The problem appears here with a solution: https://dcownden.files.wordpress.com...esolutions.pdf

I cant seem to understand the question.

The game is as follows:

Each player selects a positive integer $\displaystyle (1,2,3...)$. To win a player must choose the lowest integer n with the property that fewer than n

other players have selected the integer $\displaystyle n$.

For example,

if four people are playing, and one player chooses 1, two players choose 2 and 1 player

chooses 3, the player who chooses 1 is the winner and everyone else loses. If on

the other hand two players chose 1 and 2 players choose 2 the players who chose

2 would be winners.

Find two different Nash Equilibrium for this game, for which all players win.

The answer given are the situations where all players chooses N and N+1.

but if there are 4 people playing and all chose 4, anybody willing to chose 3 would have no person choosing 3, and anybody still playing 4 would have 2 other players choosing 4. both 4 and 3 satisfies the property, and 3 being lower wins. So 4 would never be a best response for any player, providing an incentive to deviate. So, How come (4,4,4,4) is a NE?

There is obviously something I am misunderstanding.