# Thread: Game Theory Nash Equilibrium| Help in understanding question

1. ## Game Theory Nash Equilibrium| Help in understanding question

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.

2. ## Re: Game Theory Nash Equilibrium| Help in understanding question

Suppose everyone has the same thought and no one chose 4, picking 3 instead. Then everyone loses.

3. ## Re: Game Theory Nash Equilibrium| Help in understanding question

Originally Posted by SlipEternal
Suppose everyone has the same thought and no one chose 4, picking 3 instead. Then everyone loses.
but would (4,4,4,4) be a NE outcome?
the definition of Nash equilibrium entails every player plays their best response with respect to the actions chosen by the other players. If all the others have played 4, my best response would be choosing 3, given what others are already choosing. We would all lose if everybody goes through the same reasoning, but (4,4,4,4) is not where the best responses intersect, so it would not be an equilibrium. I cant think why it would be an equilibrium ?

4. ## Re: Game Theory Nash Equilibrium| Help in understanding question

In a Nash Equilibrium, every player has the same exact strategy. If you come up with a reason to choose 3, them every player has that same exact logic and also chooses 3. So, you think, "well, everyone is going to be trying to go for higher numbers thinking that more people can choose them, so I bet no one will choose 1." Them everyone has that same exact strategy and chooses 1. Everyone loses.

Next, you think, "if I had just chosen 2, i would have won. Next round, I'll choose 2 while everyone else chooses 1." Only, everyone chooses 2, so everyone still loses. How can you win if everyone has the exact same strategy as you?