The Sally Criterion: Satisficing Meets Kelly

The Kelly criterion optimally accumulates wealth in the long term, but is very aggressive for a bettor nearly satisfied with their wealth. We consider a linearly-satisficing utility $u(W)=\min(W,W_{\max})$ for terminal wealth $W$ and a maximum $W_{\max}$. In statistical testing-by-betting, with $W_{\max}=1/\alpha$, expected utility is proportional to the probability of rejecting the null in a level-$\alpha$ test. The ensuing *Sally bet* simply multiplies the Kelly bet by a closed-form function of the current and maximum wealths. By incorporating this multiplier into existing algorithms, we easily obtain tighter confidence intervals in multiple applications.