Optimal multivariate stopping rules

Abstract

For fixediletX(i) = (X1(i), …,Xd(i)) be ad-dimensional random vector with some known joint distribution. Hereishould be considered a time variable. LetX(i),i= 1, …,nbe a sequence ofnindependent vectors, wherenis the total horizon. In many examplesXj(i) can be thought of as the return to partnerj, when there ared≥ 2 partners, and one stops with theith observation. If thejth partner alone could decide on a (random) stopping rulet, his goal would be to maximizeEXj(t) over all possible stopping rulest≤n. In the present ‘multivariate’ setup thedpartners must however cooperate and stop at thesamestopping timet, so as to maximize some agreed functionh(∙) of the individual expected returns. The goal is thus to find a stopping rulet*for whichh(EX1(t), …,EXd(t)) =h(EX(t)) is maximized. For continuous and monotonehwe describe the class of optimal stopping rulest*. With some additional symmetry assumptions we show that the optimal rule is one which (also) maximizesEZtwhereZi= ∑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 whenX(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) choicegon the full sequenceX(1), …,X(n), with corresponding return suph(EX(g)).