Equilibrium solution to the lowest unique positive integer game

We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests.

💡 Research Summary

The paper tackles the Nash‑equilibrium analysis of the “lowest unique positive integer” (LUPI) game, a reverse‑auction style contest in which each of (k) participants independently selects a positive integer and the player who picks the smallest integer that no one else has chosen wins a prize. Although the rules are simple, determining a strategy that no single player can improve upon—given that all others follow the same mixed strategy—has been analytically intractable for more than a handful of participants because the number of possible outcome profiles grows exponentially with the number of players and the range of admissible integers.

The authors first formalize the game. Let the set of admissible integers be ({1,\dots,M}). Under the assumption of a symmetric mixed strategy, each player chooses integer (j) with probability (p_j), where (\sum_{j=1}^{M}p_j=1). The probability that integer (j) is uniquely chosen is \