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Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game.
- Source :
-
International Journal of Game Theory . Jun2024, Vol. 53 Issue 2, p653-671. 19p. - Publication Year :
- 2024
-
Abstract
- The Lowest Unique Positive Integer game, a.k.a. Limbo, is among the simplest games that can be played by any number of players and has a nontrivial strategic component. Players independently pick positive integers, and the winner is the player that picks the smallest number nobody else picks. The Nash equilibrium for this game is a mixed strategy, (p (1) , p (2) , ...) , where p(k) is the probability you pick k. A recursion for the Nash equilibrium has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified when there is a large pool of potential players. Here, we summarize previous results and prove that as the (expected) number of players, n, goes to infinity, a properly scaled version of the Nash equilibrium random variable converges in distribution to a Unif(0, 1) random variable. The result implies that for large n, players should choose a number uniformly between 1 and ϕ n ∼ O (n / ln (n)) . Convergence to the uniform is rather slow, so we also investigate a continuous analog of the Nash equilibrium using a differential equation derived from the recursion. The resulting approximation is unexpectedly accurate and is interesting in its own right. Studying the differential equation yields some useful analytical results, including a precise expression for ϕ n , and efficient ways to sample from the continuous approximation. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00207276
- Volume :
- 53
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- International Journal of Game Theory
- Publication Type :
- Academic Journal
- Accession number :
- 178027607
- Full Text :
- https://doi.org/10.1007/s00182-023-00881-0