Two people take turns selecting from an even number of items. Their relative preferences over the items can be described as a permutation, then tools from algebraic combinatorics can be used to answer various questions. We describe each person's optimal selection strategies including how each could make use of knowing the other's preferences. We determine those permutations for which both players, under any strategy, get their best possible outcomes, and the same for worst possible outcomes. We also count the number of possible outcomes under optimal play, finding the ubiquitous Catalan numbers.