Econ 196 Topic: Behavioral Economics Quasi-Endowment in Private-Value Auctions Introduction The analysis of symmetric, private-value auctions pioneered by Vickrey (1961) and Myerson (1981) assumes bidders are risk neutral, a crucial assumption upon which the famous revenue-equivalence principle relies. However, despite their elegance, these models have seen little empirical validation. Neither these models nor subsequent extensions to the case of risk-averse bidders have been able to account for overbidding in auctions, a phenomenon widely documented in empirical research. Overbidding in auctions has long been an object of interest, even to non-economists. For instance, most are familiar with the colloquial explanation of overbidding known as “auction fever,” which holds that bidders become emotionally caught up in the bidding process. One of the more novel explanations for overbidding was recently proposed by Ariely et al (2004). They provide experimental evidence for a “quasi-endowment” effect, whereby bidders develop a sense of attachment to the object during the bidding process and come to see the object as an item that already belongs to them. This suggests that bidders influenced by this effect would bid higher than they otherwise would have, in order to retain what they perceive as an endowment. This result confounds most of the theoretical literature on auctions, under which no such behavior should occur. In this literature, auctions are modeled as games of imperfect information, and bidders are assumed to be rational, their strategies dictated by the game’s Bayesian-Nash equilibrium. In particular, the payoff for a particular bidder is assumed to be a function of only the value of the object and the expected payment, the function linear for the case of risk neutrality and concave for risk aversion. This paper departs from the literature on auctions first by incorporating loss aversion into specifications of bidder utility. In doing so, we provide some theoretical basis for the quasi-endowment effect, which can be framed as loss aversion. For first- and second-price auctions,1 1 In a first-price auction, the winner pays the highest bid (her bid). In a second-price auction, the winner pays the second-highest bid. English auctions are the familiar oral auctions in which an auctioneer calls out ascending prices, and bidders signal their willingness to pay.we present equilibrium bidding strategies for a fixed, exogenous reference point. We also extend the revenue-equivalence principle to bidders with loss aversion and thus determine the expected payment of such bidders for a wide class of auction formats. We next attempt to incorporate some of the insights from the empirical research on quasi-endowment in dynamic auctions into two models of English auctions. One model follows the standard model of Milgrom and Weber (MW) (1982) for interdependent-value auctions, and the second employs an “alternating recognition” (AR) model developed by Harstad and Rothkopf (2000). Unlike the MW model, the AR model gives theoretical meaning to the idea of being a leading bidder. In both models, rather than assuming loss aversion, we let each bidder’s valuation increase as other bidders exit the auction, but in the AR model, we allow only the “leading” bidder’s valuation to increase the “longer” she is in the lead. This is an attempt to incorporate the intuition behind the result found in Ariely et al (2004) that bidders submit higher rebids the longer they are in the lead. That is, the quasi-endowment effect is stronger for bidders that remain in the lead for a greater amount of time. We also discuss the technical difficulties of solving for equilibrium in these auctions. The outline of the paper is as follows. Section one presents results from the rational model of bidders in symmetric, private-value auctions. Section two surveys relevant empirical results on overbidding. Section three presents the equilibrium solutions for first- and second-price auctions. Section four discusses models of quasi-endowment for English auctions. 1. Rational Bidders Following Vickrey (1961), private-value auctions are modeled as games of imperfect information. There are N risk-neutral bidders, each bidder denoted by i, each with type which represents the bidder’s valuation of the object. The Xi’s are independently and identically distributed on [0, ω] with density f. Every bidder knows her realized value xi and has no information about the values of the other bidders, except their densities f. The payoff functions are contingent on the particular auction format. In a second-price, sealed-bid auction+ℜ⊆iX2, 2 In a sealed-bid auction, all bidders submit a single bid, and the winner is determined from these bids. In this paper, we only consider first-price and second-price auctions that are sealed-bid. In a dynamic auction, there are multiple rounds of bidding, the most well-known example being the English or oral ascending auction. 2the player who submits the highest bid pays the second highest bid, so ignoring zero-probability ties, the payoff function is ⎩⎨⎧−=Π≠−0max),(jijiiiIIbxxx if (1) jijijijibbbb≠≠<>maxmaxwhere bi is player i’s bid and x-i = (x1, …, xi-1, xi+1, …, xN). Players bid according to a strategy βi: [0, ω] Æ which is a function that assigns a bid bi to every private value xi. These auctions typically have a multitude of equilibria, but papers often restrict attention to symmetric equilibria, since bidders are symmetric. In this paper, we will follow this convention. +ℜ One can prove that in a second-price, sealed-bid auction, truth-telling, or bidding one’s private value, is a weakly dominant strategy. Additionally, it turns out that English auctions are equivalent to second-price auctions in a weak sense. To see this, notice that it cannot be profitable for a bidder to drop out before the auctioneer reaches her value or to stay in after the posted price exceeds her value. Thus, as soon as the bidder with the second-highest value drops out, which is precisely when the posted price meets her value, the bidder with the highest value wins the object at that price, which is exactly the second-highest bid in a second-price auction. Hence, bidding one’s value is also a weakly dominant strategy for English auctions. The bidding strategies in first-price auctions are slightly more involved and are not weakly