A Beauty Cont est Each student chooses a number between 1 and 100. The student whose number is closest to 2/3 of the average of the chosen numbers wins the game. a I: 106G students 0 Sq: = {1}... ,100} o u,- (3,,sj) = ls, | where s = 2 3'5! 51/ #1 How should one play this game? lOne could start with the belief that the other students randomize over the numbers. Then the average is 50 and 2/3 of that is 33. If one believes that all of the others are picking 100, 2/3 of the average is 67. If one believes that all of the other players are picking 1, the best response is to pick 1 oneself. Thus, all strategies 3,; = 1, ...,67 are best responses, but all strategies .3; = 68, ..., 100 are strictly dominated by 3,; = 67. 2 2 = |s67| |67 53 67 is strictly dominated by 33; = 67 and as rational players play best responses, he should constrain his beliefs about other players' strategies to s, g 67. = 15,;(67, 33-) 1a (3;,sj) = 0 ~33 S 67, so every 3; > 67 is strictly dominated by s,- = 67 o If 3,: g 67 for all 1', then gs g 45, so if 2' thinks that everybody else is rational, then she rationally picks 3,, S 45 o If 3,: g 45 for all i, then %5 g 30, so if i thinks that everybody else is rational and that everybody else thinks that everybody else is rational, then she rationally picks 3,; S 30 0 Everybody picks 3,: = 1 3. Simpler Beauty Contest: Consider the version of the Beauty Contest game from the class, in which the players are trying to guess the average guess minus one. More precisely, the utilities now are: u,- (s,,s,-) = |s,- (E 1)|, where E = Zsj/hl jEI (a) Which strategies are strictly dominated in this game, and by which alternative strategies? And so, what is the set of strategies BRg? (b) What is the set BRE? What is the set BR;1 for any n > 0