Behind the game
Game Theory Background
Mini Match is based on the Lowest Unique Positive Integer game, often shortened to LUPI. A large-scale Swedish version was offered under the name Limbo. The rules are simple, but every choice depends on what you think everyone else will choose.
The classic rules
- Every player privately chooses a positive integer.
- All choices are revealed at the same time.
- Numbers chosen more than once are eliminated.
- The lowest number chosen exactly once wins.
If no number is unique, nobody wins. Mini Match follows the same positive-integer rule, so the smallest valid choice is 1.
Why the obvious answer is not obvious
A very low choice can beat almost anything, but it loses its power as soon as another player makes the same choice. A higher number may avoid a collision, yet it still loses whenever a lower unique choice survives.
In commonly studied symmetric equilibria, this tradeoff leads to a mixed strategy: players randomize across several choices instead of relying on one best number. In some large-player asymptotic models, that range has an effective cutoff on the order of N / log(N). This is not a universal rule; the exact distribution changes with the player count, allowed range, and model. In the unbounded, exactly-three-player formulation, the unique symmetric equilibrium has full support and a constant hazard rate.
Research papers
- Testing Game Theory in the Field: Swedish LUPI Lottery Games — Robert Östling, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F. Camerer
- Equilibrium Solution to the Lowest Unique Positive Integer Game — Seung Ki Baek and Sebastian Bernhardsson
- Exact Asymptotics and Continuous Approximations for the Lowest Unique Positive Integer Game — Arvind Srinivasan and Burton Simon
- The Three-player Lowest Unique Number Game — Balázs Szentes