Question

Player 1 and Player 2 choose and integer between 0 and 100 (including 0 and 100). Their choice is made simultaneously and independently. Suppose Player 1 chooses x and Player 2 chooses y. If x < y Player 1 obtains x and Player 2 obtains zero. Similarly, if y < x Player 2 obtains y and Player 1 obtains zero. If x = y then each one obtains x/2 = y/2. Find all pure strategy Nash equilibrium in this game.

Answer 1

There are 3 pure strategy Nash equilibrium:

(1) Both players bid 0. In this case both fo them will get 0. There is no incentive for anyone to deviate as even if one player bids higher, he will not get anything.

(2) Both players bid 1. In this case, both of them will get 1/2 each. Now let us see if any player has the incentive to deviate. If a player bids higher than 1, he gets 0 so he will not bid higher. If he bids less, the only value available to bid is 0, in which case he gets 0, so there is no incentive to bid less as well. Thus, (1,1) is an equilibrium.

(3) Both players bid 2. In this case, both of them will get 1 each. Now let us see if any player has the incentive to deviate. If a player bids higher than 2, he gets 0 so he will not bid higher. If he bids lower to 1, he will still only get 1 that he is already getting. He will not bid 0 as that gives him 0. Thus, there is no strict incentive to deviate and thus (2,2) is an equilibrium.

It is easy to see that no other outcome is an equilibrium.

Bidding the same value above 2 is not an equilibrium. Suppose both bid 3 and get 1.5 each. The player has an incentive to bid 2 and get 2 > 1.5.

Bidding different numbers can not be equilibrium as the higher bidder can always gain by cutting down the other other player.