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

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