OPTIMAL MULTIVARIATE STOPPING RULES

For fixed i let X(i) = (X-1(i),...,X-d(i)) be a d-dimensional random V ector with some known joint distribution. Here i should be considered a time variable. Let X(i), i = 1,..., n be a sequence of n independent vectors, where n is the total horizon. In many examples X-j (i) can b e thought of as the return to partner j, when there are d greater than or equal to 2 partners, and one stops with the ith observation. If th e jth partner alone could decide on a (random) stopping rule t, his go al would be to maximize EXj (t) over all possible stopping rules t les s than or equal to n. In the present 'multivariate' setup the d partne rs must however cooperate and stop at the same stopping time t, so as to maximize some agreed function h(.) of the individual expected retur ns. The goal is thus to find a stopping rule t for which h(EX1(t),... , EXd(t)) = h(EX(t)) is maximized. For continuous and monotone h we de scribe the class of optimal stopping rules t. With some additional sy mmetry assumptions we show that the optimal rule is one which (also) m aximizes EZ(t) where Z(i) = Sigma(j=1)(d) X-j(i), and hence has a part icularly simple structure. Examples are included, and the results are extended both to the infinite horizon case and to the case when X(1),. .., X(n) are dependent. Asymptotic comparisons between the present pro blem of finding sup h (EX(t)) and the 'classical' problem of finding s up Eh (X(t)) are given. Comparisons between the optimal return to the statistician and to a 'prophet' are also included. In the present cont ext a 'prophet' is someone who can base his (random) choice g on the f ull sequence X(1),..., X(n), with corresponding return sup h(EX(g)).