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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.

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  • $\begingroup$ Do you have to pick an integer amount? $\endgroup$
    – jlammy
    Jan 5, 2021 at 18:42
  • $\begingroup$ Yes, thanks, edited the question. $\endgroup$
    – Morgan Lu
    Jan 5, 2021 at 18:43
  • $\begingroup$ For both the nonrandom players to choose $1$ is a stable equilibrium though (ie any of them loses money if they change strategies). $\endgroup$
    – Aphelli
    Jan 5, 2021 at 18:48
  • $\begingroup$ I completely agree with the other responses and wish to point out the great mathematical advantage that collusion would be in this situation. This is why, in the business world, collusion is so attractive. $\endgroup$ Jan 5, 2021 at 19:13

1 Answer 1

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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!

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  • $\begingroup$ I agree that in equilibrium the two intelligent players will play 1. There are a lot of games where the equilibrium is not the best outcome in terms of welfare. Check the prisoners dilemma. However, in this game the random guy is not a winner. In expectation he pays some money to the other two. The intelligent players pay 1 to each other (cancels out) and get something from the random guy (unless he plays 1). They could maximize their expected income by agreeing to play the 1 vs 1-optimal strategy but this is not an equilibrium. $\endgroup$
    – DFL
    Jan 12, 2021 at 18:41

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