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ECON2214 Games and Decisions Tutorial 9 1 Evolutionary Games 1.1 Question 1 There are two types of racers - tortoises and hares - who race against one another in randomly drawn pairs. In this world, hares beat tortoises every time without fail. If two hares race they tie, and they are completely exhausted by the race. When two tortoises race they also tie, but they enjoy a pleasant conversation along the way. The payo ff table is as follows: (a) Assume that the proportion of tortoises in the population, t , is 0.5. Should tortoise or hare have greater fitness? (b) Should tortoise or hare have greater fitness if t = 0 . 1 ? (c) Find all equilibria of this game. For each equilibrium, state whether it is monomorphic or polymorphic and whether it is stable. If the payo ff for a tortoise to race with another tortoise is c (where c > 0 ) instead of 5 (d) Assume that the proportion of tortoise in the population, t , is 0.5. For what values of c will tortoises have greater fitness than hares? (e) For what values of c will tortoises be fitter than hares if t = 0 . 1 ? (f) If c = 1 , will a single hare successfully invade a population of pure tortoises? Explain why or why not. (g) In terms of t , how large must c be for tortoises to have greater fitness than hares? (h) In terms of c , what is the level of t in a polymorphic equilibrium? For what values of c will such an equilibrium exist? Explain. 1

1.2 Textbook Page 507, Chapter 12, Question S8 Consider a population with two types, X and Y , with a payo ff table as follows: (a) Find the fitness for X as a function of x , the proportion of X in the population, and the fitness for Y as a function of x . Assume that the population dynamics from generation to generation conform to the following model: x t +1 = x t ⇥ F Xt x t ⇥ F Xt + (1 - x t ) ⇥ F Y t , where x t is the proportion of X in the population in period t , x t +1 is the proportion of X in the population in period t + 1 , F Xt is the fitness of X in period t , and F Y t is the fitness of Y in period t . (b) Assume that x 0 , the proportion of X in the population in period 0, is 0.2. What are F X 0 and F Y 0 ? (c) Find x 1 , using x 0 , F X 0 , F Y 0 , and the model given above. (d) What are F X 1 and F Y 1 ? (e) Find x 2 (rounded to five decimal places). (f) What are F X 2 and F Y 2 (rounded to five decimal places)? 2

The t would then become 0 . 2 + ✏ , where ✏ is a an arbitrary small positive number. Then, Expexted payo ff of tortoise = (0 . 2 + ✏ ) ⇥ 5 + (1 - (0 . 2 + ✏ )) ⇥ ( - 1) =1 + 5 ✏ - 0 . 8 + ✏ =0 . 2 + 6 ✏ Expexted payo ff of hare = (0 . 2 + ✏ ) ⇥ 1 + (1 - (0 . 2 + ✏ )) ⇥ 0 =0 . 2 + ✏ So tortoises would be fitter than hares after the mutant has invaded the population. Therefore, the polymorphism ( t = 0 . 2) is not evolutionary stable. For the monomorphism of tortoise ( t = 1) , the expected payo ff of tortoise is still greater than the expected payo ff of hare even if the monomorphism is invaded by hare so it is evolutionary stable. Similarly, for the monomorphism of hare ( t = 0) , the expected payo ff of hare is still greater than the expected payo ff of tortoise even if the monomorphism is invaded by tortoise so it is evolutionary stable. (d) If t = 0 . 5 , then Expexted payo ff of tortoise > Expexted payo ff of hare , 0 . 5 c + (1 - 0 . 5) ( - 1) > 0 . 5 ⇥ 1 + (1 - 0 . 5) ⇥ 0 , 0 . 5 c - 0 . 5 > 0 . 5 , c > 2 (e) If t = 0 . 1 , then Expexted payo ff of tortoise > Expexted payo ff of hare , 0 . 1 c + (1 - 0 . 1) ( - 1) > 0 . 1 ⇥ 1 + (1 - 0 . 1) ⇥ 0 , 0 . 1 c - 0 . 9 > 0 . 1 , c > 10 (f) When a hare has invaded, t becomes 1 - ✏ , where is a small positive number. Given c = 1 , the fitness of tortoises is (1 - ✏ ) ⇥ 1 + (1 - (1 - ✏ )) ⇥ ( - 1) = 1 - 2 ✏ , and the fitness of hares is (1 - ✏ ) ⇥ 1+(1 - (1 - ✏ )) ⇥ 0 = 1 - ✏ . Since 1 - ✏ > 1 - 2 ✏ , hares are then strictly fitter than tortoises after a single hare has invaded. Therefore, if c = 1 , a single hare will indeed successfully invade a population of pure tortoises. (g) Expexted payo ff of tortoise > Expexted payo ff of hare , ct + (1 - t ) ( - 1) > t ⇥ 1 + (1 - t ) ⇥ 0 , ct - 1 + t > t , ct > 1 , c > 1 /t Indeed, as we saw in parts (d), (e), and (f), for tortoises to be fitter the value of c needs to be larger as t declines. Note that as t approaches 0, c would need to be infinitely large for a tortoise to be fitter. Regrettably, a single tortoise in a population of pure hares would never have a chance to have a pleasant conversation with another tortoise. Thus, no matter what the value of c is, a single tortoise could never successfully invade a population of pure hares. 2

(h) In a polymorphic equilibrium, the fitness of tortoises must equal the fitness of hares: Expexted payo ff of tortoise = Expexted payo ff of hare , ct + (1 - t ) ( - 1) = t ⇥ 1 + (1 - t ) ⇥ 0 , ct - 1 + t = t , ct = 1 , c = 1 /t , t = 1 /c Since t is a proportion, it can only possibly take on values between 0 and 1 (exclusively). When t = 1 , c = 1 ; and when t approaches 0, c approaches infinity. Polymorphic equilibria are thus possible when 1 < c < 1 . (Note, however, that any polymorphic equilibrium of this evolutionary game will be unstable.) 1.2 Textbook Page 507, Chapter 12, Question S8 (a) F X = 2 x + 5 (1 - x ) = 5 - 3 x F Y = 3 x + 1 (1 - x ) = 1 + 2 x (b) F X 0 = 2 ⇥ 0 . 2 + 5 (1 - 0 . 2) = 4 . 4 F Y 0 = 3 ⇥ 0 . 2 + 1 (1 - 0 . 2) = 1 . 4 (c) x 1 = x 0 ⇥ F X 0 x 0 ⇥ F X 0 + (1 - x 0 ) ⇥ F Y 0 = 0 . 2 ⇥ 4 . 4 0 . 2 ⇥ 4 . 4 + (1 - 0 . 2) ⇥ 1 . 4 = 0 . 44 (d) F X 1 = 2 x 1 + 5 (1 - x 1 ) = 3 . 68 F Y 1 = 3 x 1 + 1 (1 - x 1 ) = 1 . 88 (e) x 2 = x 1 ⇥ F X 1 x 1 ⇥ F X 1 + (1 - x 1 ) ⇥ F Y 1 = 0 . 44 ⇥ 3 . 68 0 . 44 ⇥ 3 . 68 + (1 - 0 . 44) ⇥ 1 . 88 = 0 . 60599 (f) F X 2 = 2 x 2 + 5 (1 - x 2 ) = 3 . 18204 F Y 2 = 3 x 2 + 1 (1 - x 2 ) = 2 . 21198 3