A generalized coupon collecting model as a parsimonious optimal stochastic assignment model

摘要

There is a given set of n boxes, numbered 1 thru n. Coupons are collected one at a time. Each coupon has a binary vector x_1,..., x_n, attached to it, with the interpretation being that the coupon is eligible to be put in box i if x_i. = 1, i = 1..., n. After a coupon is collected, it is put in a box for which it is eligible. Assuming the successive coupon vectors are independent and identically distributed from a specified joint distribution, the initial problem of interest is to decide where to put successive coupons so as to stochastically minimize N, the number of coupons needed until all boxes have at least one coupon. When the coupon vector X_1,...,X_n is a vector of independent random variables, we show, if P(X_i = 1) is nondecreasing in i, that the policy π that always puts an arriving coupon in the smallest numbered empty box for which it is eligible is optimal. Efficient simulation procedures for estimating P_π (N > r) and E_π [N] are presented; and analytic bounds are determined in the independent case. We also consider the problem where rearrangements are allowed.