Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game

Arvind Srinivasan and Burton Simon ()
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Arvind Srinivasan: University of Colorado Denver
Burton Simon: University of Colorado Denver

International Journal of Game Theory, 2024, vol. 53, issue 2, No 15, 653-671

Abstract: 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, $$(p(1),p(2),\ldots )$$ ( 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 $$\phi _n \sim O(n/\ln (n))$$ 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 $$\phi _n$$ ϕ n , and efficient ways to sample from the continuous approximation.

Keywords: Poisson games; Nash equilibrium; Evolutionary Stable Strategy; Multi player games; Noncooperative games (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00182-023-00881-0

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