Innovative Applications of O.R.Competing trade mechanisms and monotone mechanism choice
Introduction
Anyone who wishes to sell via an (online) trading platform has to decide upon two issues: what type of trade mechanism to choose and how to specify this mechanism. At eBay, for instance, sellers can decide to run an auction or to offer a transaction at a posted price and have to fix a reserve price for the auction or the posted price.1 A first glance at actual eBay transactions typically hints at a trade-off: Auctions are more likely to be successful but yield a lower average revenue than posted-price transactions.2 This seems to be in contrast to the textbook advice that auctions are better for sellers than posted-price offers because they permit sellers to price discriminate with respect to bidders’ valuations.
Recently, Einav, Farronato, Levin, and Sundaresan (2018) suggested an explanation for this empirical observation: If a monopolistic seller has high opportunity costs of selling the item, it is unlikely that there are at least two bidders with valuations above the seller’s opportunity costs. If buyers face hassle costs when participating at an auction, then such a monopolistic seller may strictly prefer offering the item at a posted price rather than at an auction. As a result, posted prices tend to be optimal for a seller with high opportunity costs, whereas auctions are optimal for a seller with low opportunity costs.
In this paper, we extend this analysis to the mechanism choice of competing sellers. As suggested by the literature on competing mechanism designers (see the literature review below), it is anything but straightforward to translate results for monopolistic sellers to sellers who compete with the design of their trade mechanism. E.g., it has been shown by Eeckhout and Kircher (2010) that competing sellers may only choose posted prices (rather than auctions) whenever meetings between buyers and sellers are rival.3 Our model, however, establishes a translation of the findings by Einav et al. (2018) to a setting with competing sellers: As long as buyers do not face auction-specific hassle costs, competing sellers will only offer auctions. But if sellers’ valuations are continuously distributed and buyers face auction-specific hassle costs, it is an equilibrium that sellers with low opportunity costs of selling offer auctions while sellers with high opportunity costs offer posted prices.
We model the strategic choice between posted prices and auctions by a set of sellers as a finite action game with incomplete information as analyzed in Athey (2001). Sellers have quasi-linear preferences with a private valuation for one unit of a homogenous good drawn independently from (not necessarily identical) continuous probability distributions with full and identical support. Each seller is endowed with one unit of the homogenous good and chooses between posted prices and auctions with start prices. For a given profile of mechanisms chosen by the sellers, buyers act as price takers and the market clears. In other words, buyers who do not trade in equilibrium cannot benefit from trading at a price offered by a seller who does not trade in equilibrium, and buyers who do trade in equilibrium cannot benefit from either not trading or trading with some other seller who does not trade in equilibrium.
For a given strategy by the other sellers, any mechanism can be fully characterized by the induced probability of trade P and the expected revenue in case of trade R. The set of mechanisms at a seller’s disposal can therefore be depicted by a set of points in the plane, and we will refer to this set of points as a (P, R)–plot of mechanisms. For a given strategy of other sellers, a seller will never choose a mechanism that is dominated in the sense that another mechanism would either yield a higher selling probability with at least the same revenue in case of trade or a larger revenue with at least the same selling probability.
We first demonstrate that, without auction-specific hassle costs, a posted price f is always dominated by an auction with start price f and, as also shown with models of competitive mechanism design by McAfee (1993) or Peters (1997), sellers will only offer auctions with start prices that are monotone increasing in their valuation. Part of the literature (see e.g., Carare, Rothkopf, 2005, Einav, Farronato, Levin, Sundaresan, 2018, Wang, Montgomery, Srinivasan, 2008), however, has reasonably emphasized that posted prices may be preferred by at least some buyers due to lower hassle costs that can be attributed to waiting times, the time needed to enter an auction several times (Leszczyc, Qiu, & He, 2009) and the costs of monitoring different auctions when searching for options promising a greater surplus (Chan, Kadiyali, & Park, 2007).4 Even when some people enjoy taking part in auctions, assuming that hassle costs are, on average, larger compared to posted prices seems reasonable. When taking these auction-specific hassle costs into account, we find that (P, R)–plots, and thereby equilibrium mechanism choices, exhibit single-crossing in the sense that sellers offer posted prices if and only if they have a sufficiently high valuation.
Our model yields a set of hypotheses regarding the shape and relative position of (P, R)–plots for posted prices and auctions. First of all, undominated mechanisms resemble a downward sloping graph in the (P, R) –plot as an undominated mechanism with lower selling probability yields a higher revenue in case of trade. Together with the single-crossing of undominated mechanisms in (P, R)–plots for posted prices and auctions, this permits us to compare aggregate performance of posted prices and auctions: selling probabilities for posted prices are, in equilibrium, lower than selling probabilities for auctions, but successfully posted prices are above final auction prices. Moreover, we can characterize equilibrium mechanism choices of individual sellers: single-crossing of (P, R)–plots of posted prices and auctions implies that there is an excess revenue of auctions relative to posted prices for large selling probabilities, but an excess revenue of posted prices over auctions for small selling probabilities. Hence, a seller’s equilibrium mechanism choice is monotone in her valuation along the set of undominated mechanisms. She will choose an auction with a low start price (a high posted price) if her valuation is low (high).
In order to test the hypotheses derived from our model, we use data for tickets to matches of the 2008 UEFA European Football Championship, because the perishable nature and the lack of a competitive fringe guarantee sufficient heterogeneity in buyers’ and sellers’ valuations, and therefore in equilibrium mechanism choices. We provide support for the aforementioned insights on aggregate performances of posted prices and auctions by simple regression analysis. Furthermore, our data suggest that posted prices are sold with a higher probability than auctions with the same start price, which supports the assumption of bidders’ auction-specific hassle cost. To test the main hypothesis from our model that auctions yield lower expected final prices conditional on sale than posted prices for sufficiently small identical selling probabilities, we first estimate the selling probability both for auctions and posted prices. We then use this predicted selling probability in order to explain the excess revenue of an auction over a hypothetical posted price at which this item would have needed to be offered in order to be sold with the same probability. In line with our model, we then find that auctions outperform posted prices for large identical selling probabilities and vice versa.
Our analysis regarding the existence of a monotone pure strategy equilibrium adds to the literature on competing mechanism designers that establishes the optimality of auctions and addresses the convergence of optimal start prices to the sellers’ costs in a competitive equilibrium setting (see McAfee, 1993 or Peters, 1997) or for competing auctions (see Burguet, Sakovics, 1999, Hernando-Veciana, 2005, Peters, Severinov, 1997, Peters, Severinov, 2006, or Virag, 2010). As this literature focuses on the emergence of efficient trade institutions as the result of competition between sellers, it is typically assumed that sellers have identical or publicly observable costs (for an exception see Peters, 1997). By contrast, our paper analyzes the impact of unobservable seller heterogeneity on mechanism choice and thereby addresses the question of optimal mechanism design for different types of sellers. Specifically, the representation of a seller’s choice set by (P, R) – plots visualizes how straightforward trade-offs between selling probability and revenue in case of trade ensure the existence of pure strategy equilibria. The crucial role of this revenue-probability trade-off for equilibrium existence has been emphasized in the literature on competitive search where sellers who offer a smaller share of the surplus (and thereby keep a larger revenue for themselves) are visited less frequently by buyers; see, e.g., Moen (1997) or, more recently, Guerrieri, Shirmer, and Wright (2010) and Chang (2017).
Some of our empirical results are in line with previous empirical work on the comparison between auctions and posted prices: The aforementioned papers by Halcoussis and Mathews (2007), Hammond (2010) and Einav et al. (2018) also find that auctions are unconditionally more likely to be successful but yield a lower price conditional on sale than posted prices. Our theoretical model gives an explanation for this finding in the context of a platform with competing sellers by showing that, in equilibrium, auctions (posted prices) are typically chosen in combination with a low start price (high posted price), which implies a high (low) selling probability and low (high) expected revenue conditional on sale. In a dataset that includes almost all kinds of items sold on eBay, Einav et al. (2018) use variation in the same sellers’ mechanism choices to empirically estimate single (P, R) – plots. We develop (P, R) – plots by exploiting heterogeneity of different sellers in a sample of homogenous items, controlling for observable item characteristics. In a different vein, Hammond (2013) and Bauner (2015) estimate a structural model of sellers’ mechanism choices in order to make predictions about counterfactual markets. In their data, posted prices and auctions also co-exist, and sellers for whom they estimate higher valuations are more likely to choose posted prices. In contrast to all studies reviewed in this paragraph, we theoretically demonstrate the co-existence of auctions and posted prices in equilibrium, and apply an estimation strategy that is designed to test our theoretical hypotheses regarding (P, R)–plots.
In our context, all items are offered online and each seller has only very few items due to the initial sales mechanism for tickets by UEFA. By contrast, Sun (2008) considers a seller with multiple items that are simultaneously offered with posted prices and in auctions. Comparable to our hassle costs, he adds a disadvantage of auctions due to monitoring costs or the necessity to wait until the auction has closed. He then shows that the two mechanisms serve as a screening device. Kuruzovich and Etzion (2018) consider sellers who utilize online auctions and offline channels with posted prices simultaneously. They then analyze theoretically and empirically how offline demand impacts the characteristics of auctions (reserve price, sales probability and prices). While we compare auctions using optimal start prices with posted prices, Wang (2017) analyzes a seller’s choice between auctions with zero start prices and posted prices and finds that a wider dispersion of bidder valuations works in favor of auctions only if the web traffic is sufficiently large. Last, the choice between auctions and posted prices by buyers is considered by Katehakis and Puranam (2012) in the context of sequential purchases of multiple units and Jiang, Fang, Fan, and Dingwei (2013) under the assumption of bounded rationality, but neither of these papers consider the mechanism choice of (competing) sellers.
The remainder of the paper is organized as follows: We will develop and analyze our theoretical model of mechanism choice by competing sellers and derive empirically testable hypotheses in Section 2. Section 3 presents our empirical analysis. We conclude in Section 4.
Section snippets
The model
Consider the following set-up modelling online trade. s ≥ 1 risk-neutral sellers are endowed with one unit of an indivisible, homogenous good. Seller has reservation value for her unit of the good. For each ri is distributed with continuous density hi(ri) with full support on [0, 1].
b > s risk-neutral buyers like to purchase one unit of the indivisible, homogenous good. Buyer has valuation for one unit of the good. For each , vj is distributed
Data
We use data from secondary ticket sales for the EURO 2008, the European Football (Soccer) Championship for national teams. 16 teams participated in this major European sport event, which took place in Austria and Switzerland from June 7th to June 29th. Tickets were valid for a particular match of the championship. Altogether, 31 matches were played, including 24 matches in the preliminary round of four teams each in four groups playing round robin. The best two teams of each group qualified for
Concluding remarks
Our model of competing sellers’ choices of mechanism confirms the superiority of auctions in the absence of hassle costs and demonstrates the single-crossing of optimal mechanisms in the presence of hassle costs. We derive these results theoretically by assuming that competing auctions retrieve market clearing prices, which has been shown to emerge as an equilibrium of cross-bidding between auctions by Peters and Severinov (2006). In this sense, our model gives a “better shot” at auctions than
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