When only the last one will do

TL;DR

This paper analyzes an online selection problem where a selector aims to choose the last arriving item among an unknown number of items arriving uniformly over [0,1], establishing a game-theoretic equilibrium and estimating the success probability.

Contribution

It introduces a game-theoretic framework for the last-item selection problem with an adversarially chosen number of items and computes the optimal success probability.

Findings

Existence of a game-theoretic equilibrium.

Optimal success probability estimated at approximately 0.353.

Both selector and adversary have derived optimal strategies.

Abstract

An unknown positive number of items arrive at independent uniformly distributed times in the interval [0,1] to a selector, whose task is to pick online the last one. We show that under the assumption of an adversary determining the number of items, there exists a game-theoretical equilibrium, in other words the selector and the adversary both possess optimal strategies. The probability of success of the selector with the optimal strategy is estimated numerically to 0.352917000207196.