# Valuation structure in first-price and least-revenue auctions: an experimental investigation

## 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.

### Keywords

Auctions Winner’s curse Allocative efficiency Bidding### JEL Classification

D44 C72## 1 Introduction

Auctions for infrastructure concession contracts may be modeled as having both private and common components in the valuation structure. The winner of such an auction receives the revenue generated by the contract (e.g. tolls from highway concessions, energy transmission fees over a high-power grid, generation capacity payments, etc.) which has a common value (Bain and Polakovic 2005; Flyvbjerg et al. 2005). The winner will also incur the cost of fulfilling the contract (e.g. building the highway, constructing the infrastructure for power lines, building and maintaining power plants, etc.). If bidders’ costs of providing the infrastructure differ, these costs may represent a private component of the valuation structure.

In auctions in which there is a common value component, it is well known that bidders are prone to the winner’s curse.^{1} That is, bidders bid such that they guarantee themselves negative payoffs in expectation. This is of particular concern in auctions for infrastructure concession contracts, as bidders who go bankrupt may cause costly delays in infrastructure development.

In this paper we experimentally compare the standard first-price sealed-bid auction with an alternative auction format which may reduce the prevalence of the winner’s curse, the Least-Revenue Auction (LRA).^{2} We compare these two auction formats in two valuation structures with an uncertain common value. In the first, bidders face a private cost. In the second, bidders face a common cost which is common knowledge.

In an LRA, bidders simultaneously make sealed-bid offers which consist of the minimum amount (from the common value of the good) the bidder is willing to accept upon winning the auction.^{3} The bidder who submits the lowest amount wins the auction, and obtains that amount, provided the amount is less than the common value of the good. In the event that the winning amount is greater than the value of the good, the winning bidder only obtains this value. Thus, the winner implicitly pays the difference between the realized common-value of the good and the amount they submitted. This mechanism renders private information bidders may hold regarding the common value of the good strategically irrelevant, provided this information is not correlated with the private information of other bidders. The LRA is then strategically equivalent to a first-price procurement auction. In a purely common-value LRA equilibrium bids are not a function of the private common-value signals that the bidders observe prior to placing their bids. The game is, in effect, a game of complete information. Similarly, in auctions with private and common values, the equilibrium bid function of an LRA maps bidders’ private costs into bids, ignoring privately observed estimates of the common value. The LRA mechanism, in effect, transforms an auction with private and common values into an auction with purely private values.

It is important to note that in an LRA, uncertainty regarding the common value of the good is borne by the auctioneer rather than by the bidders. An LRA represents a contract in which the price the winning bidder pays is contingent on the realized value of the good; the auctioneer guarantees the winning bidder that she will earn her bid (provided the winning bid does not exceed the common value of the good). This transfer of risk may be desirable, and provides the original motivation for LRAs: Engel et al. (1997, 2001) first proposed the Least Present Value of Revenue Auction (LPVRA), in which bidders submit the smallest present value of revenue they would require for a contract in which they build, operate and then transfer a highway to the government at the conclusion of the contract term. In an LPVRA, the duration of a contract is contingent on the stream of revenue which is generated by tolls collected on the highway. In particular, the contract lasts until the winning bidder obtains the present value (at a pre-determined discount rate) of the toll revenue that she bid. This flexible-term contract shifts most of the risk resulting from uncertain traffic patterns to the government, relative to a standard fixed-term contract. Engel et al. (1997) estimates that the value of switching to LPVR auctions is about 33 % of the value of the infrastructure investment.

More generally, by eliminating ex ante uncertainty regarding payoffs conditional on winning the auction LRAs and LPVRAs reduces the possibility that the winning bidder, having failed to account for the informational content of winning the auction when formulating her bid, will subsequently go bankrupt.

An additional potential benefit of using LRAs is that if bidders anticipate that the contract is open to renegotiation ex post, they may bid more aggressively in a first price auction, with no intention of adhering to the contract ex post. Uncertainty regarding the common value component of the good may provide cover for such strategic behavior, since the winning bidder can claim that they simply fell victim to the winner’s curse when demanding that the contract be renegotiated. Indeed, Guasch (2004) reports that over 50 % of concession contracts for transportation infrastructure are renegotiated. Athias and Nuñez (2008) find evidence that is consistent with bidders displaying more strategically opportunistic behavior in auctions for toll-road concessions in weaker institutional settings, presumably due to a higher probability of contract renegotiation. The LRA removes this cover, and thus may reduce reported bankruptcies in practice. We leave this interesting case to future research, and in the current paper focus on the case in which the contract is binding.

It is important to note that in both the LRA and the LPVRA, the winning bidder does not have an incentive to maintain the value of the good because winning the auction guarantees the winning bidder her bid, and no more. That is, the benefits of maintaining or improving the value of the good ex post do not accrue to her. Monitoring the ex post behavior of the winning bidder, or imposing an enforceable contract, would be necessary to mitigate this problem. If neither of these are possible, LRAs and LPVRAs may not be ideal.

Our work differs from that of Engel et al. (2001) in at least two important ways. First, their focus is on optimal risk-sharing contracts and not on bidding behavior or auction performance. Second, we allow for the possibility of private costs, and we analyze the common value of the good as the realization of a random variable in a single period rather than as a stream of revenue over time (with a high or low realized value in each period). However, the underlying intuition is the same. As such, the main contribution of this paper is to formally analyze and experimentally test bidding behavior and auction performance in an environment consistent with the motivation underlying LPVRAs. Although Chile has implemented LPVRAs on more than one occasion (Vassallo 2006), to the best of our knowledge this is the first formal and empirical analysis of the allocative properties of this auction format and bidding behavior within it. In this paper we test to see if an LRA can reduce the prevalence with which bidder’s guarantee themselves negative profits in expectation. We leave for subsequent experimental research the question of optimal risk sharing between the auctioneer and bidders.

In addition, our paper contributes to the small but growing literature regarding auctions with private and common values (see e.g. Goeree and Offerman 2002; Boone et al. 2009). The theoretical analysis of such auctions begins with Goeree and Offerman (2003). We extend this analysis by deriving the cursed equilibrium in first price auctions. Cursed equilibrium was introduced in Eyster and Rabin (2005), and allows for the possibility that bidders do not fully take into account the link between the private information of other bidders and their when formulating their own bid. This model of bounded rationality includes Nash equilibrium and a naive bidding model as special cases. Goeree and Offerman (2002) (henceforth GO) present experimental evidence that first-price auctions with private and common values tend to be inefficient. The intuition behind this inefficiency is that subjects have to combine the information of two signals (the private value and the signal regarding the common value). If subjects were to ignore the common value signal, the auction would be fully efficient. This is precisely what the LRA offers. Ignoring the common-value signal presents a coordination problem for auction participants in a standard auction with private and common values. The LRA avoids this coordination problem by rendering common-value signals strategically irrelevant (provided signals are independent).^{4}

Auctions with purely common value have been studied extensively in the experimental literature. It is typically observed that inexperienced bidders are prone to fall victim to the winner’s curse. That is, inexperienced bidders often bid above a break-even bidding threshold. This observation is robust across numerous auction mechanisms, and these results cannot be explained by risk aversion, limited liability of losses or a non-monetary utility of winning.^{5} This paper provides, to the best of our knowledge, the first attempt to link analysis of the winner’s curse in auctions with only common values to auctions with both private and common value structures.

Our most dramatic result is a stark decrease in the frequency with which bidders bid above a break-even bidding threshold in LRAs relative to first-price auctions. Indeed, inexperienced bidders in LRAs very rarely bid above this threshold. Since these bidders do not face any uncertainty regarding their payoff conditional on winning the auction, this is perhaps not surprising. We also find that, when the value of the good has both private and common value components, there is a significant increase in efficiency in LRAs relative to first-price auctions. This is important because, as previously mentioned, efficiency is low in first-price auctions with this valuation structure. Thus, we demonstrate that in this environment increases in efficiency and a reduction in bidding above the break-even bidding threshold can be obtained by changing the auction mechanism. These findings support the use of LRAs or LPVRAs as a way to allocate concession contracts for infrastructure.

Contrary to theory, LRA generates less revenue than first-price auctions, regardless of valuation structure. This is largely due to the fact that bidders in first-price auctions tend to overbid aggressively, often to the point of guaranteeing negative profits in expectation. Correspondingly, bidders are better off in an LRA than in a first-price auction.

The remainder of the paper is organized as follows. Section 2 provides the theoretical background. Section 3 describes our experimental design. Section 4 provides our results. Section 5 contains the conclusion. Electronic supplementary material contains a sample set of instructions as well as derivations of theoretical predictions.^{6}

## 2 Theoretical predictions

A set of risk neutral players *N*≡{1,…,*n*} compete for a good with a common but uncertain value, *V*, by simultaneously placing bids. If a bidder wins the auction, we assume that the bidder must purchase the good at the price determined by her bid, and that she is able to do so.^{7} Prior to placing her bid, bidder *i*∈**N** privately observes a signal *v*_{i} regarding the value of the good. Each of these signals is an independently drawn realization of the random variable *v*, which is distributed according to *F* and has support [*v*_{L},*v*_{H}]. The value of the good is the average of the signals. That is, \(V=\sum_{i\in\mathbf{N}}\frac{v_{i}}{n}\). Also, bidder *i* faces a cost *c*_{i} that must be paid if she wins the auction and obtains the good; bidders know their cost prior to placing bids, but may not know the value of *c*_{j} where *j*≠*i*. In particular, we consider environments in which the *c*_{i}=*c*_{j} for ∀*i*,*j*∈**N**, and this is common knowledge. We also consider environments in which the cost of each bidder is an independent draw from a distribution that is commonly known. Each bidder observes their own cost, but not that of the other bidders. Bidder *i*∈*N* chooses a bid, \(b_{i}\in\mathbb{R}_{+}\) in an attempt to obtain the good. Bidders are not budget constrained; the strategy space of each player is \(\mathbb{R}_{+}\). The vector of bids is *b*≡*b*_{1},…,*b*_{n}. Further, *b*_{−i}≡*b*/*b*_{i} and *N*_{−i}≡*N*/*i*. We allow for the possibility that when a bidder is formulating her bid she does not fully take into account the information conveyed by the behavior of other bidders. In particular, we consider cursed equilibrium, as described in Eyster and Rabin (2005).^{8} In a cursed equilibrium bidders only partially take into account the correlation between the bids and signals of their opponents. If bids are increasing in the common value signal, this can result in unrealistically high beliefs about the value of the good conditional on winning the auction. Following Eyster and Rabin (2005), we assume that each bidder correctly anticipates the probability distribution of other bids, but mistakenly believes that with probability *χ* each of the other bidders employ a strategy that is equal to their average bid. The crucial point is that with probability *χ* each bidder believes that the behavior of the other bidders is not correlated with their private signals. For simplicity we assume that *χ* is the same for all bidders. Note that when *χ*=0 predictions will correspond to the symmetric Bayesian Nash equilibrium. Further, when *χ*=1 bidders are naive in the sense that they do not account for the informational content of winning the auction. The special case where *χ*=1 is similar to the naive bidding model studied in GO.

### 2.1 First-price auctions with private and common values

*c*

_{i}, where

*i*∈

*N*, is an independent draw of the random variable

*c*which is distributed according to

*G*with support [

*c*

_{L},

*c*

_{H}]. Thus, the value of the good has both private and common value components. To ensure that all bidders will participate in the auction, it is assumed that

*c*

_{H}<

*v*

_{L}. The net value of the good to bidder

*i*is thus

*V*−

*c*

_{i}. Note that each bidder privately observes two separate pieces of information regarding this net value, and that these pieces of information are independent. This information structure is analyzed in Goeree and Offerman (2003), and they demonstrate that the one dimensional summary statistic \(s_{i}=\frac{v_{i}}{n}-c_{i}\) can be used to map both pieces of information into equilibrium bids in a first-price auction. We denote the random variable from which these summary statistics are (independently) drawn as

*s*, with corresponding density and distribution functions

*f*

_{s}and

*F*

_{s}respectively. We denote the interval on which

*s*is distributed as \([s_{L},s_{H}]= [\frac{v_{L}}{n}-c_{H},\frac{v_{H}}{n}-c_{L}]\). The symmetric cursed equilibrium bid function is

*y*

_{1}is the highest

*s*

_{i}of the other

*n*−1 bidders. That is, \(y_{1}=\max_{j\in N_{-i}}\frac{v_{j}}{n}- c_{j}\).

^{9}

*i*in this cursed equilibrium is given by where

*Y*

_{1}is the first order statistic of the

*n*draws of

*s*, and

*Y*

_{2}is the corresponding second order statistic. The corresponding ex ante expected profit of the winner is then

*χ*.

To find the expected revenue in an FP-PC auction we first note that the winner’s net value of the good is *W*=*E*(*V*)−*E*(*c*|*s*=*Y*_{1}). Thus, expected revenue in this cursed equilibrium is \(R^{FP\mbox{\scriptsize-}PC}=W-E (\varPi_{winner}^{FP\mbox{\scriptsize-}PC})\). This means that *R*^{FP-PC} is increasing in *χ*.

#### 2.1.1 Break-even bidding threshold

### 2.2 First-price auctions with common values

*z*

_{1}is the highest signal of the other

*n*−1 bidders. That is, \(z_{1}=\max_{j\in N_{-i}}v_{j}\).

^{10}

*i*is given by where

*Z*

_{1}is the first order statistic of the

*n*draws of

*v*, and

*Z*

_{2}is the corresponding second order statistic.

Total expected surplus in this auction is given by \(X=E (V )-\bar{c}\). Thus, expected revenue in this cursed equilibrium is \(R^{FP\mbox{\scriptsize-}C}=X-E (\varPi_{winner}^{FP\mbox{\scriptsize-}C} )\).

#### 2.2.1 Break-even bidding threshold

### 2.3 Least-revenue auctions with private and common values

In a least-revenue auction with private and common values (LR-PC), bidders simultaneously submit bids, the lowest of which wins the auction. Bids consist of the minimum amount (which would come from the common-value of the good, *V*) that a bidder is willing to receive, given that she wins the auction. The winner obtains the minimum of the realization of *V* and her bid. If the winning bid is less than the realized common value, the winning bidder implicitly pays the difference between the common-value and her bid. Recall that we assume *c*_{H}<*v*_{L}. This implies that the common value will always be sufficient to cover a bidder’s cost.

When there are common and private values, the valuation structure is exactly the same as in FP-PC auctions. However, the price the winning bidder pays is contingent on the realized value of *V*. Provided her bid does not exceed *v*_{L}, the uncertainty regarding the common-value of the good does not affect the winning bidder’s payoff conditional on winning. Note that in equilibrium predicted bids will fall below *v*_{L}.^{11} Further, since we have assumed that private information is independent, a bidder’s beliefs about the private information held by opponents will not be contingent on her own private information in equilibrium. Thus, the draws of *v* that each bidder observes are strategically irrelevant. Since bidders each face a cost (should they win the auction) which is an independent draw from a common distribution, the problem that each bidder faces is strategically equivalent to a first-price procurement auction with independent private costs. Additionally, we assume that each bidder incorrectly believe that with probability *χ* her opponents will bid their average bid, rather than their type contingent strategy. However, as long as the resulting cursed equilibrium bidding strategy is monotonically decreasing in type, this will not affect expected payoffs. This is because the bidder with the lowest cost will win in equilibrium, and the probability of winning the auction is the same as the probability of having the lowest cost. As such, provided *χ*<1, the cursed equilibrium bid function will correspond to the Nash equilibrium bid function. Note that if bidders are cursed (*χ*>0), then using a LRA in place of a first price auction is predicted to reduce the frequency with which the winning bidder loses money.

*u*

_{n−1}is the smallest of

*n*−1 draws of

*c*.

The ex ante expected profit of bidder *i* is given by \(E (\varPi _{i}^{LR\mbox{\scriptsize-}PC} (\zeta (c ),c ) )= (\frac {1}{n} )E (U_{n-1} )- (\frac{1}{n} )E (U_{n} )\), where *U*_{n−1} is the second lowest and *U*_{n} is the lowest of the *n* draws of *c*. Thus the ex ante expected profit of the winner is given by \(E (\varPi_{winner}^{LR\mbox{\scriptsize-}PC} )=E (U_{n-1} )-E (U_{n} )\). Since the total expected surplus in this auction is given by *D*=*E*(*V*)−*E*(*c*|*c*=*U*_{n}), expected revenue in this equilibrium is \(R^{LR\mbox{\scriptsize-}PC}=D-E (\varPi _{winner}^{LR\mbox{\scriptsize-}PC} )\).

#### 2.3.1 Break-even bidding threshold

*V*is not relevant to the payoff of the bidder and the common-value signal does not enter into the equilibrium bid function. Any bid which is above the privately observed cost will guarantee the bidder positive profit upon winning the auction. Similarly, any bid that drops below the cost will guarantee negative profits. Thus, the break-even bidding threshold for a bidder in an LR-PC auction is

### 2.4 Least-revenue auctions with common values

In a least-revenue auction with common values and a common cost (LR-C), the game is, in effect, one of complete information. As a result, the value of *χ* is irrelevant. Since bidders place bids that do not depend on their private information (every type employs the same pure strategy), the average bid is the same. The unique symmetric (cursed) equilibrium of this game is to bid \(\bar{c}\). To see this, note that if any bidder were to bid below \(\bar{c}\), they would earn negative profits upon winning. For any bid \(b_{i}>\bar{c}\), bidder *j*∈*N*_{−i} would have an incentive to bid \(b_{j}\in(\bar {c},b_{i})\) and earn a positive profit. Notice that the ex ante equilibrium profit of bidder *i* is zero. Further, the equilibrium revenue in this game is \(R^{LR\mbox{\scriptsize-}C}=E(V)-\bar{c}\).

#### 2.4.1 Break-even bidding threshold

## 3 Experimental design

In every experimental session, twelve participants are randomly and anonymously matched into groups of three. In each round, every group participates in an auction. Each bidder submits a bid. The bidder who submits the winning bid obtains the good (ties are broken randomly). The other bidders receive payoffs of zero. After each round, all participants within a session are randomly and anonymously re-matched. The randomized group assignment was kept constant across all sessions. This process is repeated for thirty rounds.^{12}

In each auction the value of the good to each bidder is the difference between the common value and the cost the bidder faces if she were to win the auction. The common value of the good has an uncertain value. Each bidder *i*∈{1,2,3} privately observes a signal, *v*_{i}, regarding this common value. Each of these signals is an independent draw from the uniform distribution with support [100,200].^{13} The common value, *V*, is the average of the private signals. That is, \(V=\frac{1}{3}\sum_{i=1}^{3}v_{i}\). The realized value of the good is not observed by bidders before placing their bids, although bidders know the cost they must pay if they win the auction beforehand. The distribution from which the signals are drawn is common knowledge.

- 1.
*First-price auctions with private and common values*(*FP-PC*): In addition to the private signal that bidders observe regarding the common value of the good, each bidder privately observes the cost she must pay if she were to win the auction. Each of these costs is an independent draw from a uniform distribution with support [0,50]. These costs represent the private value portion of the valuation structure. The auction format in this treatment is a standard first-price sealed-bid auction. - 2.
*First-price auctions with common values*(*FP-C*): In this treatment each bidder faces the same cost if she were to win the auction. This cost is equal to the expected value of the cost distribution in the FP-PC treatment (\(\bar{c}=25\)). The auction format in this treatment is a standard first-price sealed-bid auction. - 3.
*Least-revenue auctions with private and common values*(*LR-PC*): In addition to the private signal that bidders observe regarding the common value of the good, each bidder privately observes the cost she must pay if she were to win the auction. Each of these costs is independently drawn from a uniform distribution with support [0,50]. These costs represent the private value portion of the valuation structure. The auction format in this treatment is an LRA. - 4.
*Least-revenue auctions with common values*(*LR-C*): In this treatment each bidder faces the same cost if they were to win the auction. This cost is equal to the expected value of the cost distribution in the FP-PC treatment (\(\bar{c}=25\)). The auction format in this treatment is an LRA.

*V*, all bids, her earnings from the auction, and the price paid by the winner.

All sessions were run at the Centro Vernon Smith de Economía Experimental at the Universidad Francisco Marroquín, and our participants were primarily matriculated undergraduates of the institution. The sessions were computerized using z-Tree (Fischbacher 2007). Participants were separated by dividers such that they could not interact outside of the computerized interface. They were provided with instructions and were also shown a video which read these instructions aloud. Each participant then individually answered a set of questions to ensure understanding of the experimental procedures. We elicited risk attitudes using a measure similar to that of Holt and Laury (2002).^{14} We varied the order in which subjects participated in the risk attitude elicitation procedure and the series of auctions. Each session lasted approximately one and a half hours. In half of the reported sessions, each participant began with a starting balance of Q62.5 (1 Quetzal≈US$0.125) to cover any losses; in the other half participants began with a starting balance of Q125.^{15} At the end of all thirty rounds, each participant was paid her balance which included a show-up fee of 20 Quetzales. If the balance of a participant became negative, she was permitted to continue provided she invested her show-up fee. If the show-up fee was also lost, she was permitted to continue, and received a payment of zero.^{16}

Within the 16 reported sessions, there were two participants who went bankrupt (≈1 % of participants) before the end of the experiment. The bids, signals and values were all denominated in Experimental Pesos (EP), which were exchanged for cash at a rate of 4E$=Q1≈US$0.125. The average payoff was Q105, with a minimum of Q0 and a maximum of Q165.

## 4 Results

### 4.1 Efficiency levels

When the valuation structure is pure common value, any allocation of the good is efficient. As such, efficiency is not a concern in this valuation structure. When there are private costs, however, allocating the good to the bidder with the lowest cost is the efficient allocation.

Interestingly, when there are private and common value components in the first-price auction (FP-PC), the equilibrium allocation may not be efficient (Goeree and Offerman 2003). This is because the equilibrium bid function is monotonically increasing in the summary statistic \(s_{i}=\frac{v_{i}}{n}-c_{i}\). A bidder may have a high cost relative to the other bidders in the auction, but if she also has a relatively high common-value private signal (such that \(s_{i}=\frac{v_{i}}{n}-c_{i}\) is larger than those of the other bidders) she is predicted to win the auction, which would result in an inefficient allocation.

However, in LR-PC auctions, equilibrium bids are monotonically decreasing in *c*_{i}. This implies that, in equilibrium, the bidder with the lowest cost will win with certainty. As such, the predicted efficiency level is 100 %. This points to an important property of the LR-PC auction. Namely, by rendering the common value signal strategically irrelevant, inefficiency concerns that arise in valuation structures with private and common values are, in theory, eliminated. That is, LR-PC auctions are predicted to be more efficient than FP-PC auctions.

*c*

_{winner}is the private cost of the winning bidder and

*c*

_{max}(

*c*

_{min}) is the maximal (minimal) private cost of the three bidders. This can be interpreted as the realized proportion of the difference between the most efficient and least efficient allocation.

*U*=−4.484,

*p*=0.029).

^{17}Figure 1 illustrates this difference by comparing the observed efficiency level to two benchmarks: the efficiency level predicted by equilibrium bidding behavior, and the efficiency level resulting from a random allocation of the good. Notice that in FP-PC auctions, the predicted efficiency level is much larger than that of the random allocation, while still being less than 100 % efficient. Observed efficiency falls between predicted efficiency and that of a random allocation. While observed efficiency is much higher in LR-PC auctions than in FP-PC auctions, contrary to theory LR-PC auctions are not perfectly efficient. The difference in efficiency between LR-PC and FP-PC auctions can be largely attributed to the fact that the uncertainty regarding the common value of the good has been shifted to the auctioneer in LRAs.

Summary statistics for efficiency

Efficiency measure | FP-PC | LR-PC | ||||
---|---|---|---|---|---|---|

Periods 1–10 | Periods 11–20 | All periods | Periods 1–10 | Periods 11–20 | All periods | |

Observed | 0.660 | 0.652 | 0.656 | 0.849 | 0.879 | 0.864 |

(0.413) | (0.411) | (0.411) | (0.301) | (0.271) | (0.286) | |

Random allocation | 0.504 | 0.484 | 0.494 | 0.504 | 0.484 | 0.494 |

(0.102) | (0.084) | (0.094) | (0.102) | (0.084) | (0.094) | |

Nash bidding | 0.878 | 0.839 | 0.858 | 1.000 | 1.000 | 1.000 |

(0.258) | (0.297) | (0.278) | (0.000) | (0.000) | (0.000) |

### 4.2 Bidder profits

Bidder payoffs and revenue

Variable | FP-C | FP-PC | LR-C | LR-PC |
---|---|---|---|---|

Observed revenue | 135.492 | 138.634 | 124.267 | 132.024 |

(17.379) | (23.300) | (14.975) | (17.070) | |

Predicted revenue | 116.319 | 120.367 | 124.558 | 124.028 |

(9.170) | (8.858) | (14.832) | (15.241) | |

Observed profits | −3.645 | −3.105 | 0.097 | 0.565 |

(11.365) | (14.683) | (0.896) | (6.570) | |

Predicted profits | 2.452 | 9.043 | 0.000 | 4.398 |

(2.838) | (16.140) | (0.000) | (4.905) | |

Fraction of auctions with positive payoffs | 0.250 | 0.341 | 0.984 | 0.834 |

(0.434) | (0.475) | (0.124) | (0.372) |

Theory predicts that bidders will be better off when there are private and common values than they would be in pure common value environments because the privately observed costs earn positive information rents. We are unable to find evidence that valuation structure significantly affect payoffs in first-price auctions (robust rank order test, *U*=−0.776, n.s.)^{18} or in LRAs (robust rank order test, *U*=−1.033, n.s.).^{19}

We also find that bidders are better off in LRAs. When there are private and common values, this result is marginally significant (robust rank order test, *U*=−1.586, *p*=0.1).^{20} However, this result is highly significant in the pure common values environment (robust rank order test, n.d., *p*=0.014).^{21} The intuition behind this is that in first-price auctions bidders tend to substantially overbid relative to Nash predictions, often resulting in negative payoffs. In first-price auctions bidders must estimate the common value of the good, conditional on winning. By eliminating the uncertainty regarding bidder profit conditional on winning, LRAs eliminate the need for bidders to estimate this conditional expected value. A bidder in a LRA need only bid above her cost to ensure (weakly) positive profits.

Revenue is, of course, closely related to bidder profits. As such, our results regarding auction revenue closely mirror those of bidder profits. Contrary to theory, FP-PC auctions, on average, generate more revenue than LR-PC, although this result is only marginally significant (robust rank order test, *U*=1.586, *p*=0.1).^{22} FP-C auctions generate more revenue than LR-C revenue (robust rank order test, n.d., *p*=0.014). The intuition underlying this result mirrors the analogous finding for profits. We also find that valuation structure does not significantly affect revenue in first-price auctions (robust rank order test, *U*=0.000, n.s.). However, in LRAs the private and common value valuation structure generates more revenue than the pure common value valuation structure (robust rank order test, n.d., *p*=0.014).

### 4.3 Break-even bidding threshold

*U*=11.314,

*p*<0.029), as well as in the pure common value environment (robust rank order test,

*U*=11.314,

*p*<0.029). Figure 3 illustrates this result by showing the proportion of bids above the break-even bidding threshold for all four treatments. Figure 3 breaks this into five period blocks. Notice that bidding above the break-even threshold is almost entirely eliminated in LR-C auctions. The relative dearth of bidding above the break-even threshold in LRAs is largely attributable to the fact that the uncertain common value of the good does not translate into uncertainty regarding bidder profits. Indeed, conditional on winning the auction, there is no uncertainty regarding bidder profits in LRAs. The risk regarding this uncertain value has been completely shifted to the auctioneer.

Bidding above the break-even bidding threshold (proportion of bids)

Variable | FP-C | FP-PC | LR-C | LR-PC |
---|---|---|---|---|

Bids above break-even threshold | 0.475 | 0.447 | 0.005 | 0.085 |

(0.500) | (0.497) | (0.072) | (0.280) | |

Bids above break-even threshold among winning bids | 0.691 | 0.650 | 0.016 | 0.166 |

(0.463) | (0.478) | (0.124) | (0.372) |

We also find that the valuation structure does not significantly affect the frequency with which bidders bid above the break-even bidding threshold in FPAs (robust rank order test, *U*=1.016, n.s.) or in LRAs (robust rank order test, *U*=−1.206, n.s.). This is not surprising because, holding the auction format constant, moving from the pure common-value environment to the private and common value environment does not change the level of uncertainty the bidder faces regarding the net value of the good.

### 4.4 Cursed equilibrium

*χ*=0), and for FPAs (where the equilibrium bid function depends on

*χ*) the fully cursed equilibrium (

*χ*=1). Of note is the fact that, on average, bidders overbid relative to the Nash equilibrium in every treatment except LR-C. In FPAs this overbidding is, on average, less than the fully cursed bidding strategy, indicating that bidding may be explained by an intermediate level of cursedness.

Observed bids relative to Nash and fully cursed bids

Variable | FP-C | FP-PC | LR-C | LR-PC |
---|---|---|---|---|

Observed bids | 114.085 | 111.929 | 34.160 | 29.959 |

(27.243) | (32.455) | (24.606) | (18.064) | |

Nash bids | 102.532 | 104.936 | 25.000 | 33.357 |

(15.568) | (19.739) | (0.000) | (9.927) | |

Fully cursed bids | 119.346 | 114.961 | – | – |

(6.227) | (12.287) |

Figure 4 illustrates how observed bids in FP-C auctions compare to the Nash predictions, the fully cursed bidding strategy, and the break-even bidding threshold. Notice that bids tend to be well in excess of the Nash equilibrium prediction. Indeed, bids well above the break-even bidding threshold, and above the fully cursed biding strategy are common. In these FP-C auctions, we find that bids are greater than Nash predictions (sign test, *w*=41, *p*<0.001).^{23} Further, FP-C bids are less than the fully cursed bidding strategy (sign test, *w*=32, *p*=0.015).

*w*=39,

*p*<0.001) but less than the fully cursed bidding strategy (sign test,

*w*=30 ,

*p*=0.056).

*w*=47,

*p*<0.001). Equilibrium payoffs are zero for any possible

*v*

_{i}in the LR-C treatment. Further, expected payoffs conditional on Nash beliefs are also zero for any possible

*v*

_{i}, if the bidder bids above the equilibrium bid. This is a case were bidding above the Nash predictions is very inexpensive. Indeed, average profits for winning bidders in LR-C were E0.32 (=Q0.08≈US$0.01), and about half of that if we only look at the last 10 periods. Although statistically, payoffs are different from zero, the magnitude is economically irrelevant. The observed bidding above Nash predictions as the experiment progresses could be due to throw-away bidding as a way for bidders to express their frustration about competing for very low profits, or a very low cost strategy of trying to signal to others to tacitly collude. We are unable to distinguish between these possible motivations, but it should be noted that tacit collusion would be very difficult as bidders are randomly and anonymously rematched every period.

*w*=38,

*p*<0.001). This is consistent with observed bidding behavior in FPAs with independent private values.

The cursed equilibrium model that we consider depends on some behavioral parameters. In FPAs, the cursedness parameter *χ* predicts the extent to which bidders do not account for the correlation between their opponents private information and their behavior. In LRAs payoffs do not depend on *χ*. This provides a compelling theoretical reason to adopt the LRA in favor of the FPA; the LRA provides an environment in which the cursedness of bidders is predicted to not affect behavior.

*χ*in FPAs and logit precision parameter in all four treatments. These logit precision parameters, roughly speaking, give us a measure of how much noise must be added to the model in order for it to fit the data. Denoting the expected payoff that bidder

*i*believes she has when she observes a common value signal

*v*

_{it}and a cost of

*c*

_{it}(note that in the pure common value environments \(c_{it}=\bar{c}\)) and who bids

*b*

_{it}in auction type

*k*(where

*k*is either FP-PC, FP-C, LR-PC or LR-C) as \(\prod_{it}^{k} (b_{it}|v_{it},c_{it})\). It is assumed that bidders make errors when formulating their bids. Every bid in the strategy space is played with positive probability, and this probability is increasing in the expected payoff that a bidder believes a bid would yield. In particular, the probability of observing bid

*b*is assumed to be

*λ*is a logit precision parameter, which determines how sensitive the probabilities are to relative expected payoffs. Note that if

*λ*=0, then each strategy is played with equal probability, and that as

*λ*→∞, bidders no longer make mistakes. We assume that errors are independent. Further, we assume that

*λ*and

*χ*do not vary over time or across subjects. For subject

*i*in auction type

*k*we observe a sequence of bids \(b_{i}^{k}= (b_{i1}^{k},b_{i2}^{k},\ldots,b_{i20}^{k})\). This subject observes a corresponding sequence of common value signals \(v_{i}^{k}=(v_{i1}^{k},v_{i2}^{k},\ldots,v_{i20}^{k})\) and costs \(c_{i}^{k}=(c_{i1}^{k},c_{i2}^{k},\ldots,c_{i20}^{k})\). Further, we have a total of 48 subjects in each auction type. The likelihood of observing our sample in auction type

*k*is then given by

^{24}In addition to showing the average and standard deviation for observed and predicted (Nash and fully cursed) bidding, the table also reports the

*p*-values for one sided sign tests between observed and predicted bids according to each model.

^{25}Notice that splitting the data into thirds allows us to check for differences in bidding away from the center of the distribution of relevant signals.

^{26}As the table shows, in FPAs, average observed bids tend to fall in between the two polar cases of the cursed equilibrium model. However, once we break it into blocks we see that when bidders have relevant signals in the upper third of the distribution, these bids tend to be statistically indistinguishable from the bids predicted by the fully cursed equilibrium. The table also illustrates that there is more dispersion among observed bids than among either theoretical prediction.

^{27}

Bidding relative to Nash and fully cursed, by signal magnitude

C | P&C | |||||||
---|---|---|---|---|---|---|---|---|

low | mid | high | all | low | mid | high | all | |

First-price auctions | ||||||||

Observed bid | ||||||||

Mean | 93.48 | 115.54 | 131.58 | 114.08 | 85.76 | 113.19 | 135.80 | 111.93 |

Std. Dev. | (24.12) | (21.29) | (22.43) | (27.24) | (30.57) | (27.50) | (23.68) | (32.46) |

Nash bid | ||||||||

Mean | 83.65 | 102.50 | 120.04 | 102.53 | 75.70 | 108.69 | 126.45 | 104.94 |

Std. Dev. | (5.93) | (5.48) | (5.07) | (15.57) | (9.94) | (8.55) | (3.91) | (19.74) |

| 0.001 | 0.000 | 0.000 | 0.000 | 0.007 | 0.001 | 0.000 | 0.000 |

Fully cursed bid | ||||||||

Mean | 111.79 | 119.33 | 126.35 | 119.35 | 97.35 | 116.62 | 129.25 | 114.96 |

Std. Dev. | (2.37) | (2.19) | (2.03) | (6.23) | (5.42) | (5.76) | (2.66) | (12.29) |

| 0.000 | 0.097 | 1.000 | 0.015 | 0.000 | 0.015 | 1.000 | 0.0557 |

Least-revenue auctions | ||||||||

Observed bid | ||||||||

Mean | 34.02 | 33.13 | 35.39 | 34.16 | 15.64 | 30.78 | 44.38 | 29.96 |

Std. Dev. | (24.14) | (22.15) | (27.41) | (24.61) | (10.05) | (16.65) | (13.71) | (18.06) |

Nash bid | ||||||||

Mean | 25 | 25 | 25 | 25 | 22.21 | 33.65 | 44.93 | 33.36 |

Std. Dev. | (0) | (0) | (0) | (0) | (3.33) | (3.33) | (3.24) | (9.93) |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.015 | 0.000 |

^{28}Recall that in LRAs, \(\prod_{it}^{k}(b_{it}|v_{it},c_{it})\) does not depend on

*χ*. As such, in the LRA treatments we only estimate

*λ*. There are several interesting results. First,

*χ*is significant in both FP-PC and FP-C auctions. The degree of cursedness is higher in the pure common values environment. This indicates that the introduction of the private cost in first price auctions reduces cursedness. This is intuitive, because when the private cost is introduced, a subjects private information is more important in determining payoffs conditional on winning the auction. It is then not surprising that subjects are less likely to fail to recognize that private information and subsequent bids are related. Second, notice that the precision parameter

*λ*is smaller in the FPA treatments than in the LRA treatments. This indicates that there is less error in formulating bids in LRAs. This, combined with the fact that cursedness does affect perceived payoffs conditional on winning, offers support for using LRAs in favor of FPAs.

Maximum likelihood estimates of behavioral parameters

Treatment | FP-PC | FP-C | LR-PC | LR-C |
---|---|---|---|---|

| 0.663 | 0.942 | – | – |

(0.312) | (0.106) | |||

| 0.042 | 0.044 | 0.129 | 0.418 |

(0.003) | (0.003) | (0.011) | (0.084) | |

Log likelihood | −466.407 | −466.600 | −470.360 | −470.256 |

Our estimates of cursedness are lower than the most comparable estimates from other studies. Eyster and Rabin (2005) estimate a *χ* greater than one in first-price common value auctions with three bidders. However, when they consider first-price common value auctions with more than three bidders, they find that *χ* varies between 0.4 and 0.9. In second-price common value auctions with inexperienced subjects (but with seven practice rounds), Eyster and Rabin (2005) also estimate *χ* greater than one. In second-price common value auctions with twenty-five periods of experience, however, they find that *χ*=0.63. Note that the subjects in our experiment had ten practice periods (one of which was chosen randomly for compensation).

Crawford and Iriberri (2007) also estimate levels of cursedness in first-price common value auctions. They constrain *χ* to be between zero and one, and find that high proportions have either *χ*=1 or *χ*=0. In their most comparable specification, they estimate *χ*=1 for 50 % and *χ*=0 for the other 50 % of the data.

### 4.5 Estimated bid functions

When estimating bid functions for the four treatments, we employ a random effects (at the individual level) specification, and cluster the standard errors to allow for intra-session correlation.^{29} We control for the statistic upon which equilibrium bids are based as well as experience (ln(*t*+1)).^{30} In LRAs, we also control for *v*_{i}, to test the hypothesis that the privately observed common-value signal does not enter into the bid function. We also estimate specifications which control for gender (*F*_{i}=1 if the bidder is female, 0 otherwise), the interaction of gender and experience *F*_{i}⋅ln(*t*+1), the order of the risk attitude elicitation procedure (*O*_{i}), whether or not bidders started with an endowment of E$500 (*E*_{i}), the number of safe choices in the risk elicitation procedure (*R*_{i}), and subject dummies.^{31}

*t*+1). Notice that when we control for gender and the interaction between gender and ln(

*t*+1) the respective coefficients are insignificant (although when we only control for gender, the coefficient is positive and significant). This is in contrast to the result of Casari et al. (2007), which finds that women tend to initially overbid more than men, but also learn to reduce their bids faster than men in first-price common-value auctions.

Estimated bid functions for FP-C auctions

(1) | (2) | (3) | (4) | |
---|---|---|---|---|

| 0.595 | 0.594 | 0.594 | 0.595 |

(0.033) | (0.033) | (0.033) | (0.035) | |

ln( | −4.135 | −4.135 | −3.527 | −3.529 |

(2.638) | (2.639) | (4.352) | (4.386) | |

| 2.488 | 5.498 | 3.408 | |

(0.959) | (9.356) | (9.353) | ||

ln( | −1.327 | −1.326 | ||

(4.007) | (4.037) | |||

| −1.55 | |||

(0.955) | ||||

| −4.899 | |||

(2.361) | ||||

| 3.431 | |||

(2.032) | ||||

Subject dummies | No | No | No | Yes |

– | – | – | – | |

Constant | 34.547 | 33.416 | 32.043 | 28.714 |

(9.285) | (8.958) | (10.870) | (14.515) | |

Observations | 960 | 960 | 960 | 960 |

^{32}Of interest is the fact that the coefficient on

*s*

_{i}is highly significant in all specifications, and approximately equal to one. Note that in the most inclusive specification the interaction between gender and ln(

*t*+1) is significant and positive, and that gender is (marginally) significant. This indicates that women are initially bidding less than men, but that as they gain experience they increase their bids more than men. This result is in stark contrast to that of Casari et al. (2007).

Estimated bid functions for FC-PC auctions

(1) | (2) | (3) | (4) | |
---|---|---|---|---|

| 1.091 | 1.091 | 1.091 | 1.091 |

(0.071) | (0.071) | (0.068) | (0.069) | |

ln( | −1.361 | −1.361 | −2.716 | −2.715 |

(1.869) | (1.870) | (1.926) | (1.941) | |

| 1.045 | −8.171 | −14.357 | |

(5.252) | (7.309) | (8.138) | ||

ln( | 4.062 | 4.061 | ||

(1.331) | (1.341) | |||

| 4.276 | |||

(1.897) | ||||

| 3.586 | |||

(1.708) | ||||

| 2.938 | |||

(1.984) | ||||

Subject dummies | No | No | No | Yes |

– | – | – | – | |

Constant | 87.952 | 87.604 | 90.659 | 64.064 |

(6.336) | (5.081) | (5.227) | (6.808) | |

Observations | 960 | 960 | 960 | 960 |

*t*+1) is highly significant, and positive. That is, bidders are moving away from equilibrium, on average, as they gain experience. This may be an attempt by some bidders to send signals in order to tacitly collude with other bidders on a higher price. Since bidders were randomly and anonymously re-matched every period, it would have been extremely difficult for this type of coordination to happen. At the same time, it would have been a very low-cost strategy, given the low profits observed in this auction. Alternatively, it might have been a case of throw-away bidding in which bidders simply express their frustration over competing for extremely low profits, conditional on winning. Additionally, in the most inclusive specification, the interaction between gender and ln(

*t*+1) is significant and negative. This implies that over time, the bids of male participants are increasing, and moving away from equilibrium. Once again, however, gender alone is not significant.

Estimated bid functions for LR-C auctions

(1) | (2) | (3) | (4) | |
---|---|---|---|---|

| 0.008 | 0.008 | 0.01 | 0.009 |

(0.028) | (0.027) | (0.028) | (0.028) | |

ln( | 3.750 | 3.750 | 7.110 | 7.111 |

(1.135) | (1.135) | (2.074) | (2.090) | |

| −7.657 | 5.911 | 5.853 | |

(6.796) | (2.134) | (3.791) | ||

ln( | −5.980 | −5.979 | ||

(2.458) | (2.475) | |||

| 5.818 | |||

(0.582) | ||||

| −0.513 | |||

(5.665) | ||||

| −5.225 | |||

(5.505) | ||||

Subject dummies | No | No | No | Yes |

– | – | – | – | |

Constant | 24.396 | 28.698 | 20.847 | 3.634 |

(3.011) | (4.938) | (1.573) | (7.330) | |

Observations | 960 | 960 | 960 | 960 |

Estimated bid functions for LR-PC auctions

(1) | (2) | (3) | (4) | |
---|---|---|---|---|

| 0.007 | 0.007 | 0.007 | 0.007 |

(0.012) | (0.012) | (0.012) | (0.012) | |

| 0.865 | 0.863 | 0.863 | 0.868 |

(0.038) | (0.039) | (0.039) | (0.037) | |

ln( | 0.353 | 0.353 | 0.276 | 0.274 |

(0.234) | (0.235) | (0.322) | (0.322) | |

| −4.938 | −5.432 | −8.424 | |

(3.696) | (4.235) | (5.273) | ||

ln( | 0.218 | 0.219 | ||

(0.241) | (0.242) | |||

| 0.170 | |||

(0.491) | ||||

| −4.651 | |||

(2.831) | ||||

| 0.276 | |||

(1.137) | ||||

Subject dummies | No | No | No | Yes |

– | – | – | – | |

Constant | 6.533 | 8.337 | 8.505 | 7.261 |

(1.784) | (2.504) | (2.641) | (1.576) | |

Observations | 960 | 960 | 960 | 960 |

## 5 Conclusion

In this paper we experimentally examine first-price and LRAs in two environments: one with private and common values, and other with pure common values. In an LRA, a bidder’s bid consists of the fixed amount of revenue from the common value of the good the bidder is willing to accept upon winning the auction. The lowest of these bids wins the auction. The winning bidder then incurs her cost.

Note that the uncertainty regarding the common value of the good is borne by the auctioneer in LRAs. The concept of such a risk sharing arrangement for infrastructure concession contracts has been theoretically studied in the past (Engel et al. 1997, 2001). Theory predicts that the allocative efficiency of LRAs will be higher than in first-price auctions.

Despite this advantage, a caveat regarding the general applicability of this format is in order: LRAs do not provide any incentive for the winner to invest in maintaining and enhancing the value of the good. This problem is mitigated if the ex-post value of the good is independent of the ex-post performance of the winning bidder. Alternatively, if the value of the good depends on easily monitored ex-post performance, a contract can be created which rewards and/or penalizes the winner contingent on ex-post performance. For instance, some energy markets (e.g. the PJM market in the US, the strategic reserve market in Sweden and Finland, the French energy market, etc.) employ capacity markets to secure adequate level of generation capabilities for the demand peaks and system outages. Since these markets are organized by a third party—typically the independent system operator or the transmission system operator—it is relatively easy to monitor both the revenues and the quality of the procured back-up generation service. If a company fails to provide the service during the outage, the system operator observes the fact the very same second. A contract can be easily designed to penalize such occurrences.

This paper is the first to examine, both theoretically and experimentally, allocative efficiency, bidding behavior and auction performance in LRAs. This paper is also, to the best of our knowledge, the first direct comparison of bidding behavior in first-price auctions with these two valuation structures. We do not find any significant effect of the valuation structure on the prevalence of the bidding above the break-even bidding threshold, the revenue generated, or bidder profits. This is surprising, given that theory predicts that the additional private information held by bidders when there are private and common value components of the valuation structure will lower revenue and make bidders better off. However, when estimating a cursed equilibrium bidding model, we find that the degree of cursedness is higher in the private and common valuation structure.

Perhaps the most interesting result is that, when there are private and common values, there are large increases in efficiency to be obtained by moving from a first-price auction to an LRA. The intuition underlying this result is clear: when there are private and common values, a bidder puts some weight on her common value signal when deciding her bid, while the efficiency is entirely determined by the private cost. As a result, the winning bidder may not have the lowest private cost, and thus the allocation may be inefficient. In an LRA, however, the common value signal is strategically irrelevant, and thus does not introduce inefficiency as in first-price auctions. This is, in effect, a limiting case of the finding in GO that a reduction in uncertainty regarding the common value component of the good reduces inefficiency.

The other noteworthy result is that, regardless of the valuation structure, bidding above the break-even threshold is significantly less prevalent in LRAs than in first-price auctions. Again, the intuition is due to the reduction of uncertainty in LRAs. In particular, in LRAs bidders do not need to estimate the expected common value of the good conditional on winning the auction in order to determine their expected profit. This is an important practical advantage in auctions for infrastructure concession contracts, as it allows bidders to focus on their cost, as opposed to estimating the uncertain common value and accounting for the information conveyed by winning the auction. Given the high rate of reported bankruptcy in infrastructure concessions allocated via traditional auction mechanisms (and the renegotiation that subsequently occurs), this result suggests that the use of LRAs may be preferred by policymakers.

## Footnotes

- 1.
- 2.
- 3.
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.

- 4.
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.

- 5.
See Kagel and Levin (2002) for an introduction to this literature.

- 6.
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.

- 7.
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.

- 8.
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).

- 9.
Derivations of this cursed equilibrium as well as the corresponding expected profit and auctioneer revenue can be found in the Electronic supplementary material.

- 10.
The derivations of this cursed equilibrium bid function, equilibrium bidder profits, equilibrium auctioneer revenue can be found in the Electronic supplementary material.

- 11.
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}. - 12.
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.

- 13.
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.

- 14.
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.

- 15.
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.

- 16.
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.

- 17.
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). - 18.
n.s. indicates that the test is not significant at conventional levels.

- 19.
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.). - 20.
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. - 21.
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.

- 22.
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.

- 23.
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.

- 24.
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. - 25.
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.

- 26.
We thank an anonymous referee for suggesting this analysis.

- 27.
This suggests that either bid formulation is a noisy process, there is bidder heterogeneity, or both.

- 28.
The Gauss code we used to obtain these results is available upon request.

- 29.
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.

- 30.
Recall that the equilibrium bid of LR-C bidders does not depend on the private information held by bidders.

- 31.
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. - 32.
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.

## Supplementary material

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