Question

Consider the following game played between 100 people. Each person i chooses a number si between 20 and 60 (inclusive). Let a-i be defined as the average selection of the players other than player i ; that is, a-i = summation (j not equal to i) of sj all divided by 99. Player I’s payoff is ui(s) = 100 – (si – (3/2)a-i)2 For instance, if the average of the –i players’ choices is 40 and player i chose 56, then player i would receive a payoff of 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-I , which can be denoted as abar-I . What is the best response to abar-I = 40?

B. Use your answer to part (A) to determine the set of un-dominated strategies for each player. Note that the dominated strategies are those that are not best responses (across all beliefs).

C. Find the set of rationalizable strategies. Show what strategies are removed in each round of the deletion procedure.

D. 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?

Answer 1

No. Playing 20 cannot be a dominated strategy as the mean can be averagely close to the number which every player selects..

Yes playing the same number 20 is a Nash equilibrium as the average will come to the same number and it would be a dominant strategy for every player.

If half players select a number and other half select the next then the mean would be the middle number of the two hence all players would be equally close to mean. Hence this outcome is a Nash equilibrium.