1.1 Motivation and outline

The technology for the design of economic mechanisms has grown rapidly in the last few decades. Practical applications of economic mechanisms now include spectrum auctions (Milgrom Reference Milgrom2000), school choice (Abdulkadiroğlu and Sönmez Reference Abdulkadiroğlu and Sönmez2003), and kidney exchange (Roth et al. Reference Roth, Sömez and Ünver2004), among many others. A preferred property of such mechanisms is strategy-proofness, which ensures that truthfully reporting his/her own type is always a weakly dominant strategy for every participant. Provided that a strategy-proof mechanism is employed, everyone has an incentive to report his/her true type to ensure that a social goal is realized through the collection of accurate information.

However, in the experimental literature on strategy-proof mechanisms, it has often been observed that telling the truth is far from universal (e.g., Attiyeh et al. (Reference Attiyeh, Franciosi and Isaac2000), Kawagoe and Mori (Reference Kawagoe and Mori2001), Cason et al. (Reference Cason, Saijo, Sjöström and Yamato2006), and Chen and Sönmez (Reference Chen and Sönmez2006)).Footnote ¹ These observations have been considered an anomaly to the theory, encouraging skepticism of the practical performance of strategy-proof mechanisms. A prominent example is the Vickrey auction (Vickrey Reference Vickrey1961), under which many experimental studies have observed overbidding.Footnote ² However, this does not imply that the strategy-proofness of the Vickrey auction is useless in reality. Simply put, it suggests the necessity of developing some device that enables strategy-proofness to work, making the Vickrey auction useful in real-life applications.

For an average person, it is not easy to realize that the Vickrey auction is strategy-proof.Footnote ³ That is, the Vickrey auction does not exhibit the expected performance by merely explaining the auction rule to participants. A simple method to address this issue is to announce the strategy-proofness of the Vickrey auction to users, which is what we shall attempt in this experimental research.

In our experiment, we provide a piece of simple advice on the strategy-proofness of the Vickrey auction to the subjects and examine its effect on the subjects’ behavior and the efficiency of outcomes. Our experiment focuses on a case where multiple units of identical objects are to be sold under the Vickrey auction. We observe that, although the rate of sincere bidding is 20.6% without advice, it increases to 46.9% once subjects are advised regarding the strategy-proofness of the Vickrey auction.

One might suspect that this increase is due to the so-called experimenter demand effect (Zizzo Reference Zizzo2009). Thus, we attempt to detect whether an effect of advice exists beyond the experimenter demand effect, which we call the net advice effect. To this end, we conducted similar experiments for the pay-your-bid auction, which is not strategy-proof.Footnote ⁴ We used the same advice across auction rules to cause similar experimenter demand effects in both auctions.Footnote ⁵ The comparison results for the pay-your-bid auction reveal that the Vickrey auction has a large net advice effect. Moreover, for the Vickrey auction, the effect of the advice is stronger for the subjects with a perfect score in a quiz on the auction rules than for the other subjects.Footnote ⁶

In the literature on auction theory, overbidding in the Vickrey/second price auction has been attributed to various factors such as the joy of winning (Cooper and Fang Reference Cooper and Fang2008), the presence of a cognitive limit on contingent reasoning (Li Reference Li2017), the underestimation of possible losses (Georganas et al. Reference Georganas, Levin and McGee2017), and so on. However, our results show the possibility that merely providing a piece of simple advice decreases overbidding by promoting a better understanding of the strategy-proofness of the Vickrey auction. Our experimental results suggest that, before operating strategy-proof mechanisms, explaining the property of strategy-proofness helps those who are unaware of its property to behave “correctly.”

Finally, we conducted an experiment to test the effect of advice for the Ausubel auction (Ausubel Reference Ausubel2004), which is known as a dynamic counterpart of the Vickrey auction.Footnote ⁷ This auction rule is not strategy-proof because in it, sincere bidding, where bidders may reduce their demand for units at prices equal to their valuations for these units, is not a weakly dominant strategy. Meanwhile, it is well known that in the Ausubel auction, sincere bidding is an ex-post equilibrium (Ausubel Reference Ausubel2004; Okamoto Reference Okamoto2018). Thus, based on our experimental results above, one would expect that providing simple advice on this property would increase the rate of sincere bidding in the Ausubel auction, similar to that in the Vickrey auction. Although the Ausubel auction is a dynamic mechanism, we set the instruction and advice in the Ausubel auction treatment as similar to the main experiments to the extent possible, but without making deception. We find that the advice effect also exists in the Ausubel auction. This finding suggests that providing “appropriate” advice could serve as a tool to improve the performance of even non-strategy-proof mechanisms.

1.2 Related literature

The effect of providing advice on strategy-proof mechanisms has recently been investigated by several authors (Guillen and Hing Reference Guillen and Hing2014; Ding and Schotter Reference Ding and Schotter2019). In these studies, two-sided matching models are used to verify whether providing advice will affect the rate of truth-telling. A closely related work by Guillen and Hakimov (Reference Guillen and Hakimov2018) finds that announcing the strategy-proofness of strategy-proof matching mechanisms increases the rate of truth-telling, while providing a detailed explanation of the mechanism has the opposite effect.Footnote ⁸ These results suggest that advice helps participants confirm individual optimality of truth-telling. To the best of our knowledge, our study is the first to test the role of advice in the literature of auction studies.Footnote ⁹

To improve individuals’ understanding on incentive properties, Saijo et al. (Reference Saijo, Sjöström and Yamato2007) and Li (Reference Li2017) attempt to strengthen the notion of strategy-proofness. Saijo et al. (Reference Saijo, Sjöström and Yamato2007) emphasize that, even if a mechanism is strategy-proof, it often admits the presence of an inefficient Nash equilibrium composed of mis-reporting. Saijo et al. (Reference Saijo, Sjöström and Yamato2007) strengthen strategy-proofness to “secure implementability” by additionally requiring that no Nash equilibrium achieves an outcome that is not the “true” outcome. An experimental study by Cason et al. (Reference Cason, Saijo, Sjöström and Yamato2006) compares bidding behavior under a securely implementable (and so strategy-proof) mechanism and that of other strategy-proof mechanisms. They observe that the rate of truth-telling in the former is higher than in the latter.

Li (Reference Li2017) proposes a stronger version of strategy-proofness called “obvious strategy-proofness” that could be applied to dynamic mechanisms, such as ascending auctions or matching algorithms. In a problem where a single object is auctioned, the ascending auction is obviously strategy-proof, but the Vickrey auction is not. In an experiment where a single object is auctioned, Li observes that the ratio of dominant strategies played by subjects is significantly higher in the ascending clock auction than in the Vickrey auction.

However, given that the class of strategy-proof mechanisms is already narrow (e.g., Holmström (Reference Holmström1979)), strengthening strategy-proofness severely restricts the admissible class of mechanisms. Conversely, our approach of providing advice can be applied to any strategy-proof mechanism in any environment. The Vickrey auction is neither securely implementable nor obviously strategy-proof, but providing advice drastically improves its truth-telling performance.

1.3 Organization

The remainder of the paper is organized as follows. Section 2 describes the theoretical background and our experimental design and proposes several hypotheses. Section 3 presents our experimental results. Section 4 describes the Ausubel auction treatment in detail and discusses our experimental results on the Ausubel auction. Finally, Sect. 5 provides the concluding remarks.

2.1 Theoretical considerations

There are three bidders, $\{1,2,3\}$ , and two indivisible, identical objects to be auctioned. Each bidder is admitted to demand two units. Bidder i’s valuation for the objects is denoted by $v_i=(v_i^1,v_i^2)$ , where $v_i^1\ge v_i^2\ge 0$ and $v_i^j$ denotes the value that bidder i assigns to the j-th unit. Given any $v_i$ , bidder i’s utility of obtaining k units of objects and paying $m_i$ units of money is:Footnote ¹⁰

$\begin{array}{c}U(k,m_i;v_i)\equiv \left(\begin{array}{cc}{v}_i^1+v_i^2-m_i& \text{if}k=2\\ v_i^1-m_i& \text{if}k=1\\ -m_i& \text{if}k=0.\end{array}\right)\end{array}$

A list $v=(v_1,v_2,v_3)$ is a valuation profile. A bid submitted by bidder i is denoted by $b_i=(b_i^1,b_i^2)$ , where $b_i^1\ge b_i^2\ge 0$ and $b_i^j$ denotes the bid of bidder i for the j-th unit. Let $b=(b_1,b_2,b_3)$ be a bid profile.

An assignment function is a function $d=(d_1,d_2,d_3)$ that specifies, for each bid profile b, the number of the objects bidder i obtains $d_i\left(b\right)\in \{0,1,2\}$ under the following resource constraint: $d_1\left(b\right)+d_2\left(b\right)+d_3\left(b\right)=2$ . A payment function is a function $m=(m_1,m_2,m_3)$ that specifies, for each bid profile b, the amount of money bidder i pays $m_i\left(b\right)$ . An rule is a pair of assignment and payment functions (d, m).

We are interested in rules in which sincere bidding (i.e., bidding the true valuations) is a weakly dominant strategy for everyone. Formally:

Strategy-proofness:

For each bidder i, each bidder i’s valuation $v_i$ , each bidder i’s bid $b_i$ , and each other bidders’ bids $b_{-i}$ ,

$\begin{array}{c}U(d_i(v_i,b_{-i}),m_i(v_i,b_{-i});v_i)\ge U(d_i(b_i,b_{-i}),m_i(b_i,b_{-i}));v_i).\end{array}$

The following rules are central in the literature on auction theory.

• • Vickrey auction: Each bidder i simultaneously submits his/her bid $b_i=(b_i^1,b_i^2)$ . After the seller collects all bidders’ bids, the seller ranks them from the highest to the lowest bid and allocates the two units to the two highest bids. If bidder i wins one (respectively, two) unit, then the bidder has to pay the highest bids (respectively, the sum of the highest and the second-highest bids) from among the other bidders’ losing bids.

• • Pay-your-bid auction: Each bidder i simultaneously submits hies/her bid $b_i=(b_i^1,b_i^2)$ . After the seller collects all bidders’ bids, the seller ranks them from the highest to the lowest bid and allocates the two units to the two highest bids. If bidder i wins one (respectively, two) unit, then the bidder has to pay $b_i^1$ (respectively, $b_i^1+b_i^2$ ).

The most important feature of the Vickrey auction is that it is strategy-proof. However, many previous experimental studies report that bidders tend to overbid in Vickrey auctions. The pay-your-bid auction is widely used in real life, while the Vickrey auction is not so. However, in contrast to the Vickrey auction, the pay-your-bid auction is not strategy-proof.

2.2 Experimental procedures

We conducted an experiment to test the effect of advice for the Vickrey auction. We used between-subjects design. To distinguish between whether subjects follow the advice because they understand it or obey the advice without understanding it, we also conducted corresponding experiments for the pay-your-bid auction. We have four types of treatments in total:

• • Treatment VA: the Vickrey auction with advice

• • Treatment VN: the Vickrey auction without advice

• • Treatment PA: the pay-your-bid auction with advice

• • Treatment PN: the pay-your-bid auction without advice

In each treatment, three bidders compete for two units of identical objects. For each bidder, two integer valuations are drawn from the uniform distribution with the interval 0 to 1000 with increments of 10. Then, the larger (respectively, smaller) integer is assigned the valuation for the first (respectively, second) unit. All valuations are displayed in Japanese yen (JPY).

VA, PA, and PN each had three experimental sessions, whereas VN had two sessions. We conducted these 11 sessions at Osaka University in March 2015 and July 2016. We recruited student subjects from Osaka University through campus-wide advertisements. None of the students was experienced in this particular type of experiment. No subject attended more than one session. Our experiment was computerized using the experimental software z-Tree (Fischbacher Reference Fischbacher2007). Twenty-one or twenty-four subjects participated in each session. Table 1 summarizes the number of observations. Figure 1 illustrates the timeline of one session.

Table 1

Summary of treatments

Treatment	Auction rule	Advice	Date	# of Sessions	# of Subjects (Groups)
VA	Vickrey	Yes	Mar-15	3	63 (21)
VN	Vickrey	No	Mar-15	2	48 (16)
PA	Pay-your-bid	Yes	Jul-16	3	69 (23)
PN	Pay-your-bid	No	Jul-16	3	72 (24)

Fig. 1

Timeline of one session

Each subject was seated at a computer terminal assigned by a lottery. All terminals were separated by partitions. No communication among subjects was allowed. Each subject had a set of printed instructions and a recording sheet.Footnote ¹¹ The experimenter read aloud the instructions. Then, subjects answered a 17-question quiz that tested whether they understood the auction rule that they had been informed of a short time ago. Every correct answer was worth $0.3 ($1 $=$ JPY 100). The experimenter read aloud the answers to the quiz. Subsequently, only in VA and PA, the experimenter distributed a paper with written advice and also read it aloud. The text of the advice is as below:

Note that the advice involves no deception problem for VA and PA. The subjects were given time to ask questions before proceeding to two practice periods and to the 25 successive payment periods under the random matching protocol. At the beginning of each period, all subjects were separated into groups of three. At the bidding stage, each subject was asked to enter his/her two bids as nonnegative integers—such that the first unit bid is weakly greater than that of the second unit bid—into a box on the display screen. We set the maximum feasible first unit bid to 2000.

After the 25 payment periods, the subjects completed a questionnaire and were immediately paid in cash. Each subject was privately paid the sum of his/her earnings over the 25 periods. Individual payments including (or excluding) participation fees and quiz points ranged from $19.1 to $85.4 (or from $5.9 to $70.3).

2.3 Hypotheses

3.1 Level of understanding of the auction rules

Fig. 2

Distribution of quiz scores

3.2 Bidding behavior

For a given unit, we say that a bid is sincere if it matches exactly the valuation drawn for the unit. Similarly, we say that a bid is over (or under) if the bid is more (or less) than the valuation. For example, if valuations and bids are $v_1=(800,300)$ , $v_2=(600,400)$ , $v_3=(900,500)$ , $b_1=(850,240)$ , $b_2=(600,400)$ , and $b_3=(900,0)$ , then the sincere bids are the first and second units of bidder 2 and the first unit of bidder 3, respectively. The unique overbid is bidder 1’s first unit, while the remaining two bids are underbids. Hence, the overall average sincere bid rate is 0.5 (i.e., we divide the number of sincere bids by the total number of bids by the three bidders), whereas the average sincere bid rate of bidders 1, 2, and 3 is 0, 1, and 0.5, respectively.

Fig. 3

Scatter plots of bids. Green $\times$ and purple $\bullet$ indicate the bids for the former 13 periods and latter 12 periods, respectively

Figure 3 shows the scatter plots for valuations and bids in each treatment. Initially, the graphs for the Vickrey treatments suggest that, for unit 1, bidders with higher valuations are more likely to overbid. When valuations are close to the maximum, 1000, bids jump as high as 2000. For unit 2, bidding behavior is polarized with an increase in underbidding: overbidding is prevalent regardless of the realized valuations. Further, we observe a cluster of zero biddings when the valuation for unit 2 is no more than 500. Table 2 summarizes the classification of bids into three categories by treatments and periods.Footnote ¹⁵

Table 2

Bid category by treatments and periods

Bid category	Vickrey		VA − VN	Pay-your-bid		PA − PN
	VA	VN		PA	PN
(a) All
Sincere	0.469	0.206	0.263 ${}^{\ast \ast \ast }$	0.083	0.058	0.025 ${}^{\ast \ast \ast }$
	(0.499)	(0.404)		(0.276)	(0.234)
Over	0.453	0.634	− 0.181 ${}^{\ast \ast \ast }$	0.006	0.009	− 0.004 ${}^{\ast \ast }$
	(0.498)	(0.482)		(0.074)	(0.095)
Under	0.078	0.160	− 0.083 ${}^{\ast \ast \ast }$	0.911	0.933	− 0.021 ${}^{\ast \ast \ast }$
	(0.268)	(0.367)		(0.284)	(0.250)
(b) Former 13 periods
Sincere	0.467	0.185	0.282 ${}^{\ast \ast \ast }$	0.093	0.058	0.034 ${}^{\ast \ast \ast }$
	(0.499)	(0.389)		(0.290)	(0.234)
Over	0.453	0.629	− 0.176 ${}^{\ast \ast \ast }$	0.004	0.015	− 0.010 ${}^{\ast \ast \ast }$
	(0.498)	(0.483)		(0.067)	(0.121)
Under	0.080	0.186	− 0.106 ${}^{\ast \ast \ast }$	0.903	0.927	− 0.024 ${}^{\ast \ast \ast }$
	(0.271)	(0.389)		(0.296)	(0.261)
(c) Latter 12 periods
Sincere	0.472	0.228	0.243 ${}^{\ast \ast \ast }$	0.073	0.058	0.015 ${}^{\ast \ast }$
	(0.499)	(0.420)		(0.260)	(0.234)
Over	0.453	0.639	− 0.186 ${}^{\ast \ast \ast }$	0.007	0.003	0.004 ${}^{\ast }$
	(0.498)	(0.481)		(0.081)	(0.054)
Under	0.075	0.133	− 0.057 ${}^{\ast \ast \ast }$	0.920	0.939	− 0.019 ${}^{\ast \ast }$
	(0.264)	(0.340)		(0.271)	(0.239)

a) ^* denotes significant at the 10% level, ^** at the 5% level, and ^*** at the 1% level. b) Numbers in parentheses are standard deviations

As presented in Panel (a) of Table 2 (VA−VN column), sincere bidding increases by 26.3% with the statement of strategy-proofness in the Vickrey auction. In particular, we observe, from Panels (b) and (c) of Table 2, that the rate of sincere bidding increases further in the former 13 periods than in the latter 12 periods. Specifically, the rate of sincere bidding increases by 28.2% in the former 13 periods, while the rate increases by 24.3% in the latter 12 periods. Notice that overbidding is prevalent in the Vickrey auction without advice (VN), which amounts to 63.4% of all bids. This observation is consistent with findings in the experiments of the multi-unit Vickrey auction by Manelli et al. (Reference Manelli, Sefton and Wilner2006), Engelmann and Grimm (Reference Engelmann and Grimm2009), and Kagel and Levin (Reference Kagel and Levin2009).

With an explicit statement on the strategy-proofness of the Vickrey auction (VA), 45.3% of the case is overbidding; this implies that our advice decreases overbidding by 18.1%. Moreover, we find that underbidding also decreases when advice is given to the subjects. Specifically, our advice decreases underbidding by 8.3%. To compare the effect of the advice on overbids and underbids, we calculate the ratio of the reduction rate of overbidding to the rate of overbidding in VN, as well as the ratio of the reduction rate of underbidding to the rate of underbidding in VN. Thus, we find that the advice has more effect on underbids (8.26/16.04) than overbids (18.07/63.38). Even when we break down the data into the former 13 periods and the latter 12 periods, (Panels (b) and (c) of Table 2), the same tendency holds. Conversely, approximately 90% of the subjects in PA and PN are likely to underbid. We also observe a 3%-increase of sincere bidding although the advice is not true in this case.Footnote ¹⁶

3.3 Advice effect

Table 3

Frequency of sincere bidding by treatments, quiz scores, and units

Data	Vickrey		(1)	Pay-your-bid		(2)	$\left(1\right)-\left(2\right)$
	VA	VN	VA − VN	PA	PN	PA − PN
(a) Both units
All	0.469	0.206	0.263 ${}^{\ast \ast \ast }$	0.083	0.058	0.025 ${}^{\ast \ast \ast }$	0.238 ${}^{\ast \ast \ast }$
	(0.499)	(0.404)		(0.276)	(0.234)
Perfect Score	0.523	0.211	0.313 ${}^{\ast \ast \ast }$	0.079	0.046	0.034 ${}^{\ast \ast \ast }$	0.279 ${}^{\ast \ast \ast }$
	(0.500)	(0.408)		(0.270)	(0.209)
Imperfect Score	0.361	0.198	0.163 ${}^{\ast \ast \ast }$	0.104	0.105	− 0.002	0.165 ${}^{\ast \ast \ast }$
	(0.481)	(0.399)		(0.305)	(0.307)
(b) Unit 1
All	0.416	0.175	0.241 ${}^{\ast \ast \ast }$	0.020	0.006	0.014 ${}^{\ast \ast \ast }$	0.227 ${}^{\ast \ast \ast }$
	(0.493)	(0.380)		(0.141)	(0.078)
Perfect Score	0.463	0.176	0.287 ${}^{\ast \ast \ast }$	0.018	0.004	0.014 ${}^{\ast \ast \ast }$	0.273 ${}^{\ast \ast \ast }$
	(0.499)	(0.381)		(0.133)	(0.065)
Imperfect Score	0.322	0.173	0.149 ${}^{\ast \ast \ast }$	0.033	0.013	0.019 ${}^{\ast }$	0.129 ${}^{\ast \ast \ast }$
	(0.468)	(0.379)		(0.178)	(0.115)
(c) Unit 2
All	0.523	0.237	0.286 ${}^{\ast \ast \ast }$	0.146	0.110	0.036 ${}^{\ast \ast \ast }$	0.250 ${}^{\ast \ast \ast }$
	(0.500)	(0.425)		(0.353)	(0.313)
Perfect Score	0.584	0.245	0.338 ${}^{\ast \ast \ast }$	0.141	0.087	0.054 ${}^{\ast \ast \ast }$	0.285 ${}^{\ast \ast \ast }$
	(0.493)	(0.431)		(0.348)	(0.282)
Imperfect Score	0.400	0.222	0.178 ${}^{\ast \ast \ast }$	0.175	0.197	− 0.023	0.201 ${}^{\ast \ast \ast }$
	(0.490)	(0.416)		(0.380)	(0.399)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations

Table 4

Frequency of sincere bidding by treatments, quiz scores, and periods

Data	Vickrey		(1)	Pay-your-bid		(2)	$\left(1\right)-\left(2\right)$
	VA	VN	VA − VN	PA	PN	PA − PN
(a) Former 13 periods
All	0.467	0.185	$0.{282}^{\ast \ast \ast }$	0.093	0.058	0.034 ${}^{\ast \ast \ast }$	0.248 ${}^{\ast \ast \ast }$
	(0.499)	(0.389)		(0.290)	(0.234)
Perfect Score	0.511	0.182	0.329 ${}^{\ast \ast \ast }$	0.088	0.045	0.043 ${}^{\ast \ast \ast }$	$0.{286}^{\ast \ast \ast }$
	(0.500)	(0.386)		(0.284)	(0.208)
Imperfect Score	0.379	0.190	$0.{189}^{\ast \ast \ast }$	0.115	0.108	0.008	$0.{181}^{\ast \ast \ast }$
	(0.486)	(0.393)		(0.320)	(0.310)
(b) Latter 12 periods
All	0.472	0.228	0.243 ${}^{\ast \ast \ast }$	0.073	0.058	0.015 ${}^{\ast \ast }$	$0.{228}^{\ast \ast \ast }$
	(0.499)	(0.420)		(0.260)	(0.234)
Perfect Score	0.537	0.242	$0.{295}^{\ast \ast \ast }$	0.070	0.046	0.024 ${}^{\ast \ast \ast }$	$0.{271}^{\ast \ast \ast }$
	(0.499)	(0.428)		(0.255)	(0.210)
Imperfect Score	0.341	0.206	$0.{135}^{\ast \ast \ast }$	0.091	0.103	−0.012	$0.{147}^{\ast \ast \ast }$
	(0.475)	(0.405)		(0.288)	(0.304)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations

3.4 Regression analysis of sincere bidding

We perform a regression analysis to check the robustness of the results, and almost all results are confirmed. We employ a logit model with a random effect at the individual level. The dependent variable is a dummy variable of sincere bidding, which takes the value of 1 if subject i in period t bids sincerely. The three independent variables are the same ones we examined independently above: Advice is a dummy for treatments with advice; Latter is a dummy for periods more than 13; and Perfect Score is a dummy for subjects with a perfect score in the quiz. We also include the interaction terms of these three variables. Table 5 summarizes the results by auction rules and units. For each auction rule and each unit, we run three regressions with different specifications.

Table 5

Regression analysis of sincere bidding

(a) Vickrey
	Unit 1			Unit 2
	(1)	(2)	(3)	(4)	(5)	(6)
Advice	1.986 ${}^{\ast }$	3.289 ${}^{\ast \ast \ast }$	2.291 ${}^{\ast \ast }$	1.829 ${}^{\ast \ast }$	2.466 ${}^{\ast \ast \ast }$	2.044 ${}^{\ast \ast }$
	(1.103)	(0.709)	(1.135)	(0.888)	(0.556)	(0.916)
Perfect Score	−0.135		−0.583	0.613		0.424
	(1.059)		(1.100)	(0.831)		(0.861)
Advice $\times$ Perfect Score	1.308		1.563	0.732		0.592
	(1.375)		(1.420)	(1.105)		(1.141)
Latter		0.652 ${}^{\ast \ast \ast }$	0.187		0.452 ${}^{\ast \ast }$	0.211
		(0.225)	(0.347)		(0.182)	(0.309)
Advice $\times$ Latter		−0.802 ${}^{\ast \ast \ast }$	−0.622		−0.236	−0.433
		(0.273)	(0.438)		(0.231)	(0.389)
Latter $\times$ Perfect Score			0.798 ${}^{\ast }$			0.363
			(0.459)			(0.382)
Advice $\times$ Latter $\times$ Perfect Score			−0.373			0.337
			(0.564)			(0.485)
Constant	−3.253 ${}^{\ast \ast \ast }$	−3.707 ${}^{\ast \ast \ast }$	−3.360 ${}^{\ast \ast \ast }$	−2.323 ${}^{\ast \ast \ast }$	−2.173 ${}^{\ast \ast \ast }$	−2.432 ${}^{\ast \ast \ast }$
	(0.845)	(0.565)	(0.872)	(0.662)	(0.422)	(0.685)
Observations	2,775	2,775	2,775	2,775	2,775	2,775
Akaike Inf. Crit.	1,909.857	1,902.138	1,903.642	2,193.692	2,189.629	2,187.752

(b) Pay-your-bid
	Unit 1			Unit 2
	(7)	(8)	(9)	(10)	(11)	(12)
Advice	1.177	1.100 ${}^{\ast }$	0.215	−0.005	0.673 ${}^{\ast \ast }$	0.331
	(1.111)	(0.660)	(1.240)	(0.664)	(0.317)	(0.699)
Perfect Score	−1.091		−1.594	−0.936 ${}^{\ast }$		− 0.924 ${}^{\ast }$
	(0.970)		(1.079)	(0.496)		(0.529)
Advice $\times$ Perfect Score	0.447		1.405	0.609		0.469
	(1.290)		(1.476)	(0.740)		(0.782)
Latter		−0.506	−1.443		0.028	0.046
		(0.643)	(1.160)		(0.172)	(0.308)
Advice $\times$ Latter		0.783	2.334 ${}^{\ast }$		−0.499 ${}^{\ast \ast }$	−0.778
		(0.735)	(1.381)		(0.234)	(0.497)
Latter $\times$ Perfect Score			1.541			−0.025
			(1.425)			(0.371)
Advice $\times$ Latter $\times$ Perfect Score			−2.343			0.341
			(1.660)			(0.566)
Constant	− 5.841 ${}^{\ast \ast \ast }$	− 6.549 ${}^{\ast \ast \ast }$	− 5.379 ${}^{\ast \ast \ast }$	− 2.144 ${}^{\ast \ast \ast }$	− 2.901 ${}^{\ast \ast \ast }$	− 2.169 ${}^{\ast \ast \ast }$
	(0.949)	(0.761)	(0.974)	(0.439)	(0.237)	(0.465)
Observations	3,525	3,525	3,525	3,525	3,525	3,525
Akaike Inf. Crit.	445.609	446.079	450.302	2,231.654	2,226.571	2,230.219

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard errors

The highlights of the logit model regression results are as follows.

• • Vickrey: Advice in specifications from (1) to (6) has positive coefficients for both units, supporting Result Footnote ². Meanwhile, Advice $\times$ Perfect Score has an insignificant impact. This is seemingly inconsistent with Result Footnote ⁴, but seems a natural consequence of allowing a random effect, where the impact is regarded as individual characteristics. Moreover, the insignificant effect in Latter and Advice $\times$ Latter in specifications (3) and (6) suggests that the rate of sincere bidding does not change over time and the advice quickly improves the rate of sincere bidding in earlier periods.Footnote ¹⁸ This quick effect of advice is consistent with Table 4.

• • Pay-your-bid: Specification (9) suggests that, with advice, the sincere bidding rate marginally increases over time. Specification (12) with the negative coefficient of Perfect Score confirms that the subjects become less likely to bid sincerely as they understand the auction rule correctly. These are consistent with Tables 2 and 3.

The literature reports mixed results on whether the second price auction subjects learn to bid sincerely from experiencing losses owing to overbidding. For example, Cooper and Fang (Reference Cooper and Fang2008) observe learning, while Harstad (Reference Harstad2000) does not. Using an additional regression, we do not find such evidence in any of the two auction rules, and this is consistent with Harstad (Reference Harstad2000).

3.5 Efficiency, bidders’ payoffs, and seller’s revenue

Table 6 summarizes the results on efficiency with a focus on experience.Footnote ²⁰ In Table 6, the upper (or middle, bottom) panel collects the results obtained for all periods (periods 1–13 and periods 14–25, respectively).

Table 6

Efficiency by treatments and periods

Data	Vickrey		(1)	Pay-your-bid		(2)	$\left(1\right)-\left(2\right)$
	VA	VN	VA − VN	PA	PN	PA − PN
All periods	0.977	0.967	0.010 ${}^{\ast \ast }$	0.990	0.988	0.002	0.009 ${}^{\ast }$
	(0.069)	(0.081)		(0.028)	(0.029)
Periods 1–13	0.970	0.956	0.015 ${}^{\ast \ast }$	0.991	0.989	0.002	0.012 ${}^{\ast }$
	(0.080)	(0.098)		(0.026)	(0.029)
Periods 14–25	0.984	0.978	0.006	0.989	0.988	0.001	0.004
	(0.053)	(0.054)		(0.030)	(0.028)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations

Result Footnote ⁶ suggests that a relative improvement in the efficiency of the Vickrey auction compared with the pay-your-bid auction occurs mainly when the subjects are less experienced.Footnote ²¹

Table 7

Average bidder’s payoff and average seller’s revenue per auction

	Vickrey		VA − VN	Pay-your-bid		PA − PN
	VA	VN		PA	PN
(a) Bidder’s payoff
All periods	136.844	122.292	14.553 ${}^{\ast \ast }$	82.829	89.928	− 7.099 ${}^{\ast \ast }$
	(225.109)	(221.572)		(105.467)	(111.454)
Periods 1–13	132.247	115.000	17.247 ${}^{\ast }$	78.997	83.301	− 4.305
	(235.584)	(231.757)		(101.469)	(107.679)
Periods 14–25	141.825	130.191	11.634	86.981	97.106	− 10.126 ${}^{\ast \ast }$
	(213.225)	(209.896)		(109.545)	(115.037)
(b) Seller’s revenue
All periods	1139.410	1125.875	13.535	1303.704	1288.700	15.004 ${}^{\ast }$
	(397.611)	(431.408)		(180.917)	(196.875)
Periods 1–13	1144.689	1142.933	1.756	1313.211	1303.397	9.813
	(406.210)	(428.434)		(190.073)	(211.432)
Periods 14–25	1133.690	1107.396	26.295	1293.406	1272.778	20.628 ${}^{\ast }$
	(388.806)	(434.969)		(170.189)	(178.786)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations

4.1 Experimental procedures

To compare the results of the Ausubel auction with those of the Vickrey auction, the experimental settings for the Ausubel auction are the same as the experimental settings for the Vickrey auction. This follow-up experiment consists of two treatments:

• • Treatment AA: the Ausubel auction with advice

• • Treatment AN: the Ausubel auction without advice

We conducted three experimental sessions in each of AA and AN at Osaka University in February 2020. A total of 9, 21, or 24 subjects participated in each session. We recruited student subjects from Osaka University through campus-wide advertisements. None of the students were experienced in this particular type of experiment, and no subject attended more than one session. Table 8 summarizes the number of observations.

Table 8

Summary of treatments

Treatment	Auction rule	Advice	Date	# of Sessions	# of Subjects (Groups)
AA	Ausubel	Yes	Feb-20	3	54 (18)
AN	Ausubel	No	Feb-20	3	57 (19)

The experimental procedure is exactly the same as that described in Sect. 2.2 except for the content of advice.Footnote ²⁴ Specifically, the text of the advice is as follows:

Note that the advice involves no deception problem for AA. One period of the Ausubel auction proceeds as follows. The price is gradually increased from 0 to 2000, which is equal to the maximum possible bid in our Vickrey treatments. The price rises in increments of 10 JPY every two seconds.Footnote ²⁵ Each subject clicks a button on the screen to reduce his/her demand by one unit. The auction ends when the aggregate demand becomes two (i.e., the button is clicked four times), unless the price reaches 2000.

After the 25 payment periods, the subjects completed a questionnaire and were immediately paid in cash. Each subject was privately paid the sum of his/her earnings over the 25 periods. Individual payments including (or excluding) participation fees and quiz points ranged from $32.6 to $91.5 (or from $18.1 to $76.4).

4.2 Hypotheses

4.3 Level of understanding and bidding behavior

Fig. 4

Distribution of quiz scores for each treatment prior to the auctions

Fig. 5

Scatter plots of bids. Green $\times$ and purple $\bullet$ indicate bids for the former 13 periods and latter 12 periods, respectively

Figure 5 illustrates how bidding behavior contrasts between units in the Ausubel auction. The subjects typically bid sincerely for the first unit, while they overbid for the second unit. This behavior explains 59.0% and 52.9%, respectively, of the first and second unit bids, when combining the data of the AA and AN. A full categorization of each bid into “Sincere,” “Over,” and “Under” is summarized in Table 9. Bids are more likely to be sincere over time when the advice is provided.Footnote ^26, Footnote ²⁷

Table 9

Bid category by treatments and periods in the Ausubel auction

Bid category	AA	AN	AA − AN
(a) All
Sincere	0.389	0.305	0.084 ${}^{\ast \ast \ast }$
	(0.488)	(0.460)
Over	0.415	0.427	− 0.012
	(0.493)	(0.495)
Under	0.196	0.268	− 0.072 ${}^{\ast \ast \ast }$
	(0.397)	(0.443)
(b) Former 13 periods
Sincere	0.339	0.295	0.044 ${}^{\ast \ast }$
	(0.474)	(0.456)
Over	0.451	0.418	0.033 ${}^{\ast }$
	(0.498)	(0.493)
Under	0.210	0.287	− 0.077 ${}^{\ast \ast \ast }$
	(0.408)	(0.453)
(c) Latter 12 periods
Sincere	0.443	0.317	0.126 ${}^{\ast \ast \ast }$
	(0.497)	(0.466)
Over	0.376	0.437	− 0.060 ${}^{\ast \ast \ast }$
	(0.485)	(0.496)
Under	0.181	0.246	− 0.066 ${}^{\ast \ast \ast }$
	(0.385)	(0.431)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations

4.4 Advice effect

Table 10 summarizes the frequencies of sincere bidding by treatments, whether the quiz score is perfect, and by units. Table 11 summarizes the frequency of sincere bidding in the former period (periods 1–13) and in the latter period (periods 14–25).Footnote ²⁸

Table 10

Frequency of sincere bidding, by treatments, quiz scores, and units

Data	Ausubel		(1)	Vickrey		(2)	$\left(1\right)-\left(2\right)$
	AA	AN	AA − AN	VA	VN	VA − VN
(a) Both units
All	0.389	0.305	0.084 ${}^{\ast \ast \ast }$	0.550	0.240	0.310 ${}^{\ast \ast \ast }$	− 0.226 ${}^{\ast \ast \ast }$
	(0.488)	(0.460)		(0.498)	(0.427)
Perfect Score	0.427	0.295	0.132 ${}^{\ast \ast \ast }$	0.602	0.255	0.347 ${}^{\ast \ast \ast }$	− 0.215 ${}^{\ast \ast \ast }$
	(0.495)	(0.456)		(0.490)	(0.436)
Imperfect Score	0.351	0.314	0.037 ${}^{\ast \ast }$	0.434	0.216	0.218 ${}^{\ast \ast \ast }$	− 0.181 ${}^{\ast \ast \ast }$
	(0.477)	(0.464)		(0.496)	(0.412)
(b) Unit 1
All	0.672	0.512	0.160 ${}^{\ast \ast \ast }$	0.573	0.226	0.347 ${}^{\ast \ast \ast }$	− 0.187 ${}^{\ast \ast \ast }$
	(0.470)	(0.500)		(0.495)	(0.419)
Perfect Score	0.685	0.541	0.144 ${}^{\ast \ast \ast }$	0.616	0.252	0.365 ${}^{\ast \ast \ast }$	− 0.220 ${}^{\ast \ast \ast }$
	(0.465)	(0.499)		(0.487)	(0.435)
Imperfect Score	0.659	0.486	0.173 ${}^{\ast \ast \ast }$	0.457	0.189	0.268 ${}^{\ast \ast \ast }$	− 0.095 ${}^{\ast }$
	(0.475)	(0.501)		(0.500)	(0.393)
(c) Unit 2
All	0.268	0.213	0.055 ${}^{\ast \ast \ast }$	0.540	0.246	0.294 ${}^{\ast \ast \ast }$	− 0.239 ${}^{\ast \ast \ast }$
	(0.443)	(0.410)		(0.499)	(0.431)
Perfect Score	0.316	0.189	0.127 ${}^{\ast \ast \ast }$	0.595	0.257	0.339 ${}^{\ast \ast \ast }$	− 0.212 ${}^{\ast \ast \ast }$
	(0.465)	(0.392)		(0.491)	(0.437)
Imperfect Score	0.220	0.236	−0.016	0.425	0.229	0.197 ${}^{\ast \ast \ast }$	− 0.212 ${}^{\ast \ast \ast }$
	(0.415)	(0.425)		(0.495)	(0.421)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations. c) In the Vickrey treatments, this table focuses on non-winning bids

Table 11

Frequency of sincere bidding by quiz scores and periods

	Ausubel		(1)	Vickrey		(2)	$\left(1\right)-\left(2\right)$
	AA	AN	AA − AN	VA	VN	VA − VN
(a) Former 13 periods
All	0.339	0.295	0.044 ${}^{\ast \ast }$	0.560	0.218	0.342 ${}^{\ast \ast \ast }$	− 0.298 ${}^{\ast \ast \ast }$
	(0.474)	(0.456)		(0.497)	(0.413)
Perfect Score	0.390	0.289	0.101 ${}^{\ast \ast \ast }$	0.603	0.214	0.389 ${}^{\ast \ast \ast }$	−0.288 ${}^{\ast \ast \ast }$
	(0.488)	(0.454)		(0.490)	(0.411)
Imperfect Score	0.284	0.300	− 0.016	0.459	0.223	0.237 ${}^{\ast \ast \ast }$	−0.253 ${}^{\ast \ast \ast }$
	(0.451)	(0.459)		(0.499)	(0.417)
(b) Latter 12 periods
All	0.443	0.317	0.126 ${}^{\ast \ast \ast }$	0.540	0.264	0.275 ${}^{\ast \ast \ast }$	−0.149 ${}^{\ast \ast \ast }$
	(0.497)	(0.466)		(0.499)	(0.441)
Perfect Score	0.470	0.302	0.168 ${}^{\ast \ast \ast }$	0.601	0.299	0.302 ${}^{\ast \ast \ast }$	−0.134 ${}^{\ast \ast \ast }$
	(0.500)	(0.460)		(0.490)	(0.458)
Imperfect Score	0.418	0.331	0.087 ${}^{\ast \ast \ast }$	0.407	0.208	0.199 ${}^{\ast \ast \ast }$	−0.111 ${}^{\ast \ast }$
	(0.494)	(0.471)		(0.492)	(0.407)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations. c) In the Vickrey treatments, this table focuses on non-winning bids

Support. Table 12 summarizes the non-winning bids in the Vickrey and Ausubel auctions. Support for (i) is provided in Panel (a) of Table 12. We observe that the rates of sincere bidding in $\mathrm{AA}$ and $\mathrm{VA}$ are 38.9% and 55%, respectively. Support for (ii) is given by finding the majority bid categories in Table 12 (shown in bold). In Panel (c) for the second unit, the rate of overbidding is nearly equal between $\mathrm{AA}$ and $\mathrm{AN}$ , while it decreased by about 18% in the Vickrey treatments. Support for (iii) is given by testing Hypothesis 3′, and we have $p<0.001$ . $\square$

Table 12

Non-winning bid category by treatments and units

Bid category	Ausubel		AA − AN	Vickrey		VA − VN
	AA	AN		VA	VN
(a) Both units
Sincere	0.389	0.305	0.084 ${}^{\ast \ast \ast }$	0.550	0.240	0.310 ${}^{\ast \ast \ast }$
	(0.488)	(0.460)		(0.498)	(0.427)
Over	0.415	0.427	−0.012	0.340	0.536	−0.196 ${}^{\ast \ast \ast }$
	(0.493)	(0.495)		(0.474)	(0.499)
Under	0.196	0.268	−0.072 ${}^{\ast \ast \ast }$	0.110	0.224	−0.114 ${}^{\ast \ast \ast }$
	(0.397)	(0.443)		(0.312)	(0.417)
(b) Unit 1
Sincere	0.672	0.512	0.160 ${}^{\ast \ast \ast }$	0.573	0.226	0.347 ${}^{\ast \ast \ast }$
	(0.470)	(0.500)		(0.495)	(0.419)
Over	0.145	0.195	− 0.050 ${}^{\ast \ast }$	0.375	0.599	− 0.224 ${}^{\ast \ast \ast }$
	(0.353)	(0.397)		(0.485)	(0.491)
Under	0.182	0.293	− 0.110 ${}^{\ast \ast \ast }$	0.052	0.174	− 0.123 ${}^{\ast \ast \ast }$
	(0.387)	(0.455)		(0.222)	(0.380)
(c) Unit 2
Sincere	0.268	0.213	0.055 ${}^{\ast \ast \ast }$	0.540	0.246	0.294 ${}^{\ast \ast \ast }$
	(0.443)	(0.410)		(0.499)	(0.431)
Over	0.530	0.529	0.001	0.325	0.508	− 0.182 ${}^{\ast \ast \ast }$
	(0.499)	(0.499)		(0.469)	(0.500)
Under	0.202	0.258	− 0.056 ${}^{\ast \ast \ast }$	0.135	0.246	− 0.111 ${}^{\ast \ast \ast }$
	(0.402)	(0.438)		(0.341)	(0.431)

a) ${}^{\ast }$ denotes significant at the 10% level, ${}^{\ast \ast }$ at the 5% level, and ${}^{\ast \ast \ast }$ at the 1% level. b) Numbers in parentheses are standard deviations. c) In the Vickrey treatments, this table focuses on non-winning bids