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• W3122727119 abstract "Burdett, Shi and Wright (2001) o¤er a directed search model where the buyers decide which seller to visit after observing the price each seller posts, and showed that there exists a unique symmetric equilibrium. Coles and Eeckhout (2003) showed that there is a continuum of symmetric equilibria, if the sellers are allowed to post general mechanisms and not only …xed prices. I show that many of the equilibria that Coles and Eeckhout identify, including the …xed price equilibria suggested by Burdett, Shi and Wright (2001), are not robust to introducing heterogeneous buyers with two possible types. In this case only ex-post e¢ cient equilibria exist, i.e. a buyer with a lower valuation can never win against a buyer with a higher valuation if they visit the same seller. This suggests that mechanisms like auctions that utilize buyer competition in an e¢ cient manner may endogenously arise when sellers post competing mechanisms. When the type space is continuous instead of having two possible types, the ex-post e¢ ciency result is not maintained. If the sellers’ strategy space is unrestricted, then any allocation respecting sequential rationality of the buyers, can be implemented in equilibrium. However, posting simple …xed price mechanisms never constitute an equilibrium, while posting (second-price or …rst-price) auctions may if the distribution function is convex. I am grateful to Paulo Barelli for his useful comments. All remaining errors are mine. yUniversity of Rochester, Economics Department, 228 Harkness Hall, NY14627, e-mail: gvirag@troi.cc.rochester.edu 1 Introduction Burdett, Shi and Wright (2001) o¤er a directed search model where the buyers decide which seller to visit after observing the price each seller posts. If a shop is visited by more than one buyer, then each of those buyers obtains the good with the same probability. While they obtain a unique symmetric equilibrium, Coles and Eeckhout (2003) show that there is a continuum of symmetric equilibria if the sellers can post general mechanisms and not only prices. Some of these equilibria resemble the posted price case considered by Burdett, Shi and Wright (2001), because the price a buyer pays does not depend on the number of other buyers in the same store. Other equilibria are more similar to auctions: if more than one buyer showed up, then the price is close to the value of the object. Coles and Eeckhout (2003) argue that without further restrictions it is impossible to decide which of these equilibria prevails. This paper assumes that the sellers do not know the reserve values of the buyers that can be either low (l) or high (h). In this case, many of the equilibria described by Coles and Eeckhout (2003) disappear and in equilibrium each store sells to the buyer with the highest valuation who visited the store (ex-post e¢ ciency). Therefore, many of the equilibria (including the one with …xed price mechanisms) identi…ed by Coles and Eeckhout (2003) are not robust to small changes in valuations. The intuition for this result is simple. Suppose that seller j posts an ex-post ine¢ cient mechanism in equilibrium. Then an ex-post e¢ cient mechanism, conducted in three steps, increases the revenue of j. In Step 0 the buyers who visited seller j report their types. In Step 1 the original mechanism is played ignoring the announcements in Step 0. In Step 2 there is a possible retrading depending on the reports from Step 0: if in Step 1, a low type obtains the object when a buyer with high type is also present at j, then the low type buyer (re)sells the object to j for an amount l and the high type buyer buys it (from j) for an amount h, giving the seller an extra pro…t of h l > 0. It is a (weakly) dominant strategy for each buyer to report truthfully in Step 0 and then the buyers make the same utility at seller j as in the case when j used the original mechanism. Therefore, it remains an equilibrium in the the buyers’stage game to use the same actions (together with reporting the type truthfully if a buyer visited j) as in the case when seller j did not deviate. Then the suggested deviation is obviously pro…table, because j obtains an extra revenue of h l when ex-post ine¢ ciency would have occurred without the deviation of j. However, there may be other equilibria in the stage game of the buyers and the buyers may coordinate in a way that hurts the deviator. We show that the deviator can o¤er a more complicated mechanism that eliminates all the equilibria of the buyers’stage game that are unfavorable for him. To derive this mechanism, we follow the approach of the unique implementation literature and for each buyer de…ne lotteries whose payo¤s depend on the decisions of the other buyers.1 Virag (2007) argues that auctions constitute natural equilibria even in the original complete information model. He shows that if the sellers can collude on equilibria that maximize their pro…ts, then only such equilibria exist where a buyer’s payment is increasing in the number of other buyers visiting the same seller. This suggests that posting auction-like mechanisms may be a means to achieve collusion between sellers. However, there is a crucial di¤erence between the collusion based argument of that paper and the incomplete information analysis presented here. Under collusion, the key is that the price must increase in the number of bidders who visited the same store, while under incomplete information the key is that the mechanism is ex-post e¢ cient. Obviously, both features can be obtained by using an open auction, but there are many non-auction mechanisms that satisfy one of the two features. But an equilibrium in auctions is compelling, because it combines both of these properties (higher price when more buyers visit and favoring a consumer with higher valuation). When the analysis is extended to the case of a continuous type space the results change dramatically. Suppose that the sellers post arbitrary (but identical) mechanisms 1 = 2 = ::: = n and the buyers play a symmetric Bayesian equilibrium in the stage game induced by those mechanisms. Then the resulting allocation can be induced as an equilibrium outcome of the entire game. The intuition is that by posting appropriate lotteries, seller j can pin down the equilibrium visiting probabilities he receives from all types 1An alternative equilibrium selection device is to require that when the buyers face two situations in which there is a one-toone correspondence between the two sets of equilibria, and equilibrium number k in situation one yields the same payo¤s (for all buyers and sellers) as equilibrium number k in situation two, then the buyers coordinate on equilibria that correspond to each other in the two situations. Once this requirement is made, the simple mechanism described above is a pro…table deviation. In other words, all that is required is that once a deviation is made the buyers are not out to punish the deviator." @default.
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• W3122727119 date "2008-01-01" @default.
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• W3122727119 title "Buyer heterogeneity and competing mechanism" @default.
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