# Guess 2/3 of the average with integers - mixed strategy equilibria?

The "Guess 2/3 of the average" game (Wikipedia) is a game in which n people guess what 2/3 of the average of their guesses will be, and where the numbers are restricted to the integer (in this case) numbers between 0 and 100, inclusive. The winner is the one closest to the 2/3 average.

By iterated deletion of weakly dominated strategies, it is easy to show that all integers but 0 and 1 vanish because it is possible to rule out iteratively those numbers who cannot be 2/3 of the average of the played numbers. For example, the first step is to eliminate all the integers above 66 because even though everybody will play 100, 2/3*100 = 66.67, thus this is the maximum value the right number could reach.

Now, my question is which are the pure and mixed strategy Nash equilibria of this game?

Winning the game gives a payoff of 1, losing a payoff of 0. If there is a tie, the payoff is distributed uniformly among the winners. The pure NE should be playing 0, while I am totally lost about the mixed strategy one.

Does anybody know how to proceed?

• What do you mean by "how to proceed"? Proceed to find all the mixed strategy NEs? I think the Wikipedia article already says what the pure NEs are: "In this case, all integers except 0 and 1 vanish; it becomes advantageous to select 0 if one expects that at least 1/4 of all players will do so, and 1 otherwise." – Herr K. Feb 21 '19 at 1:09

Quick note: in pure strategies for sufficiently many players, shouldn't all 1 be NE as well?

Onto mixed strategies:

To have a mixed strategy Nash equilibrium, each player needs to be indifferent between the strategies they are mixing and they must prefer all those strategies to the ones they never play.

That means players must mix only between strategies between which they have an equal probability of winning (when given all other players' strategies). I suspect you can use this to show that players must mix between 0 and 1. Then it remains to solve for whatever mixed strategies give which players indifference between 0 and 1.