8. Consider the following game played between 100 people. Each person i chooses a number s! between 20 and 60 (inclusive). Let a. _1 be defmed as the average selection of the players other than player i; that is, a _r. = (Sum. sf.) 99. Person i' s payoff is \":(3) = 100 - (s, - 3a _,)2. For instance, if/ the average of the - i players' choices is 40 and playert' chose 56, then player i would receive a payoff of 100 - (56 - 3' 40)2 = 100 - 4-2 = 84. (a) Find an expression for player i' s best response to her belief about the other players' strategies as a function of the expected value of a_'., which we can denote 54. What is the best response to E_ . = 40? l' I) Use our answer to art a to determine the set of undominated strate- 3' P gies for each player. Note that dominated strategies are those that are not best responses (across all beliefs). (1:) Find the set of rationalizable strategies. Show what strategies are removed in each round of the deletion procedure. ((1) If there were just two players rather than 100., but the definition of payoffs remained the same, would your answers to parts (a) and (b) change? (e) Suppose that instead of picking numbers between 20 and 60, the play- ers can select numbers between y and 60, where y is a fixed parameter between 0 and 20. Calculate the set of rationalizable strategies for any xed 3:. Note how the result depends on whethery L'> 0 or y =