The Least Unique Game

From The Grey Labyrinth
by Kevin Lin

The rules to the Least Unique Game are simple: Three players each pay $1 to play. They then choose, in secret, a positive integer. The numbers are revealed and the player with the lowest unique integer wins the $3. If they all choose the same number, the round is a draw and their dollars are returned.

Imagine that you find yourself playing this game online. Assume the following conditions apply:

  1. The game is repeated indefinitely.
  2. The players may not quit.
  3. Your opponents are anonymous and there is no mechanism to communicate with one another (i.e., you may rule out secret offline collusions).
Finally, assume that your opponents are at least as rational and capable as you. They understand the rules of the game, game theory, and will act to maximize their return. Like you, they will assume that they are playing against two intelligent opponents unless proven otherwise.

What is your best strategy?