version of the game: report $100 (i.e., act like a High type) with probability 0.4. (b) To calculate expected payoffs for pairs that include a Mixer type, remember that the Mixer reports $100 with probability 0.4 and $50 with probability $40. So when a Mixer plays against a Hi

16、gh type, there is a 40% chance that both report $100 and a 60% chance that there is one report of $00 and one report of $50. The expected payoff of the pairing (Mixer, High) is thus 0.4*(100, 100) + 0.6*(70, 30) = (82, 58). Similarly, the expected payoff of the pairing (Mixer, Low) is 0.4*(30, 70) +

17、 0.6*(50, 50) = (42, 58). The expected payoff when two Mixer types meet is a little more complicated, since there are four cases to consider. With probability 0.4*0.4, both Mixers report $100. With probability 0.4*0.6, the first reports $100 and the second $50. With the same probability, the first r

18、eports $50 and the second $100. Finally, with probability 0.6*0.6, they both report $50. The expected payoff for the pairing (Mixer, Mixer) is thus 0.4*0.4*(100, 100) + 0.4*0.6*(30, 70) + 0.6*0.4*(70, 30) + 0.6*0.6*(50, 50) = (58, 58). The three-by-three table of expected payoffs with the Mixer type

19、 is then:HighMixerLowHigh100, 10058, 8230, 70Mixer82, 5858, 5842, 58Low70, 3058, 4250, 50 (c) Consider the case when a single High type attempts to invade a population entirely composed of Mixer types. In this situation we need only the upper left corner of the payoff table in part (b):HighMixerHigh

20、100, 10058, 82Mixer82, 5858, 58 In terms of mthe proportion of Mixer types in the populationthe fitness of the High type is 100(1 m) + 58m, whereas the fitness of the Mixer type is 82(1 m) + 58m. When m = 1, it is true that the fitnesses are equal: 100(0) + 58(1) = 82(0) + 58(1). However, whenever m

21、 82(1 m) + 58m 18(1 m) 0 whenever m 1. A single High type will be more fit than the Mixer types in the rest of the population, so the High type will successfully invade. Now consider the case when a single Low type attempts to invade a population entirely composed of Mixer types. Now we need only th

22、e lower right corner of the payoff table in part (b):MixerLowMixer58, 5842, 58Low58, 4250, 50 In terms of mthe proportion of Mixer types in the populationthe fitness of the Low type is 58m + 50(1 m), whereas the fitness of the Mixer type is 58m + 42(1 m). When m = 1, the fitnesses of the two types a

23、re equal: 58(1) + 50(0) = 58(1) + 42(0). However, whenever m 58m + 42(1 m) 8(1 m) 0 whenever m 1. A single Low type will be more fit than the Mixer types in the rest of the population, so the Low type will successfully invade.Since either a High type or a Low type could successfully invade a populat

24、ion of Mixer types, the Mixer phenotype is not an ESS of this game. S6. (a) The fitness of the solar type, FS, is 2s + 3(1 s) = 3 s, and the fitness of the fossil fuel type, FFF, is 4s + 2(1 s) = 2 + 2s. Graphing the fitness curves of the solar type and the fossil fuel type with respect to s, we hav

25、e: There are three possible equilibria: an unstable monomorphic equilibrium where everyone uses fossil fuels (s = 0), an unstable monomorphic equilibrium where everyone uses solar (s = 1), and a stable polymorphic equilibrium where one-third of the population uses solar (s = 1/3). The monomorphic eq

26、uilibria exist as possibilities because in the absence of any alternative types the lone existing type will simply reproduce in perpetuity. (b) With the change in the (solar, solar) payoff, FS = ys + 3(1 s) = 3 + (y 3)s. There is no change in FFF. At a polymorphic equilibrium, the fitness of the solar type is equal to the fitness of the fossil fuel type, so 3 + (y 3)s