How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman

TL;DR

This paper disproves a long-standing conjecture by L. Breiman, showing that the proposed threshold strategy does not always minimize the expected number of rounds to reach a target bankroll in a gambling game.

Contribution

It provides a counterexample to Breiman's 50-year-old conjecture, demonstrating that the threshold strategy is not universally optimal for minimizing expected rounds.

Findings

Counterexample disproves Breiman's conjecture

Threshold strategy is not always optimal

Optimal strategies may differ from simple threshold rules

Abstract

Consider a gambling game in which we are allowed to repeatedly bet a portion of our bankroll at favorable odds. We investigate the question of how to minimize the expected number of rounds needed to increase our bankroll to a given target amount. Specifically, we disprove a 50-year old conjecture of L. Breiman, that there exists a threshold strategy that optimizes the expected number of rounds; that is, a strategy that always bets to try to win in one round whenever the bankroll is at least a certain threshold, and that makes Kelly bets (a simple proportional betting scheme) whenever the bankroll is below the threshold.