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Article Excerpt
1. Introduction
In the last few years, Revenue Management (RM) has widened its focus from capacity control and dynamic pricing to alternative selling mechanisms proposed by electronic commerce, such as group purchasing, online negotiations, and auctions. (See Talluri and van Ryzin 2004 for a reference on RM methods and applications, or the survey by Bitran and Caldentey 2003 for an overview of dynamic pricing models.) Although list pricing is probably still the most familiar and used pricing mechanism, online auctions are certainly an increasing phenomenon.
Nowadays, a huge variety of products is sold simultaneously through online posted price and auction channels, allowing consumers to compare prices and bid states easily across different channels in real time. This boost in market information and the corresponding reduction in search costs have a significant impact on consumers' purchasing behavior and should be considered by a seller when designing online sales mechanisms.
In this paper, we address the problem of an online seller who is endowed with a fixed initial inventory and faces a stochastic arriving stream of strategic consumers. To capture different e-business environments, we consider two alternative formulations. First, we consider the case in which the seller controls an online auction exclusively and competes with a third party that manages a list price channel. We refer to this case as the single-auction channel model. In the second formulation, the dual-channel model, the seller is a monopolist and controls both the auction and list price channels.
For these two scenarios, we are interested in answering the following questions: How should strategic consumers behave, that is, which channel should each consumer choose? What should the bidding strategy be of those consumers that enter the auction? Given consumers' behavior, how should the seller manage an online auction to maximize revenue? What should the length of the auction and the reservation price (i.e., minimum acceptable bid) be? How should the seller manage parallel online auction and list price channels to maximize revenue? In particular, the dual-channel case motivates an important managerial question: How should the seller design both channels to segment the population of consumers to extract as much revenue as possible from each segment? A seller does not want to offer a business model that cannibalizes itself, that is, if she offers a high posted price, she narrows the list price channel, and middle-to-high valuation consumers are tempted to join the auction, which could eventually close at a low price. On the other hand, if she posts a low list price, she widens the list price channel, pooling together low- and high-valuation consumers. In both cases, the seller runs the risk of decreasing revenues. Our purpose is to shed some light on the trade-offs that are inherent to this business environment.
More formally, we analyze a single-period model in which a seller operates a multiunit, uniform price, online auction, offering multiple units of a homogeneous good. The seller announces the inventory put up for sale [Q.sub.0], the auction duration T, and the auction reservation price [v.sup.R]. Consumers with single-unit demand arrive according to a Poisson process. They have a private value for the product, independently drawn from a continuous distribution. They must decide whether to bid and wait for the auction outcome at time T, or buy at the posted price [^.P], and get the unit instantaneously. As mentioned above, we consider two variants of this problem:
* In the single-auction channel case, the fixed-price channel is external and run by another firm, which we assume has unlimited inventory. Hence, if a bidder is not among the winners of the auction, he can always buy the item at the posted price at a later time T, although his utility is discounted.
* In the dual-auction and list price channel case, the seller is a monopolist who manages both the auction and the list price channels. In this case, an auction takes place at time T only if there are units left unsold by then. Hence, supply is limited, and bidders that lose in the auction have no alternative market in which to buy the product.
The consumers' strategy consists of two decisions: (i) whether or not to join the auction, and (ii) in the case of joining the auction, what bidding strategy to use. The supply size under both scenarios produces different bidding behaviors. Regarding the first decision, we prove that a symmetric equilibrium strategy exists in both variations of the problem, and it is characterized by a threshold function in the space (valuation, time): For a consumer arriving at time t with valuation v, there is a threshold H(v) such that if H(v) [less than or equal to] t, then he will participate in the auction. Otherwise, he will buy at the posted price [^.P]. This participation strategy can be computed using an iterative algorithm (in an appropriate function space) provably convergent under some special conditions. Unfortunately, this procedure is computationally intensive and does not lead to simple managerial insights. To overcome this limitation, we formulate an asymptotic version of the problem, in which the demand rate and the initial number of units grow proportionally large. We get a simple closed-form expression for the equilibrium strategy in this limiting regime, which is then used as an approximated solution for the original problem. Numerical computations show that this heuristic is very accurate.
Finally, we analyze the seller's optimization problem in both the single- and dual-channel settings, and plug the consumer's asymptotic participation strategy into them to compute the optimal values of the parameters [Q.sub.0], T, [v.sup.R] (and [^.P] in the dual channel case), that the seller must announce to maximize revenues. We can then assert that the asymptotic solution culminates in precise and simple guidelines for how bidders should behave and how the seller should design the auction and list price channels.
The main insights that we obtained are the following. For the single-auction channel case, we find that the optimal number of units to offer is a nonmonotonic function of the external fixed price, [^.P], and that it is bounded above by 80% of the average demand. In addition, the optimal duration of the auction is an increasing function of [^.P]. In the dual auction and list price channel case, we find that if the seller's initial inventory endowment is small or if her discount factor is large, then she does not have enough economic incentives to run a terminal auction; a single fixed-price channel is the optimal selling mechanism. On the other hand, if the endowment is large, or the seller discount factor is small, or buyers are impatient, then running both channels in parallel is optimal. In any of these cases, we show that a dual-channel operation can have a significant impact on revenues compared to a single fixed-price channel. The magnitude of the increase in revenues can be as large as 33% for the case of uniformly distributed valuations.
1.1. Literature Review
Auctions have been extensively studied in the economic literature (e.g., see the survey by Klemperer 1999 or the recent book by Krishna 2002). Price discrimination has been argued as one of the main reasons for using them (see Bulow and Roberts 1989). Maskin and Riley (1989) proved the optimality of the uniform price mechanism for the single-period multiunit auction.
Few papers have put auctions in an operational perspective. Specifically, regarding its connection with RM, Pinker et al. (2000) study how to run a sequence of standard multiunit auctions, using bidding information to learn about the consumer's valuation distribution. Vulcano et al. (2002) characterize an optimal dynamic auction for a firm selling a fixed capacity over a finite horizon.
The firm's choice between auctions and posted prices for the single-channel case has also been addressed (e.g., see Vany 1987, Wang 1993, Harstad 1990).
The problem of jointly managing auction and list price channels has not received much attention in the literature. New features like the buy now prices have been addressed by Budish and Takeyama (2001), although their model is limited to two bidders and two valuation types. Within the business-to-consumer (B2C) framework, the empirical study of Vakrat and Seidmann (1999) compares prices paid through online auctions and catalogs for the same product. They observe that auctions result in average prices 25% below the catalog ones. They build a simple model of single-unit auctions with a deterministic number of bidders, but ignoring consumer choice behavior. In the infinite-horizon model of van Ryzin and Vulcano (2004, [section]3.3), the seller operates auctions and posted prices simultaneously, and replenishes her stock in every period. However, the streams of consumers for both channels are independent, and the seller decides how many units to allocate to each of the channels separately.
Our research is mainly motivated by the work of Etzion et al. (2006). They analyze simultaneous online auctions and list price channels in a multiperiod B2C framework, where a seller with infinite supply maximizes her average expected revenue. Consumers arrive according to a Poisson process, and decide which channel to join. They found two optimal auction design strategies: short single-unit auctions and long multiunit auctions.
Our work differs from theirs in the way we model the supply side, because in our dual channel case scarcity plays a critical role, as is usually the case in RM: Given the risk that potentially no item could remain available for the auction by time T (which occurs when all the inventory is depleted through the list price channel), what should the consumer's participation strategy be? In the case of going for the auction, scarcity induces the standard dominant "bid your own value" strategy for multiunit uniform price auctions (see Krishna 2002, [section]2.2 for a comprehensive study of bidding behavior). The situation is different in the single-auction channel case, where the infinite supply (as is the case in Etzion et al. 2006) induces no bidder to bid higher than the posted price: In case he loses, he always has the chance to pay that price at time T. Now, given both settings, how should the seller structure the business, accounting for consumers' strategic behavior?
Etzion et al. (2006) work with additive consumer utility functions, and characterize the consumer equilibrium bidding strategy with a single value [bar.t], such that all consumers with valuation below the posted price, and those arriving later than the threshold [bar.t] with valuation above the posted price, will join the auction. All other consumers will go to the online catalog. Our equilibrium participation strategy turns out to be more complex because it is based on a multiplicative utility function, and as we said above, is defined by a continuous threshold function in the space (valuation, time). Furthermore, when computing the participation strategy in their paper, Etzion et al. assume that the total number of competing consumers is deterministic. We instead embed the random nature of the arrival process in the computation of the consumer's participation strategy. (1)
A distinguishing characteristic of our research is the asymptotic analysis of the game. We show that the complex threshold that describes the strategic behavior of the consumers can be easily computed in the limiting regime where the consumers' arrival rate and the number of units offered grow proportionally large, without missing the predictive power of the model.
Overall, we believe that the two models share some features, but contrast in these important dimensions, which are worth exploring.
Finally, the problem of analyzing the equilibrium of a system where consumers arrive during a time window has been addressed by few papers, but they are oriented to the characterization of the arrival pattern (e.g., Glazer and Hassin 1983 or Lariviere and Van Mieghem 2004). In our setting, the arrival process is exogenous, and we concentrate on characterizing the Nash equilibrium (in pure strategies) of the participation behavior of the consumers.
The remainder of this paper is organized as follows. We introduce the model for both variants of the problem in [section]2. In [section][section]3 and 4, we study the consumers' problem of selecting an optimal participation and bidding strategy for both the single-auction and dual-channel models, respectively. We prove the existence of a symmetric equilibrium within a large class of participation strategies and use asymptotic analysis to characterize this equilibrium. The analysis in [section][section]3 and 4 assumes that the initial inventory [Q.sub.0], the auction bidding period T, the reservation price [v.sup.R], and the posted price [^.P] are fixed. We turn to the seller's revenue maximization problem of optimally choosing [Q.sub.0], [v.sup.R], T, and [^.P] in [section]5. Finally, [section]6 summarizes our concluding remarks.
The paper includes two online appendices (provided in the e-companion). (2) Appendix A contains all the proofs. Appendix B provides a detailed mathematical description of the dual-channel model.
2. Model Description
We study the problem faced by a firm (seller) endowed with an initial inventory [bar.Q] of a homogeneous product. We take an RM point of view and assume that the seller cannot replenish her inventory throughout the selling season. We do, however, allow the seller to ration by choosing the quantity [Q.sub.0] [less than or equal to] [bar.Q] to put up for sale. The remaining quantity [bar.Q] - [Q.sub.0] is discarded at no extra cost or salvage value.
We discuss two variants of this problem. In the first variant, the seller manages a single-auction channel, and there is an external market with infinite supply where the same product is available at a fixed price [^.P]. In the second one, the seller is a...
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