1 Introduction

A wealth of evidence has accumulated questioning some of the foundations of expected utility theory, and behavioral theorists have shown how these challenges can be accommodated (Wakker Reference Wakker2010). At the core of standard and behavioral economic modelling remains the assumption that people are endowed with well articulated and stable preferences. This fundamental assumption, however, has been challenged amongst others by Ariely et al. (Reference Ariely, Loewenstein and Prelec2003), who have shown that preferences are initially malleable by normatively irrelevant anchors. People subsequently choose consistently with these initial preferences, and thereby end up with preferences that are characterized by what Ariely et al. (Reference Ariely, Loewenstein and Prelec2003) call “coherent arbitrariness”. For a series of products that range from familiar (like an average bottle of wine) to unfamiliar (like listening to an unpleasant sound), they find substantial anchoring effects.

Economists often assign less weight to behavioral anomalies when they are obtained in non-repeated individual decision-making tasks. The line of reasoning is that anomalies may be eroded when people have relevant experience, for instance, as a result of trading in markets. To counter such skepticism, Ariely et al. (Reference Ariely, Loewenstein and Prelec2003) included a treatment where subjects, after being exposed to an anchor, submitted a bid to avoid listening to an annoying sound. In the uniform-price sealed-bid auction, the three lowest bidders had to listen to the sound and each of them received a payment equal to the fourth lowest bid. Like in the individual decision-making treatment, sizable (and lasting) anchoring effects were observed in this treatment.

This paper aims to make two contributions. A first contribution is that we investigate the effects of uninformative anchoring on valuations for a familiar good in a large sample. This is important because previous papers have provided mixed evidence, from sizable anchoring effects (Ariely et al. Reference Ariely, Loewenstein and Prelec2003) to no anchoring effects (Fudenberg et al. Reference Fudenberg, Levine and Maniadis2012). Our paper stands out because of the combination of two features. First, we have a large sample of 316 subjects who are all exposed to the same anchoring protocol, while previous studies have often been based on rather small samples. Second, we use a transparently random anchor that subjects know to be uninformative because they generate it themselves with a ten-sided die.

A second contribution of our paper is that we investigate how elicited preferences are affected in a richer market setting than the one of Ariely et al. (Reference Ariely, Loewenstein and Prelec2003), where subjects could not learn from others’ bids during the auction. We employ a standard double auction where traders are continuously updated about other traders’ bids and asks. We believe that a double auction provides a much better chance for market forces to erode initial traces of anchoring.

Our experiment consists of three phases. In the first phase, we apply a typical anchoring protocol: we ask whether subjects are willing to sell a bottle of wine for an individually drawn, random price. Then we elicit their valuation (Willingness-To-Accept) for the bottle of wine with the Becker–DeGroot–Marschak (BDM) procedure (Becker et al. Reference Becker, DeGroot and Marschak1964). In the second phase, we randomly assign subjects to either a small double auction market (n = 2) or a large double auction market (n = 8). Subjects participate in two trading periods, once as a buyer and once as a seller. In the third phase, we elicit each subject’s valuation once more.

The first phase of the experiment allows us to test whether a random anchor influences elicited valuations. We hypothesize that subjects’ valuations correlate positively with their anchors. We further conjecture that market experience will affect subjects’ elicited preferences. Subjects who are not completely sure about their preference may move into the direction of the preferences exhibited by other traders. This way, anchoring effects may diminish or even disappear. Thus, we hypothesize that the valuations elicited in the third phase will exhibit smaller (if any) anchoring effects. We also hypothesize that the large market will have a stronger effect on subjects’ preferences than the small market, and that anchoring effects are eroded more efficiently in the former.

Contrary to our first hypothesis, we observe no effect of the random anchor on subjects’ valuations. We believe our null result contributes to the literature on the robustness of anchoring effects. In Sect. 4, we position our paper in the literature and elaborate on what we can learn from our null result. There, we discuss the results of a concise meta-analysis of experimental papers that cite Ariely et al. (Reference Ariely, Loewenstein and Prelec2003) and investigate the effects of anchoring on preferences.

We do find support for the idea that market participation affects how people value the bottle of wine. The variance in subjects’ elicited valuations after the market shrinks within trading groups. As expected, the effect of other traders’ behavior on a subject’s preference is stronger in the large market. These results underline the potential power that markets may play in eroding individual biases and noise. However, in this study, the double auction is not needed to avoid anchoring effects on valuations.

The remainder of the paper is organized as follows. Section 2 describes our experimental design and the hypotheses to be tested. Section 3 presents the results of the experiment. Section 4 provides a discussion of how our results fit in the literature.

2 Experimental design and implementation

4 Concluding discussion

In this paper, we find supportive evidence for the potential of markets to reduce the effects of anchoring on valuations.Footnote ¹¹ However, in our study, the market was not needed to correct a potential individual bias due to anchoring. We find no effect of anchoring on reported valuations. This result raises doubt on the robustness of the anchoring effect on people’s preferences. Before we discuss differences between our design and other similar studies, we emphasize that our null result is not a consequence of an under-powered study. The observations of the first two sessions were used as a pilot to conduct a power analysis. We conducted a power analysis with an aim to obtain a significant result at the 5% level with 80% power in each of our market treatments separately. The power analysis resulted in an estimated sample size of 148 subjects per market treatment, so 296 subjects in total. To be on the safe side, we aimed for 320. Given that the anchoring hypothesis is based on the total sample (as no treatment has been introduced yet), we have a very high power of 99%.

The existing literature provides a mixed picture of whether anchors affect people’s valuations. Studies differ in detail in how they were run, and it is possible that anchoring effects on preferences occur in some circumstances but not in others. When trying to make sense of previous results on anchoring, a complicating factor is that many of these are based on rather small samples which makes it impossible to distinguish between true results, false positives and false negatives. However, even the large studies provide mixed results.

To make sense of previous results, we carried out a limited meta-analysis. In this analysis, we restricted our attention to the 1493 papers that cite Ariely et al. (Reference Ariely, Loewenstein and Prelec2003). Among those, we selected the ones that reported an incentivized experiment investigating the effect of anchoring on elicited valuations. We left out results based on hypothetical payments, which includes a very large literature on the effects of anchoring on contingent valuations.Footnote ¹² Table 3 lists the 19 selected studies on the anchoring of preferences, together with their main features and the reported effect. Each study result is summarized in two ways: (i) as a ratio of valuations of High- over Low-anchor group, and (ii) as Hedge’s g, defined as the difference in valuations between High- and Low-anchor group divided by the pooled standard deviation.

Table 3

Anchoring on valuation studies (ordered by publication year)

(1)	(2)		(3)		(4)		(5)	(6)	(7)	(8)
Study	Good		Anchor		WTP/WTA		#Results	Sample	Ratio(H/L)	Hedge’s g
(a) Informative and questionable anchors (corresponds to Fig. 3a)
Ariely et al. (Reference Ariely, Loewenstein and Prelec2003)	Familiar	[0.2%]	Questionable	[2.4%]	WTP	[0.2%]	1	55	1.71	0.841
Ariely et al. (Reference Ariely, Loewenstein and Prelec2003)	Unfamiliar	[15.0%]	Informative	[0.9%]	WTA	[8.9%]	4	61	2.16	1.311
Ariely et al. (Reference Ariely, Loewenstein and Prelec2003)	Unfamiliar	[5.7%]	Questionable	[3.7%]	WTA	[3.4%]	1	90	1.62	1.137
Simonson and Drolet (Reference Simonson and Drolet2004)	Familiar	[1.2%]	Questionable	[13.2%]	WTA	[12.0%]	2	139	1.23	0.331
Ariely et al. (Reference Ariely, Loewenstein and Prelec2006)	Unfamiliar	[11.7%]	Informative	[0.7%]	WTP	[0.7%]	2	164	1.36	2.905
Ariely et al. (Reference Ariely, Loewenstein and Prelec2006)	Unfamiliar	[5.2%]	Questionable	[3.4%]	WTP	[0.3%]	1	81	3.95	1.016
Bergman et al. (Reference Bergman, Ellingsen, Johannesson and Svensson2010)	Familiar	[0.5%]	Questionable	[5.2%]	WTP	[0.5%]	1	116	1.43	0.794
Tufano (Reference Tufano2010)	Unfamiliar	[9.8%]	Informative	[0.6%]	WTA	[5.9%]	1	134	1.05	0.061
Sugden et al. (Reference Sugden, Zheng and Zizzo2013)	Familiar	[3.9%]	Informative	[4.1%]	WTA	[39.1%]	9	100	1.11	0.165
Sugden et al. (Reference Sugden, Zheng and Zizzo2013)	Familiar	[4.4%]	Informative	[4.5%]	WTP	[4.6%]	9	111	1.09	0.071
Sugden et al. (Reference Sugden, Zheng and Zizzo2013)	Unfamiliar	[13.1%]	Informative	[0.8%]	WTA	[7.8%]	9	20	1.06	0.166
Sugden et al. (Reference Sugden, Zheng and Zizzo2013)	Unfamiliar	[14.6%]	Informative	[0.9%]	WTP	[0.9%]	9	22	1.09	0.086
Koçaş and Demir (Reference Koçaş and Demir2014)	Familiar	[0.1%]	Informative	[0.2%]	WTP	[0.2%]	1	46	5.00	1.766
Maniadis et al. (Reference Maniadis, Tufano and List2014)	Unfamiliar	[8.5%]	Informative	[0.5%]	WTA	[5.0%]	1	116	1.29	0.221
Alevy et al. (Reference Alevy, Landry and List2015)	Familiar	[0.8%]	Questionable	[8.9%]	WTA	[8.2%]	1	187	1.06	0.120
Ma et al. (Reference Ma, Li, Shen and Qiu2015)	Unfamiliar	[1.5%]	Informative	[0.1%]	WTA	[0.9%]	1	48	1.34	3.228
Shah et al. (Reference Shah, Shafir and Mullainathan2015)	Familiar	[0.4%]	Questionable	[4.1%]	WTP	[0.4%]	1	95	1.53	0.934
Isoni et al. (Reference Isoni, Brooks, Loomes and Sugden2016)	Unfamiliar	[14.9%]	Informative	[0.9%]	WTA	[8.9%]	1	204	1.19	0.186
Jung et al. (Reference Jung, Perfecto and Nelson2016)	Familiar	[82.2%]	Informative	[84.9%]	WTP	[85.7%]	14	1383	1.13	0.140
Li et al. (Reference Li, Yin, Li, Liu, Wang and Qu2017)	Familiar	[0.4 %]	Questionable	[4.0 %]	WTP	[0.4 %]	1	88	1.21	0.624
Ifcher and Zarghamee (Reference Ifcher and Zarghamee2020)	Familiar	[0.8%]	Informative	[0.8%]	WTP	[0.9%]	1	190	1.31	0.315
Yoon et al. (Reference Yoon, Fong and Dimoka2019)	Familiar	[1.5%]	Questionable	[16.0%]	WTP	[1.5%]	3	117	1.30	0.615
Yoon and Fong (Reference Yoon and Fong2019)	Familiar	[3.6%]	Questionable	[39.1%]	WTP	[3.7%]	4	215	1.33	0.581
(b) Uninformative anchors (corresponds to Fig. 3b)
Fudenberg et al. (Reference Fudenberg, Levine and Maniadis2012)	Familiar	[28.6%]	Uninformative	[24.0%]	WTA	[27.2%]	2	79	1.00	0.001
Fudenberg et al. (Reference Fudenberg, Levine and Maniadis2012)	Familiar	[14.1%]	Uninformative	[11.8%]	WTP	[100%]	1	78	0.99	0.009
Fudenberg et al. (Reference Fudenberg, Levine and Maniadis2012)	Unfamiliar	[100%]	Uninformative	[16.4%]	WTA	[18.6%]	1	108	0.96	0.086
Current study (2020)	Familiar	[57.2%]	Uninformative	[47.9%]	WTA	[54.3%]	1	316	1.05	0.089

Notes to columns (column number in parentheses):

(2–4): The percentages indicate the weight each study receives in the analyses we carry out along each of the three dimensions (see footnote Footnote 14 for formulas). The percentages are normalized to add up to 100% within each category. For aggregated results, we present the sum of weights.

(5): #Results display the number of treatments in a paper that share the same features for the variables listed in columns (2), (3) and (4) and that differ in features that are not included in the table.

(6): For aggregated results, we present the average sample size.

(7): The ratio is computed on the basis of a median split. For aggregated results, we present the sample-weighted average ratio.

(8): For aggregated results, we present the sample-weighted average Hedge’s g

The first feature describes the type of good for which a valuation was elicited. Familiar goods are ordinary goods that most people now and then consume, like wine, chocolate and books. Unfamiliar goods are goods for which people lack daily life experience, like consuming badly tasting liquids and listening to unpleasant sounds. People’s preferences may be more affected by anchors when they report their value for an unfamiliar product for which they do not have well-articulated preferences.

The second feature describes the extent to which the anchor may have been perceived as being informative about the price of the good. Some studies use informative anchors. One such example is provided by Jung et al. (Reference Jung, Perfecto and Nelson2016), who investigate the effect of a default anchor on people’s donation in a Pay-What-You-Want pricing scheme. Naturally, a default may be perceived as a recommended donation. There are also studies that intended to provide an uninformative anchor, but which may unintentionally have been interpreted as informative by subjects. We categorize these anchors as questionable. One such approach is to let a subject’s anchor be determined by the last two digits of their social security number (SSN) (e.g., Ariely et al. Reference Ariely, Loewenstein and Prelec2003; Bergman et al. Reference Bergman, Ellingsen, Johannesson and Svensson2010). About one-third of the subjects of Chapman and Johnson (Reference Chapman and Johnson1999) mention that they thought that the SSN anchor was informative. Likewise, Yoon and Fong (Reference Yoon and Fong2019) and Yoon et al. (Reference Yoon, Fong and Dimoka2019) use randomly generated uninformative prices, but leave subjects in the dark about the nature of the random number. Their instructions do not exclude the possibility that the random number is somehow correlated to the true price.Footnote ¹³ Studies that use a randomly generated anchor, and clearly communicate the whole procedure to the subjects, are categorized as using an uninformative anchor.

The third feature in which studies differ is whether subjects’ willingness-to-accept (WTA) or willingness-to-pay (WTP) is elicited. We include this variable because initially there was some support for the idea that anchoring effects are more easily observed for WTP than for WTA (Simonson and Drolet Reference Simonson and Drolet2004).

Before presenting the results of this concise meta-analysis, we motivate some methodological choices. First, all results need to be weighted appropriately with respect to their precision. Studies are typically weighted by the inverse of the variance of the estimated effect size. We use this approach here and weigh each study according to the variance of Hedge’s g.Footnote ¹⁴ Second, to test whether the effect size varies across different subgroups, we use the Q statistic. The Q statistic is a measure of the weighted variance of the effect sizes and is compared with the variance that would have been observed if all effect sizes where sampled from a population with the same mean.Footnote ¹⁵ Third, we use a random-effects model. Given that we are accumulating data from different studies that were carried out in different ways, we believe that the random-effects model is more appropriate than the fixed-effects model.

Figure 3 provides an overview of how these three dimensions affect the effect of anchoring on elicited valuations. We present the results in two separate forest plots, the lower one for studies that use uninformative anchors and the upper one for studies that use informative or questionable anchors. Overall, whether an anchor is uninformative or not has a strong effect on whether anchoring affects elicited valuations or not.Footnote ¹⁶ With uninformative anchors, we find a precise null-result for the effect of anchoring on elicited valuations. In contrast, with informative or questionable anchors there is a sizable and significant effect of anchoring. The difference in anchoring effects between studies with uninformative anchors and the other studies is significant $(Q=27.67,p<0.001,N=83)$ .

Fig. 3

Forest plots of results by anchor type

Within the class of studies that use informative or questionable anchors, we find the following results of the mediating variables on the empirical relevance of anchoring: (i) anchors have a significantly stronger effect for unfamiliar goods for which people do not have a clear initial preference to start with than for familiar goods, whereas (ii) whether WTA or WTP is used to elicit subjects’ preferences does not matter; (iii) somewhat surprisingly, we find a significantly stronger effect of questionable anchors than familiar anchors, which supports the view that many of the questionable anchors were actually interpreted as informative by subjects.

Interestingly, in studies that use transparently uninformative anchors, anchoring never has an effect on elicited valuations. So far, whether a familiar or unfamiliar good is used does not matter when the anchor is clearly uninformative. Although these studies are based on relatively many data, there are only few of them, and clearly more studies in this category are welcome.

In the class of studies that use familiar goods, a final interesting comparison is between studies that use clearly uninformative anchors and those that do not. Anchoring effects on elicited valuations are only observed in the latter category, and the difference in the effect is significant $(Q=23.21,p<0.001,N=52)$ . So, preferences for familiar goods can be anchored, but this requires the use of an anchor that is perceived as informative.

Overall, this meta-analysis yields the following picture: (i) if anchors are informative or perceived to be informative, then (unsurprisingly) anchoring has an effect, and mediating variables play mostly a sensible role, that is, no difference in the effect of anchoring on WTA and in the effect on WTP, while anchors have a stronger effect for valuations of unfamiliar products than familiar products; (ii) in the few studies in which anchors are uninformative, there is a quite precise null-effect, and so far none of the mediating variables plays any role.

In many cases with real-world relevance, evaluations may be made in the presence of seemingly (though not truly) informative anchors. In our view, our study sheds light on what makes anchoring of valuations actually have an impact. Originally, Tversky and Kahneman (Reference Tversky and Kahneman1974) demonstrated the effect of anchors on people’s judgments with transparently uninformative numbers. On page 1128, they write “subjects were asked to estimate various quantities, stated in percentages (for example, the percentage of African countries in the United Nations). For each quantity, a number between 0 and 100 was determined by spinning a wheel of fortune in the subjects’ presence. The subjects were instructed to indicate first whether that number was higher or lower than the value of the quantity, and then to estimate the value of the quantity by moving upward or downward from the given number. Different groups were given different numbers for each quantity, and these arbitrary numbers had a marked effect on estimates.” Our study suggests that the source of the anchoring of valuations may not be that a random uninformative number is imprinted in a subject’s mind. Instead, the problem seems to be that people can be tricked into believing that an uninformative piece of information is actually a relevant piece of information. In this sense, the anchoring of valuations is more about people being too gullible when they process information. If the anchoring of preferences only reliably appears when subjects perceive the anchor as informative, then it may be less appropriate to think of the anchoring of preferences as an anchoring bias. Instead, it seems to be driven by a perception bias.