Rewarded Bayesian box search Bellman equation

If discovery in box $i$ earns reward $R_i$ and stopping earns zero, the Bellman equation isIf ${\sum }_i{c}_i/({\alpha }_i{R}_i)<1$, every posterior state has some $i$ with ${\alpha }_i{p}_i{R}_i>c_i$, so stopping is never optimal.