Optimal quota allocation for a revenue-maximizing auction holder facing a random number of bidders

This paper is motivated by the following practical multi-unit auction. Before purchasing a vehicle in Singapore, a prospective buyer must bid for a certificate that entitles him to purchase the vehicle. Each month a certain number, say Q, of ‘certificates of entitlement’ are made available and the highest Q bidders receive the right to purchase. Each successful bidder pays a ‘quota premium’ equal to the bid submitted by the Qth highest bidder. We construct a stochastic optimization model using results from the theory of order statistics, where the objective is to determine the optimal number of units to be released in order to maximize the auction holders expected revenue. As the amounts bid and the number of bidders are both random, there may be wide – and, undesirable – fluctuations in the quota premia from one month to the next. We perform the optimization subject to a probabilistic constraint that limits the month-to-month fluctuations in quota premia. We show, assuming specific distributions for the bid amounts and the number of bidders, that in order to satisfy the probabilistic constraint one may have to reduce the available quota. For different bid random variables, the feasible set arising from the probabilistic constraint may assume different shapes. We also present a sufficient condition for comparing the size of the two feasible sets corresponding to different bid random variables. The results are illustrated by sensitivity analyses that examine the effect of varying parameter values on the optimal solution.