Optimality results for coupon collection

Abstract

We consider the coupon collection problem, where each coupon is one of the types 1,…,s with probabilities given by a vector 𝒑. For specified numbers r₁,…,r_s, we are interested in finding 𝒑 that minimizes the expected time to obtain at least r_i type-i coupons for all i=1,…,s. For example, for s=2, r₁=1, and r₂=r, we show that p₁=(logr−log(logr))∕r is close to optimal.