Isotonicity of minimizers in polychotomous discrete interval search via lattice programming

We consider several sequential search problems for an object which is hidden in a discrete interval with an arbitrary prior distribution. The searcher decomposes the momentary search interval into a fixed number of subintervals and obtains the information in which of the intervals the object is hidden. From the well-known lattice programming results of Topkis we develop a unifying method for proving in a transparent way the computationally very useful property of isotonicity of (largest and smallest) minimizers. We obtain rather natural sufficient conditions on the prior distribution and the cost structure, some of them weakening corresponding ones in Hassin/Henig. We also show how these assumptions can be checked in particular cases. Some of our auxiliary results on lattices and submodular functions may be of independent interest.