We present a simple operationalization of the environment in Akbarpour and Li (Reference Akbarpour and Li2020). There is a single, indivisible item for sale. Let $N\cup \left\{0\right\}$ be a finite set of agents, consisting of $\left|N\right|=n$ bidders and a seller (player 0). Let X be a set of outcomes, where an outcome $x=(y,t)$ consists of a winner $y\in N$ and a profile of payments $t\in R^n$ such that $t_i=0$ for $i\ne y$ . A type space is $\Theta ={\times }_{i\in N}{\Theta }_i$ , where ${\Theta }_i=[\underline{v},\overline{v}]$ . Let $F:{\Theta }_i⟶[0,1]$ denote a marginal distribution over types.Footnote ⁴ The distribution F is common knowledge. The bidding function of player i is given by $b_i:{\Theta }_i⟶R_{+}$ .

After observing the profile of bids $b=(b_1,b_2,\cdots ,b_n)$ , the seller selects the winner $y\in N$ and the price of the item $t_y\in [0,b_y]$ .Footnote ⁵ The feedback that bidders receive is captured by a partition ${\Omega }_i$ of X for each $i\in N$ , representing what bidder i directly observes about the outcome.Footnote ⁶ In particular, $(y,t),(y^{\prime },t^{\prime })\in {\omega }_i$ if and only if one of the following two conditions holds: (1) $y\ne i$ and $y^{\prime }\ne i$ , or (2) $y=y^{\prime }=i$ and $t_i=t_i^{\prime }$ . That is, each bidder observes whether she wins the item and observes her own payment.

We now describe the preferences of the players. Bidders have private values: $u_i(x,\theta )=1_{\left\{i=y\right\}}({\theta }_i-t_i)$ for player $i\in N$ . The seller receives the revenue from the auction: $u_0(t_y,b)=t_y$ . We focus attention on the symmetric Perfect Bayesian equilibrium in undominated strategies. Let $b^{\mathrm{FP}}$ , $b^{\mathrm{NCSP}}$ , $b^{\mathrm{CSP}}$ denote the symmetric and strictly increasing equilibrium bidding functions in the first-price, non-credible second-price, and credible second-price auctions, respectively.

Proposition 1

In equilibrium,

1. (i) the seller selects the highest bidder as the winner.

2. (ii) chooses a price equal to the highest bid. That is,

   $\begin{array}{c}{t}_y^{\ast }=max\left\{b_1,b_2,\cdots ,b_n\right\}.\end{array}$

3. (iii) for each bidder $i\in N$ and each value ${\theta }_i\in {\Theta }_i$ , we have that

   $\begin{array}{c}E\left(\underset{j\ne i}{max},{\theta }_j,|,\underset{j\ne i}{max},{\theta }_j,<,{\theta }_i\right)=b^{\mathrm{FP}}\left({\theta }_i\right)=b^{\mathrm{NCSP}}\left({\theta }_i\right)<b^{\mathrm{CSP}}\left({\theta }_i\right)={\theta }_i.\end{array}$

We omit the proof of Proposition Footnote ¹ since it follows directly from the results presented in Akbarpour and Li (Reference Akbarpour and Li2020). However, we note that Proposition Footnote ¹ provides a clear testable prediction that the NCSP auction is theoretically equivalent to the FP auction.

3.1 Implementation

The experimental sessions were run at the Laboratory for Economic and Decision Research (LEDR) at the University of East Anglia. For the main set of experiments, we recruited a total of 285 subjects across 13 sessions (with either 18, 21, or 24 subjects per session). Each session lasted approximately 60 min. The experiment was programmed and conducted with the z-Tree software (Fischbacher, Reference Fischbacher2007). Table 1 provides more detailed summary statistics.

Table 1 Overview of experimental design

Treatment	Active seller	# of sessions	# of subjects	Average earnings
First-price (FP)	✗	3	72	£19.59
Credible second-price (CSP)	✗	3	69	£20.13
Non-credible second-price (NCSP)	$✓$	7	144	£20.34
All		13	285	£20.10

4.1 Bidder behavior

We begin by investigating bidding behavior. For the case of two risk-neutral bidders with values independently and uniformly drawn from [0, 100], the symmetric Perfect Bayesian Equilibrium (PBE) predictions in undominated strategies are given by

$\begin{array}{c}{b}^{\mathrm{FP}}\left({\theta }_i\right)=b^{\mathrm{NCSP}}\left({\theta }_i\right)=0.5{\theta }_i<{\theta }_i=b^{\mathrm{CSP}}\left({\theta }_i\right).\end{array}$

Fig. 1 Observed and theoretical bids

Figure 1 shows scatter plots of observed bids/values in the three treatments, along with the theoretical predictions in solid lines. It is clear that overbidding relative to the equilibrium prediction is common in the FP and NCSP auctions, occurring in 86% (411/480) and 89% (853/960) of bid-value pairs, respectively. In the CSP auction, however, subjects have a dominant strategy of bidding their true values. Although overbidding still occurs in 39% (181/460) of cases, 18% (85/460) of cases correspond to dominant-strategy play. If we relax this condition to include subjects who bid either 1 token below or 1 token above their true values, then 28% (127/460) of cases correspond to dominant-strategy play.

As a next step, we conduct linear estimations of the bidding functions (forcing the intercepts to pass through zero).Footnote ¹¹ Regression 1 in Table 2 shows the estimated bidding functions for the three treatments, with the NCSP auction as the baseline. In particular, we have that

$\underset{0.74{\theta }_i}{\underset{\Vert }{{\hat{b}}^{\mathrm{FP}}\left({\theta }_i\right)}}{<}^{\ast \ast \ast }\underset{0.83{\theta }_i}{\underset{\Vert }{{\hat{b}}^{\mathrm{NCSP}}\left({\theta }_i\right)}}{<}^{\ast \ast \ast }\underset{1.01{\theta }_i}{\underset{\Vert }{{\hat{b}}^{\mathrm{CSP}}\left({\theta }_i\right)}}$

Notice that the estimated slopes of the bidding functions are significantly different at the 1% level. This generates two main insights. First, contrary to theoretical predictions, the NCSP auction fails to converge to the FP auction. Second, the NCSP auction is still behaviorally distinct from the CSP auction. Overall, bidders’ behavior is consistent with the belief that sellers in the NCSP auction are choosing an intermediate price between the highest and second-highest bids. We will return to this intuition later when we investigate seller behavior.

Table 2 Linear estimates of the bidding functions (passing through the origin). Standard errors are clustered at the subject level

Dependent variable: Bid	All treatments		FP	CSP	NCSP
	Rounds 1–10	Rounds 6–10
	(1)	(2)	(3)	(4)	(5)
Value	0.834***	0.845***	0.757***	0.984***	0.845***
	(0.015)	(0.019)	(0.035)	(0.038)	(0.028)
Value* $1_{\left\{\mathrm{FP}\right\}}$	− 0.097***	− 0.106**
	(0.027)	(0.031)
Value* $1_{\left\{\mathrm{CSP}\right\}}$	0.180***	0.252***
	(0.029)	(0.036)
Value* $1_{\left\{Risk-Averse\right\}}$			− 0.025	0.045	− 0.015
			(0.044)	(0.050)	(0.033)
Observations	1900	950	480	460	960
Number of subjects	190	190	48	46	96

Significance levels are indicated as follows: ** $p<0.05$ , *** $p<0.01$

We conduct several robustness checks. First, we account for the possibility of subject learning over time. Regression 2 in Table 2 estimates the bidding functions only using data from the last five rounds of the experiment. Our qualitative results remain unchanged. Second, we investigate the effect of risk preference on bidding behavior.Footnote ¹² We do this separately for each treatment, since the interplay between risk preference and bidding behavior varies across the different auction formats. The results are shown in Regressions 3–5 in Table 2. In all three auctions, we find that risk-averse subjects do not bid significantly differently than other subjects.Footnote ¹³ This finding is consistent with the theoretical prediction for the CSP auction (where subjects have a dominant strategy), but inconsistent with the theoretical predictions for the FP and NCSP auctions (where risk-averse subjects should bid more aggressively).

Finally, we analyze bidding behavior at the subject level. The data for each bidder consists of 10 pairs of bids and values (one for each round of the experiment). For each bidder, we calculate her bid/value coefficient as the average of her 10 bid/value ratios. The cumulative distribution functions (CDFs) of the bid/value coefficients are shown in Fig. 2. Several patterns are clear by inspection. First, the fact that bid/value coefficients exceed 1 for many subjects reveals the prevalence of dominated strategies in the data. Second, we observe a first-order stochastic dominance relationship when comparing the CDFs of either the FP or NCSP treatments to the CSP treatment. Using a Kolmogorov–Smirnov test, we can also reject the null hypothesis that bid/value coefficients are drawn from the same underlying distribution in all three pairwise comparisons ( $p<0.01$ in all cases). This further underscores the point that bidding behavior is significantly different across the three auction institutions.

Fig. 2 CDFs of bid/value coefficients

4.2 Seller Behavior

We now investigate the behavior of sellers. In the credible auctions (FP and CSP treatments), the seller’s role is passive. Thus, our data consists only of the 48 sellers who participate in the NCSP treatment. Recall that each seller makes two choices in each round: she decides which bidder wins the item and the price of the item. We find that the seller selects the highest bidder as the winner in 99% (476/480) of cases. This behavior is consistent with both the dominant strategy of the seller and the rules of the auction. Furthermore, the fact that sellers nearly always allocate the item to the highest bidder provides a basic rationality check and rules out subject confusion as an explanation for our results.

We now explore the degree to which sellers break the rules of the auction when making pricing decisions. To do so, we define the “overcharging ratio” $s_{\mathrm{it}}$ for seller i in round t of the experiment as

$\begin{array}{c}{s}_{\mathrm{it}}=\frac{{\mathrm{price}}_{\mathrm{it}}-b_{\mathrm{it}}^{\min }}{b_{\mathrm{it}}^{\max }-b_{\mathrm{it}}^{\min }},\end{array}$

where $pric{e}_{\mathrm{it}}$ is seller i’s chosen price in round t, $b_{\mathrm{it}}^{\max }$ is the highest bid in seller i’s group in round t, and $b_{\mathrm{it}}^{\min }$ is the lowest bid in seller i’s group in round t.Footnote ¹⁴ The ratio $s_{\mathrm{it}}$ captures the share of surplus that is extracted by seller i in round t. If $s_{\mathrm{it}}=0$ , then the seller chooses a price equal to the lowest bid (i.e., she extracts no additional surplus beyond the amount prescribed by the rules of the auction). If $s_{\mathrm{it}}=1$ , then the seller chooses a price equal to the highest bid (i.e., she extracts the maximum possible surplus).Footnote ¹⁵

We first investigate whether sellers’ overcharging behavior changes with experience, and whether overcharging is correlated with a traditional measure of lying aversion taken from the literature.Footnote ¹⁶ The results are shown in Regressions 1–2 of Table 3. We find that sellers overcharge 29 percentage points more in the second half of the experiment than in the first half, and that lie-averse sellers overcharge 33 percentage points less than sellers who are not lie-averse. Although both of these results are consistent with our intuition, only the latter effect is statistically significant at conventional levels.

Table 3 OLS regressions of seller behavior in the NCSP auction. Standard errors are clustered at the subject level.

	(1)	(2)	(3)
	Overcharging ratio	Overcharging ratio	Decision to overcharge
$1_{\left\{Round>5\right\}}$	0.289
	(0.210)
$1_{\left\{Lie-Averse\right\}}$		− 0.333**
		(0.135)
Bid spread			0.004***
			(0.001)
Constant	0.456**	0.901***	0.690***
	(0.218)	(0.039)	(0.058)
Observations	470	470	470
Number of subjects	48	48	48

Significance levels are indicated as follows: ** $p<0.05$ , *** $p<0.01$

Table 4 Distribution of seller types in the NCSP auction

Seller type	# of subjects	Average $s_i$	Average $s_i|s_{\mathrm{it}}>0$
Never overcharge	3	0	–
Sometimes overcharge	22	0.36	0.77
Always overcharge	23	0.91	0.91
All	48	0.60	0.84

We now investigate the prevalence of different seller types in our experiment. In particular, we calculate the “overcharging coefficient” $s_i$ of seller i as the average of her 10 overcharging ratios (one for each round of the experiment). This allows us to classify sellers as one of three possible types: sellers who never overcharge, sellers who sometimes overcharge, and sellers who always overcharge. Table 4 shows the distribution of seller types in our experiment. It is clear that overcharging is the norm: only three sellers never overcharge in all rounds of the experiment (i.e., $s_i=0$ ). We find that the 22 sellers who sometimes overcharge extract an average of 36% of the available surplus, while the 23 sellers who always overcharge extract an average of 91% of the available surplus. This difference is both economically and statistically significant (Mann–Whitney test, $p<0.001$ ). Finally, of the 23 sellers who always overcharge, only six sellers consistently extract the maximum surplus (i.e., $s_i=1$ ).

Taken together, our results suggest that most sellers in the NCSP auction are not profit-maximizing. We now analyze whether there are any systematic patterns in sellers’ deviations from profit-maximizing behavior. We focus on two main questions. First, do sellers who sometimes overcharge behave similarly to sellers who always overcharge, conditional on the decision to break the rules of the auction? We find that this is not the case. Conditional on overcharging, sellers who sometimes overcharge extract an average of 77% of the available surplus, which is still significantly different than the rate among sellers who always overcharge (Mann–Whitney test, $p=0.010$ ). Second, are sellers responsive to the size of the pecuniary gains associated with breaking the rules of the auction? In principle, the seller’s likelihood of overcharging the winner might be increasing in the difference between the maximum and minimum bids (i.e., the bid spread). To test this hypothesis, we estimate an OLS regression of a dummy variable for overcharging (i.e., $s_{\mathrm{it}}>0$ ) on the bid spread.Footnote ¹⁷ The results are shown in Regression 3 of Table 3. We find a statistically significant effect: in particular, every 10 token increase in the bid spread increases the seller’s probability of overcharging the winner by four percentage points.

4.3 Aggregate outcomes

We now compare efficiency and revenue across the different auction formats. We use standard definitions: the outcome of an auction is efficient if the bidder with the highest value wins the item, and the revenue from an auction is the price of the item. If subjects use symmetric and strictly increasing bidding functions, then all three auctions are efficient. If these bidding functions are also an equilibrium, then all three auctions yield the same expected revenue for the seller.Footnote ¹⁸ In the case of two bidders with values independently and uniformly distributed on [0, 100], the expected revenue is 33.

Table 5 Average efficiency and revenue

Property	Rounds 1–10					Rounds 6–10
	Treatment					Treatment
	CSP		FP		NCSP	CSP		FP		NCSP
Efficiency	0.80	<	0.82	<	0.84	0.85	${>}^{\ast \ast }$	0.79	<	0.84
Theor. Efficiency	1	$=$	1	$=$	1	1	$=$	1	$=$	1
Revenue	36	${<}^{\ast \ast \ast }$	50	${<}^{\ast }$	53	38	${<}^{\ast \ast }$	49	<	54
Theor. Revenue	33	$=$	33	$=$	33	33	$=$	33	$=$	33

Significance levels are indicated as follows: * $p<0.10$ , ** $p<0.05$ , *** $p<0.01$

Table 5 documents the average efficiency and revenue in each treatment.Footnote ¹⁹ We observe high rates of efficiency in our experimental auction markets: the fraction of efficient outcomes is at least 80% in all three treatments.Footnote ²⁰ Moreover, the differences in efficiency across the three treatments are not statistically significant. This finding is also robust to subject experience. When restricting attention to the last five rounds of the experiment, we observe similarly high rates of efficiency and only one significant treatment effect among the three pairwise comparisons.

We do observe significant differences in revenue across the three treatments. In particular, we find that the CSP auction generates the least revenue while the NCSP auction generates the most revenue. However, the only robust finding is the revenue inferiority of the CSP auction. The additional revenue generated by the NCSP auction over the FP auction is not economically significant and also fails to be statistically significant in the last five rounds of the experiment.

5.1 Behavioral model

We consider a behavioral seller who is averse to rule-breaking. Since we restrict attention to equilibria in undominated strategies, the seller would never choose a price below the second-highest bid. Thus, we can characterize the price $t_y$ by a parameter $\alpha \in [0,1]$ as follows:

$\begin{array}{c}{t}_y=\alpha b_{\left(1\right)}+(1-\alpha )b_{\left(2\right)},\end{array}$

where $b_{\left(1\right)}$ is the highest bid and $b_{\left(2\right)}$ is the second-highest bid.

For the seller, we allow for both a fixed cost and a variable cost of rule-breaking. The fixed cost of rule-breaking is represented by the parameter $\gamma >0$ . The variable cost of rule-breaking is represented by the function

$\begin{array}{c}c\left(\alpha \right)(b_{\left(1\right)}-b_{\left(2\right)}),\end{array}$

where $c\left(\alpha \right)$ is a continuous, monotone, and concave function such that $c\left(0\right)=0$ , $c^{\prime }\left(0\right)\le 1$ , and $c^{\prime }\left(1\right)\ge 1$ . Then, the seller’s utility function is as follows:

$\begin{array}{c}u\left(\alpha \right)=\alpha b_{\left(1\right)}+(1-\alpha )b_{\left(2\right)}-c\left(\alpha \right)(b_{\left(1\right)}-b_{\left(2\right)})-\gamma 1_{\{\alpha >0\}}.\end{array}$

We can now update the theoretical predictions for the case of a behavioral seller. Recall that $b^{\mathrm{FP}}$ , $b^{\mathrm{NCSP}}$ , $b^{\mathrm{CSP}}$ denote the symmetric and strictly increasing equilibrium bidding functions in the first-price, non-credible second-price, and credible second-price auctions, respectively.

5.2 Revisiting the NCSP auction

5.3 Experimental design

While the predictions of the behavioral model are consistent with our experimental results, there are other possible forces at play. For example, if the seller has other-regarding preferences, then she would also choose an intermediate price between the highest and second-highest bids. In this section, we design a new experimental treatment to directly test the mechanism proposed in the behavioral model. In particular, we define a “no-rules” auction (NR) that is procedurally identical to the NCSP auction but that differs in framing. In both auctions, the seller has agency to determine the outcome of the auction (i.e., the winner and the price of the item). However, the experimental instructions vary across the two treatments. In the NCSP auction, both the rules of the auction and the seller’s strategy space are described to subjects. In the NR auction, the rules are omitted and only the seller’s strategy space is described to subjects.Footnote ²¹ Comparing sellers’ pricing behavior across the NR and NCSP auctions allows us to pin down whether sellers’ aversion to rule-breaking is the driving force behind our experimental results.

5.4 Experimental results

We first conduct a linear estimation of the bidding function in the NR auction (forcing the intercept to pass through zero). Ordering the estimated bidding functions, we find that

$\underset{0.74{\theta }_i}{\underset{\Vert }{{\hat{b}}^{\mathrm{FP}}\left({\theta }_i\right)}}{<}^{\ast \ast \ast }\underset{0.75{\theta }_i}{\underset{\Vert }{{\hat{b}}^{\mathrm{NR}}\left({\theta }_i\right)}}{<}^{\ast \ast \ast }\underset{0.83{\theta }_i}{\underset{\Vert }{{\hat{b}}^{\mathrm{NCSP}}\left({\theta }_i\right)}}$

Several insights emerge from these comparisons. First, bidding behavior is significantly different between the NCSP and NR auctions. This suggests that bidders believe that sellers will behave differently in the absence of auction rules. Second, there is no significant difference in bidding behavior between the FP and NR auctions. This allows us to draw a further inference: in the absence of auction rules, bidders’ behavior is consistent with the belief that sellers will extract the maximum possible surplus.

We now compare the behavior of sellers between the NCSP and NR auctions. To do so, we estimate OLS regressions of both the overcharging ratio ( $s_{\mathrm{it}}$ ) and a dummy variable for overcharging ( $s_{\mathrm{it}}>0$ ) on a treatment dummy variable for the NR auction. The results are shown in Table 6. We observe statistically significant differences in seller behavior on both the intensive and extensive margins. In particular, sellers in the NR auction extract 27 percentage points more of the available surplus and are 14 percentage points more likely to overcharge the winner than in the NCSP auction.

Clearly, from the perspectives of both sides of the market, the NCSP and NR auctions are distinct mechanisms. In the absence of auction rules, sellers’ pricing behavior is closer to the FP auction. Bidders, in turn, correctly anticipate this and behave as if they are participating in a FP auction. Overall, the NR treatment demonstrates that sellers’ aversion to rule-breaking plays a large role in explaining the results of the NCSP auction.

Table 6 OLS regressions of seller behavior in the NCSP and NR auctions. Standard errors are clustered at the subject level

	(1)	(2)
	Overcharging ratio	Decision to overcharge
$1_{\left\{\mathrm{NR}\right\}}$	0.269**	0.137***
	(0.122)	(0.044)
Constant	0.603***	0.817***
	(0.116)	(0.041)
Observations	707	707
Number of subjects	72	72

Significance levels are indicated as follows: ** $p<0.05$ , *** $p<0.01$