Wednesday, November 9 2022

3:30pm - 5:30pm

3:30pm - 5:30pm

Masters Presentation

Master’s Defense for Arvind Srinivasan - Asymptotic Analysis of the Least Unique Positive Integer Game

Master’s Defense for Arvind Srinivasan

Wednesday November 9, 2022, 3:30 pm – 5:30 pm in Student Commons Building 4119

Title: Asymptotic Analysis of the Least Unique Positive Integer Game

Committee: Burt Simon, Erin Austin, Yaning Liu

Abstract: The Least 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 win if they pick 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 (p_k) has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified with 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 converges to a uniform distribution. The result implies that for large N, players should choose a number uniformly between 1 and N/ln(N). The convergence to the uniform is rather slow, so we also investigate approximations of equilibrium strategies using a certain differential equation derived from the recursion and provide some results about solutions to the differential equations. The results allow players to easily play strategies that closely approximate the Nash equilibrium for games of any size.

Wednesday November 9, 2022, 3:30 pm – 5:30 pm in Student Commons Building 4119

Title: Asymptotic Analysis of the Least Unique Positive Integer Game

Committee: Burt Simon, Erin Austin, Yaning Liu

Abstract: The Least 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 win if they pick 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 (p_k) has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified with 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 converges to a uniform distribution. The result implies that for large N, players should choose a number uniformly between 1 and N/ln(N). The convergence to the uniform is rather slow, so we also investigate approximations of equilibrium strategies using a certain differential equation derived from the recursion and provide some results about solutions to the differential equations. The results allow players to easily play strategies that closely approximate the Nash equilibrium for games of any size.

Speaker: | Arvind Srinivasan |

Affiliation: | Department of Mathematical and Statistical Sciences, University of Colorado Denver |

Location: | ACAD4119 |

Done