Optimal Selection of the Most Probable Multinomial Alternative

Multinomial selection is concerned with selecting the most probable (best) multinomial alternative. The alternatives compete in a number of independent trials. In each trial, each alternative wins with an unknown probability specific to that alternative. A long-standing research goal has been to find a procedure that minimizes the expected number of trials subject to a lower bound on the probability of correct selection (P(CS)). Numerous procedures have been proposed over the past 55 years, all of them suboptimal, for the version where the number of trials is bounded. We achieve the goal in the following sense: For a given multinomial probability vector, lower bound on P(CS), and upper bound on trials, we use linear programming (LP) to construct a procedure that is guaranteed to minimize the expected number of trials. This optimal procedure may necessarily be randomized. We also present a mixed-integer linear program (MIP) that produces an optimal deterministic procedure. In our computational studies, the MIP always outperforms previously existing methods from the literature, with a modest additional benefit arising from the LP's randomized procedure.