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Article Excerpt
We analyze the problem of a seller of multiple identical units of a good who faces a set of buyers with unit demands, private information, and identity-dependent externalities. We derive the seller's optimal mechanism and characterize its main properties. We show that the probability that a buyer obtains a unit is an increasing function of the externalities he generates and enjoys. Also, the seller's allocation of the units of the good need not be ex post efficient. As an illustration, we apply the model to the problem faced by a developer of a shopping mall who wants to allocate and price its retail space among anchor and non-anchor stores. We show that a commonly used sequential mechanism is not optimal unless externalities are large enough.
1. Introduction
* This article solves an optimal multi-unit auction design problem with private information and identity-dependent externalities. The model we study is best understood as a model of space allocation in a mall (although we shall see below that it subsumes other applications as well). Consider a developer of a new shopping mall who wants to sell its retail space to a set of stores. An important constraint in this allocation problem is that each store has private information about its profit function, such as the cost of production or the demand for its product. Another key feature of this problem is the existence of interstore externalities: the identity of the stores located in the mall determines its customer traffic, which in turn affects the stores' volume of sales. Thus, a store's willingness to pay for retail space depends on the identity of the other stores that locate in the mall. If the developer wants to maximize her profits, what is the optimal selling procedure?
We develop a simple model that captures the most salient aspects of the problem. We consider a seller who has two identical units of a good and faces a set of potential buyers with unit demands. A buyer's valuation for the good depends on his privately known type and on an externality parameter that depends upon the identity of the buyer who obtains the other unit.
In this setting, we characterize the revenue-maximizing mechanism for the seller. We find that the seller should allocate the units of the good to the pair of buyers that generates the largest sum of virtual surpluses, weighted by the external effects they enjoy. The probability that a buyer obtains a unit of the good is increasing in both the externality he imposes on other buyers and the one that he enjoys. More importantly, the allocation that ensues need not be expost efficient for the following reasons: (i) as in the case without externalities, the seller sometimes keeps one or both units of the good; (ii) because the presence of external effects makes buyers asymmetric, those who receive the good need not have the largest sum of valuations; and (iii) when externalities are negative, the seller may sell a unit when it is ex post efficient for her to keep it.
We also characterize an optimal payment rule that internalizes the externalities generated and enjoyed by a buyer, and makes the optimal mechanism a dominant strategy one. In particular, we show that a buyer's payment is a decreasing function of the externalities he generates.
As an application, we elaborate on the shopping mall problem. A standard procedure in practice is to sign the anchor stores first (i.e., department stores), which are the main externality generators, and then approach the remaining interested stores. The empirical evidence also suggests that anchors receive large discounts that are increasing in the externalities they generate. (1) In mm, the non-anchor stores that enjoy these externalities pay a premium that is increasing in their magnitude. We characterize the properties of this sequential selling procedure, and show that it is not an optimal one for the seller unless externalities are large enough.
In order to simplify the exposition, we focus on the case of two units, positive externalities, and buyers' payoff functions that are multiplicatively separable in types and externality parameters. We later show that all the results extend to the case of N units, negative externalities, and also to a more general class of complementarities in the buyers' payoff functions.
[] Related literature. To the best of our knowledge, this is the first article to analyze an optimal multi-unit auction problem with private information and identity-dependent externalities.
The closest related papers are Jehiel, Moldovanu, and Stacchetti (1996), Das Varma (2002), Figueroa and Skreta (2008), and Brocas (2007). Jehiel, Moldovanu, and Stacchetti (1996) and Das Varma (2002) both analyze auctions with externalities. Jehiel, Moldovanu, and Stacchetti (1996) characterize the optimal mechanism and allow the external effects to be private information, albeit buyers are ex ante identical in their model. Das Varma (2002) studies open ascending-bid auctions with commonly known identity-dependent externalities, and shows that when they are nonreciprocal the open auction yields a higher expected revenue than a sealed-bid auction. Unlike our article, these references deal with single-unit auctions and focus on the case in which the winner imposes a negative externality on the losers through their reservation utility. (2) Both Figueroa and Skreta (2008) and Brocas (2007) analyze optimal auctions with externalities, with the emphasis placed on the role played by (privately known) outside options in the optimal allocation of a single unit of a good. (3)
Finally, the article also relates to Segal (1999), who analyzes contracting situations with externalities under complete information. Our model is a contracting problem with externalities, but unlike Segal, we study the effects that private information has on the optimal contract.
The rest of the article proceeds as follows. Section 2 presents the model and some preliminary results. Section 3 contains the derivation of the optimal mechanism and its main properties. Section 4 applies the model to the shopping mall problem. Section 5 presents several extensions of the analysis. Section 6 concludes. The Appendix contains proofs omitted from the text.
2. The model
* There are I + 1 risk-neutral agents: a seller, whom we call agent 0, and I [greater than or equal to] 2 potential buyers, numbered 1, 2, ..., I. The seller owns two identical units of an indivisible good, and buyers have unit demands (i.e., each one demands at most one unit of the good). (4)
[] Valuations and external effects. Without loss of generality (henceforth wlog), we assume that the seller derives no value from the two units. The valuation of buyer i, i = 1, 2, ..., I, for a unit of the good depends on two factors. First, it depends on a parameter (type) [[theta].sub.i] that is private information and is distributed on [[THETA].sub.i] = [[[[theta].bar].sub.i], [[bar].[theta]].sub.i]], [less than or equal to] [[[theta].bar].sub.i] < [[bar.[theta]].sub.i], with positive and atomless density [[phi].sub.i](x) and cumulative distribution function [[phi].sub.i](x). Let [J.sub.i]([[theta].sub.i]) = [[theta].sub.i] - 1 - [[PHI].sub.i]([[theta].sub.i])/[[phi].sub.i]([[theta].sub.i]); we assume that [J.sub.i](x) is a strictly increasing function. Moreover, buyers' types are independently distributed. Second, a buyer's valuation depends also on who obtains the other unit of the good. We model this feature by introducing a matrix of external effects [{[[alpha].sub.ij]}.sub.1[greater than or equal to] i[greater than or equal to]I,0[greater than or equal to]j[greater than or equal to]I], which is assumed to be common knowledge among the agents. (5)
To simplify the presentation of the main results, we assume that externalities are positive and that a buyer's type and the externality parameter interact multiplicatively in his payoff function. (6) Formally, if buyers i and j each obtains a unit of the good, and i pays the seller - [t.sub.i], then buyer i's payoff is [[alpha].sub.ij] [[theta].sub.i] + [t.sub.i]; moreover, 1 [less than or equal to] [[alpha].sub.ij] [less than or equal to] [bar.[alpha]] < [infinity], [[alpha].sub.ii] = [[alpha].sub.i0] = 1. For simplicity, the reservation utility of a buyer is assumed to be equal to zero. Notice that i does not derive value from a second unit of the good ([[alpha].sub.ii] = 1), and he does not enjoy a positive externality if the seller keeps it ([[alpha].sub.i0] = 1).
[] The seller's problem. The goal of the seller is to design a mechanism that maximizes her expected revenue, taking into account that (i) buyers have private information, (ii) ownership entails external effects, and (iii) participation is voluntary.
By the revelation principle, we can restrict the search for the optimal selling scheme to direct...
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