Decision making and card collecting

Collecting a set of different, yet similar, items is a popular hobby. The ‘coupon collector's problem’ is concerned with the number of items which must be obtained, one at a time, in order to complete the set, assuming that the collector is sampling from an infinite population where each item has a known probability of being found. In recent years, manufacturers have chosen to produce the items in sealed packets which contain more than one item, and possibly items from more than one set. This paper considers the problem of collecting items in packets, and the decision problem faced by a collector who is offered the chance to buy several packets at a discount. Dynamic programming is used to determine when it is worthwhile to purchase in bulk, for various sets and packets, as a function of the discount rate. Finally, mention is made of the other player in the transaction, the card manufacturer, who is also a decision-maker.