## Garrett van Ryzin

*Optimal Auctioning and Ordering in an Infinite Horizon Inventory-Pricing System*

Coauthor(s): Gustavo Vulcano.

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**Abstract:**

We consider a joint inventory-pricing problem in which buyers act strategically and bid for units of a firm's product over
an infinite horizon. The number of bidders in each period as well as the individual bidders' valuations are random but
stationary over time. There is a holding cost for inventory and a unit cost for ordering more stock from an outside supplier.
Backordering is not allowed. 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 following an orderup-
to (basestock) policy for replenishing inventory at the end of each period. Moreover, the optimal basestock level can be
easily computed. We then compare this optimal basestock, reserve-price-auction policy to a traditional basestock, list-price
policy. We prove that in the limiting case of one buyer per period and in the limiting case of a large number of buyers
per period and linear holding cost, list pricing is optimal. List pricing also becomes optimal as the holding cost tends to
zero. Numerical comparisons confirm these theoretical results and show that auctions provide significant benefits when: (1)
the number of buyers is moderate, (2) holding costs are high, or (3) there is high variability in the number of buyers per
period.

**Source:** *Operations Research*

**Exact Citation:**

van Ryzin, Garrett, and Gustavo Vulcano. "Optimal dynamic auctions for production and inventory systems." *Operations Research* 52, no. 3 (2004): 346-367.

**Volume:** 52

**Number:** 3

**Pages:** 346-367

**Date:**
2004