Trapping the Ultimate Success

TL;DR

This paper introduces a betting game related to predicting the last success in a sequence, analyzing optimal stopping strategies using a Polya-Lundberg process with a log-series distribution.

Contribution

It formulates a novel betting game linked to last success prediction and analyzes optimal stopping rules for trials modeled by a Polya-Lundberg process.

Findings

Optimal stopping strategies derived for the game.

Connection established between betting game and last success prediction.

Analysis specific to Polya-Lundberg process with log-series distribution.

Abstract

We introduce a betting game, where the gambler aims to guess the last success epoch from past observed data. The player may bet on the event that no further successes occur, or choose a `trap' which is any span of future times. In the latter case winning is achieved if the last success turns out to be the only one falling in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success in a finite sequence of trials. We use this connection to analyse the problem of stopping at the last record for trials paced by a Polya-Lundberg process with log-series distribution of the total number of trials.