Retirement threshold with multiplicative capture risk
= Retirement threshold with multiplicative capture risk
If a successful step in action $i$ has expected reward $a_i\geq0$ and survival probability $q_i=1-p_i$, a finite-horizon retirement problem with zero wealth after capture has threshold $\max_i a_iq_i/p_i$. Above it, induction makes retirement optimal; below it, a single further step already beats retirement. The threshold does not depend on the remaining positive horizon.