Optimal multivariate stopping rules

Abstract

For fixed i let X(i) = (X1(i), …, Xd(i)) be a d-dimensional random vector 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 Xj(i) can be thought of as the return to partner j, when there are d ≥ 2 partners, and one stops with the ith observation. If the jth partner alone could decide on a (random) stopping rule t, his goal would be to maximize EXj(t) over all possible stopping rules t ≤ n. In the present ‘multivariate’ setup the d partners must however cooperate and stop at the same stopping time t, so as to maximize some agreed function h(∙) of the individual expected returns. 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 describe the class of optimal stopping rules t*. With some additional symmetry assumptions we show that the optimal rule is one which (also) maximizes EZt where Zi = ∑dj=1Xj(i), and hence has a particularly 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 problem of finding suph(EX(t)) and the ‘classical’ problem of finding supEh(X(t)) are given. Comparisons between the optimal return to the statistician and to a ‘prophet’ are also included. In the present context a ‘prophet’ is someone who can base his (random) choice g on the full sequence X(1), …, X(n), with corresponding return suph(EX(g)).