Figure 2 plots the mean absolute deviations of bids from values over time, for our experiment and Li’s. Initial levels are similar across all static auction formats (2P, 2PAC, 2PAC-B), 12–15 Euros, and also similar across both dynamic ones (AC, AC-B), 6–10 Euros, indicating that our additional treatments did not introduce “instruction effects,” and they generally decline swiftly over the very next few rounds.

At the same time as we closely replicate Li’s results on 2P and AC, differences across the static and also across the dynamic formats emerge after the first two rounds. 2PAC and 2PAC-B quickly outperform 2P, and AC quickly outperforms AC-B, with all of our three novel formats (2PAC, 2PAC-B, AC-B) performing very similarly at intermediate levels from round 3 onwards.

The results indicate that subjects find the dominant strategy in AC easiest to identify. To understand the evolution of behavior, we evaluate the null hypothesis of equality of any given auction format with AC for six different time intervals, rounds 1–3, 4–6, 7–10, which are standard auctions, and rounds 11–13, 14–16, 17–20, which are X-auctions. We account for the panel character of the data using (two-sided) tests controlling for unobserved heterogeneity, at the session level and also at the subject-level within session. Despite the large differences initially, the difference between AC and 2PAC becomes (and remains) insignificant from the final four auctions of the first stage onwards, periods 7–10 of the standard auctions. The other auction formats catch up successively: AC-B auctions stop differing significantly from AC with the start of the X-auctions (periods 11–13), and the two other variants of static auctions stop differing significantly toward the end of the X-auctions (periods 17–20). The difference between plain AC and 2P auctions remains close to being significant, with a p value of 0.064 in two-sided tests over the last four rounds. Hence, the contrast to Li’s result—where the difference remains significant also in the final rounds of the X-auctions—may be considered marginal (see Tables 5 and 6 in Appendix B for details).

Result 1

In all auction formats, absolute differences between bids and values decline over time, converging to small and eventually insignificantly different values.

Figure 3 shows the distributions of bids in detail, plotting histograms of the differences between bids and values. This highlights that behavior in our experiment resembles behavior in Li’s at an enormous level of detail: in both 2P and AC auctions the initial bid distributions are virtually equivalent to Li’s, and subjects initially tend to underbid in all auction formats (as in comparable experiments, e.g., Noussair et al., Reference Noussair, Robin and Ruffieux2004). The tendency to overbid develops almost exclusively in 2P, both in our experiment and in Li’s. In 2P, eventually around 40% of bids exceed the respective bidders’ values by more than €1 (a conventional threshold for overbidding), and in all other auction formats, this value remains below 20%. Apparently, the simple modification of showing a passive clock after bid submission (2PAC, 2PAC-B) substantially helps subjects not to develop the tendency to overbid – even if the auction itself formally maintains sealed bids.Footnote ¹⁴

Result 2

Initially, subjects tend to underbid, and only in 2P auctions, subjects learn to systematically overbid. Exposing them to an ascending clock, be it actively (during bidding) or passively (after bidding), largely prevents the shift toward overbidding.

The pure presentation effect of the ascending clock concerns the difference between the classic 2P and the novel format 2PAC-B, and since round-1 behavior is very similar, sharing a pronounced tendency towards underbidding, it is due exclusively to differences in learning. The effect sets in very quickly: Bidders generally bid closer to their values, but the clock presentation in 2PAC-B prevents the shift towards overbidding in 2P and keeps mean absolute deviations of bids from values (MADs) lower throughout. The effect is also long-lived. While differences become insignificant in terms of MADs by the final standard auction of round 10, the nature of the remaining bias is markedly different: In 2P, there is three times more overbidding than underbidding, whereas in 2PAC-B this is reversed, with roughly twice as much underbidding than overbidding (see Figures 13 and 14 in Appendix B).

Notably, the evolution of bidding we observe under classic 2P (here and in Li’s data) is well in line with the large number of prior experiments on this format;Footnote ¹⁵ e.g., revisiting the data from the most closely related experiments of Kagel et al. (Reference Kagel, Harstad and Levin1987), Harstad (Reference Harstad2000, p. 267) observes: “(...) subjects upon encountering a second-price auction respond with tremendous heterogeneity, most bidding below their values initially. Bidding rapidly becomes more aggressive, quickly breaking through the threshold of bidding equal to value.” Harstad also documents carry-over effects on 2P behavior from prior experience with AC or “price acceptance list” auctions, which relates well with the effect of presentation we find in comparing 2P with 2PAC-B.

On top of the pure effect of clock presentation, we seem to be observing a second effect due to actual dynamic bidding and a third effect due to drop-out information (equivalently, blindness). Recall that actual dynamic bidding implies OSP in our context, and thus we hypothesized an effect, whereas drop-out information is theoretically irrelevant. The pure effect of dynamic bidding on the clock concerns the difference between our two novel formats 2PAC-B and AC-B. When both are combined with drop-out information, it concerns the difference between 2PAC and AC. Figure 2 suggests that this effect is significant in round 1, while 2PAC and 2PAC-B (when combined referred to as 2PAC*) subsequently seem to catch up with AC and AC-B (when combined referred to as AC*), respectively. Potentially, however, learning in the 2PAC* auctions simply is delayed by one round, i.e., it is comparable to learning in the actual dynamic auctions AC* once we account for the mechanical one-round delay implied by displaying the clock only after bids have been submitted in the 2PAC* auctions rather than live. In Fig. 4 and Tables 2a, 2b, we have put this exploratory hypothesis to a simple test focusing on the standard auctions of rounds 1 through 10 where the main differences in learning occur. The results in Table 2a suggest that the mechanical one-round delay explains the differences between 2PAC-B and AC-B but not quite between 2PAC and AC: The effect of actual dynamic bidding (“Dynamic”) becomes insignificant between 2PAC-B and AC-B when bidding in AC-B is lagged by one round, yet remains significant between 2PAC and AC with a one-round lag. Table 2b shows that overall, a two-round delay captures the difference in the evolution of behavior towards truthful bidding fairly exhaustively.

Result 3

Actual dynamic bidding on the ascending clock accelerates learning to bid truthfully by approximately one round in the absence of drop-out information, and two rounds in the presence of drop-out information, as well as overall.

Finally, blindness to drop-out information seems to matter by itself. As observed with Fig. 2, bidders deviate more from dominance play in AC-B than in AC. Table 2c reports on the significance of this effect of blindness (“Blind”), again focusing on the standard auctions, and demonstrates that it is indeed significant under dynamic bidding, between AC and AC-B, but not under static bidding, between 2PAC and 2PAC-B. Overall, when aggregating AC and 2PAC (denoted as *C) as well as AC-B and 2PAC-B (denoted as *C-B), respectively, the effect is once again significant.

Result 4

Drop-out information on the ascending clock has no significant effect on truthful bidding under static bidding (passive information), whereas it significantly reduces deviations under dynamic bidding (live information) as well as overall.

Figure 1 in the introduction summarizes our results on these effects, aggregating over all auctions following the first three rounds and decomposing the total difference in MADs between 2P and AC into the percentage contributions of clock presentation, dynamic bidding and drop-out information, respectively. Clock presentation contributes the largest share to reducing deviations from truthful bidding, whereas dynamic bidding only significantly adds to this in conjunction with (live rather than passive) drop-out information.

The effect of drop-out information we observe is puzzling. We are not aware of any theory that would generate a hypothesis as to how this effect comes about. However, since our experiment provides the first instance of a clean treatment comparison regarding such information, we add here a suggestive exploratory analysis devoted to this effect, which we hope may be picked up in future research.

In hindsight, we suspected that it would be relevant for learning dominance from seeing opponents’ bids whether there are such bids near one’s own value (differing by no more than two monetary units). Table 3 reports the results of a test of this post-hoc intuition, again focusing on standard auctions, and they suggest that this may indeed be important for explaining why bidding in AC is significantly closer to dominance play than in AC-B. More precisely, the indicator “prevNearMyVal” equals 1 if in the previous auction at least one opponent submitted a bid deviating from one’s value (in that auction) by no more than 2 Euros. It has a rather large, negative and significant coefficient in AC, reducing deviations from truthful bidding, but is not significant in any other auction format, including AC-B. Notably, this is also in line with Result Footnote ⁴, establishing the absence of a significant effect of drop-out information in the static formats 2PAC and 2PAC-B. Seeing opponents drop out near one’s value therefore seems to trigger a better understanding of dominance when this is observed live during the actual auction, but not when observed only as feedback after bidding.

Table 3 Determinants of deviations from sincere bidding (BSK)

	Dependent variable: Absolute value of difference between bid and value
	2P	2PAC-B	2PAC	AC	AC-B
(Intercept)	11.50***	3.16	3.87**	2.99***	2.11
	(3.23)	(2.65)	(1.24)	(0.84)	(1.16)
Period	− 1.15***	− 0.45*	− 0.57***	− 0.07	− 0.33***
	(0.23)	(0.21)	(0.09)	(0.06)	(0.08)
Value	0.03	0.06**	0.02**	0.00	0.02**
	(0.02)	(0.02)	(0.01)	(0.01)	(0.01)
DecisionTime	− 0.01	0.01	0.03
	(0.03)	(0.03)	(0.02)
PreviousDeviation	0.10**	0.19***	0.08***	− 0.02**	− 0.00
	(0.04)	(0.04)	(0.02)	(0.01)	(0.01)
prevNearMyVal	− 0.64	− 2.02	− 0.34	− 1.55**	0.35
	(1.20)	(1.04)	(0.44)	(0.52)	(0.55)
prevAboveMyVal	1.14	− 0.43	− 0.21	0.02	1.53*
	(1.28)	(1.21)	(0.49)	(0.41)	(0.72)
AIC	3902.15	3468.50	2952.45	2292.59	2568.00
BIC	3939.82	3505.66	2990.13	2325.14	2600.55
Log likelihood	− 1942.07	− 1725.25	− 1467.23	− 1138.29	− 1276.00
Num. obs.	486	459	486	432	432
Num. groups	72	68	72	64	64
Var: subj	70.37	1.59	9.76	1.29	5.98
Var: Residual	138.01	104.84	19.06	10.00	17.60

For each of the treatments, we regress the absolute value of the difference between bid and value on “DecisionTime” (the time taken by a subject to submit the bid in sealed bid auctions”), “prevNearMyVal” (a binary indicator equal to one if at least one of the opponents submitted a bid in the previous auction deviating from my value in that auction by no more than 2 Euro), “prevAboveMyVal” (a binary indicator equal to one if at least one of the opponents submitted a bid in the previous auction above my value in that auction by more than 2 Euro), controlling for the number of the auction within the session (“Period”), the value of the subject, and the subject’s absolute value of the difference between bid and value in the previous auction (“PreviousDeviation”). As above, we allow for random effects at the subject level, where “Num. groups” indicates the number of subjects, “Var: subj” indicates the variance of the random effect, and “Var: Residual” indicates the variance of the residual

$\ast \ast \ast p<0.001$ ; $\ast \ast p<0.01$ ; $\ast p<0.05$

Finally, Table 3 also explores potential further determinants of deviations from truthful bidding, to round off this overview of basic results. It shows that the “Period” itself, i.e., the number of the auction within the session, seems to matter highly significantly in reducing deviations, and that the deviations tend to be auto-correlated (see the significance of “PreviousDeviation” in most treatments), while “DecisionTime” which is the time taken by subjects prior to bid submission in the static auctions does not directly correlate with the extent of truthful bidding.

In order to better understand this observed behavior, let us relate it to economic incentives and “optimization premia” in the sense of Battalio et al. (Reference Battalio, Samuelson and van Huyck2001). Specifically, we seek to evaluate the monetary costs associated with deviations from the dominant strategy, loosely following Harrison (Reference Harrison1989), and based on this, we shall set up a simple structural model evaluating to which extent and also how subjects’ behavior reflects these incentives.

Given a subject with value v and any bid b, define the “relative bid” $a=b-v$ as the amount by which this bid exceeds the subject’s value. Let A denote the set of possible relative bids. The highest opponent bid is a random variable $B^{\ast }$ , and the conditional probability that it equals $b^{\ast }$ is $Pr(B^{\ast }=b^{\ast }|v)$ . We make the conventional assumption that subjects abstract from distortions at the bounds of the signal space, so we can capture subjective beliefs about $B^{\ast }$ in terms of relative bids (i.e., independently of their own signal). Formally, a belief is then a function ${\sigma }_b\in \Delta \left(A\right)$ , and ${\sigma }_b\left(a^{\ast }\right)$ denotes the probability that the highest relative bid of an opponent is $a^{\ast }$ , where relative refers to v (i.e., the highest actual opponent bid is $b^{\ast }=v+a^{\ast }$ ).

Expected payoffs of bids

The subjective probability that bidding a wins the auction is ${\sum }_{a^{\prime }<a}{\sigma }_b\left(a^{\prime }\right)$ . The winning probabilities faced by subjects in our experiment are presented in panel (A) of Figure 5 for rounds 4–10 of the standard auctions (for further plots see Appendix B).Footnote ¹⁶ For example, bidding $a=0$ yields values very close to the theoretically predicted winning probability of 0.25 in all treatments except 2P, where it is slightly lower, at 0.19, due to subjects’ overbidding. Given belief ${\sigma }_b$ , the ex-ante expectation of the profit from bidding a is

(1) $\begin{array}{c}\Pi \left(a\right)=-\sum _{a^{\prime }<a}{a}^{\prime }·{\sigma }_b\left(a^{\prime }\right),\end{array}$

which, for our data, is displayed in panel (B) of Fig. 5. The (weakly) dominant strategy of bidding truthfully, $a=0$ , generally maximizes expected payoffs. Close to bidding truthfully, the empirical payoffs are fairly symmetric and flat (see also Harrison, Reference Harrison1989; Georganas and Nagel Reference Georganas and Nagel2011), but further away from truthful bidding, overbidding is substantially more costly than underbidding. This suggests that subjects would tend to underbid in static auctions.

Fig. 5 Beliefs, profits and incentives under rational expectations (periods 4–10 of standard auctions)

Expected payoffs of bid increments

In ascending-clock auctions, a subject is sequentially presented various values of a as the clock ascends, starting at relatively low bids $a<0$ , up to potentially high bids $a>0$ . Here, we directly quantify the costs of enacting one-shot deviations from the dominant strategy. At bids $a<0$ , the one-shot deviation requires the subject to stop bidding, thus forfeiting the expected profits associated with the dominant strategy. At bids $a\ge 0$ , the one-shot deviation requires the subject to keep bidding for exactly one bid increment, and thus to risk winning the auction at a price above her value v. We denote the expected costs of such one-shot deviations, conditional on the current bid a, as $L_{\mathrm{AC}}^{-}\left(a\right)$ and $L_{\mathrm{AC}}^{+}\left(a\right)$ for the cases $a<0$ and $a\ge 0$ , respectively.

(2) $\begin{array}{cccc}a<0:L_{\mathrm{AC}}^{-}\left(a\right)& =\frac{{\sum }_{a^{\prime }=a}^0{a}^{\prime }·{\sigma }_b\left(a^{\prime }\right)}{{\sum }_{a^{\prime }=a}^{\infty }{\sigma }_b\left(a^{\prime }\right)}& a\ge 0:L_{\mathrm{AC}}^{+}\left(a\right)& =\frac{-a·{\sigma }_b\left(a\right)}{{\sum }_{a^{\prime }=a}^{\infty }{\sigma }_b\left(a^{\prime }\right)}\end{array}$

We can evaluate analogously such expected costs of “one-shot deviations” in static auctions. Intuitively, we will think of a subject engaging in a hypothetical thought process that mimicks the ascending clock: Starting at low bids, she iteratively evaluates whether to increment her bid or not, up to a point where she decides to stop, which yields the bid she eventually submits in the auction. Since this thought process takes place ex ante, so would be expectations, implying

(3) $\begin{array}{cccc}a<0:L_{2P}^{-}\left(a\right)& =\sum _{a^{\prime }=a}^0{a}^{\prime }·{\sigma }_b\left(a^{\prime }\right)& a\ge 0:L_{2P}^{+}\left(a\right)& =-a·{\sigma }_b\left(a\right).\end{array}$

The expected costs of deviating from the dominant strategy for our experimental subjects are provided in panel (C) of Fig. 5. The differences across treatments are striking. Ex-ante, the probability that a one-shot bid increment beyond one’s value turns out costly is very low. Hence, subjects have little reason not to increment their bid by a tick when deliberating their choice based on unconditional expected costs $L_{2P}$ in static auctions. Ex post—i.e., conditional on the standing price being above their own value—further increments likely win the auction, thereby yielding a substantial loss even in expectation and providing subjects with strong incentives to play the dominant strategy and exit. This relates closely to the intuition of Cooper and Fang (Reference Cooper and Fang2008), that bidders perceive the benefits and costs of raising their bids differently in sealed-bid and ascending-clock auctions, and indeed clarifies that this intuition is not solely an implication of bounded rationality (as conjectured by Cooper and Fang). It represents the difference between unconditional and conditional monetary incentives.

Both unconditional and conditional expected costs are contained as special cases in

(4) $\begin{array}{cccc}{L}^{-}\left(a\right)& =\frac{{\sum }_{a^{\prime }=a}^0{a}^{\prime }·{\sigma }_b\left(a^{\prime }\right)}{({\sum }_{a^{\prime }=a}^{\infty }{\sigma }_b\left(a^{\prime }\right){)}^{\beta }}& L^{+}\left(a\right)& =\frac{-a·{\sigma }_b\left(a\right)}{({\sum }_{a^{\prime }=a}^{\infty }{\sigma }_b\left(a^{\prime }\right){)}^{\beta }},\end{array}$

where the unconditional expectation obtains for $\beta =0$ and the conditional expectation obtains for $\beta =1$ . Since $\beta =1$ refers to the case that subjects compute expected costs (of one-shot deviations) contingent on having reached the information set where this deviation is implemented, and $\beta =0$ refers to the case where subjects do not account for this contingency, we refer to $\beta$ as the degree of contingent reasoning. By estimating $\beta$ , we will thus be able to assess if there are differences in the degree of contingent reasoning between treatments, potentially as a function of whether the auction is OSP—thus testing if obviousness amplifies contingent reasoning.

Given the previous definitions, we can use a structural model to analyze how behavior reflects these incentives, and in particular, whether it corresponds to the static perspective in Eq. (1), the ex-post incremental perspective in Eq. (2), or the ex-ante incremental perspective in Eq. (3). Our prior hypothesis is that subjects bid according to incentives from the static perspective in static auctions and according to the ex-post incremental perspective in dynamic auctions.

In addition, our analysis of incentives allows us to examine further the cognitive channels through which theoretical obviousness (might) affect behavior. One possible channel was introduced already: it could help subjects in reasoning contingently such that they compute conditional rather than unconditional expected payoffs. Another channel might be that it heightens the perception of payoff differences across the board, which we can capture through allowing for differences in the usual precision parameter $\lambda$ below. A third, slightly more subtle but also more direct channel is that it helps subjects find the dominant action such that it is chosen disproportionately often, i.e., more often than payoff differences predict. Here, we follow Huck et al. (Reference Huck, Rasul and Shephard2015) and Breitmoser (Reference Breitmoser2021), amongst others, who capture choice effects due to roundedness of numbers by allowing for additive increments to choice propensities (or, utilities) when numbers are round. If these increments are significantly positive, then the numbers are chosen more often than say utility differences predict. We will estimate similar additive increments for the dominant action, which may be considered “focal” in the sense of Breitmoser (Reference Breitmoser2021), to then test for their significance and also for differences across auction formats. Our hypothesis is, naturally, that subjects react to dominance more when it is formally obvious.

Our structural model directly implements the monetary incentives quantified above assuming logistic errors. Allowing for logistic errors follows McKelvey and Palfrey (Reference McKelvey and Palfrey1995, Reference McKelvey and Palfrey1998) and is standard practice in behavioral analyses of laboratory experiments (Goeree et al., Reference Goeree, Holt, Palfrey, Durlauf and Blume2008), in particular also in analyses of auctions (Goeree et al., Reference Goeree, Holt and Palfrey2002; Crawford & Iriberri, Reference Crawford and Iriberri2007; Turocy, Reference Turocy2008). In addition to the monetary incentives, our specification will include terms ${\nu }^{-}$ and ${\nu }^{+}\ge 0$ that denote the aforementioned additional weight awarded to the dominant action to continue if $a<0$ and not to continue if $a\ge 0$ (or, in static auctions, to not deviate from bidding one’s value toward either $a<0$ or $a>0$ , respectively). These weights capture the degree to which dominance as such is choice-relevant—beyond elevating expected payoffs.

Thus, given the current price is a, a subject holding the “dynamic perspective” in Eq. (4) does not exit (or, continues bidding) with probability

$\begin{array}{cc}a<0:{Pr}_{\mathrm{cont}}\left(a\right)& =\frac{exp\{-\lambda ·L^{-}\left(a\right)+{\nu }^{-}\}}{exp\left\{0\right\}+exp\{-\lambda ·L^{-}\left(a\right)+{\nu }^{-}\}},\\ a\ge 0:{Pr}_{\mathrm{cont}}\left(a\right)& =\frac{exp\{\lambda ·L^{+}\left(a\right)-{\nu }^{+}\}}{exp\left\{0\right\}+exp\{\lambda ·L^{+}\left(a\right)-{\nu }^{+}\}}.\end{array}$

here $\lambda \ge 0$ denotes the weight of monetary incentives in decision making. The probability that the subject ends up bidding $a\in A$ isFootnote ¹⁷

(5) $\begin{array}{c}Pr\left(a\right)=(1-{Pr}_{\mathrm{cont}}\left(a\right))·\prod _{a^{\prime }<a}{Pr}_{\mathrm{cont}}\left(a^{\prime }\right).\end{array}$

A subject with a static perspective chooses her bid simply based on expected payoffs, implying

(6) $\begin{array}{c}Pr\left(a\right)=\frac{exp\{\lambda ·\Pi \left(a\right)-{\nu }^{-}·I_{a<0}-{\nu }^{+}·I_{a>0}\}}{{\sum }_{a^{\prime }}exp\{\lambda ·\Pi \left(a^{\prime }\right)-{\nu }^{-}·I_{a^{\prime }<0}-{\nu }^{+}·I_{a^{\prime }>0}\}},\end{array}$

using the same parameters $(\lambda ,{\nu }^{+},{\nu }^{-})$ as before. We estimate these parameters by maximum likelihood and then evaluate our statistical hypotheses using the robust likelihood-ratio tests of Schennach and Wilhelm (Reference Schennach and Wilhelm2017). The Schennach-Wilhelm test is robust to misspecification and arbitrary nesting structures, while allowing us to cluster at the subject level.

The results of this analysis are provided in Table 4. In the upper panel of the table, we report the estimates for the full model and for basic tests of treatment effects, on “Monetary Incentives” ( $\lambda$ ), “Obviousness” ( ${\nu }^{-},{\nu }^{+}$ ) and “Contingent Reasoning” (exponent $\beta$ in Eq. (4)). We find that treatment differences seem to show up only with respect to obviousness of dominance regarding underbidding ( ${\nu }^{-}$ ). The respective p values are reported for each test and each phase of the experiment. The second panel then evaluates a refined model that we obtain after ruling out treatment differences in ${\nu }^{+}$ and $\beta$ , so we can focus on treatment differences in the weight on monetary incentives ( $\lambda$ ) and obviousness that underbidding is dominated ( ${\nu }^{-}$ ). We will focus on the results obtained for this refined model. Let us first summarize the results before we discuss the statistical support.

Table 4 Results of the structural analysis

Parameter/label	Null hypothesis	Alternative	Standard auctions				X-auctions
			1–3		4–10		1–3		4–10
Estimates (standard errors)
${(\lambda ,\beta )}_{2P}$			0 (0.04)	0 (–)	0.66 (0.19)	0 (–)	1.39 (0.35)	0 (–)	1.21 (0.31)	0 (–)
${(\lambda ,\beta )}_{2PAC-B}$			0.28 (0.14)	6.77 (0.43)	1.15 (0.67)	0 (0.43)	2.79 (2.72)	0 (0.74)	0.73 (0.49)	0 (0.69)
${(\lambda ,\beta )}_{2PAC}$			0 (0)	4.17 (0.1)	1.55 (1.14)	0 (0.53)	3.44 (1.84)	0 (0.32)	2.98 (1.67)	0 (0.44)
${(\lambda ,\beta )}_{\mathrm{AC}}$			0 (0)	4.45 (0.1)	0 (0)	0 (0.48)	2.81 (2.29)	0 (0.52)	0.65 (0.36)	0 (0.53)
${(\lambda ,\beta )}_{AC-B}$			0.02 (0.01)	3.54 (1.51)	0.71 (0.4)	0 (0.29)	3.8 (0.93)	0 (0.18)	3.19 (1.28)	0 (0.31)
${({\nu }^{+},{\nu }^{-})}_{2P}$			2.36 (0.22)	2.44 (0.21)	0.98 (0.22)	2.8 (0.28)	0.57 (0.25)	2.07 (0.34)	0.62 (0.23)	2.72 (0.33)
${({\nu }^{+},{\nu }^{-})}_{2PAC-B}$			0.58 (7.7)	0.44 (0.57)	0 (0.17)	2.3 (0.33)	0 (0.2)	1.85 (0.82)	0 (0.23)	3.02 (0.17)
${({\nu }^{+},{\nu }^{-})}_{2PAC}$			0 (0.46)	1.73 (0.11)	0 (0.27)	2.42 (0.47)	0 (0.32)	1.75 (0.52)	0 (0.22)	2.28 (0.51)
${({\nu }^{+},{\nu }^{-})}_{\mathrm{AC}}$			0 (0.27)	3.22 (0.14)	0 (0.19)	4.87 (0.29)	0 (0.22)	1.96 (0.7)	0 (0.24)	3.28 (0.13)
${({\nu }^{+},{\nu }^{-})}_{AC-B}$			2.03 (0.11)	0.72 (0.64)	2.43 (0.27)	0 (0.27)	1.23 (0.29)	0 (0.78)	1.86 (0.39)	0 (0.93)
BIC			1889.74		4026.35		1929.99		4177.37
Basic tests of treatment effects (p values)
(Monetary Inc)	$H_A:\text{all}\lambda \text{equal}$	$H_{\mathrm{Base}}$	0.925		0.769		0.826		0.772
(Obviousness +)	$H_B:\text{all}{\nu }^{+}\text{equal}$	$H_{\mathrm{Base}}$	0.39		0.236		0.927		0.81
(Obviousness −)	$H_C:\text{all}{\nu }^{-}\text{equal}$	$H_{\mathrm{Base}}$	0		0.014		0.752		0.808
(Cont Reason)	$H_D:\text{all}\beta \text{equal}$	$H_{\mathrm{Base}}$	0.76		0.826		0.986		0.699
(Combined)	$H_E:H_B\wedge H_D$	$H_B$	0.729		0.84		0.986		0.812
${(\lambda ,{\nu }^{-})}_{2P}$			0.02 (0.12)	1.91 (0.32)	0.72 (0.19)	2.14 (0.28)	1.38 (0.34)	1.7 (0.34)	1.28 (0.32)	2.28 (0.54)
${(\lambda ,{\nu }^{-})}_{2PAC-B}$			0.41 (0.16)	0.2 (0.61)	1.25 (0.64)	2.24 (0.31)	2.66 (1.32)	1.9 (0.46)	0.49 (6.66)	3.11 (2.39)
${(\lambda ,{\nu }^{-})}_{2PAC}$			0 (0)	1.71 (0.11)	1.59 (1.1)	2.4 (0.46)	3.39 (1.56)	1.75 (0.46)	2.87 (6.56)	2.32 (1.5)
${(\lambda ,{\nu }^{-})}_{\mathrm{AC}}$			0 (0)	3.22 (0.14)	0 (1.13)	4.88 (0.66)	2.82 (1.43)	1.99 (0.49)	0.46 (7.33)	3.35 (2.99)
${(\lambda ,{\nu }^{-})}_{AC-B}$			0.16 (0.43)	1.65 (1)	0.68 (0.38)	2.45 (0.27)	3.74 (0.97)	1.25 (0.31)	3.12 (5.62)	1.88 (1.24)
$\beta ,{\nu }^{+}$			1.04 (0.34)	6.16 (0.17)	0 (0.1)	0 (0.5)	0 (0.11)	0 (0.21)	0 (0.59)	0 (5.16)
BIC			1924.39		4034.86		1912.07		4166.14
Treatment effects after ruling out effects on ${\nu }^{+},\beta$ (Hypothesis $H_E$ )
(Monetary Inc)	$H_F:H_E\wedge \text{all}\lambda \text{equal}$	$H_E$	0.966		0.767		0.964		0.383
(Obviousness −)	$H_G:H_F\wedge \text{all}{\nu }^{-}\text{equal}$	$H_F$	0.002		0.003		0.582		0.655
(Obviousness −)	$H_H:H_F\wedge \text{all}{\nu }_{-AC}^{-}\text{equal}$	$H_F$	0.648		0.679		0.716		0.809
Are 2PAC and 2PAC-B played as if dynamic or sealed-bid? (BIC of refined model $H_H$ )
2P sealed bid, Rest dynamic			1929.41		4031.55		1909.92		4176.63
2P* sealed bid, AC* dynamic			1921.28		4161.51		1944.03		4289.78
All dynamic			1959.49		4153.51		1979.35		4291.31

The table contains “label columns” on the left-hand side and “data columns” on the right-hand side. These data columns distinguish the four phases of the experiment discussed in the text. The table reports all parameter estimates, Huber-Sandwich standard errors (clustering at subject level), Bayesian Information Criteria (BIC, Schwarz, Reference Schwarz1978) of model adequacy, and the results of the Schennach-Wilhelm likelihood-ratio tests for model discrimination. For each likelihood-ratio test, we specifically report which null hypothesis is tested against which alternative and all p values

Result 5

(Behavior in relation to incentives) Initially (rounds 1–3), bidding is not correlated with monetary incentives ( $\lambda =0$ ). Subsequently:

1. 1. Bidding is significantly correlated with monetary incentives in all formats ( $\lambda >0$ ), without significant differences between formats.

2. 2. Bidding is dynamic in all formats featuring a price clock, including 2PAC and 2PAC-B. It is static only in 2P.

3. 3. Bidders evaluate incentives unconditionally ( $\beta =0$ ) in all formats, including AC and AC-B.

4. 4. In all formats, dominance is recognized only regarding underbidding, not overbidding ( ${\nu }^{-}>{\nu }^{+}\approx 0$ ).

5. 5. There are no significant differences between formats in the extent to which subjects recognize dominance, with the sole exception of AC’s standard auctions, where dominance regarding underbidding is recognized significantly better than in the others.

Point 1 reviews the estimates of the weight on monetary incentives, $\lambda$ , which is zero initially and then increases substantially. In Table 4, the lines labeled “(Monetary Inc)” report the p values of tests for significance of differences in $\lambda$ between treatments. The p values are always above 0.3 and mostly even above 0.7, showing that differences are statistically minor. That is, winning an additional Euro is considered similarly valuable across treatments, and in this sense subjects are similarly rational in all conditions.

Point 2 summarizes the results reported in the bottom panel of Table 4. We say that bidding in a format is static if it is best explained by static expected payoffs, Eq. (6), and we say that bidding is dynamic if it is best explained by incentives under the incremental choice process in Eq. (5). The bottom panel evaluates the hypothesis that only 2P is played as static against two alternatives: either that bidding is static in all formally static formats (2P, 2PAC and 2PAC-B, summarized as “2P*”) and dynamic in both dynamic ones, or that bidding is dynamic in all formats. The results are fairly clear-cut: Aside from the first three auctions, where monetary incentives bear no weight in any treatment ( $\lambda =0$ ), the differences in the Bayesian Information Criterion (BIC) are substantial in all phases of the experiment.

Point 3 follows immediately from the fact that $\beta =0$ is estimated in all phases of the experiment where monetary incentives carry positive weight (i.e., after round 3). It means that subjects fail to update winning probabilities as they increase their own bid, even in the dynamic auctions. In Fig. 3, we can directly see the main implication of it: Conditional on overbidding in AC auctions, we observe a rather flat, uniform distribution in both our AC treatment and Li’s. That is, once subjects move above their values, they seem to believe in a low probability of winning the auction with the next bid increment; otherwise, they would face strong incentives to exit, as shown in Fig. 5.

Point 4 follows from the observation that the weight ${\nu }^{+}$ , which captures the extent to which subjects account for overbidding being dominated in excess of the payoff difference, does not differ between treatments and is estimated to be zero after round 3. The underlying statistical test is reported in line “(Obviousness $+$ )” in the top panel of the table. In turn, the estimates for ${\nu }^{-}$ are generally positive and large in relation to their standard errors.

Point 5 follows from the observation that the corresponding weights for underbidding, ${\nu }^{-}$ , differ highly significantly between treatments in “standard” auctions—although there are no differences between auction designs other than AC. The results (p values) of the underlying statistical tests are reported in the two lines “(Obviousness $-$ )” in the middle panel of the table (with ${\nu }_{-AC}^{-}$ the vector of all ${\nu }^{-}$ excluding AC).

Discussion

The structural analysis, Result Footnote ⁵, offers a specific explanation of the differences in behavior between AC and 2P auctions. After an initial learning phase of three auctions, where underbidding is prevalent, subjects understand better that underbidding is dominated than that overbidding is dominated, quantified as ${\nu }^{-}>{\nu }^{+}$ . Considering that expected payoffs are rather symmetric around truthful bidding, see Fig. 5, this implies that subjects will tend to overbid when they approach bidding from a static perspective, Eq. (6). This is the case only for plain 2P auctions, however. Subjects approach all other auctions, including 2PAC and 2PAC-B, from a dynamic perspective. That is, presentation with the ascending clock instils an iterative reasoning process, where subjects iterate through possible bids in ascending order, even when bidding itself is ultimately static. Similarly to 2P auctions, also in these other auctions subjects understand dominance better with regards to underbidding ( ${\nu }^{-}>0$ ) than overbidding ( ${\nu }^{+}\approx 0$ ), but there is a substantial difference: Walking through multiple prices/bids below one’s value in ascending order mechanically implies the observed bias toward underbidding, given that choice is stochastic.Footnote ¹⁸

We observe that dominance as such statistically affects behavior, and contrary to monetary incentives, it does so from the very start of the experiment. Further, its effect is stronger in AC than in the other formats, as the ${\nu }^{-}$ differ. However, we observe this difference only for the dominatedness of underbidding, not overbidding, not for the unfamiliar X-auctions, and not for the other format in which dominance should be obvious as well (AC-B). Thus, overall, the structural analysis confirms our earlier basic results by indicating that theoretical obviousness of dominance does not robustly help predicting when dominated actions are avoided by subjects.