Auction fever: Rising revenue in second-price auction formats

Games and Economic Behavior 92 (2015) 206–227

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Auction fever: Rising revenue in second-price auction formats Karl-Martin Ehrhart a , Marion Ott b,∗,1 , Susanne Abele c a b c

Karlsruhe Institute of Technology (KIT), ETS, Neuer Zirkel 3, Building 20.53, 76131 Karlsruhe, Germany Karlsruhe Institute of Technology (KIT), ETS, Zirkel 2, Building 20.21, 76131 Karlsruhe, Germany Miami University, Oxford, OH, USA

a r t i c l e

i n f o

Article history: Received 19 June 2013 Available online 3 July 2015 JEL classification: D44 D03 Keywords: Auction fever Experiment Private uncertain values Pseudo-endowment effect

a b s t r a c t The prevalent term “auction fever” visualizes that ascending auctions – inconsistent with theory – are likely to provoke higher bids than one-shot auctions. To explore and isolate causes of auction fever experimentally, we design four different strategy-proof auction formats and order these according to expected rising bids based on pseudo-endowment effect arguments (psychological ownership and disparity between willingness to pay and willingness to accept). Observed revenues in the experiment in the four formats rank as expected if bidders have private uncertain values (the private information of a bidder is the distribution of her value). A control treatment supports our view that the traditional private certain values approach prevents auction fever in the laboratory. Another control treatment with a procurement auction relates the auction fever bids to bids in a one-shot auction with real endowments. We conclude that, when bidders are uncertain about their valuations, auctions that foster pseudo-endowment may raise bids and revenues. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Fierce bidding particularly in ascending auctions is often associated with auction fever. The auctions appear to induce people to bid higher than they would if they were afebrile.2 Web sites on Internet auctions advise bidders to avoid auction fever, indicating its prevalence.3 Jones (2011) defines bidding fever, or auction fever, as “the expectation of extra utility from winning an auction that is inspired during the bidding process.” Several factors might inspire auction fever. We focus on perceived ownership of the item as a factor, which increases a bidder’s willingness to pay for the auctioned item over the course of the auction. This is termed the pseudo-endowment effect. We hypothesize that different auction formats differ in the strength of the pseudo-endowment they evoke. In a laboratory experiment, we compare four different auction formats: a second-price sealed-bid auction (A1) and three ascending auctions (A2–A4). The ascending auction formats differ only in their high-bidder selection. In A2, at any price,

*

Corresponding author. E-mail addresses: [email protected] (K.-M. Ehrhart), [email protected] (M. Ott), [email protected] (S. Abele). 1 Present permanent address: RWTH Aachen University, Templergraben 64, 52056 Aachen, Germany. 2 See, for example, Ku (2000), Ku et al. (2005), Ockenfels et al. (2006), Delgado et al. (2008), Haruvy and Popkowski Leszczyc (2010). 3 For example: “Set your buying limit before the auction starts; DON’T get bidding fever, if the item goes above your limit, FORGET it!” (http://www. ukauctionguides.co.uk/hints_tips.htm, 03-06-2012), “It’s true that ‘auction fever’ sometimes grips a gallery and propels bidding levels beyond the real value of certain coins. This, of course, is a plus if you happen to be the person who consigned those coins for sale, but something to be avoided if you’re a buyer.” (http://www.usgoldexpert.com/articles/the-unspoken-truth-about-rare-coin-auctions/, 03-06-2012). http://dx.doi.org/10.1016/j.geb.2015.06.006 0899-8256/© 2015 Elsevier Inc. All rights reserved.

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all bidders that are still in the auction share the high bidder position. In A3, at any price one of the bidders is randomly selected as the current high bidder, and in A4, the bidder who first accepts the price is the high bidder. We consider a private values setting and refer to a value as private, if the bidder holds all existing information about her value. In particular, no other bidder holds information that, if known to the bidder, would affect her value. We conduct the four auctions with induced private certain values, i.e., fixed values, and with induced private uncertain values, i.e., value lotteries. In the latter case, a bidder’s private information is the distribution of her value. In our auctions, the item is presented as a good for commercial use (a ship) whose exact value for the bidder in the uncertain values setting is revealed after the auction.4 Both with private certain values and with private uncertain values the auctions are strategically equivalent5 and strategyproof, that is, they provide weakly dominant strategies to bid truthfully.6 ,7 Private uncertain values in combination with weakly dominant strategies allow separating valuation uncertainty from strategic uncertainty. Strategic uncertainty may be caused by the auction format, such as a first-price auction, or by value interdependencies. Our private values setting with strategy-proof auctions eliminates strategic uncertainty. All risk involved in the decision arises from individual valuation uncertainty, which does not depend on others’ information. We expect auction fever to occur in the auctions for lotteries (our main treatments) but not in the fixed value auctions (our control treatments). When values are uncertain, we expect increasing final auction prices from auction format A1 to A4. In our control treatments, we expect no differences between auction formats. The hypothesis of increasing prices bases on the pseudo-endowment effect and the assumption that the strength of this effect differs between our auction formats. Pseudo-endowment is an increased attachment to the auctioned item that may occur if the bidder feels like already owning the item to some extent or with some probability. The pseudo- or quasiendowment effect resembles the endowment effect (Thaler, 1980) but with no real ownership. The endowment effect, which bases on the theory of reference-dependent utility (Kahneman and Tversky, 1979), refers to the observation that the minimum compensation that people demand to give up a good is higher than the maximum amount they are willing to pay to acquire the same good (the disparity between willingness to pay (WTP) and willingness to accept (WTA)). The early experimental literature on the WTP–WTA disparity shows that the average WTP for buying a lottery ticket is less than the average WTA for selling the same lottery ticket, and that this difference is higher than what wealth effects could explain (e.g., Knetsch and Sinden, 1984; Marshall et al., 1986). Besides lotteries, WTP–WTA studies have also used real goods but not fixed induced values, arguing that it is implausible that someone’s WTP and WTA for some certain monetary value would differ.8 Kahneman et al. (1990) apply this argument and use induced certain values to test whether participants understand their market mechanism, while they use real goods to test for an endowment effect.9 Our setup follows the same idea for comparisons between certain-value auctions and uncertain-value auctions. In all our auction formats, bidding (up to) the WTP is weakly dominant. Bidders that are attached to the item (i.e., the lottery) may have a higher WTP: if quitting the auction is experienced as giving up the item to the extent that the bidder feels like owning it, WTA considerations enter the WTP determination and a pseudo-endowment effect occurs. In ascending auctions, at every stage, quitting and the associated sensations from giving up the item can be avoided by a further bid. Differences in the WTP and, thus, differences in bids and prices between auction formats may arise due to the different quality of the sensations from giving up the item caused by a different quality or awareness of the high bidder position (shared in A2, achieved by chance in A3, achieved by own action in A4) or by the lack of the option to give up the item (in A1). Our hypothesis of increasing auction prices from A1 to A4 in the private uncertain values setting thus follows, if the design differences strengthen pseudo-endowment, thereby increasing the WTP. Another treatment with a sealed-bid procurement auction and private uncertain values seeks to support the WTP–WTA and the loss aversion arguments. Our approach complements other experimental and empirical studies on auction or bidding fever, pseudo- or quasiendowment, and overbidding. In the literature, beside the pseudo-endowment effect, further hypotheses on the origin

4 Uncertain values might represent more general settings. Research on human behavior suggests that people’s evaluation of objects is usually not certain but rather uncertain and subjectively constructed (e.g., Edwards, 1954, 1962; Kahneman and Tversky, 1984; Fiske and Taylor, 1991; Dubourg et al., 1994; Ariely et al., 2003). 5 We call two auctions strategically equivalent if their sets of strategies (the bids in a one-shot auction or the maximum bids, which characterize the strategies in an ascending auction) are the same and if each strategy profile leads to the same outcome in both auctions. Then, the auctions have the same winners and prices in equilibrium. 6 In all auctions the weakly dominant strategy of an expected utility maximizer is to bid (up to) the private certain value or the certainty equivalent of the private value lottery if values are uncertain. More specifically, in the latter case, an expected utility maximizer bids such that her current wealth is the certainty equivalent of owning the private value lottery at the price equal to the bid (see Equation (A.1)). Deviating from theory, due to discrete increments and random tie-breaking, two treatments in the experiment are not exactly strategy-proof but slightly lower or higher bids may be optimal given particular beliefs about ties. We take this into account in our design and analyses (see Section 2.2). 7 If payoffs are quasi-linear, strategy-proofness in second-price auctions is given for private values but not if values are interdependent, i.e., if other bidders hold information relevant for a bidder’s value, or if values depend on the others’ performance, i.e., if bidders impose value externalities on each other. 8 For an overview of WTP–WTA studies, see Kahneman et al. (1991) and Horowitz and McConnell (2002). Arguments on no WTP–WTA disparity with induced certain values are, e.g., by Kahneman et al. (1990) and Coursey et al. (1987) (see Section 3.2 for details). 9 Indeed, they do not find indications for a WTP–WTA disparity with induced certain values but with real goods.

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of auction fever have been studied. We are the first to address auction fever by a comparison across auction formats. This approach allows to exclude other factors as potential drivers of our results and to interpret our findings as supporting the pseudo-endowment arguments. Many of the factors identified in the literature either may not play a role in our experiment or, those that may play a role, do not capture the expected differences between our second-price auction formats. Several papers analyze a longer duration of being the high bidder as the source of higher bids due to a stronger pseudoendowment effect. Heyman et al. (2004) vary the period of being the high bidder and find higher bids by the group with the longer high-bidder period in a survey-based experiment with fictitious second-price auctions and in an ascending auction of real goods (with a fixed number of bidding rounds and, unknown to the subjects, bidding against five computerized agents). Similarly, people who are high bidders in eBay Motors’ auctions for a longer time are more likely to re-bid (Wolf et al., 2005). When items on eBay are available in auctions and at a fixed (buy-it-now) price, Malmendier and Lee (2011) find overbidding by 17% of bidders but do not find a positive relationship between the bid and the time in the auction as a high bidder or in general. In our study, the duration of being the high bidder or the number of times as a high bidder are, caused by the design, highly correlated with the height of the bid in all ascending auctions. Bidding stage lengths hardly differ, and from the time when two (of three) bidders remain in the auction, they will have similar times as high bidder. Thus, the duration of being high bidder cannot capture differences between our ascending auctions. Competitive arousal and escalation of commitment may cause overbidding (e.g., Ku et al., 2005; Haruvy and Popkowski Leszczyc, 2010). The competitive arousal hypothesis states that diverse factors, such as social facilitation, rivalry, and time pressure may increase arousal in an auction and, thereby, affect the decision making of bidders (Ku et al., 2005). In our study, participants decide privately and anonymously; thus, social facilitation should not be relevant. Even if it were relevant, it would be the same in all treatments. Rivalry, which, in contrast to the standard competition argument, means that the fewer bidders that are left, the more a bidder will bid,10 should be the same in our ascending auctions. More importantly, in our ascending auctions, bidders are not informed about the number of bidders left. Time pressure is reduced by ensuring that before leaving the auction, every bidder has 50 seconds to consider the decision not to accept a price level. Escalation of commitment results from the justification of previous decisions by investing more in the auction instead of leaving it. Even if there were a high investment in coming to our lab and reading the instructions, the differences between the treatments, if any, should be minor. Risk aversion, loser regret, fear of losing, charitable motives, and spite (e.g., Engelbrecht-Wiggans, 2009; Filiz-Ozbay and Ozbay, 2007; Delgado et al., 2008; Cramton and Sujarittanonta, 2010; Goeree et al., 2005; Cooper and Fang, 2008) may serve as additional hypotheses for higher bids. The effect of risk aversion should be the same in our auctions because we only consider second-price auctions with the same private uncertain values. Experiments on loser regret and fear of losing implement repeated first-price auctions, and their hypotheses refer to a bidder’s recent experience of not winning an auction although her valuation was higher than the winning bid. By letting the subjects participate in only one auction, we avoid such experience-based effects. In charity auctions (as used, e.g., in the study by Ku et al., 2005), bidders may assign value to each monetary unit spent, for example due to altruism or because a public good is up for sale, which in turn induces higher bids. There is no reason to expect such motives in our experiment. Spiteful bidding by losing bidders to drive up the winner’s payment in second-price formats does not predict the expected differences between our auctions. Joy of winning or another extra utility from winning leads to higher bids (e.g., Kagel, 1995; Cox et al., 2002; Goeree and Offerman, 2003; Jones, 2011; Malmendier and Lee, 2011). However, extra utility from winning leads to higher bids independent of the auction format, and, thus, is not suited to capture differences between our auctions. Jones (2011) and Malmendier and Lee (2011) investigate overbidding on eBay relative to a more or less prevalent fixed price outside option. Malmendier and Lee (2011) find 42% final prices above the available fixed price in their main study and 48% in a cross-section analysis on many products. They conclude that limited attention and limited memory of outside options underlies this overbidding.11 Jones (2011) finds a similar 41% of prices above the face value for Amazon vouchers on eBay. Since the monetary value is obvious and the vouchers are available at Amazon.com at the face value, Jones excludes limited attention as the source of the observed overbidding. Limited attention to outside prices does not play a role in our study. Adjustments in the willingness to bid can also be due to learning during the auction about the own value in an interdependent values setting. We investigate a private values setting, so this does not apply. Note however, although strategy-proofness breaks down with interdependent values, our setting is also suited to test auction fever when values are interdependent. Exit information about bidders is withheld until the auction is over and so strategic differences between the auctions are minor also for interdependent values.12

10

The rationale for more aggressive bidding when less competitors are left is the perceived higher chance of winning. Malmendier and Lee (2011) refute extra utility from winning as underlying their data because their simulated auction outcome distributions match properties of outcomes from a model of limited attention better than properties of a model with a restricted extra utility from winning. 12 Some inference is possible. For example, in A3, becoming high bidder often is more likely if more competitors have quit. A bidder who becomes high bidder every other stage may conclude that one bidder has left at some earlier stage. 11

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Table 1 Number of groups (subjects) in the different treatments. Main treatments

Control treatments

A1u: A2u: A3u: A4u:

A1c: A2c: A4c: P1u:

12 12 12 12

(36) (36) (36) (36)

48 (144)

12 12 12 12

(36) (36) (36) (36)

48 (144)

Sum: 96 (288)

Summarizing, while the literature identifies many interesting effects that may induce bidding fever, overbidding, or a change in the willingness to bid, the effects different from the pseudo-endowment effect do not capture our predicted treatment differences. 2. Experimental design This section describes the experimental procedures, the implemented auctions, and the treatments. 2.1. Organization and implementation of the experiment The experiment was conducted at the University of Karlsruhe with randomly selected students from various disciplines. Every subject competed with two other bidders in only one single auction. We preferred this design over a setting with several consecutive auctions because the subjects should fully focus on one payoff-relevant auction and because our auctions are very simple and easy to understand. Many auctions in practice have a one-shot nature, e.g., if a unique item, like a house, is up for sale. Each of eight treatments with twelve auctions (groups) per treatment was conducted in two sessions with six groups (18 subjects) in each session. Thus, 36 subjects participated in each treatment and a total of 288 subjects participated in the experiment. The treatments are separated into two classes, “private values with uncertain valuations” and “private values with certain valuations,” which are indicated in the treatment names by a “u” or a “c” (see Table 1). Auction formats are labeled by A1 to A4 and P1. In the temporal sequence, the main treatments A1u to A4u were conducted before the control treatments A1c, A2c, A4c, and P1u. Based on the experience of the treatments with uncertain valuations and our hypotheses about certain valuations, we concluded that auction format A3 would provide no new insights and omitted a possible treatment A3c. The procurement auction P1u is described in Section 5. The experiment was computerized. Each subject was seated at a computer terminal separated from the other subjects. The subjects received written instructions, which were read aloud by an experimental assistant (see Appendix C). Before the experiment began, each subject answered several questions regarding the instructions at her/his computer terminal. After all subjects had answered all questions correctly, the subjects were given their private information, and then the auction began. Communication was not permitted. Subjects could not identify the members of the session with whom they actually interacted. The experimental sessions lasted less than one hour. At the end of an experimental session, the subjects were paid in cash according to their profits in the game. The conversion rate was 5 Euro for 100 Experimental Currency Units (ExCU). The average, minimum, and maximum payments realized were e 10.82, e 3.85, and e 18.10, respectively. 2.2. Sales auction formats in the experiment In each of the four different sales auction formats A1–A4, a virtual good (a ship) is offered to three bidders, each of whom is assigned the role of a shipping company owner.13 We distinguish between a static auction (sealed-bid auction) and three ascending auctions (Japanese- and English auctions), which differ with respect to the selection of the current high bidder. A minimum bid bmin of 500 ExCU and a constant increment of 5 ExCU are set. When the auction ends, the winner is paid her private value minus the price. All subjects receive a lump-sum payment of 200 ExCU. Prices are determined as second price plus one increment or a small variation thereof. Auction format A1. Subjects are asked to submit their upper bidding limit only once. A bidding mechanism then outbids the bids against each other, as in an English auction. That is, bidders face a second-price or Vickrey auction (Vickrey, 1961). The bidder with the highest bid receives the item. We apply two pricing rules for A1, which dif-

13 We consider a ship well suited because it is unlikely that the students who participated in our experiment would have prior experience with ships or would be particularly interested in ships. Naming the good a ship makes it also easy to explain the uncertain values environment (owning the ship would produce uncertain revenues).

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fer if there is no tie. In half of the groups, the price equals the second-highest bid plus one increment (pricing rule a). The other groups face a pure second-price rule (pricing rule b). This differentiation assures comparability of our four auction formats when discrete increments are considered (see Appendix A.1). In case of a tie, under both pricing rules the winner is randomly drawn from the set of the highest bidders and pays the highest bid. It is well known that subjects in Vickrey-auction experiments with private certain values tend to bid higher than weakly dominant strategies predict (e.g., Kagel et al., 1987; Kagel and Levin, 1993; Kagel, 1995; Harstad, 2000). To prevent such distortions, our subjects are told that they enter the bidding limit to a computerized agent who bids on their behalf in an ascending (English) proxy auction (Seifert, 2006).14 Auction format A2. In the Japanese auction (or single-item clock auction) A2, we introduce dynamics. The price increases incrementally, and bidders are asked at every stage whether they accept the current price. If a bidder does not accept the price within 50 seconds, she quits the auction. Bidders are not informed about the number of remaining bidders. The bidding process stops when all but one bidder (or all bidders) have quit the auction. If only one bidder remains, this bidder receives the item and pays the price of the last stage (pricing rule a) or the penultimate stage (pricing rule b). If all remaining bidders quit the auction at the same stage, the winning bidder is randomly drawn from this set of bidders and pays the price of the last stage. Thus, the pricing rules are identical to those of auction A1. Auction format A3. A3 is a variation of A2. As in auction A2, the price increases incrementally, and bidders are asked at every stage if they accept the current price. In addition, at every price level, a current high bidder is randomly chosen from the set of accepting bidders. The process stops when none of the other bidders accepts the next price level within 50 seconds. The last high bidder receives the item and pays the price of the stage before the auction stops, that is, the stage at which she was designated the high bidder.15 ,16 Auction format A4. The ascending auction A4 is a computerized form of an English clock auction. As before, the price increases incrementally. At every price level, the bidder who first accepts the price is designated the current high bidder. Then, the price increases by one increment and is shown for five seconds (counted down on the screen). After five seconds, the button to accept the new price is enabled for the other bidders. The process stops when the new price is not accepted by any other bidder within 45 seconds. We use the two pricing rules a and b because in A3 and A4 only rule a is feasible, which in A1 and A2, however, allows for incentives to bid lower than in A3 and A4. Hence, using only rule a might work in favor of our hypothesis. Rule b allows for incentives to bid higher in A1 and A2. The incentives may arise for a bidder who assigns high probability to a tie in our setting with discrete increments.17 To ensure that subjects are able to read the screen and recognize the current price in the ascending auctions A2, A3, and A4, each price level is shown for at least five seconds. In an A2 or A3 auction, even if all bidders accept a price level immediately, the screen is not updated before five seconds have passed.18 For more details, see the translated instructions in Appendix C.

14 The data suggests that this description helps overcome problems caused by unfamiliarity with the mechanism. With private certain values, we observe 19% over- and 42% underbidding in A1. The lab experiment by Garratt et al. (2012) finds 38% over- and 41% underbidding by experienced subjects, compared to 70% and 13% by unexperienced subjects (data from Kagel and Levin, 1993). Field studies by Bajari and Hortaçsu (2003) and Malmendier and Lee (2011) do not find a relationship between experience and overbidding by winners in eBay auctions, whereas Jones (2011) finds a negative relationship. In all studies, experienced bidders do overbid. 15 Note that the difference between auction formats A2 and A3 is rather small because the only difference is the random selection of the current high bidder in A3. 16 A multi-item, uniform-price version of this auction, with a selection of provisional winners in each round for each item, was applied in a spectrum auction in India. Cramton and Sujarittanonta (2010) compare such multi-item auctions in terms of two pricing rules. In contrast to our auction, their rules require the high bidder to bid in the following round to stay in the auction. 17 Discrete bid increments cause a difference between a standard second-price rule (rule b) and pricing equal to the winning bid in a dynamic process (rule a), which will usually be one increment above the second-highest bid. In A3 and A4, only pricing rule a is feasible because applying rule b requires to observe ties between bidding limits, and bids in A3 and A4 do not provide the necessary information (in A3 and A4 the current high bidder is not allowed to outbid himself and in A4, in addition, only one bid per stage is accepted). In A3 and A4, bidders’ weakly dominant strategy is to bid up to the highest price below their value (or certainty equivalent). However, in A1 and A2 with rule a, bidding the highest price below the value may be worse than bidding an increment lower for a bidder who assigns high probability to a tie with the latter bid (provoking a tie and trading off lower winning probability with a lower payment, see Appendix A.1). With rule b, bidding the highest price below the value may be worse than bidding an increment higher for a bidder who assigns high probability to a tie with the former bid (to escape a tie and win with certainty with the same payment, see Appendix A.1). We use pricing rule a for half of the groups to apply the same pricing rule in all treatments and pricing rule b to the other half of the groups to see whether bids differ between the two rules. 18 Strahilevitz and Loewenstein (1998) find a positive influence of the duration of ownership on the endowment effect, and Heyman et al. (2004) find the same in an auction environment with respect to the duration of the high-bidder status. However, the main point of the short five seconds delay in our experiment is not the duration of ownership but the possibility of becoming aware of the fact of being the current high bidder.

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Table 2 Bidders’ private information about their individual valuations. Bidder

A1u–A4u

A1c, A2c, A4c

B1 B2 B3

U [512, 712] U [517, 717] U [552, 752]

612 617 652

2.3. Treatments There are seven treatments with sales auctions. In the four main treatments A1u–A4u, we test our auction fever hypothesis. Treatments A1c, A2c, A4c are control treatments to ascertain whether the effect depends on the use of uncertain values and whether subjects understand the strategic equivalence of the auction formats. 2.3.1. Uncertainty treatments A1u, A2u, A3u, and A4u Bidders have private but incomplete information about their valuations. Note that although a bidder is uncertain about her valuation, there is no common value component. The winner’s value is realized after the auction. In all groups of three, bidders’ values are given by discrete uniform distributions on the same three value intervals, as displayed in Table 2. Because we only conduct strategy-proof auctions in which we want subjects to concentrate on their individual values, we deliberately withhold information about the distributions of others. The subjects only know that others’ value distributions are different. There are three reasons for choosing the value configuration of Table 2. First, the two distributions with low expected valuations of 612 and 617 are very close (differing by one increment) and thus create a competitive situation for bidders B1 and B2, who are expected to determine the price. Second, the large gap between B3’s expected valuation of 652 and the two low expected values admits that subjects of type B1 or B2 that suffer from auction fever raise their bid far above their valuation, impelled by bidder B3. Third, each expected value lies between two admissible bids or price steps, which are multiples of five. Consequently, a bidder who wants to bid her expected value must decide to bid below or above her expected valuation and cannot simply use it as an anchor. 2.3.2. Certainty treatments A1c, A2c, and A4c In the setting with certain valuations, we set aside the somewhat artificial auction format A3 and restrict our analysis to the three established formats A1c, A2c, and A4c, in which each bidder is aware of her private valuation at the time of the auction. Again, we withhold information about others’ values. Subjects know only that their values are different. For comparability, the certain valuations correspond to the expected values (i.e., the means of the intervals) in the treatments A1u–A4u (see Table 2). 3. Theory and hypotheses In this section, we present theoretical benchmarks for the different auctions and our hypotheses. 3.1. Theoretical benchmarks As a benchmark, consider auction formats A1–A4 with continuous bidding and pricing spaces. In what follows, the term “bid” refers to the bid in the sealed-bid auction and to the bidding limit in the ascending auctions. For comparability, we ensure that from a theoretical point of view the auctions A1u–A4u are strategically maximum bid equivalent. That is, an expected-utility maximizing bidder chooses the same bid in A1u–A4u. The same holds for auctions A1c, A2c, and A4c. Proposition 1. An expected-utility maximizing bidder has a weakly dominant strategy to bid her WTP in the auctions A1–A4 with private certain and with private uncertain values. A bidder with a private certain value bids her value and a bidder with a private uncertain value bids her certainty equivalent of her private value lottery. While an expected-utility maximizer’s WTP is fixed throughout the auction, the WTP of a bidder that falls prone to the pseudo-endowment effect may increase in the course of the auction. Consequently, we denote such a bidder’s WTP at any given time during the auction by her current WTP. The strategic considerations are unaffected by her adjustment of her WTP. Proposition 2. For a bidder that falls prone to the pseudo-endowment effect and whose WTP is unaffected by her beliefs about the others’ strategies and weakly increases during the auction it is at each price level weakly dominant to bid her current WTP in the auctions A1–A4 with private certain and with private uncertain values. We will argue that only if values are uncertain, a bidder’s current WTP may strictly increase during the auction.

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At the time when the auction ends, the bidder with the highest (current) WTP receives the good and pays the secondhighest (current) WTP. Thus, in all auction formats A1–A4 with private certain and with private uncertain values, a price equal to the second highest (current) WTP at the ending time is expected. The proofs are given in Appendix A.1, together with a discussion of the impact of discrete bid increments and tiebreaking. 3.2. Hypotheses Our hypotheses on ranking auction formats by revenue are based on the pseudo-endowment effect in combination with private uncertain values. In auctions, the pseudo-endowment effect may occur when bidders develop psychological ownership of the item. If bidders are loss averse, this may affect their WTP and, thus, their bids.19 We hypothesize that in our treatments higher bids due to the pseudo-endowment effect, first, require uncertain values and, second, are stimulated by auction dynamics. We expect that the pseudo-endowment effect does not matter in the private certain values setting. A bidder’s WTP is externally given by her private certain monetary value, which leaves no scope for WTP construction. In the words of Kahneman et al. (1990): “There are some cases in which no endowment effect would be expected, such as when goods are purchased for resale rather than for utilization. A particularly clear case of a good held exclusively for resale is the notional token typically traded in experimental markets commonly used to test the efficiency of market institutions [. . .]. Such experiments employ the induced-value technique in which the objects of trade are tokens to which private redemption values that vary among individual participants have been assigned by the experimenter [. . .]. No endowment effect would be expected for such tokens, which are valued only because they can be redeemed for cash.” and those of Coursey et al. (1987): “The commodity in induced-value experiments can simply be a coupon that can be redeemed at the end of the experiment by the subject for its face value in cash. Thus, the value to the subject of the commodity is controlled by the experimenter. It is precisely this framework in which the strong demand-revealing nature of the Vickrey auction has been demonstrated. However, this controlled value framework does not plausibly allow for a divergence between WTA and WTP to exist for subjects. Thus, a commodity with an initially unknown value to the subjects must be used.” Hence, we expect that in the auctions A1c–A4c, bidders’ private certain values determine their WTP and, thus, their bids (Propositions 1 and 2). Hypothesis 1. There is no difference in prices in the treatments A1c, A2c, and A4c. With our uncertain values setting we follow Coursey et al. (1987), who argues that uncertainty about the value is necessary for a WTP–WTA disparity to occur (see the quote above), and Knetsch and Sinden (1984) and Marshall et al. (1986), who experimentally analyze the endowment effect and detect the WTP–WTA disparity using lottery tickets. In the auctions with private uncertain values, A1u–A4u, a bidder subjectively constructs her (current) WTP for her private uncertain value lottery. According to Propositions 1 and 2, in all four auctions A1u–A4u, we expect bidders to bid their (current) WTP. The three ascending auctions A2 to A4 are multi-stage processes in which bidders, as the price increases, repeatedly have the opportunity to raise their bids or to leave the auction. We hypothesize that for private uncertain values the pseudo-endowment effect matters in the ascending auctions because ascending auctions promote a sense of psychological ownership of the item (lottery), i.e., pseudo-endowment, which in turn leads to reluctance to leave the auction. Bidders get attached to the item, and quitting the auction is experienced as losing the item to the extent that the bidder feels like owning it. That is, the aversion to ultimately giving up the item increases the WTP, which leads to higher bids (Proposition 2). We also hypothesize that under private uncertain values the effects of pseudo-endowment and loss aversion are stronger in ascending auctions in which a bidder can become the current high bidder, as in A3u and A4u. Being the high bidder intensifies the idea of owning the item. Being outbid then is experienced as losing the item, increasing the bidder’s WTP above the level prior to becoming the high bidder. The underlying mechanism can be increased loss awareness or loss aversion, evoked by losing the high bidder position, or a stronger shift of the reference point toward owning the item while filling the high-bidder position. Strahilevitz and Loewenstein (1998) state “It is not ownership per se, but awareness of ownership that causes reference point shifts.” We assume that the pseudo-endowment effect is additionally strengthened by the source-dependence effect if the bidder achieves the current high-bidder position by virtue of her own actions in A4u. The source-dependence effect goes back to Loewenstein and Issacharoff (1994), who find that people tend to attach higher value to an item that they receive because of good performance on some task than to an item that they receive by chance. In our setting, a bidder who becomes high bidder because she is the unique, first bidder to grasp the high bidder position, may feel stronger attached to the item

19 Ariely and Simonson (2003) argue similarly by supposing that bidders may develop a higher valuation for an item during an auction than they had before the auction.

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213

Table 3 Average auction prices (in ExCU). Treatment

A1u

A2u

A3u

A4u

Average price Median price

607.9 615

620.8 620

630.8 630

652.9 645

Table 4 Deviation of the auction price from the equilibrium price with risk-neutral bidders. Treatment

<

A1u A2u A3u A4u

5 3 3 0

Sum

= (41.7%) (25.0%) (25.0%) (0%)

11 (22.9%)

3 4 2 2

Sum

> (25.0%) (33.3%) (16.7%) (16.7%)

11 (22.9%)

4 5 7 10

(33.3%) (41.7%) (58.3%) (83.3%)

12 12 12 12

26 (54.2%)

48

compared to a setting, like A3, in which becoming the high bidder is the result of a lottery draw out of several bidders. The stronger attachment further increases the current WTP. These arguments, by Proposition 2, lead to the following hypothesis. Hypothesis 2. Auction prices increase from A1u to A4u. Summarizing, we hypothesize that bidders bid differently in strategically maximum bid equivalent auctions because of the pseudo-endowment effect and the source-dependence effect. However, we assume that these effects only occur in ascending auctions and if a lottery (i.e., an item with an uncertain value) is auctioned. 4. Experimental results We investigate Hypotheses 1 and 2 on auction fever: Do prices increase from A1u to A4u but not increase in the treatments A1c, A2c, A4c? In addition, we compare A1u, A2u, and A4u with A1c, A2c, and A4c as far as possible, considering the potential effects of risk aversion. In all tests, we apply a significance level of 5%. 4.1. Results of the uncertainty treatments A1u–A4u Because the two pricing rules a and b lead to similar prices,20 we take the liberty of pooling the a and b samples for Treatment A1u as well as for Treatment A2u. The results show a clear trend (see Table 3): the average auction price increases from treatment to treatment. In an overall comparison, the nonparametric Kruskal–Wallis test reveals differences between the four treatments, and the Jonckheere–Terpstra test supports the auction fever hypothesis that auction prices rise from Treatment A1u to A4u.21 The range of the average prices covers 45 ExCU, which amounts to 22.5% of the individual value intervals’ spread of 200 ExCU. Result 1. The auction prices increase significantly from A1u to A4u. We compare auction prices pairwise using the Tukey HSD (Honestly Significant Difference) test. A1u and A4u as well as A2u and A4u differ statistically significantly. Hence, the effect of auction dynamics alone appears to be rather small, whereas the strongest effect appears to result from giving bidders the possibility of becoming the current high bidder by their own effort. The ranked prices of all groups, which are listed in Table 11 in Appendix B, support these results. For example, four of the five highest prices belong to Treatment A4u. The highest-price A2u-group is ranked in the sixth position, and the highest-price A1u-group is ranked in the 13th position. In contrast, seven of the ten lowest prices belong to A1u and A2u. The impact of the auction format is also evident when comparing the auction prices with the two equilibrium prices 615 and 620 if bidders are risk-neutral utility maximizers. In A1u, five auctions end below and four above the risk-neutral prediction, whereas in A4u, the ratio is zero to ten (see Table 4).

20 Applying Mann–Whitney U -tests, we find no differences between the results of the a-groups and the b-groups in A1u and A2u: Treatment A1u: sample sizes: 6, 6; average prices: 611.67, 604.17; test statistics: 17.5; p-value > 0.5 (2-tailed). Treatment A2u: sample sizes: 6, 6; average prices: 627.50, 618.33; test statistics: 13.5; p-value > 0.4 (2-tailed). 21 Kruskal–Wallis one-way analysis of variance by ranks: test statistics: 11.577; p-value = 0.009. Jonckheere–Terpstra test (e.g., Hollander and Wolfe, 1973): test statistics: 615; asymptotically normally distributed test statistics: 3.379; p-value = 0.001.

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Table 5 Comparison of quitting bidders with active bidders: distributions over bidders of the same bidder type in the same auction format with higher/equal/lower number of having been high bidder than a bidder who quits the auction. Bidder to quit

A3u [in %]

A4u [in %]

>

=

<



>

=

<



B1 B2 B3

62 42 52

23 29 9

15 29 39

47 13 13

64 70 90

25 5 3

11 25 7

53 45 83

Avg. over all bidders

53

22

25

28

73

12

14

59

Table 6 Average auction prices (in ExCU). Treatment

A1c

A2c

A4c

Average price Median price

617.1 615

620.4 615

621.3 620

In the uncertain values setting, 54% of the auctions are won by bidder B3, which indicates heterogeneity among bidders (see Table 10). Although this share increases from 41.7% (A1u) to 66.7% (A4u), the χ 2 -test does not reveal significant differences between the four treatments.22 We do not observe irrational bidding in the sense that subjects bid above the upper bound of their values’ distribution. Affection by auction fever in the ascending auctions appears to differ between subjects. In A2u, only five subjects bid above their expected valuation (i.e., they bid more than the next price step). In A3u, this number increased to ten subjects and in A4u to 14 subjects. This amounts to 19%, 34%, and 45% of the subjects for whom we know whether they bid over the expected value, because for ten, seven, and five winners the auctions ended below their expected values. (See also Footnote 33.) Does assigning high bidder positions indeed impact exit decisions? If yes, a bidder may be less affected if she was high bidder less often. We investigate whether a bidder of a certain type in a certain auction format is more likely to quit the auction at a given price if she held the high bidder position less often. For the three bidder types B1, B2, and B3 and for the auctions A3u and A4u we separately count, at each price level at which a bidder is observed to quit, how many bidders of the same bidder type in the same auction format are still active (i.e., accept the price) and have been high bidder more/equally/less often than the quitting bidder (see Table 15). The averaged distributions per bidder type are given in Table 5. We test whether the difference between the percentage of bidders with higher and those with lower numbers of having been the high bidder is above zero. One-sample Wilcoxon signed rank tests reveal that the differences, which are on average 28% and 59% for A3u and A4u, both significantly differ from zero. A Mann–Whitney U -test does not reveal a significant difference between A3u and A4u.23 Result 2. Bidders that have held the high bidder position less often than comparable bidders in the same auction format are more likely to quit at given price levels in A3u and A4u. 4.2. Results of the certainty treatments A1c, A2c, and A4c Does auction fever occur within a typical private certain values framework? First, there is no evidence that the two pricing rules a and b, which we use for six groups each in both A1c and A2c, lead to different prices.24 Thus, we pool the samples with rules a and b in each of the Treatments A1c and A2c. Table 6 reflects the clear result: the average auction prices in the three treatments are close together, within a oneincrement interval, which includes the theoretically predicted result. There are no statistical differences.25 Thus, there is no evidence of auction fever.

22

χ 2 -test: degree of freedom: 6; test statistics: 9.20; p-value > 0.1. Wilcoxon signed rank test, A3u: sample size: 20; test statistics: 92.5; standardized: 1.859; p-value < 0.04 (1-tailed). Wilcoxon signed rank test, A4u: sample size: 10; test statistics: 36; standardized: 2.555; p-value < 0.01 (1-tailed). Mann–Whitney U -test, A3u vs. A4u: sample sizes: 20, 10; averages: 28, 59; test statistics: 67.5; p-value > 0.15 (2-tailed). 24 Applying Mann–Whitney U -tests, we find no differences between the results of the a-groups and the b-groups in A1c and A2c: Treatment A1c: sample sizes: 6, 6; average prices: 618.33, 615.83; test statistics: 10; p-value > 0.24 (2-tailed). Treatment A2c: sample sizes: 6, 6; average prices: 622.50, 618.33; test statistics: 10.5; p-value > 0.24 (2-tailed). 25 Kruskal–Wallis one-way analysis of variance by ranks: test statistics: 2.444; p-value = 0.295. 23

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215

Table 7 Deviation of the auction price from the equilibrium price. Treatment

<

=

>

Sum

A1c A2c A4c

4 (33.3%) 4 (33.3%) 0 (0%)

6 (50.0%) 6 (50.0%) 10 (83.3%)

2 (16.7%) 2 (16.7%) 2 (16.7%)

12 12 12

Sum

8 (22.2%)

22 (61.1%)

6 (16.7%)

36

Table 7 reveals that 22 of 36 auctions end according to the game-theoretical prediction, with a price of 615 or 620. Moreover, nine other auctions lead to a price of either 610 or 625. Hence, the results of 31 of 36 auctions are considered to be (approximately) in line with the theory of utility-maximizing bidders. This observation suggests that most of the subjects are guided by their individual valuation regardless of the auction format. This finding is also supported by Tables 12 and 13 in Appendix B. Note that 28 of 36 auctions (78%), much more than the 54% in the treatments with uncertain values, are won by bidders of type B3, as theoretically predicted. This applies particularly to the twelve A4c auctions, in which all but one auction was won by B3 and ten auctions ended “theoretically properly” (see also Table 7). The χ 2 -test, however, does not reveal significant differences in the distribution of prices in relation to the equilibrium price between the treatments.26 Individual bids suggest that most of the subjects adopt the weakly dominant strategy. In Treatment A1c, 15 of 36 subjects bid exactly in line with their valuation.27 Fourteen subjects bid less than their valuation and seven subjects bid more than their valuation, with four of these submitting a bid on the next price level above their valuation. In contrast to the aforementioned previous studies, we do not find systematic bidding above the private certain values; this outcome may be due to the different auction design (description of the auction as an ascending proxy auction and nondisclosure of the distribution of valuations to the subjects). The two ascending auctions also do not induce subjects to bid aggressively. In Treatment A2c, only two subjects exceed their values; in Treatment A4c, two of the three subjects who bid more than their value submit their last bid at the next price step above their value. Obviously, the auction format rarely seduces subjects to bid (far) above their values. These observations do not support auction fever hypotheses. However, they are consistent with the empirical hypothesis that English auctions guide bidders to execute their weakly dominant strategy more than other formats (e.g., Cox et al., 2002; Kagel, 1995). Result 3. With private certain values, there is no evidence that strategy-proof auction formats systematically induce bidders to bid more than their valuation. Hence, there is no evidence of auction fever if bidders are aware of their individual monetary valuations. Because subjects in the treatments with certain values predominantly bid according to their private valuations, we conclude that they understand the auction formats well. This finding supports ex post our decision to have each subject participate in only one auction such that she or he focuses on that auction. 4.3. Comparison of the results of the certainty with the uncertainty treatments In addition, we compare the results of the treatments A1c, A2c, and A4c with their uncertainty counterparts A1u, A2u, and A4u.28 Under format A1, we observe a higher average auction price in A1c than in A1u (617.08 vs. 607.92). Although this difference is not significant, it can be considered in line with the hypothesis of risk-averse decision makers, which predicts higher auction prices under certainty than under uncertainty (in the expected utility approach, as presented in Section 3.1, assuming the expected valuations and the certain valuations are equal). Auction A2 induces almost the same average result in both A2c and A2u (620.42 vs. 620.83). It appears that auction fever compensates for risk aversion, whereas the significant difference between A4c and A4u (621.25 vs. 652.92) indicates a dominance of auction fever over risk aversion in the English auction A4.

26

Table 7:

χ 2 -test: degree of freedom: 4; test statistics: 5.5; p-value > 0.2.

Table 12: χ 2 -test: degree of freedom: 4; test statistics: 5.5; p-value > 0.2. 27 In the groups of Treatment A1c under pricing rule a, a bid is considered in line with the valuation if it deviates by −2 from the valuation, whereas under pricing rule b, deviations of −2 or +3 count. 28 Pairwise comparison of the auction prices of A1c, A2c, and A4c with A1u, A2u, and A4u by means of the Mann–Whitney U -test: A1c and A1u: sample sizes: 12, 12; average prices: 617.08, 607.92; test statistics: 65.5; p-value > 0.7 (2-tailed). A2c and A2u: sample sizes: 12, 12; average prices: 620.42, 620.83; test statistics: 61.5; p-value > 0.5 (2-tailed). A4c and A4u: sample sizes: 12, 12; average prices: 621.25, 652.92; test statistics: 20.5; p-value < 0.01 (2-tailed).

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Table 8 Bidders’ private information about their individual valuations in Treatment P1u. Bidder

Valuation interval

Expected valuation

B1 B2 B3

U [112, 312] U [117, 317] U [122, 322]

212 217 222

5. A procurement auction to capture the “real” endowment effect As a further indication for the hypothesis of reference point shifts, we conduct a procurement auction in which bidders own the good for sale. This maximum reference-point shift toward owning the item may cause a real endowment effect. 5.1. Treatment P1u We assume that the course of the ascending auctions induces a pseudo-endowment effect, such that a person’s intrinsic WTP is only measured in the one-shot auctions of Treatment A1u. Hence, in order to record the WTA, we design the strategy-proof procurement counterpart P1u of A1u. Auction format P1. In auction P1, bidders may sell a virtual item to the auctioneer. Bidders are asked to submit their lower bidding limit once. A bidding mechanism then runs the bids against each other, starting from a high price and decreasing it by the constant decrement. Bidders face a second-price procurement auction in which the bidder with the lowest bid sells her item and is paid the second-lowest bid.29 In P1u (as in A1u), bidders have incomplete information about their private valuation for the virtual good. The distributions of subjects’ valuations are given in Table 8 and they are chosen such that P1u is comparable to A1u (for a detailed argument, see Appendix A.2). Since we conduct a sealed-bid auction only, we need not be concerned about the differences between the values in P1u, and we decided on intervals whose expected values are different but close. These values promise comparable payoffs (approximately 200 ExCU) as the sales auctions. At the end of the auction, the winner sells her good at the auction price, and the other bidders’ valuations are drawn from their individual distributions. Because the bidders are endowed with the good, there is no additional lump-sum payment. We ran twelve auctions (groups) in Treatment P1u, which was conducted in two sessions each with six groups (18 subjects). Thus, 36 subjects participated in Treatment P1u. 5.2. Hypothesis An expected-utility maximizer submits a bid b so that her certainty equivalent of owning the good equals her wealth after selling it at b, i.e., she bids her WTA. We use the observed deviation from expected valuations, b = b − E [ V ], to compare bids in A1u and P1u. Denote the deviations in the two auctions by bA and bP . For risk-neutral or constant absolute risk-averse expected-utility maximizers bP = bA . A risk-averse (risk-loving) bidder’s bP and bA are negative (positive). However, a bidder affected by the endowment effect has a WTA that is higher than her WTP, i.e., bP > bA .

Hypothesis 3. bP in P1u is higher than bA in A1u. That is, subjects submit higher bids (relative to the expected value of the item) in the procurement auction P1u than in the corresponding sales auction A1u. 5.3. Comparison of the results of P1u with A1u–A4u First, we compare the prices of P1u with those of Treatment A1u.30 For each bidder, we compute b = b − E [ V ] and summarize the results in Table 9.31 Applying the Mann–Whitney U -test leads to the rejection of the test hypothesis of equal b in A1u and P1u.32

29 Because we did not find any differences between the groups under pricing rule a and pricing rule b in the sales auctions (see Section 2.2 for the description and Appendix A.1 for the motivation), we decided to implement only the second-price rule b in Treatment P1u. 30 Table 14 in Appendix B lists the results of the groups of P1u. 31 Computing b makes treatments comparable. We prefer E [ V ] over other points of comparison, e.g., the upper or lower bound of the interval, because b = b − E [ V ] describes the deviation from the risk-neutral expected-utility maximizers’ equilibrium bid. 32 A1u and P1u: sample sizes: 36, 36; average (median) deviations: −17.56 (−14.5), 20.78 (15.5); test statistics: 376; asymptotically normally distributed test statistics: 3.07; p-value < 0.01. (2-tailed).

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217

Table 9 Deviation b = b − E [ V ] of all 36 bidders in A1u and P1u and of the twelve decisive bidders in A1u–A4u and P1u (#: number of). Treatment

# observations

# (b < 0)

# (b > 0)

Mean b

Median b

A1u P1u

36 36

20 12

16 24

−17.6 20.8

−14.5 15.5

A1u A2u A3u A4u P1u

12 12 12 12 12

7 7 7 4 3

5 5 5 8 9

−24.3 −8.3 −4.9 24.7 18.4

−24.5 − 2 .0 − 2 .0 13.0 18.0

Result 4. Average deviations of subjects’ bids from their expected valuations differ significantly between A1u and P1u. In the A1u sales auctions, subjects tend to bid below E [ V ], whereas in the P1u procurement auctions, they predominantly bid above E [ V ]. We conclude that WTP < WTA. In addition, we compare b for treatments A1u–A4u and P1u. In A4u, we can only observe the dropout price of the twelve decisive bidders who determine the auction price. For comparison, we therefore consider only the twelve pricedetermining bidders in all treatments. Their average b is shown in the lower part of Table 9. Again, we observe a clear trend: the average and median deviations increase from negative values in A1u to positive values in A4u,33 where they reach the level of P1u.34 Moreover, in A1u, A2u, and A3u only five of twelve decisive bidders exceed their expected valuations, whereas eight decisive bidders do so in A4u and nine in P1u. With regard to WTP–WTA considerations, we interpret the observed increase in decisive bidders’ b as an increase in their WTP from A1u to A4u, reaching the level of the decisive bidders’ WTA in P1u. This suggests that the pseudoendowment effect for our decisive bidders in A4u is about as strong as the endowment effect. 6. Conclusion Our results shed light on potential revenue differences of auction designs. Although the auctions in our experiment are identical with respect to (maximum) bids in weakly dominant strategies and with respect to outcomes (winning bidder and price) under the standard model with expected-utility-maximizing bidders, we find evidence for a revenue ranking, which we attribute to loss aversion and the different impacts of auction formats on the development of a sense of ownership during the auction. We conclude that reference-dependent preferences may affect the revenue ranking of auction formats. In particular, our results indicate that repeated exposure to pseudo-losses by having to decide about giving up the item and by losing the high-bidder position lead to higher bids. The effect is stronger, if the high-bidder achieves his position by own action (in addition to the action necessary to stay an active bidder), for example, by being the first to accept a price level as in A4. A seller interested in high revenue might therefore prefer ascending auctions with perceptible exit decisions and competitive high-bidder determination. None of these effects occurs in an environment with private certain valuations. Thus, our results support the view that auction fever occurs particularly with items for which the WTP–WTA disparity has been observed, lotteries and real goods. Another interesting view on pseudo-endowment in auctions arises from recent models on bidders with reference˝ dependent preferences and expectations-based endogenous reference points (Koszegi and Rabin, 2006, 2007; Lange and Ratan, 2010; Belica and Ehrhart, 2014; Ehrhart and Ott, 2014). In these models, in equilibrium a bidder’s reference-point depends on her own strategies and her beliefs about others’ strategies. The pseudo-endowment effect matters if bidders evaluate gains and losses relative to the reference point separately for the item for sale and for money. If item and money are evaluated in one (monetary rent) dimension – which is the model that fits our setting with certain values (Lange and Ratan, 2010) – auction fever does not occur and bidders bid truthfully (or do not participate). Our results should be especially valuable for auctioneers who are interested in raising their revenue. We advise bidders in a dynamic auction to be aware of the possible change in their WTP before deciding to participate in an auction and to consider this effect when developing a bidding strategy. In contrast, it is possible that in a sealed-bid auction, a bidder will begin to think about owning the item only after she has submitted her bid, and she may realize that she has bid too low when it is already too late to raise her bid. Acknowledgments Financial support from the Deutsche Forschungsgemeinschaft, SFB 504, at the University of Mannheim, is gratefully acknowledged. We thank Giulio Bottazzi, Clemens Puppe, Reinhard Selten, Muriel Niederle, Thomas Kittsteiner, and seminar

33

Jonckheere–Terpstra test: test statistics: 602; asymptotically normally distributed test statistics: 2.95; p-value < 0.01. Comparison of the decisive bidders’ b in A4u and P1u by means of the Mann–Whitney U -test: sample sizes: 12, 12; average (median) deviations: 24.67 (13), 18.42 (18); test statistics: 69.5; p-value > 0.8 (2-tailed). 34

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participants at Stanford University, Université de Strasbourg, Karlsruhe Institute of Technology, and RWTH Aachen University for valuable feedback and Thomas Gottfried for programming excellent experimental software. We thank two anonymous referees and an advisory editor for helpful suggestions. Appendix A. Equilibria with expected-utility maximizers In the following we prove that a bidder bids (up to) her WTP in all four auction formats A1–A4. In addition, we prove comparability of sales and procurement auctions. A.1. Equivalence of A1–A4 Proof of Proposition 1. Consider an auction in which one item is auctioned among n bidders, whose wealth positions before the auction are given by w 1 , . . . , w n . Let u i : R → R denote bidder i’s von Neumann–Morgenstern increasing utility functions and v i her valuation of the item, i = 1, . . . , n. Bidders’ valuations are private but may be uncertain. Bidder i’s uncertainty about her valuation is modeled by the random variable V i with distribution F i and support [ v i , v¯ i ] ⊂ R. In a private values setting, F i has the following properties: F i is known to i and fully captures the value of the item to i at the time of the auction. That is, there is no information held by other bidders or gained throughout the auction that, if known to i, would change F i . The process by which the values v i or V i are determined is not decisive for the analysis (e.g., values may be independent or correlated). The distribution G ki (x) = Pr{max j =i b j ≤ x} summarizes bidder i’s beliefs about her opponents’ highest bids in the auction

of type k ∈ { A1, A2, A3, A4}.35 Bidder i may update G ki (·) in the course of A2, A3, and A4. G ki may depend on F i , but, by the assumption of private values, F i does not depend on G ki . The lowest feasible bid is denoted by bmin with bmin < v i . Bidder i’s strategy b i ∈ [bmin , ∞) in a sealed-bid or an ascending clock auction is characterized by her bid in A1 or her bidding limit in A2, A3, and A4. In what follows, we skip the index i because it applies to every parameter. Bidder i’s objective is to choose b to maximize her expected utility

b v¯ U (b) =

u ( w + v − r ) dF ( v ) dG k (r ) + (1 − G k (b))u ( w ).

bmin v

The first-order condition ∂ U (b)/∂ b = 0 gives the following equation for the optimal bid b∗ :

v¯

u ( w + v − b∗ ) dF ( v ) = u ( w ).

(A.1)

v

 v¯

The second-order condition ∂ 2 U (b)/∂ b2 = − v u ( w + v − b∗ ) dF ( v ) < 0 is fulfilled for all increasing utility functions. Note that the solution does not depend on i’s belief about her opponents. Thus, updating the beliefs in the course of an auction has no impact on the optimal bidding limit b∗ . Moreover, b∗ constitutes a weakly dominant strategy (with respect to expected utility). For example, a risk-neutral bidder, whose utility is given by u (x) = x, bids her expected valuation

 v¯

b∗ = v v dF ( v ) = E [ V ]. Thus, in the uncertain values setting, a bidder’s expected utility is maximized by the same b in all auction formats k ∈ { A1, A2, A3, A4}. The optimal b equals her WTP, which is given by her certainty equivalent at wealth w. With certain values, where a bidder’s WTP is given by v, condition (A.1) becomes u ( w + v − b∗ ) = u ( w ), which leads to the well-known solution of truthfully bidding b∗ = v for all k ∈ { A1, A2, A3, A4}. This weakly dominant strategy is independent of u (·) and, thus, the bidder’s risk attitude. Proof of Proposition 2. The proof of Proposition 1 does not depend on how the bidder’s WTP is constructed (e.g., via a reference-dependent utility function), as long as this construction is independent of her beliefs about the opponents’ strategies. Falling prone to the pseudo-endowment effect during an auction means to adjust the WTP upwards, such that bidding up to the current WTP is a feasible strategy throughout an ascending auction. At any given price level in the auction, according to the proof of Proposition 1, it is weakly dominant to bid up to the current WTP. Impact of the price increment. The price increment of 5 ExCU does not have an impact on rational bidding behavior in A3 and A4; however, it may have an impact in A1 and A2 (and similarly in P1). In A3 and A4, a bidder’s optimal bidding limit b∗ (derived from (A.1)) determines her benchmark, independent of the increment. That is, the bidder accepts every

35

For example, with independent certain private values drawn from a distribution H and opponents’ symmetric bidding functions β(·), G ki (x) =

Pr{max j =i β( v j ) ≤ x} = Pr{max j =i v j ≤ β −1 (x)} = H n−1 (β −1 (x)).

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price below b∗ and quits the auction when the price rises beyond b∗ . Because the certain values 612, 617, and 652 lie between two auction prices, bidders quit the auction at prices of 615, 620, and 655. The argument is similar when values are uncertain. However, the rules of A1 and A2 may induce bidders to deviate slightly from b∗ . This incentive is caused by the tie-breaking rule of randomly choosing the winner. Consequently, if the winner must pay the last announced price (pricing rule a), she may have an incentive to stop bidding at a price level below her valuation to evoke a tie and avoid the price increase. In contrast, if the unique highest bidder must pay the penultimate price (pricing rule b), she may have an incentive to accept the closest price level above her valuation to escape a tie.36 Although we expect these effects to be rather small, we establish both pricing rules in treatments A1c and A2c as well as in A1u and A2u by running half of the groups under pricing rule a and the other six groups under pricing rule b to control and compensate for these effects. A.2. Comparability of P1 Small WTP–WTA differences can be explained by wealth and income effects. The WTP–WTA disparity refers to the observation that the difference is higher than the difference those effects would be likely to cause (Knetsch and Sinden, 1984). We choose the lump-sum payments such that WTP–WTA differences due to income or wealth effects can be expected to be low or non-existent. In the procurement setting, bmax denotes the maximum possible bid, and H (·) denotes the distribution of bidder i’s beliefs about her opponents’ lowest bid. To determine her optimal bid b∗ , bidder i maximizes her expected utility

v¯ U (b) = H (b)

b max

u ( w + v ) dF ( v ) +

u ( w + r ) dH (r ).

v

b

The first-order condition ∂ U (b)/∂ b = 0 leads to

v¯

u ( w + v ) dF ( v ) = u ( w + b∗ ).

(A.2)

v

The second-order condition ∂ 2 U (b)/∂ b2 = −u ( w + b∗ ) < 0 is fulfilled for all increasing utility functions. According to the FOCs (A.1) and (A.2), a risk-neutral bidder (with a linear utility function u (·)) or a constant absolute risk-averse bidder (with CARA utility function u (x) = 1 − e −α x , α > 0) submits the same bid in P1u as in A1u–A4u. The bid,

 v¯ ∗ given by b∗ = E [ V ] for a risk-neutral bidder and by v e −α ( v −b ) dF ( v ) = 1 for a CARA bidder, is in all auctions independent of w. Shifting [ v , v¯ ] to [ v + z, v¯ + z] for some z results in a shift of b∗ to b∗∗ = b∗ + z.37 In P1u, as in A1u–A4u, a risk-averse

 v¯

bidder bids less than her expected value because v u ( w + v ) dF ( v ) < u ( w + E [ V ]), and the reverse holds for a risk-loving bidder. Note that in (A.2), compared to (A.1), all arguments in u (·) are shifted by b. In the experiment, the lump-sum payment of 200 ExCU in the A-treatments adjusts w such that the A- and P -treatments are more comparable even for non-CARA bidders. Let w denote the wealth before the P -auction P1u. Thus, the wealth in the A-auctions A1u–A4u is w + 200. Assuming a risk-averse or risk-loving bidder, replacing b∗ in equation (A.1) by E [ V A ] + bA and in equation (A.2) by E [ V P ] +

bP , the equations become u ( w + E [ V ] + bP ). For  300 equations are 100 u ( w A P P

 v¯ A +200− E [ V A ] v A +200− E [ V A ]

u ( w + v − bA )dF A ( v − 200 + E [ V A ]) = u ( w + 200) and

¯A

 v¯ P vP

u ( w + v )dF P ( v ) =

¯P

example, for bidder B2 with v ∈ [ v , v ] = [517, 717] in A and v ∈ [ v , v ] = [117, 317] in P , these  317 + v − bA )dF A ( v + 417) = u ( w + 200) and 117 u ( w + v )dF P ( v ) = u ( w + 217 + bP ). Thus, since A

P

F and F are both uniform distributions on an interval of width 200, even for risk-averse or risk-loving expected-utility maximizers whose risk-attitude changes with wealth, bA and bP are approximately equal.

36 The incentives to deviate from truthful bids are caused by the price increments, the location of values between price steps, and the probability of ties. Consider a bidder with value v and a sequence of increasing auction prices p 0 , p 1 , p 2 , . . . , pt −1 , pt , pt +1 , . . . with pt < v < pt +1 . Pricing rule a: Since the bidder has to pay her winning bid, she has no incentive to bid more than pt . Let qt denote the probability that the bidder wins at pt (including tie-breaking) if she bids pt . The bidder prefers to bid pt −1 over pt if qt −1 ( v − pt −1 ) > qt ( v − pt ). If this is true, the bidder has to compare pt −2 and pt −1 in the same way and so on. Pricing rule b: The bidder has no incentive to bid less than pt or more than pt +1 . Let qtT and qtN denote the probabilities that the auction ends at pt and the bidder wins by tie-breaking and not by tie-breaking. The bidder prefers the bid pt +1 to the bid pt if qtT+1 ( v − pt +1 ) + qtN+1 ( v − pt ) > qtT ( v − pt ) ⇐⇒

qtT+1 ( v − pt +1 ) + (qtN+1 − qtT )( v − pt ) > 0, otherwise she prefers to bid pt . For similar considerations, see Mathews and Sengupta (2008) and Cramton and Sujarittanonta (2010). 37

(A.1) and (A.2) are equal for CARA utility:

 v¯ v

 v¯  v¯ ∗ ∗ 1 − e −α ( w + v −b ) dF ( v ) = 1 − e −α w ⇔ v e −α ( v −b ) dF ( v ) = 1 ⇔ v 1 − e −α ( w + v ) dF ( v ) = 1 −

 v¯ +z  v¯ ∗ ∗ ∗∗ e −α ( w +b ) . Shifting [ v , v¯ ] to [ v + z, v¯ + z] results in a shift of b∗ to b∗∗ = b∗ + z : v e −α ( v −b ) dF ( v ) = 1 shifts to v +z e −α ( v −b ) dF ( v − z) = 1 ⇔  v¯ v

 v¯ ∗∗ ∗∗ e −α (( v +z)−b ) dF ( v ) = 1 ⇔ v e −α ( v −(b −z)) dF ( v ) = 1.

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Appendix B. Ranked auction results and distributions of winning bidders

Table 10 Distribution of winning bidders in the treatments A1u–A4u. Treatment

B1

B2

A1u A2u A3u A4u

5 0 2 1

2 5 4 3

Sum

8 (16.7%)

(41.7%) (0.0%) (16.7%) (8.3%)

B3

(16.7%) (41.7%) (33.3%) (25.0%)

14 (29.2%)

5 7 6 8

Sum

(41.7%) (58.3%) (50.0%) (66.7%)

12 12 12 12

26 (54.2%)

48

Table 11 Ranked auction results of all groups in the treatments A1u–A4u. Rank 1 2 3 4 5 6 8 9 10

13

16

19 21

27

33

38 39

42 43 45 46 47 48

Price [ExCU]

Treatment

Group number

Winning bidder

710 700 695 690 670 660 660 655 655 650 650 650 640 640 640 635 635 635 630 630 625 625 625 625 625 625 620 620 620 620 620 620 615 615 615 615 615 610 605 605 605 600 590 590 585 575 560 550

A4u A4u A3u A4u A4u A2u A3u A3u A4u A2u A3u A4u A1u A2u A4u A2u A3u A4u A3u A3u A1u A1u A1u A2u A4u A4u A1u A2u A2u A3u A3u A4u A1u A1u A2u A2u A4u A1u A1u A2u A2u A3u A1u A3u A3u A1u A2u A1u

4 10 4 1 6 1 10 1 9 12 7 11 11 5 5 8 5 7 11 12 5 6 9 3 3 12 10 2 9 2 3 8 4 8 4 10 2 1 3 6 7 9 2 8 6 7 11 12

B3 B3 B2 B3 B1 B2 B3 B1 B2 B3 B3 B3 B2 B2 B2 B3 B3 B3 B3 B2 B3 B3 B3 B3 B3 B3 B1 B3 B3 B3 B2 B2 B1 B1 B2 B3 B3 B2 B1 B3 B2 B3 B3 B1 B2 B3 B2 B1

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Table 12 Distribution of winning bidders in the treatments A1c, A2c, A4c. Treatment

B1

B2

B3

Sum

A1c A2c A4c

2 (16.7%) 3 (25.0%) 1 (8.3%)

2 (16.7%) 0 (0%) 0 (0%)

8 (66.7%) 9 (75.0%) 11 (91.7%)

12 12 12

Sum

6 (16.7%)

2 (5.6%)

28 (77.8%)

36

Table 13 Ranked auction results of all groups in the treatments A1c, A2c, A4c. Rank 1 3 5 7

17

29

36

Price [ExCU]

Treatment

Group number

Winning bidder

655 655 650 650 625 625 620 620 620 620 620 620 620 620 620 620 615 615 615 615 615 615 615 615 615 615 615 615 610 610 610 610 610 610 610 595

A2c A4c A1c A2c A1c A4c A1c A1c A1c A2c A2c A4c A4c A4c A4c A4c A1c A1c A1c A2c A2c A2c A2c A4c A4c A4c A4c A4c A1c A1c A1c A2c A2c A2c A2c A1c

1 11 7 12 2 4 3 1 5 3 6 1 2 3 5 9 6 9 10 4 5 9 10 6 7 8 10 12 8 4 11 2 7 8 11 12

B1 B1 B2 B1 B3 B3 B1 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B3 B2 B3 B3 B1 B3 B3 B3 B1

Table 14 Ranked auction results of all groups in the treatment P1u. Rank 1 2 3 4 5 6 7 9 10 11

Price [ExCU]

Group number

Winning bidder

285 275 255 250 240 235 225 225 220 210 200 200

2 6 9 12 11 1 7 10 5 3 4 8

B1 B2 B2 B1 B1 B3 B1 B2 B3 B2 B3 B1

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Table 15 Comparison of quitting bidders with active bidders: for each observed quitting bidder, how many of the active bidders of the same type in the same auction format have been high bidder more often/equally often/less often? A3u Bidder to quit

B1 B1 B1 B1 B1 B1 B1 B1 B1 B2 B2 B2 B2 B2 B2 B2 B3 B3 B3 B3

# Others

Percentage [%]

Difference

>

=

<

>

=

<

[%]> − [%]<

6 9 1 2 3 3 4 2 2 8 1 0 1 5 0 1 11 2 3 3

4 1 5 3 3 1 0 0 0 3 0 0 6 0 5 0 0 1 2 0

1 0 3 2 1 1 0 1 0 0 8 8 1 0 0 0 0 7 3 3

55 90 11 29 43 60 100 67 100 73 11 0 13 100 0 100 100 20 38 50

36 10 56 43 43 20 0 0 0 27 0 0 75 0 100 0 0 10 25 0

9 0 33 29 14 20 0 33 0 0 89 100 13 0 0 0 0 70 38 50

45 90 −22 0 29 40 100 33 100 73 −78 −100 0 100 0 100 100 −50 0 0

53

22

25

28

Avg. A4u Bidder to quit

B1 B1 B1 B1 B2 B2 B2 B3 B3 B3

# Others

Percentage [%]

Difference

>

=

<

>

=

<

[%]> − [%]<

4 3 2 0 3 4 2 7 5 3

0 0 0 1 1 0 0 1 0 0

1 1 0 0 3 0 1 2 0 0

80 75 100 0 43 4 67 70 100 100

0 0 0 100 14 100 0 10 0 0

20 25 0 0 43 0 33 20 0 0

60 50 100 0 0 0 33 50 100 100

73

12

14

59

Avg.

Appendix C. Translated instructions Instructions for all treatments consist of two pages each. The instructions for treatments A1u–A4u, as well as those for treatments A1c, A2c, and A4c, share a common first page. Treatments A1u and A1c, A2u and A2c, and A4u and A4c, all have the same second page. In treatments A1u, A2u, A1c, and A2c, we distinguished between two pricing rules, so in these cases we have two variants of the second page of the instructions. In Treatment P1u we conduct a procurement auction and thus the instructions differ from those of the other treatments. C.1. First page of treatments A1u–A4u Instructions You are going to participate in an experiment on auctions. In this experiment, you will make your decisions as a bidder at your computer terminal, isolated from the other participants. In the auction you may earn money in cash. How much you earn depends on your decisions and on the decisions of the other participants. The monetary units in the experiment are called currency units (CU). Point of Departure Imagine that you are the owner of a ship that offers cruises. This is why you are participating in an auction in which the cruise ship “One World” will be auctioned once. Besides you, 2 other bidders will participate in this auction. If you purchase the ship through the auction, its value W will result from its use in your fleet. However, calculations show that the value of the ship for you, W, lies uniformly distributed between W0 and W1 . This means that the ship has a value for you of at least W0 and at most W1 , where all values (integers) from W0 to W1 have equal probability.

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Please note: the boundaries W0 and W1 are different for every bidder. Your individual boundaries W0 and W1 will be communicated to you on your screen directly before the auction; the boundaries of the other bidders, however, are unknown to you. In addition, you have a lump-sum payment F of 200 CU at your disposal. Payoff 1. If you are awarded the ship for price P, the value of the ship for you will be determined immediately after the auction by drawing a value W from the uniform distribution over W0 to W1 . Your profit from the auction will then be calculated as:

G=W−P Please note: if you pay more for the ship than it is worth to you, that is, if P > W, then your profit from the auction will be negative, that is, G < 0. 2. If you are not awarded the ship, your profit from the action equals zero, i.e.:

G=0 The payoff you receive is your lump-sum payment F plus your profit G from the auction, i.e.:

Payoff = F + G Your payoff will be converted into Euro and paid to you in cash at the end of the experiment (where 1 CU corresponds to 5 Euro Cents). The payment will be individual and anonymous. C.2. First page of treatments A1c, A2c, and A4c Instructions You are going to participate in an experiment on auctions. In this experiment, you will make your decisions as a bidder at your computer terminal, isolated from the other participants. In the auction you may earn money in cash. How much you earn depends on your decisions and on the decisions of the other participants. The monetary units in the experiment are called currency units (CU). Point of Departure Imagine that you are the owner of a ship that offers cruises. This is why you are participating in an auction in which the cruise ship “One World” will be auctioned once. Besides you, 2 other bidders will participate in this auction. If you purchase the ship through the auction, its value W will result from its use in your fleet. The value W for you will be announced to you at your screen directly before the auction. Please note: the value W is different for every bidder, and you do not know the value for the other bidders, and they do not know its value for you. In addition, you have a lump-sum payment F of 200 CU at your disposal. Payoff 1. If you are awarded the ship for price P, your profit from the auction, G, is calculated using your value W as:

G=W−P Please note: if you pay more for the ship than it is worth to you, that is, if P > W, then your profit from the auction will be negative, that is, G < 0. 2. If you are not awarded the ship, your profit from the action equals zero, i.e.:

G=0 The payoff you receive is your lump-sum payment F plus your profit G from the auction, i.e.:

Payoff = F + G Your payoff will be converted into Euro and paid to you in cash at the end of the experiment (where 1 CU corresponds to 5 Euro Cents). The payment will be individual and anonymous. C.3. Second page of treatments A1u and A1c, pricing rule a Auction The auction by which the ship is to be sold has the following rules. You submit a bid B at the beginning of the auction exactly once. Your bid expresses the maximum CU you are willing to pay for the ship.

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The minimum bid Bmin for the ship is 500 CU. That means that you have to bid at least 500 CU; that is, B ≥ Bmin = 500. The bid must be a multiple of 5 CU, that is, B = 500, 505, 510, 515, 520, . . . When submitting your bid, you do not know the bids of the other bidders, and they do not know your bid. When all 3 bidders have submitted their bids, the automatic bidding process begins, which places the bids against each other and thereby determines which bidder is awarded the ship. In this auction, the price starts at the minimum bid of Bmin = 500. The auction price is gradually increased by 5 CU at any one time. If a bid is exceeded by the auction price the bid quits the auction. The bidding process may end in two ways, depending on the bids: 1. If the highest bid and the second highest bid differ, the bidding process stops when the auction price exceeds the second highest bid. We denote the remaining highest bid B∗ and the price at which the bidding process stops P where P equals the second highest bid plus 5 CU. The bidder who submitted bid B∗ is awarded the ship and must pay price P. In this case P ≤ B∗ . This means that the bidder must pay his bid B∗ or less, depending on the price P at which the bidding process stops. 2. If at least two bidders have submitted the same bid B∗ which is the highest bid, the bidding process stops at the price P = B∗ . One of these bidders is then randomly selected to be awarded the ship at the price P = B∗ . In this case the bidder who is awarded the ship has to pay his bid. Please note: in this auction you are only allowed to submit a bid once, and it may not be changed afterwards! Before the auction begins, you will be asked some questions on the screen concerning the rules. This is to ensure that all participants have understood the instructions. C.4. Second page of treatments A1u and A1c, pricing rule b Auction The auction by which the ship is to be sold has the following rules. You submit a bid B at the beginning of the auction exactly once. Your bid expresses the maximum CU you are willing to pay for the ship. The minimum bid Bmin for the ship is 500 CU. That means that you have to bid at least 500 CU; that is, B ≥ Bmin = 500. The bid must be a multiple of 5 CU, that is, B = 500, 505, 510, 515, 520, . . . When submitting your bid, you do not know the bids of the other bidders, and they do not know your bid. When all 3 bidders have submitted their bids, the automatic bidding process begins, which places the bids against each other and thereby determines which bidder is awarded the ship. In this auction, the price starts at the minimum bid of Bmin = 500. The auction price is gradually increased by 5 CU at any one time. If a bid is exceeded by the auction price the bid quits the auction. The bidding process may end in two ways, depending on the bids: 1. If the highest bid and the second highest bid differ, the bidding process stops when the auction price exceeds the second highest bid. We denote the remaining highest bid B∗ and the price at which the bidding process stops P where P equals the second highest bid. The bidder who submitted bid B∗ is awarded the ship and must pay price P. In this case P < B∗ . That means that the bidder pays less than his bid B∗ . 2. If at least two bidders have submitted the same bid B∗ which is the highest bid, the bidding process stops at the price P = B∗ . One of these bidders is then randomly selected to be awarded the ship at the price P = B∗ . In this case the bidder who is awarded the ship has to pay his bid. Please note: in this auction you are only allowed to submit a bid once, and it may not be changed afterwards! Before the auction begins, you will be asked some questions on the screen concerning the rules. This is to ensure that all participants have understood the instructions. C.5. Second page of treatments A2u and A2c, pricing rule a Auction The auction by which the ship is to be sold has the following rules. The auction starts with a price of 500 CU. You will be asked on your screen if you are willing to pay this price for the ship. If you are, click on the OK-button on your screen or press the enter key. You have 50 seconds to do this. If you are not willing to pay this price, do nothing until the 50 seconds have passed. In this case, you automatically quit the auction. On your screen you can always see how many seconds are left until the 50 seconds expire. Every bidder makes his decision without knowing the decisions of the other two bidders. If at least two bidders are willing to pay the price of 500 CU, the auction price is raised by 5 CU. The bidders who are still in the auction then have another 50 seconds to decide whether they are willing to pay 505 CU for the ship. If once again at least two bidders accept, the auction price will be raised by another 5 CU, and so on. The bidding process may end in two ways, depending on the bids:

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1. The auction price is raised stepwise by 5 CU until at price P only one bidder is still in the auction. That is, there is only one bidder who is willing to buy the ship at the actual auction price P. This bidder is awarded the ship and has to pay price P. 2. The auction price is raised stepwise by 5 CU until at price P at least two bidders are still in the auction, who, however, all quit at the next price P + 5 CU. One of these bidders is randomly selected to be awarded the ship for price P. Before the auction begins, you will be asked some questions on the screen concerning the rules. This is to ensure that all participants have understood the instructions. C.6. Second page of treatments A2u and A2c, pricing rule b Auction The auction by which the ship is to be sold has the following rules. The auction starts with a price of 500 CU. You will be asked on your screen if you are willing to pay this price for the ship. If you are, click on the OK-button on your screen or press the enter key. You have 50 seconds to do this. If you are not willing to pay this price, do nothing until the 50 seconds have passed. In this case, you automatically quit the auction. On your screen you can always see how many seconds are left until the 50 seconds expire. Every bidder makes his decision without knowing the decisions of the other two bidders. If at least two bidders are willing to pay the price of 500 CU, the auction price is raised by 5 CU. The bidders who are still in the auction then have another 50 seconds to decide whether they are willing to pay 505 CU for the ship. If once again at least two bidders accept, the auction price will be raised by another 5 CU. The auction price is raised stepwise by 5 CU until at price P at least two bidders are still in the auction, but at the next price P + 5 CU only one bidder or no bidder is left. That is, at price P + 5 CU either all or all but one bidders quit. The bidding process stops and the high-bidder or one of the high-bidders, if there are more than one, is then awarded the ship. The price for the ship in both cases equals P CU. 1. If there is only one high-bidder, that is, if at price P + 5 CU all bidders but one quit, the remaining bidder is awarded the ship and has to pay the price P. 2. If there are several high-bidders who all quit at price P + 5 CU, one of these is randomly selected to be awarded the ship for price P. Before the auction begins, you will be asked some questions on the screen concerning the rules. This is to ensure that all participants have understood the instructions. C.7. Second page of treatments A3u and A3c Auction The auction by which the ship is to be sold has the following rules. The auction starts with a price of 500 CU. You will be asked on your screen if you are willing to pay this price for the ship. If you are, click on the OK-button on your screen or press the enter key. You have 50 seconds to do this. If you are not willing to pay this price, do nothing until the 50 seconds have passed. In this case, you automatically quit the auction. On your screen you can always see how many seconds are left until the 50 seconds expire. Every bidder makes his decision without knowing the decisions of the other two bidders. If two or more bidders are willing to pay the price of 500 CU, one of these bidders is randomly selected to be current high-bidder, to whom we refer as HB500. You are always informed if you are the current high-bidder or not. The auction price is then raised by 5 CU. The bidders who are still in the auction then have 50 seconds to overbid the current high-bidder HB500 by accepting the auction price of 505 CU. The current high-bidder HB500 is at this point not allowed to bid, but he remains in the auction as a matter of course. If at the price of 505 CU no bidder signals his willingness to buy the ship, the current high-bidder HB500 is awarded the ship and has to pay 500 CU. If one or several bidders signal their willingness to pay 505 CU, one among these is randomly selected as new current high-bidder HB505. The auction price is then raised again by 5 CU to 510 CU. The bidders who remained in the auction, including the former high-bidder HB500, then once again have 50 seconds to overbid the current high-bidder HB505 by accepting the auction price of 510 CU. If at least one bidder accepts the auction price of 510 CU, the auction price then increases to 515 CU, and so on. Before the auction begins, you will be asked some questions on the screen concerning the rules. This is to ensure that all participants have understood the instructions. C.8. Second page of treatments A4u and A4c Auction The auction by which the ship is to be sold has the following rules. The auction starts with a price of 500 CU. This price is displayed on your screen for 5 seconds. You will then be asked if you are willing to pay this price for the ship. If you are, click on the OK-button on your screen or press the enter key. You have 45 seconds to do this. On your screen you can always see how many seconds are left until the 45 seconds expire.

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The bidder who accepts the price of 500 CU first becomes current high-bidder. We refer to this bidder as HB500. The other bidders may no longer accept the price of 500 CU. You are always informed if you are the current high-bidder or not. The auction price is then raised by 5 CU, even if the 45 seconds have not expired, and the new state of the auction is shown on your screen for 5 seconds. The bidders then have up to 45 seconds to overbid the current high-bidder HB500 by accepting the auction price of 505 CU. The current high-bidder HB500 is at this point not allowed to bid. If at the price of 505 CU no bidder signals his willingness to buy the ship, the current high-bidder HB500 is awarded the ship and has to pay 500 CU. If a bidder signals his willingness to pay 505 CU, this bidder becomes new current high-bidder HB505. The auction price is then raised again by 5 CU to 510 CU. Again all bidders but HB505 can bid at this price, and the one who accepts first becomes new high-bidder. If, on the other hand, no bidder accepts at 510 CU, the current high-bidder HB505 is awarded the ship and has to pay 505 CU. If a bidder bids 510 CU, he becomes new current high-bidder HB510 and the auction price increases to 515 CU, and so on. Before the auction begins, you will be asked some questions on the screen concerning the rules. This is to ensure that all participants have understood the instructions. C.9. Treatment P1u Instructions You are going to participate in an experiment on selling. In this experiment, you will make your decisions as a bidder at your computer terminal, isolated from the other participants. You may earn money in cash. How much you earn depends on your decisions and on the decisions of the other participants. The monetary units in the experiment are called currency units (CU). Point of Departure Imagine that you are the owner of a cruise ship. The use of this ship yields profits and it thus has some value for you. You also have the opportunity to sell the ship, via a selling process that will be explained to you in detail in the following. Besides you, 2 other ship owners, who own one ship each, will participate in this selling process, but only one ship can be sold. If you do not sell the ship, the value of the ship for you W will be realized only through its use. Calculations show that the value W of the ship for you lies uniformly distributed between W0 and W1 . This means, that the ship has a value for you of at least W0 and at most W1 , where all values (integers) from W0 to W1 have equal probability. Please note: the boundaries W0 and W1 are different for every ship owner. Your individual boundaries W0 and W1 will be communicated to you on your screen directly before the selling process; the boundaries of the other bidders, however, are unknown to you. Payoff 1. If you do not sell your ship, its value for you will be determined by drawing a value W from your uniform distribution over W0 to W1 , which will then correspond to your payoff. 2. If you sell your ship at price P , this selling price corresponds to your payoff. Your payoff will be converted into Euro and paid to you in cash at the end of the experiment (where 1 CU corresponds to 5 Euro Cents). The payment will be individual and anonymous. Selling process The process by which the ship is to be sold has the following rules. You submit an offer A exactly once, at the beginning of the selling process. With this offer you express the minimum CU you want to have for the ship. An offers has to be a multiple of 5 CU, that is, A = 100, 105, 110, 115, 120, . . . When submitting your offer, you do not know the offers of the other two vendors, and they do not know your offer. When all 3 vendors have submitted their offers, the automatic selling process begins, which bids the offers against each other and thereby determines which vendor sells the ship. The selling process starts with the highest offer, which is thus eliminated. The price is then gradually decreased by 5 CU at each step. If the price reaches that of another offer, this offer also quits the process. The selling process may end in two ways, depending on the offers: 1. If the lowest and the second lowest offer differ, the selling process stops when the price reaches the second lowest offer. We denote the remaining lowest offer A∗ and the price at which the selling process stops P, where P equals the second lowest offer. The vendor who submitted offer A∗ is accepted as seller and sells his ship for price P. In this case P > A∗ . That means that the seller receives for his ship more than what he asked for with his offer A∗ . 2. If at least two vendors have submitted the lowest offer A∗ , the process stops at price P = A∗ . One of these vendors is then randomly selected as seller at price P = A∗ . In this case the selling price equals the offer of the vendor who sells the ship. Please note: you are only allowed to submit once an offer at the beginning of the selling process, and it may not be changed afterwards!

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