The total variation of a simple, symmetric random walk with absorbing barrier at zero, is stochastically larger than the total variation of any other nonnegative, integer-valued supermartingale with the same initial position. This strengthens a result of David Freedman on the optimality of timid play for maximizing the time to bankruptcy in certain gambling situations.
"Timid Play when Large Bets are Profitable." Ann. Probab. 5 (4) 573 - 576, August, 1977. https://doi.org/10.1214/aop/1176995764