You are playing a game with two other players. You all get a chance to select a random integer from 1-100 (inclusive) and the player with the lowest number is paid that amount by both of the other players. In the case of a tie for the lowest number, no payment is made. One of the players you play against randomly selects a number from 1-100. The other is intelligent and will play optimally. What would be the best strategy?
For a simplified case, in which you played against only the random player, I calculated that the optimal choice was 33 or 34, in which your expected winnings were 16.83. However, for the three-person game, I am not sure what the strategy would be. I feel like you and the other intelligent player will constantly try to undercut each other, but I don't feel like the equilibrium should be 1 because there is money to be made by taking advantage of the random player.
You have to pick $1$ always. This strategy dominates every other strategy.
If the optimal guy plays $1$ everytime, then you will lose money unless you also play $1$ everytime.
The uniform random guy is the real winner here, because in this equilibrium he'll never need to pay over money!