Abstract :
A problem of relevance to the design of multiple access protocols is the following: Let X1,...,XN be i.i.d, random variables uniformly distributed in [0,1]. We have to determine the largest Xi,1 ??i ??N, as follows: Given N, we pick a set A(1) ?? [0,1] and ask "Do you belong to A(1) ?", whereupon each Xi responds with a yes or a no. Based on these responses we pick a set A(2) ?? [0,1] and ask "Do you belong to the set A(2) ?", and so on. Is there an optimal strategy to choose the sets A(1), A(2), ... so as to minimize the expected number of questions required to determine the largest Xi ? Further, what is the infimum of the expected number of questions required ? Arrow et.al. prove the existence of a strategy that is optimal in the class of strategies where every set A(i) is of the form (a(i) 1], for some a(i) ?? [0, 1). We show that this same strategy is also optimal in the larger class of strategies where the A(i) are allowed to be finite unions of right closed, left open intervals. Modulo measure theoretic technicalities this solves the general problem.
