This paper generalizes the problems of optimal selection considered by Roth, Kadane and DeGroot by allowing a set of J items to be chosen by two decision makers, the first of whom has A challenges and the second has B challenges. The two decision makers each have an opportunity to challenge each item before it is accepted, in some arbitrary fixed order. We assume that the decision makers know the utility function of the other side as well as their own over sets of J items, and that they know

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... subjective distribution, assigned by the other side, of characteristics of potential items that will be observed, as well as their own. Under these conditions the other side's response to each potential item can be predicted with certainty, and backward induction defines an optimal strategy. We study an important special case we call regular, and show that it is never disadvantageous to go first in the regular case. The use of peremptory challenges in jury trials motivates our model. The basic model in which jurors are challenged one at a time is extended to a more general class of problems that includes the group system and the struck jury system. 952