## Abstract

In many auctions the valuation structure involves both private and common value elements. Existing experimental evidence (e.g. Goeree and Offerman in Am. Econ. Rev. 92(3):625–643, 2002) demonstrates that first-price auctions with this valuation structure tend to be inefficient, and inexperienced subjects tend to bid above the break-even bidding threshold. In this paper, we compare first-price auctions with an alternative auction mechanism: the least-revenue auction. This auction mechanism shifts the risk regarding the common value of the good to the auctioneer. Such a shift is desirable when ex post negative payoffs for the winning bidder results in unfulfilled contracts, as is often the case in infrastructure concessions contracts. We directly compare these two auction formats within two valuation structures: (1) pure common value and (2) common value with a private cost. We find that, relative to first-price auctions, bidding above the break-even bidding threshold is significantly less prevalent in least-revenue auctions regardless of valuation structure. As a result, revenue in first-price auctions is higher than in least-revenue auctions, contrary to theory. Further, when there are private and common value components, least-revenue auctions are significantly more efficient than first-price auctions.

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## Notes

To put it in the context of Engel et al. (2001), the future cash flows of toll revenue are a common unknown value, and bids consist of the present value of toll revenue required by bidding firms.

GO also show that increasing competition (i.e. the number of bidders) exogenously or reducing the uncertainty (i.e. the variance) of the common value increases efficiency. Our results regarding LRAs are consistent with this finding, since LRAs eliminate the uncertainty regarding the common value of the good.

See Kagel and Levin (2002) for an introduction to this literature.

The instructions used are in Spanish. The sample instructions found in the Electronic supplementary material have been translated into English. The remaining instructions are available upon request.

Our analysis is motivated, in part, by the prevalence of unfulfilled contracts due to bidders having, or claiming to have, ex post negative payoffs in traditional auction formats. Two possible explanations for why bidder claim such negative payoffs are: (1) when formulating bids, bidders do not take into account the informational content of winning, and consequently fall victim to the winner’s curse; (2) when formulating bids, bidders take into account that they may be able to renegotiate the contract (or at least not be liable for potential losses), and consequently bid more aggressively. We focus on the case in which winning bids represent a binding contract, and seek to determine if a change in auction format can alleviate the prevalence of the winner’s curse (and the ability of bidders to claim that they are victims). The effect of allowing bidders to renegotiate on bidding behavior is an interesting and important question we leave for future experimental research. As such, we do not explicitly model the possibility of renegotiation.

We do not consider a level-k model, because in three of our four treatments this model predicts Nash bidding behavior for all but level zero bidders (the exception is the first-price auction with common values and a private cost).

Derivations of this cursed equilibrium as well as the corresponding expected profit and auctioneer revenue can be found in the Electronic supplementary material.

The derivations of this cursed equilibrium bid function, equilibrium bidder profits, equilibrium auctioneer revenue can be found in the Electronic supplementary material.

Since a bidder of type

*c*_{ H }will almost surely lose in any monotonically decreasing equilibrium, it must be the case that in equilibrium she will bid*c*_{ H }<*v*_{ L }. If a bidder of type*c*_{ H }were to bid*b*<*c*_{ H }in equilibrium, she would earn a positive profit if she won the auction. She would then have an incentive to decrease her bid in order to have a positive probability of winning the auction. Thus the equilibrium bid of type*c*_{ H }must be equal to*c*_{ H }.One of the first ten periods (referred here as periods −9 to 0) is randomly selected to be paid. Each of the remaining 20 periods (referred to as periods 1 to 20) are paid. In the analysis that follows, data from the initial ten periods is not utilized.

We used 360 (12 subjects × 30 rounds) iid draws that were kept constant across all sessions. That is, we had 12 types of subjects who saw the same sequence of signals constant across all sessions.

Our risk attitude elicitation task differs from Holt and Laury (2002) in that, instead of choosing between two lotteries, subjects choose between a certain amount and a lottery.

Low or high starting balance sessions are balanced across treatments. For each treatment we have 2 sessions with a low starting balance and two sessions with a high starting balance. The starting balance was increased due to the prevalence of bankrupt subjects with the lower starting balance. Bankruptcies only occurred in sessions with first-price auctions.

If more than one participant went bankrupt then the data from the session was not included in the reported analysis. We exclude the data from six sessions. In two of these, multiple subjects went bankrupt. In the remaining four, one subject went bankrupt in each session, and additional problems prevented us from completing the session.

Unless otherwise noted, our non-parametric tests use average results from each session as an independent observation. Thus, we have 4 independent observations per cell. Given that the asymptotic

*p*-value is not a good approximation when both samples have less than 12 observations, we rely on critical values of the test statistic for different levels of statistical significance calculated by Feltovich (2003).n.s. indicates that the test is not significant at conventional levels.

As Fig. 4 illustrates, in both auction formats, bidder profits are greater on average under private and common values than under pure common values. However, using session level data (with only four observations per cell), we cannot reject the null hypothesis of equality of means under using the robust rank order test. For FPAs, results are not robust to dropping 1 session in each treatment where a bankruptcy occurred: dropping those sessions, payoffs are significantly greater under FP-PC than under FP-C (robust rank order test,

*U*=−2.348,*p*=0.1). However, if we drop all periods after a subject went bankrupt rather than the entire session, we cannot reject equality (robust rank order test,*U*=−0.776, n.s.).If the session with bankruptcy in the FP-PC is dropped, this result is no longer significant (

*U*=−1.000). If only the periods after the bankruptcy occurred are dropped, rather than the entire session, the result is unchanged.When the lowest observation from one treatment is higher than the highest observation of the other treatment, the test statistic of the robust rank order test is undefined. We denote this highly significant case as n.d.

If the FP-PC session in which a subject went bankrupt is dropped, this result is no longer significant. However, if periods after the subject went bankrupt are dropped, rather than the entire session, the result is unchanged.

The unit of observation used in the sign test is the individual participant. That is, the average bid of a participant over all periods is compared with the average Nash equilibrium bid or the average naive bid. This unit of observation was used for all non-parametric tests regarding observed bidding relative to theory.

For instance in FP-C, we split the data according to whether individuals observed a private signal about the common value of the good that was in the lower, mid or higher third of the theoretical distribution of signals. For FP-PC, the split is regarding the observed

*s*_{ i }relative to the theoretical distribution of the surplus summary statistic. For LR-C there is no relevant signal since the weakly dominant strategy ignores the private signal about the common value, but for comparability, we use the same blocks as in FP-C. Finally, for LR-PC, we split into blocks according to whether the observed private cost falls into the lower, mid or upper third of the theoretical cost distribution.For these sign tests, we use individual level data (i.e. the average bid of each individual when the observed relevant signal was in the specific block). For the test between observed and Nash bidding, the alternative hypothesis is that bidders bid more aggressively than the theory predicts: the median of observed bids exceed the median of Nash predicted bids. For the test between observed and fully cursed, the alternative hypothesis is that individuals bid more conservatively than predicted: the median of observed bids is below the median of fully cursed predicted bids.

We thank an anonymous referee for suggesting this analysis.

This suggests that either bid formulation is a noisy process, there is bidder heterogeneity, or both.

The Gauss code we used to obtain these results is available upon request.

As a robustness check, we also estimated bid functions with dummies for subjects who went bankrupt, and a dummy indicating whether or not a bankruptcy occurred in the session. These results are available upon request.

Recall that the equilibrium bid of LR-C bidders does not depend on the private information held by bidders.

A subject is defined as the sequence of 20 draws of

*v*_{ i }and, if applicable,*c*_{ i }that a participant faced, as well as the sequence of unobserved draws that her opponents faced. That is, in each session we utilized the same set of (once random) draws as the other sessions. Thus, exactly one participant in each session observed each sequence of random draws. The dummy variable for a subject is equal to one for the set of participants who observed that sequence, and zero for the other participants.The equilibrium bid function for FP-PC auctions is not predicted to be linear. However, for some values of

*s*_{ i }this bid function cannot be separated into linear and nonlinear parts. We report linear bid functions, which we find to be a better fit for the data than nonlinear specifications. As such, the reported regressions should not be interpreted as an explicit test of the equilibrium bidding strategy.

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## Additional information

Financial support from Gettysburg College and the Facultad de Ciencias Económicas at Universidad Francisco Marroquín is gratefully acknowledged. Thanks to Pedro Monzón and Diego Fernandez for outstanding research assistance. We have benefited from comments and suggestions from participants in seminars at Chapman University, Universidad Francisco Marroquín, George Mason University, the Alhambra Experimental Workshop, the 2010 North-American ESA conference in Tuscon Arizona, the Eastern Economic Association 37th annual conference in New York City and the Sydney Experimental Seminar at UNSW. We also thank the editor and two anonymous referees for many valuable comments and suggestions.

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Aycinena, D., Baltaduonis, R. & Rentschler, L. Valuation structure in first-price and least-revenue auctions: an experimental investigation.
*Exp Econ* **17**, 100–128 (2014). https://doi.org/10.1007/s10683-013-9359-7

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DOI: https://doi.org/10.1007/s10683-013-9359-7

### Keywords

- Auctions
- Winner’s curse
- Allocative efficiency
- Bidding

### JEL Classification

- D44
- C72