For fixed i let X (i) = (X 1(i), …, X d (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 X j (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 E X j (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(E X 1 (t), …, E X d (t)) = h (E X (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 EZ t where Z i = ∑d j=1X j (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(E X (t)) and the ‘classical’ problem of finding supE h(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(E X (g)).