## Abstract

This paper studies the impact of human subjects in the role of a seller on bidding in experimental second-price auctions. Overbidding is a robust finding in second-price auctions, and spite among bidders has been advanced as an explanation. If spite extends to the seller, then the absence of human sellers who receive the auction revenue may bias upwards the bidding behavior in existing experimental auctions. We derive the equilibrium bidding function in a model where bidders have preferences regarding both the payoffs of other bidders and the seller’s revenue. Overbidding is optimal when buyers are spiteful only towards other buyers. However, optimal bids are lower and potentially even truthful when spite extends to the seller. We experimentally test the model predictions by exogenously varying the presence of human subjects in the roles of the seller and competing bidders. We do not detect a systematic effect of the presence of a human seller on overbidding. We conclude that overbidding is not an artefact of the standard experimental implementation of second-price auctions in which human sellers are absent.

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## Notes

Cramton and Ockenfels (2015) report some evidence for the presence of the spite motive in a high-stakes, real-world auction.

The two treatments without subjects in the role of sellers were previously reported in Bartling and Netzer (2016). See Rutström (1998), Lusk et al. (2004), and Flynn et al. (2016) for related papers on bidding in the SPA and the Becker–DeGroot–Marschak mechanism, which essentially constitutes an SPA against a computer bidder. Bartling and Netzer (2016) rely on the mechanism design approach by Bierbrauer and Netzer (2016) to derive an externality-robust version of the SPA and compare its performance to the standard SPA in a laboratory experiment. They conclude that externality-robustness and dominant-strategy robustness are equally important.

Restricting the parameters to a magnitude of less than one ensures that no player would pay one unit or more to increase or decrease another player’s payoff by one unit. Note that we assume, for reasons of simplicity, that it is common knowledge that bidders have identical other-regarding preferences.

In large competitive markets, by contrast, players who have interdependent preferences may act as if they had classical, self-centered preferences because they are not pivotal in determining the payoff of a seller (see e.g. Dufwenberg et al. 2011).

See Brandt et al. (2007) for a model in which the bidders may be exclusively spiteful and may not care about their own payoff at all. Interdependent preferences among the bidders also arise when bidders are firms who own shares of each other. The literature has investigated the effect of such cross-shareholdings on equilibrium bidding in the SPA and other auction formats (e.g. Ettinger 2003; Dasgupta and Tsui 2004; Chillemi 2005). In our model, symmetric cross-shareholdings correspond to the case where \(\alpha _S=0\) and \(\alpha _B < 0\).

We note that Engelbrecht-Wiggans (1994) analyzes a more general model with affiliated values. In his setting, the bidders’ concern for the revenue arises because a share of the revenue is paid out to them. Preferences in favor of a high revenue can also arise with toeholds in takeover contests (e.g. Burkart 1995; Singh 1998), in charity auctions (e.g. Engers and McManus 2007; Isaac et al. 2010), or with other financial externalities (e.g. Salmon and Isaac 2006; Maasland and Onderstal 2007).

Ettinger (2008) and Lu (2012) also allow for both types of interdependent preferences. However, Ettinger (2008) analyzes equilibrium bidding in the SPA (and other auction formats) only separately for the two types, while Lu (2012) derives optimal auctions and does not consider equilibrium bidding in standard auctions.

To avoid doubt among the subjects in the role of bidders that subjects in the role of the seller are really present in the experimental sessions, all subjects had to collect their written instructions from one of two piles upon entry into the laboratory. One pile was clearly labeled “bidder,” the other pile was clearly labeled “seller.” The bidders thus saw that there were instructions for subjects in the role of the seller and that some participants where instructed to take a copy from this pile when they entered the laboratory.

We had to conduct more sessions for SPA-S than for SPA and SPA-C, as for each pair of bidders a third subject in the role of the seller had to be present, such that fewer bidders could be placed in our laboratory. We had to conduct even more sessions for SPA-C-S, as each bidder had to be paired with a distinct seller.

We also measured subjects’ “joy of winning”, using a symbolic contest, and their “cognitive skills”, using a 12-item Raven test, but do not use these data in this paper. Details of the two measures and how they affect individual bidding behavior in SPA and SPA-C is provided in Bartling and Netzer (2016).

All Wilcoxon ranks-sum tests reported here are two-sided and compare averages of the 144 independent observations described in the design section, unless stated otherwise.

Note that average overbidding is smaller in SPA-C-S than in SPA-S (3.6 vs. 4.1), in line with the hypothesis of spite among bidders (Wilcoxon rank-sum test, \(p=0.0565\)). This difference is, however, not significant if we consider “artificial matching groups” in SPA-C-S that correspond to the matching groups that we implemented for the respectively paired subjects in SPA-S (\(p=0.9372\)). The failure to find clearer evidence for lower bids in SPA-C-S than in SPA-S might be due to the unsystematic effect of the presence of a human seller on bidding behavior, which gives rise to relatively low bids in SPA-S but relatively high bids in SPA-C-S. Using the respective artificial matching groups also for the comparison of average overbidding in SPA vs. SPA-C, SPA-C vs. SPA-C-S, and SPA & SPA-C vs. SPA-S & SPA-C-S does not change our previous results (Wilcoxon rank-sum tests, \(p=0.0152\), \(p=0.1796\), and \(p=0.4189\), respectively).

One exception is Ivanova-Stenzel and Kröger (2008), who combine an SPA with a buy-it-now offer that is made by the seller to a buyer prior to the SPA. However, they do not systematically vary the presence of the seller, as their interest is in studying the theoretical prediction that the price offer is always rejected and sales are made in the auction only. A second exception is Grebe et al. (2016), who study the effect of bargaining power and additionally consider sell-it-now offers, made by a buyer to the seller.

With the convention \(0^0=1\), the expression of \(\mathrm{EU}(v_1,\hat{v})\) is applicable to any \(n \ge 2\).

See Drichoutis et al. (2013) for an application of these tools in an auction setting related to ours.

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## Acknowledgements

We thank the editor, Nikos Nikiforakis, and two anonymous referees for valuable and constructive comments. Tobias Gesche acknowledges support from the Swiss National Science Foundation (SNSF).

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## Appendix

### Appendix

### 1.1 Proof of Proposition 1

We proceed in two steps. First, we derive a necessary optimality condition from which function \(\beta ^*\) is obtained. Second, we verify that \(\beta ^*\) is in fact a Bayes–Nash equilibrium.

*Step 1* Consider a candidate equilibrium in which all bidders adopt the strictly increasing and differentiable bidding function \(\beta\). Without loss of generality, take the perspective of bidder 1. By definition of equilibrium, he cannot strictly increase his expected utility by bidding \(\beta (\hat{v})\) for any “reported value” \(\hat{v} \ne v_1\) instead of \(\beta (v_1)\), for all \(v_1 \in [0,1]\). Expected utility as a function of \(v_1\) and \(\hat{v}\) can be written as

The term in the first line, abbreviated \(A(v_1,\hat{v})\) in the following, captures the cases in which bidder 1 wins, where multiplication by \(n-1\) ensures that the assumption of bidder 2 submitting the second-highest bid is without loss of generality. The second term, abbreviated \(B(v_1,\hat{v})\), captures the cases in which one of the \(n-1\) other bidders wins but the bid of 1 determines the price. The last term, abbreviated \(C(v_1,\hat{v})\), captures the cases in which one of the \(n-1\) other bidders wins and the price is determined by one of the remaining \(n-2\) bidders.^{Footnote 14}

We obtain the following derivatives:

The term in the second line of \(\partial B(v_1,\hat{v})/ \partial \hat{v}\) cancels with \(\partial C(v_1,\hat{v})/ \partial \hat{v}\). Simplifying then yields

Setting this expression zero at \(v_1 = \hat{v} =: v\) yields the necessary condition

for all \(v \in (0,1)\). If \(\alpha _S = \alpha _B\), this immediately implies \(\beta ^*(v)=v\), which can be extended to all \(v \in [0,1]\) by continuity of \(\beta ^*\). Otherwise, if \(\alpha _S \ne \alpha _B\), rearranging (2) yields

where \(k=(1+\alpha _B)/(\alpha _B - \alpha _S)\). Using the integrating factor \([1-F(v)]^k\), this differential equation can be solved to

for all \(v \in (0,1)\), where *K* is a constant of integration.

Suppose first that \(\alpha _S < \alpha _B\), which implies \(k >0\). It follows that \(\beta ^*(v)\) is unbounded as \(v \rightarrow 1\) except if \(K= \int _0^1 \left[ 1 - F(x) \right] ^k \mathrm{d}x\), in which case \(\lim _{v \rightarrow 1} \beta ^*(v)=1\). This determines the constant and yields

which can be extended to all \(v \in [0,1]\) by continuity.

Suppose now that \(\alpha _S > \alpha _B\), which implies \(k <0\). Since (2) then requires \(\beta ^*(v) < v\) for all \(v\in (0,1)\), we must have \(\lim _{v \rightarrow 0}\beta ^{*}(v)=K=0\). This yields

which can be extended to all \(v \in [0,1]\) by continuity.

*Step 2* It remains to be shown that \(\beta ^*\) as derived in Step 1 is in fact an equilibrium. We can first calculate the derivative

which verifies that \(\beta ^*\) is strictly increasing. If \(\alpha _S < \alpha _B\), substituting \(\beta ^*\) as well as its derivative into (1) yields after some simplifications

Hence expected utility is strictly increasing in \(\hat{v}\) when \(\hat{v} < v_1\) and strictly decreasing in \(\hat{v}\) when \(\hat{v} > v_1\), so that \(\hat{v}=v_1\) is in fact optimal. Proceeding analogously, exactly the same result is obtained for \(\alpha _S=\alpha _B\) and for \(\alpha _S>\alpha _B\). This completes the proof for the latter two cases, in which surjectiveness \(\beta ^*([0,1])=[0,1]\) implies that every possible bid can be made by some announcement \(\hat{v} \in [0,1]\). If \(\alpha _S<\alpha _B\), we have \(\beta ^*([0,1])=[K,1]\) for \(K=\int _0^1 [1-F(x)]^k \mathrm{d}x\). It is easily verified, however, that the expected utility of any bidder with any valuation is non-decreasing in the bid for \(b \in [0,K]\), provided all other bidders follow strategy \(\beta ^*\). This completes the proof also for \(\alpha _S<\alpha _B\). \(\square\)

### 1.2 Proof of Proposition 2

Fix any \(v \in (0,1)\). We are interested in the effect of \(\alpha _S\) on \(\beta ^*(v)\), keeping \(\alpha _B \in (-1,+1)\) fixed. Within each segment of the piecewise defined function \(\beta ^*\), \(\alpha _S\) affects \(\beta ^*(v)\) only through \(k=(1+\alpha _B)/(\alpha _B - \alpha _S)\). Note that \(\partial k / \partial \alpha _S = (1+\alpha _B)/(\alpha _B - \alpha _S)^2 > 0\) whenever \(\alpha _S \ne \alpha _B\). Now consider the effect of *k* on \(\beta ^*(v)\). We obtain

We claim that the expression for \(\alpha _S < \alpha _B\) is strictly negative. Indeed, this follows since

by the fact that \(\log [1-F(x)]\) is strictly decreasing in *x*. The analogous argument reveals that the expression for \(\alpha _S > \alpha _B\) is also strictly negative. We can thus conclude that \(\beta ^*(v)\) is strictly decreasing in \(\alpha _S\) whenever \(\alpha _S \ne \alpha _B\). We finally claim that \(\lim _{\alpha _S \nearrow \, \alpha _B} \beta ^*(v) = \lim _{\alpha _S \searrow \, \alpha _B} \beta ^*(v)=v\), so that \(\beta ^*(v)\) is continuous in \(\alpha _S\). Note that \(\lim _{\alpha _S \nearrow \, \alpha _B} k = + \infty\) and \(\lim _{\alpha _S \searrow \, \alpha _B} k = - \infty\). Note further that

and

This completes the proof. \(\square\)

### 1.3 Regression analyses

In the following, we present regression results which complement the non-parametric statistics in the main text. They are based on estimating a random effects model where a subject’s overbidding is the dependent variable. The independent variables are dummies for the treatments SPA-C, SPA-S, and SPA-C-S (such that SPA serves as the omitted category), together with a linear time trend over the 24 periods. We also perform the same analysis with a single dummy for the pooled treatments with a human seller (SPA-S & SPA-C-S).

A random effects model already accounts for correlation among multiple bids coming from a given subject. Hence we first perform the analysis with standard errors clustered on the subject level (“S”-clustering). However, there could also be spillovers across subjects within the same matching group. This should not be an issue in the treatments with a computer bidder, where interaction between bidders does not occur. Hence we repeat our analysis with standard errors clustered on the matching group level for treatments SPA and SPA-S (“MG”-clustering), but still on the subject level for treatments SPA-C and SPA-C-S. Finally, as a robustness check, we can also cluster on artificial matching groups for the treatments with computerized bidders, corresponding to the matching groups from which the values and bids were actually taken.

The results in Table 2 reflect the analysis in the main text. Columns 1–3 show that, while there is significant decrease in overbidding in treatment SPA-C compared to SPA, the treatments with a human seller (SPA-C and SPA-C-S) do not change overbidding in a statistically significant manner. When we pool the treatments with the presence of a seller (columns 4–6) we do not find a significant effect either. The regression framework also allows us to control for time effects, e.g. whether initial overbidding vanishes when subjects learn to play the auction over time. We observe the opposite: Overbidding increases significantly over the 24 periods.

### 1.4 Power calculations

Given our null result, it is worth to investigate whether our experimental setup had enough statistical power to detect a potential effect. In the above regression analysis, the treatment dummies’ standard errors are smallest when we have the largest number of clusters, i.e., when we cluster on the subject level. While this is the least conservative specification with respect to statistical significance, it has the highest statistical power and will thus form the basis of the following power analysis.

A power analysis asks about the minimal sample size that would be necessary to detect an effect of size *d* at significance level \(\alpha\) with at most probability \(\beta\) of committing the type-II error of falsely not rejecting the null. Liu and Wu (2005) provide the tools for this analysis for settings where the variable of interest can be correlated within sub-groups, e.g. when there are multiple observations from each subject.^{Footnote 15} Assuming two equally sized treatment groups, they derive the minimum necessary number of subjects \(\underline{S}\) as the following expression (Eq. (2) on p. 438):

In this formula, *N* is the number of observations per subject and the correlation among those observations is measured by \(\rho\). The parameter \(\sigma\) denotes the standard deviation of the dependent variable in the complete data set (with both treatment groups). The significance level \(\alpha\) (based on a two-sided test) and power level \(1-\beta\) are reflected in the values of the inverse of the standard normal distribution \(z_{1-\alpha /2}\) and \(z_{1-\beta }\).

We take the reference values for our power analysis from the comparison of the SPA to the SPA-C treatment, where we find a significant decrease in overbidding of size \(\hat{d}=3.149\). The empirical standard deviation of overbidding in these two treatments is \(\hat{\sigma }=13.172\). We also observe that the within-subject correlation is \(\hat{\rho }=0.376\), based on \(N=24\) observations per subject. Table 3 reports the solutions to (5) for parameter values around these estimates, assuming \(\alpha =0.05\) and \(1-\beta =0.8\).

We have 70 and 68 subjects in treatments SPA and SPA-S, and we have 64 and 68 subjects in treatments SPA-C and SPA-C-S. When pooling the treatments without a human seller (SPA & SPA-C) and those with a human seller (SPA-S & SPA-C-S), we can compare 134 and 136 subjects. Hence a potential effect of a size similar to the one we found between SPA and SPA-C would have been found for many plausible parameters values, either by separate or by pooled treatment comparisons.

It is important to keep in mind that this analysis addresses the issue of a false negative due to too large standard errors. This is relevant in the SPA vs. SPA-S comparison, where we find a slight but insignificant decrease in overbidding. In the SPA-C vs. SPA-C-S and the pooled treatment comparisons, by contrast, we find an *increase* when a human bidder is introduced. We can thus reject our initial hypothesis based on this directional effect alone.

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Bartling, B., Gesche, T. & Netzer, N. Does the absence of human sellers bias bidding behavior in auction experiments?.
*J Econ Sci Assoc* **3**, 44–61 (2017). https://doi.org/10.1007/s40881-017-0037-y

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DOI: https://doi.org/10.1007/s40881-017-0037-y