A New Proof and Extension of the Odds-Theorem

TL;DR

This paper presents a new, simpler proof and an extension of Bruss's Odds-Theorem, addressing a generalized last-success problem with weighted rewards in a sequential Bernoulli setting.

Contribution

It offers an alternative, more elementary proof of the Odds-Theorem and extends it to include weighted payments for predicting the last success.

Findings

Established the optimal strategy for the generalized problem

Derived the expected profit in terms of the odds

Provided a simpler proof of the original Odds-Theorem

Abstract

There are $n$ independent Bernoulli random variables $I_{k}$ with parameters $p_{k}$ that are observed sequentially. We consider a generalization of the Last-Success-Problem considering $w_{k}$ positive payments if the player successfully predicts that the last "1" occurs in the variable $I_{k}$. We establish the optimal strategy and the expected profit in similar terms to the Odds-Theorem. The proof provided here is an alternative proof to the one Bruss provides in his Odds-Theorem (case $w_{i}=1$) that is even simpler and more elementary than his proof.