  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.
