In this dissertation, we study two auction-related problems and one network capacity control problem; all of them belonging to the broad field of Revenue Management.
The auction setting for the first two problems is the same, following the traditional private values model, but embedded in a dynamic framework: The seller conducts a sequence of auctions. Within each period, a random number of potential buyers (bidders) participate in the auction. Each bidder has his own valuation, but from the seller's perspective and from other bidders' perspective, valuations follow some common knowledge distribution. The sets of bidders are independent from one period to the other.
In the first essay, a seller with C units to sell faces a sequence of buyers separated into T time periods. The problem is to find the revenue-maximizing auction. Buyers compete directly against each other within a period, as in a traditional auction, and indirectly with buyers in other periods through the opportunity cost of capacity assessed by the seller, providing a variation of the traditional single-leg, multi-period revenue management problem over a finite time horizon. For this setting, we prove that dynamic variants of the first-price and second-price auction mechanisms maximize the seller's expected revenue. We also show explicitly how to compute and implement these optimal auctions. The optimal auctions are then compared to a traditional revenue management mechanism-in which list prices are used in each period together with capacity controls-and to a simple auction heuristic that consists of allocating units to each period and running a sequence of standard, multi-unit auctions with fixed reserve prices.
The second essay is an extension of the first essay to a joint inventory-pricing system over an infinite horizon. The firm must decide how to conduct its auctions and how to replenish its stock over time to maximize its profits. We show that the optimal auction and replenishment policy for this problem is quite simple, consisting of running a standard first-price or second-price auction with a fixed reserve price in each period and using an order-up-to (basestock) policy for replenishing inventory. We also present some numerical comparisons with list pricing.
Finally, in the third essay we consider the problem of jointly managing the capacities of a network of resources through virtual nesting. In virtual nesting, itinerary-fare-class combinations are mapped (indexed) into a small number of virtual classes on each leg of the network. Leg protection levels are then applied to these virtual classes in order to control itinerary availability. We formulate a continuous demand and capacity model for this discrete problem, allowing the partial acceptance of requests. Since our model is continuous, we are able to compute subgradients exactly using a simple and efficient recursion, and then embed them within a stochastic quasigradient algorithm, leading to a locally convergent procedure. Moreover, in several test problems on realistic networks, the method produces significant performance improvements relative to the protection levels produced by heuristic virtual nesting schemes. These results suggest the method has good practical potential as well.