Optimal index for Bayesian box search (${\alpha }_i{p}_i/c_i$)

When search continues until discovery, expected search cost is minimized by searching at each step a box maximizing ${\alpha }_i{p}_i/c_i$. An adjacent-interchange argument compares two searches: placing $i$ before $j$ costs less exactly when ${\alpha }_i{p}_i/c_i\ge {\alpha }_j{p}_j/c_j$.