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Klaus Peter Kaas/Heidrun Ruprecht*

ARETHE VICKREY AUCTIONANDTHE BDM MECHANISMREALLY INCENTIVE COMPATIBLE? EMPIRICAL RESULTS

AND OPTIMAL BIDDING STRATEGIESIN CASESOF

UNCERTAIN WILLINGNESS-TO-PAY**

ABSTRACT

Willingness-to-pay (WTP) data elicited with the help of incentive compatible methods

like the Vickrey auction and the BDM mechanism promise higher validity than stated

preferences data and provide more information content than do revealed-preferences

data. However, research shows that subjects in a Vickrey auction do not always follow

the dominant strategy of bidding their WTP, but frequently bid too high or too low.

Although this phenomenon is usually attributed to irrational behavior, that need not

be the only explanation. In marketing research applications, subjects are typically

asked their WTP for a new product, and it appears that most subjects are uncertain

about their exact WTP. We present a modified bidding model to explain that in this

case, it is optimal for risk-averse but rational bidders to underestimate, rather than tooverestimate, their WTP. The model is supported by preliminary experimental data.

JEL-Classification: D81, M31.

Keywords: BDM-mechanism; Discriminal Dispersion; Incentive Compatibility; Vickrey

Auction; Willingness-to-pay.

1 INTRODUCTION

With the advent of strategies like price bundling, two-part-tariffs, and internetpricing, the whole field of pricing has been experiencing a renaissance in both prac-tical applications and in theory. This development has led to the increased need forvalid methods to measure the willingness-to-pay (WTP) of every single customer for agiven product (Cameron and James (1987, 389)). However, eliciting valid WTP data isa problem that has not yet been adequately solved. Traditionally, marketing research hasused two elicitation methods: The analysis of revealed-preference data and stated-prefer-ence data. Revealed-preference data normally do not allow the elicitation of exact WTP

* Klaus Peter Kaas, Professor of Marketing, Heidrun Ruprecht, Research Assistant, Professur fr Betriebswirtschafts-

lehre, insb. Marketing I, Goethe-Universitt Frankfurt am Main, Mertonstr. 17-25, 60054 Frankfurt, Germany,Tel.: (+49)-69-798-23925, Fax: (+49)-69-798-23402, Email: [email protected].** We would like to thank the German Research Foundation (DFG) for the financial support (KA 397/

6-1) for this research project.

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of individual customers, but are limited to aggregate information such as price-responsefunctions (Ben-Akiva et al. (1994, 344)). To the contrary, stated-preference data yield indi-

vidual WTP data. However, the validity of such data is generally assumed to be low, becausesubjects give hypothetical answers to hypothetical questions (Carson, Groves, and Machina(2000, 41)).

Since the beginning of the 1990s an additional class of methods has received increasingattention. These are the so-called incentive-compatible methods, which have beenwidely discussed in experimental economics and which may be able to overcome thelimits of the traditional marketing research methods (Hoffman et al. (1993)). The incen-tive-compatible methods include the Vickrey auction (Vickrey (1961)) and the BDMmechanism (after Becker, DeGroot, and Marschak (1964)). The validity of the WTP data

obtained with these methods should be high, due to the incentives. The incentives work assubjects are put in a real purchase situation: they are asked to submit a binding purchaseoffer for the product in question, without being able to influence the resulting price withtheir offer. Therefore, it is optimal for rational subjects to reveal their exact WTP (Shogrenet al. (2001a); Wertenbroch and Skiera (2002, 230)).

Although incentive compatibility has been proven in theory, doubts remain as to thebehavioral properties of both methods (Kagel (1995); Harstad (2000)). If the methodswere also behaviorally incentive compatible, no overbidding or underbidding should beobserved, and the resulting WTP distributions should be the same for both methods.

However, this is not always the case, as studies from experimental economics andmarketing research show. More research is needed to determine whether the incentivecompatible methods keep their theoretical promise in empirical applications. If we wereto find systematic deviations from the theoretical predictions, it would be helpful to knowtheir direction and scope.

The aim of this paper is to contribute both empirically and theoretically to this ongoingresearch. To accomplish our goal, we present empiricalresults from an experimental studythat compares the WTP for a new chocolate brand, which we measure by incentive-compatible methods and stated preferences. In addition, we obtain indicators for overbid-

ding and underbidding from a questionnaire.

The findings are striking: the bids in the incentive-compatible methods are not onlybelow the WTP measured by the stated-preference data, but also below the true WTP.This tendency towards underbidding is slightly stronger in the Vickrey auction than inthe BDM mechanism.

The theoretical contribution of our paper comprises a modified bidding model.The main idea of the model is a discriminal dispersion of WTP, which can bethought of as a perception or measurement error of the subjects when they try to

find their true WTP. The model shows that rational, risk-averse bidders who maxi-mize their utility according to expected utility theory (Von Neumann and Morgenstern(1947)) would rather underestimate their WTP than overestimate it.

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The paper continues as follows. Section 2 outlines the Vickrey auction and theBDM mechanism. We discuss the empirical results from the experimental economics

and marketing research literature, focusing on the findings on overbidding andunderbidding. Section 3 presents our design and interpretation of a new exper-imental study. In Section 4 we develop the model of bidding under discrim-inal dispersion. We demonstrate that the underbidding found in the experimentcan be explained by this model. In Section 5 we conclude with implications for thepractical measurement of WTP and further theoretical research.

2 PRIORRESEARCHONINCENTIVE-COMPATIBLEMETHODS

2.1 VICKREYAUCTIONAND BDM MECHANISM

The Vickrey auction, developed by William Vickrey (1961), is also known as a second-price sealed-bid auction. As the name implies, the bids are submitted using a sealedbid procedure in which the individual bids cannot be observed by rival bidders, and thehighest bidder wins the auction at the second-highest bid price. The incentive compat-ibility is achieved through the fact that the winners bid cannot influence the price shepays (McAfee and McMillan (1987, 708)). The dominant strategy for all bidders is to bidtheir exact WTP, independent of the number of bidders and their behavior in the auction(Vickrey (1961, 22)).

The BDM mechanism works in much the same way as the Vickrey auction, but differsin the rule determining the price a bidder has to pay (Davis and Holt (1993, 461)). Theprice is not the result of competition between the bidders in an auction, but is determinedrandomly, e.g., by a draw from a uniform distribution. Every subject whose bid equals orexceeds the random price buys the product at this random price; all other subjects are notallowed to buy the product (Wertenbroch and Skiera (2002, 230)).

Both the Vickrey auction and the BDM mechanism are theoretically incentive compat-ible, as it is optimal for a rational bidder to bid her exact WTP (McAfee and McMillan(1987, 712)). But are they alsobehaviorally incentive compatible, i.e., do subjects behaveaccording to the theoretical predictions and actually bid their exact WTP (Hofmann et al.(1993, 322); Rutstrm (1998, 428))? Behavioral incentive compatibility is obviously diffi-cult to test, since the subjects true WTP cannot be observed. To overcome this difficulty,we can either control the true WTP by inducing values for imaginary products (Smith(1976)), or we can determine and validate the WTP for real products with the help ofindicators obtained through a questionnaire and through method comparisons.

2.2 TESTSOFINCENTIVECOMPATIBILITYIN INDUCED-VALUES EXPERIMENTS

Because it allows the bidding strategy to be isolated, the experimental economics litera-ture has applied the induced-values design mainly to test the behavioral properties of theVickrey auction (e.g., Coppinger, Smith, and Titus (1980)). From the available studies with

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induced values, we can draw the following conclusion: the dominant strategy seems not tobe intuitive in the Vickrey auction, subjects react with underbidding or overbidding.

Studies by Coppinger, Smith, and Titus (1980), Cox, Roberson, and Smith (1982) andHarstad (2000) show an overall tendency towards underbidding. Underbidding may stemfrom the illusion that a lower bid may lower the price without lowering the chances ofwinning. Other studies find evidence of moderate overbidding in Vickrey auctions (Kagel,Harstad, and Levin (1987), Kagel and Levin (1993)). Kagel and Levin (1993) illustratethe appeal of overbidding as follows: () the idea that bidding modestly in excess ofxonly increases the chances of winning the auction when you do not want to win is farfrom obvious under the sealed bid procedure (p. 1286). Studies by Gth, Schmittberger,and Schwarze (1983) and Sonnegrd (1996) report strong, frequent underbidding and

some moderate overbidding. Overall, it appears that subjects react heterogeneously, butmostly with underbidding, such that the WTP distribution obtained by a Vickrey auctionshould on average produce values too low.

2.3 TESTSOFINCENTIVECOMPATIBILITYFORREALPRODUCTS

How can we evaluate the behavioral incentive compatibility of the Vickrey auction andBDM mechanism in the case of real products? Since the true WTP remain unknown,tests must rely on indicators of bidding strategy and method comparisons. These are theapproaches used in marketing research applications (Skiera and Revenstorff (1999)).

Studies involving method comparisons between incentive-compatible and stated-preferencesmethods overwhelmingly conclude that stated WTP are higher1. This result is generallyattributed to a hypothetical bias of stated preferences. Validity tests also favor the incentivecompatible methods. E.g., Wertenbroch and Skiera (2002) find that the WTP for cake andCoke is more strongly correlated with hunger or thirst and also with the evaluation of thepurchase in the case of incentive compatible methods than in the case of stated preferences.

Another reason for the higher WTP measured by the stated-preferences method could alsocome from a downward bias of WTP measured by incentive-compatible methods. This

issue has not been raised so far, and at this point the studies neither support nor contradictthis possibility. We expand on this idea later in this paper when we discuss our model.

We can further investigate the behavioral incentive compatibility by comparing theWTP that results from different incentive-compatible methods. The few studies that usethis approach show that different methods do not lead to the same WTP distributions(Shogren et al. (2001a); Rutstrm (1998)). The information content of this finding islimited. It indicates that at best, only one of the two methods provides empirically incen-tive-compatible results, but we cannot determine which one of the two it is, if any.

1 See Cummings, Harrison, and Rutstrm (1995); Schulze et al. (1996); Fox et al. (1998); Wertenbroch and Ski-era (2002); Balistreri, McClelland, and Schulze (2001); for a different result see Sattler and Nitschke (2003).

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The overall picture that can be drawn from all these results is still somewhat incomplete.The number of results is still too small and the results themselves are partly contradictory,

especially in the case of WTP values for real products, which are relevant in marketing.We designed the following experiment to help complete this picture.

3 ANEXPERIMENTTESTINGINCENTIVE-COMPATIBLEMETHODSFORTHEELICITATIONOFWTP

3.1 DESIGNANDPROCEDURE

The goal of the experiment is to test the behavioral incentive compatibility of the Vickreyauction and the BDM mechanism with the help of indicators, and to determine the effect

of the incentive-compatible methods compared to the stated-preferences method.

To accomplish our goals, we used a between-subjects design to ensure that no prior expe-rience with the other methods could lead to confusion about strategy, as might be the casein a within-subjects design. We conducted all experiments in class with first-year studentsmajoring in business or economics at the Goethe-University Frankfurt, Germany, over atwo-week period from Nov 25, 2002 to Dec 5, 2002.

The sample size was n = 101 for the Vickrey auction, n = 33 for the BDM-mechanismand n = 27 for the stated-preferences question, which took the form of an open-ended-question (OEQ). We asked for WTP for a bar of chocolate imported from France just forthis purpose and not available in Germany at the time of the experiment. Subjects weretold that it was a 200g bar of white chocolate with nuts and cracknel, manufactured bythe French traditional brand Poulain. We chose this product because it was new to thesubjects, and because it was affordable due to its relatively low price, yet supposedly inter-esting for the majority. We then gave the chocolate to the subjects to look at and to try apiece. Then we explained the respective method to every experimental group.

For the incentive-compatible methods, we gave a numerical example to illustrate the domi-nant strategy and the disadvantages of overbidding and underbidding. To avoid anchoringas much as possible, our example used a different price range from that of the chocolate

price range. We encouraged students to ask questions, and some did. We also stressed thatall bids were binding and that winners would be obliged to purchase the chocolate.

In the Vickrey auction, subjects then submitted their bids. To accommodate the largegroup, we used a multiple units generalization of the Vickrey auction and determinedthe number of units so as to obtain a share of winners of one out of seven bidders,which is an average of previous studies. While the bids were being sorted, the subjectsanswered a questionnaire that included questions about product involvement, compre-hension, and bidding strategy. Then we announced the winners and the resulting price,and the products were sold. Subjects then answered a second questionnaire with ques-

tions on the outcome of the auction.

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The procedure in the BDM group was similar. In contrast to the usual version of drawingindividual random prices, here, due to time constraints, a single random price was drawn

for the whole group, but we believed that this modification would not affect the results.The random price was drawn from a uniform distribution in the interval from 1 to 2 ,which was not known to the subjects. If they asked, they were informed that all possibleprices were realistic.

In the stated-preferences group, subjects were not given the opportunity to buy the choc-olate. They only had to answer a questionnaire with an OEQ about their WTP for thechocolate, and some other questions.

3.2 TESTSOFBEHAVIORALINCENTIVECOMPATIBILITY

We conducted tests with the help of two indicators of overbidding and underbidding. Weobtained these indicators from the second questionnaire after the announcement of thewinners and the resulting prices.

First, subjects evaluated the purchase occasion at the resulting price on an 11-point scalefrom much too expensive to a very good deal. In combination with the subjectssuccess in the auction or BDM, we were able to ascertain their bidding strategy. Winners(who should have earned a non-negative consumer surplus at the actual selling price) who

judged the product too expensive had obviously bid too much, but losers (who shouldhave earned a negative consumer surplus in case they had to buy the chocolate bar at theactual selling price) who thought the product was a good deal had apparently underbid.

We believe that this indicator yields valid results, as the situation closely resembles a realshopping situation in which the price is given and subjects need to evaluate the purchaseoccasion at this given price. Furthermore, subjects saw how the products were sold to thewinners, which renders the evaluation question highly realistic and involving.

Second, we asked whether the losers would revise their bids upwards if they had one moreopportunity to buy the chocolate. The underbidding is indicated by the share of the loserswho revise their bids, and by the amount of the revision.

3.3 RESULTS

First, we present the results of the two indicators2. The first indicator classifies 24%(22%) of the subjects in the Vickrey auction (BDM) as underbidders (negativeconsumer surplus but positive purchase evaluation) and 7% (9%) of subjects as over-bidders (see table 1 ).

2 The data and analysis are available upon request from the authors.

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Table 1: Underbidding and overbidding in the incentive compatible methods

Negative consumer surplus (WTP < p) Positive consumer surplus (WTP > p)

Evaluation: Purchaseis a good deal

UnderbiddingVickrey: 24% of subjectsBDM: 22% of subjects

Truthful bidding

Evaluation: Productis too expensive

Truthful biddingOverbiddingVickrey: 9% of subjectsBDM: 7% of subjects

For both methods, the proportion of underbidders is significantly higher than the propor-

tion of overbidders (at the 5% level). The scope of underbidding indicated by the secondindicator is also relatively high [see table 2]. The share of losers revising their bids is 47%for the Vickrey auction and 41% for the BDM mechanism, and the average revision is0.30 for the Vickrey auction and 0.24 for BDM (both means significantly larger thanzero,p=0,000 andp=0,003, respectively). So both indicators are high, which reinforcesthe results of strong underbidding.

Table 2: Revision of bids in the incentive compatible methods

Share of losers revising their bids Average revision in

Vickrey 47% 0,30

BDM 41% 0,24

To compare the WTP distributions of all three methods, we present boxplots infigure 1.

Figure 1: Boxplots of WTP for the Vickrey auction, BDM mechanism and

open-ended question (medians: Vickrey 0.6 , BDM 0.99 , OEQ 1.25 )

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The hypothetical answers to the OEQ are signficantly above the WTP from the incentivecompatible methods (p < 0.05). This finding is consistent with the results in the litera-

ture. However, we doubt that the discrepancy can be explained by only a hypothetical biasof the stated preferences. Rather, we suggest that at least part of this deviation is due to adownward bias, as indicated by the strong tendency towards underbidding in the Vickreyauction and the BDM mechanism.

We can also state that the BDM mechanism yields higher WTP values than that of theVickrey auction, which is not surprising, given the somewhat higher tendency towardsunderbidding in the latter. The difference between the results of the two methods may becaused by different speculations about the WTP of other bidders in the Vickrey auctionand about the price range of the random draw in the BDM mechanism. Instead of further

exploring these deviations from rational behaviour (Ruprecht (2005)), we concentrate onthe downward bias that seems to be inherent in both methods.

4 A MODELOFBIDDINGUNDERUNCERTAIN WTP

How can this bias, which is rather strong, be explained? In this section, we modify thestandard bidding model for the case of uncertain WTP, which yields one possible expla-nation.

4.1 THENOTIONOFUNCERTAIN WTP

The proof of theoretical incentive compatibility relies on the usual assumption of micro-economic preference theory, that subjects know their own preferences (Von Neumannand Morgenstern (1947); Savage (1972); McAfee and McMillan (1987)). However, thisassumption may not always apply. The behavioral literature stresses that in general, prefer-ences are not necessarily known by the subjects and may be unstable over time (Kahnemanand Snell (1992)), valuations are initially malleable (Ariely, Loewenstein, and Prelec(2003)). Also, WTP as a special formulation of preference is a construct that is unfamiliarto subjects, who normally only decide whether to buy a product at a given price. Appar-

ently subjects experience great cognitive difficulty reporting their exact WTP when askedan open-ended question (Brown et al. (1996); Gregory et al. (1995); Ready, Whitehead,and Blomquist (1994)). The same may be true when subjects are asked to offer a bid inthe context of an auction or BDM mechanism. Some literature on auctions acknowl-edges the need for WTP formation by subjects (Shogren et al. (2001a); Chakravarti et al.(2002, 290)). Surprisingly, this notion of uncertain WTP has not led to an analysis of theconsequences that this uncertainty has on bidding strategy. Here, we drop the assump-tion that subjects really know their own WTP and instead investigate the consequenceson bidding strategy.

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4.2 DISCRIMINALDISPERSIONASAPERCEPTIONERROR

Uncertainty about preferences by subjects can be conceptualized with the random utilitymodel framework for discrete choice analysis (Hanemann (1984); Koning and Ridder(2003)). There, the utility of an alternative A is composed of a systematic utility componentUAand a random utility component . In analogy, we can define WTP (=Z) as consistingof a systematic component__Zand a random component 3.This notion of a random partof WTP is similar to the concept of a discriminal dispersion which was introduced in thepsychometric literature by L. L. Thurstone (1927; 1959). Thurstone used it to describe thehaziness in perception that leads to a normal distribution instead of a deterministic valuewhen subjects evaluate or perceive a psychological quantity such as the beauty of a paintingor the severity of a crime. The true value of this respective quantity equals the mean of

the distribution, and errors are symmetric around the mean. In the context of WTP elic-itation, this means that subjects who aim at their true WTP can make a kind of measure-ment error and report a higher or lower value by mistake. For example, the reader may askif she is able to find the true value of her WTP for a bar of the new chocolate brand justat this moment. Would they be able to report the right number, so that they were sure thatfive cents less would be not enough and a penny more would be too much?

The subjects in a Vickrey auction or BDM face double uncertainty as a consequence ofthe discriminal dispersion. They are not only uncertain about the WTP of other biddersin the auction or about the price range in the BDM mechanism, but also about their ownWTP. We show that this uncertainty will cause rational but risk-averse bidders to system-atically underestimate their WTP.

The intuition of the formal proof is as follows: imagine a subject who approaches herWTP cautiously from below and keeps a safe distance, because if she overestimates, shemight incur a loss that can be prevented by underestimating her WTP. On the other hand,underestimation cannot result in a loss.

4.3 OPTIMALBIDDINGUNDERCERTAIN WTP

Before modelling the optimal bidding strategy under discriminal dispersion, we outlinethe case of certainty as a benchmark. The proofs of theoretical incentive compatibilityfor the two mechanisms go back to Vickrey (1961) and Becker, DeGroot, and Marschak(1964), respectively, and can be formalized within an expected-utility-framework (e.g.,Cox, Roberson, and Smith (1982)). The following expositions build on that framework.

The surplus Sof a bidder with WTP__Zwho wins the product at pricep can be written as

S=__Zp. Letf(p) denote the density function of the resulting price in the Vickrey auction

or BDM mechanism, with b being the lowest possible price. Then the expected surplusfor the subject biddingZ* is:

3 We thank Prof. Dr. Ingo Balderjahn for this analogy.

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(1) E[S(p)] = b

z*

(__Z p) f (p)dp

We assume that all bidders expect a uniform distribution for the price in the interval[b, c]. This assumption enables us to illustrate our model without restricting thevalidity of our results. Figure 2 shows the surplus of a bidder as a function of theprice if she receives the product at this price. The surplus is a straight line S(p) =__Z-p in the interval [b, c] with the slope -1. We note that a bidder who bids under her __Zcan expect, if she wins, a positive surplus as the price will be smaller than

__Z. If she over-bids, the bidder may also find herself on the right side of

__Zifp turns out to be > __Z, whereshe has to expect a loss.

Obviously, the dominant strategy is to bid ones exact WTP. Then the expected surplusequals the areas I + II, weighted with the probability 1/(c b). Because all areas in thefollowing comparisons are equally weighted, we ignore this weighting factor in the rest ofthe paper. If the bidder underbids and reports, e.g., onlyZ, the expected surplus shrinksto area I. If she overbids and reports, e.g.,Z+, the surplus diminishes as the loss area III isadded. Therefore, the best strategy of a rational bidder who wishes to maximize her expectedprofit is to bid her exact WTP. Keep in mind that in this utility formulation there is noother utility than the monetary surplus (as the utility of the product is already included inthe WTP). This also implies that if a bidder loses the product because of underbidding,then the utility is zero, and there is no disutility. We return to this issue later.

Figure 2: Surplus as a function of price

4.4 OPTIMALBIDDINGUNDERDISCRIMINALDISPERSIONANDAVERAGE WTP

How does the optimal bidding strategy change when rational bidders try to find their trueWTP but encounter a discriminal dispersion? We investigate the consequences for twopossible symmetrical perception errors,

__Z- Z= Z+ __Z. We also assume for now that

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the mean of the discriminal dispersion__Zcoincides with the mean of the uniform price

distribution ((c + b)/2) (see figure 3).

Figure 3: Discriminal dispersion and price distribution

In this case the bidder is on margin (Shogren et al. (2001b)), which means she assumesthat she has an average WTP compared to the other bidders in the auction or to thepossible price range for BDM. Note that this also includes the case in which a bidder hasno idea about the other bidders WTP or the price range, since the bidder has no reasonto assume that her WTP is higher or lower than average, due to Laplaces principle ofinsufficient reason.

First, we consider the case of underestimation that leads to a bid ofZ. The expectedsurplus then equals area I infigure 2. We can obtain the same result by computing theintegral according to equation 1 [see appendix].

Next, we consider the case of overestimation by the same absolute amount, which leadsto a bid ofZ+. The expected surplus now equals areas I + II (forp

__Z). Due to the symmetry of the perception error, areas II and III are equal, exceptfor the sign, and cancel out. The excepted surplus in the case of overestimation thereforeamounts to area (I + II III =) I and is the same as in the case of underestimation [alsosee appendixfor the integral formulation]. This means that a risk-neutral bidder is indif-ferent between a positive and a negative perception error of the same amount.

Furthermore,figure 2shows that the expected surplus is higher when the perception erroris smaller, and vice versa. We can conclude that a rational bidder should try to deter-mine her WTP as accurately as possible and keep the perception error small. This notionreflects the idea of bounded rationality as formulated by H. Simon (1957) that subjectsact intendedly rational, but only limitedly so.

What conclusions can we draw for a risk-averse bidder? We can infer the risk associ-ated with overestimation and underestimation fromfigure 2. First, the range of possible

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outcomes is larger for overestimation [__Z b, __ZZ+] than for underestimation [ __Z b,

__Z Z]. Second, underestimation produces only a positive surplus (area I), but overesti-mation can lead to losses (area III). So for both risk measures, risk is higher in the case inwhich WTP is overestimated.

Thus, a risk-averse bidder with average WTP would prefer to underestimate than overesti-mate her WTP, if she cannot name it exactly due to discriminal dispersion. If we accept riskaversion as the prevailing risk attitude4 for WTP elicitation, then we have given an explana-tion for the downward bias of the incentive compatible methods found in our experiment.

4.5 OPTIMALBIDDINGUNDER DISCRIMINALDISPERSION ANDHIGHORLOW WTP

Now, we assume an off margin bidder, i.e., one whose WTP __Zis higher or lower thanaverage. We assume that the perception errors are unchanged. Figure 4shows for a high

WTP that the surplus function moves to the right. As a result, area I is enlarged while areasII and III remain the same (as long asZ+< cwhich we assume for now). This means thatexpected surplus increases compared with that of an on margin bidder, but remains iden-tical for overestimation and underestimation.

Figure 4: Consequences of overestimation and underestimation for high WTP

How has the risk changed? Overestimation has become less risky as the share of theexpected loss (area III) to the expected positive surplus (areas I + II) has diminished. Acomparison of the range of outcomes for overestimation and underestimation producesthe same result, that overestimation is now relatively less risky. However, it is still morerisky than underestimation in absolute terms.

In the other case of a low WTP, the result is the opposite. We can now summarize bothcases: in a Vickrey auction or BDM mechanism, risk-averse bidders who feel to have a

4 This is the usual assumption in portfolio theory, where investors are said to prefer less variance in their returnsto more (e.g., Markowitz (1952); Rudolph (1979); Mehra and Prescott (1985)).

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relatively high (low) WTP have a weaker (stronger) tendency towards underestimation oftheir WTP than do bidders with average WTP.

Next, we look at bidders with extremely high or low WTP. The term extremedenotes a WTP that is situated so far from the mean of the price distribution thatthe perception error lies outside of the boundaries of the interval [b,c]. If there isan extremely high WTP

__Zclose to the upper bound c, part of the loss area is cut off[see figure 5], resulting in a higher expected surplus for the case of overestimation(I + II - III) than for the case of underestimation (I). Therefore, a risk-neutral bidder withextremely high WTP will tend to overestimate rather than underestimate her WTP.

Figure 5: Overestimation with very high WTP so that Z + > c

At the same time, the risk inherent in overestimation diminishes, but the risk associ-ated with underestimation increases, both in relation to the situation of average WTP.However, there is still more risk in overestimation. How will that affect a risk-aversebidder? The answer depends on the degree of risk aversion. A bidder with low risk aver-sion will overestimate, if the higher risk is more than compensated by the higher surplusof overestimation. Only a bidder who is strongly risk averse and has an extremely highWTP will underestimate her WTP.

We now consider the opposite and assume a bidders WTP to be extremely low. In thecase of underestimation, the bidder cannot win in the auction or in the BDM mechanism,because Z < b, the expected surplus, is zero. Overestimation even produces a negativeexpected surplus as the loss area III becomes larger than area II (area I has disappeared).As a consequence, bidders with extremely low WTP should be strongly inclined to under-estimate their WTP, and this result does not depend on the bidders risk attitude.

5 SUMMARYOFRESULTS

In section 3 we reported experimental results of strong underbidding in the Vickreyauction and BDM mechanism. As an explanation, we have developed a modi-

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fication of the standard bidding model. Our modification is based on the concept ofdiscriminal dispersion, which goes back to Thurstone (1927). It is similar to the random

utility assumption in choice models.

Under the assumption of rational and risk-averse bidders, we can make the followingpredictions:

Risk-averse bidders, who either assume they have an average WTP or have no idea abouttheir relative WTP, tend to underestimate their WTP. Due to the uncertainty about theirtrue WTP, they prefer to be careful and bid lower rather than higher values.

For risk-averse bidders who feel that their WTP is higher than average, the tendency

towards underestimation is weaker, or can even turn into overestimation if the degreeof risk aversion is low. For risk-averse bidders who assume that their WTP is lower thanaverage, the tendency towards underestimation is strongest. In this case, underestimationcan be optimal even for risk-neutral bidders.

We can confirm these relations between bidding and relative WTP with the data fromour experiment. The bidders in Vickrey and BDM who are classified as overbidders [recalltable 1] bid significantly (p < 0.001) higher than did underbidders [see table 3].

Table 3: Relationship between bidding and relative WTP

Range of bids Mean bid

Overbidders 1.69 2.99 2.11

Underbidders 0 1.50 0.81

6 CONCLUSION

In experimental economics, the Vickrey auction and the BDM mechanism areapplied to elicit WTP as indicators of the relative attractiveness of options inexperimental choice situations and market simulations. In marketing applications, thesetwo incentive-compatible methods promise to enhance the information basis for pricing,if the WTP elicited with these methods are more valid than are the WTP elicited withthe help of traditional methods.

But are the WTP data obtained with the incentive compatible methods reallymore valid? Both the Vickrey auction and the BDM mechanism implicitly assumethat subjects know their exact WTP. Then it is the dominant strategy to reveal

their WTP. Empirical findings of violations of the dominant strategy are usuallyattributed to irrational behavior. An alternative explanation, at least for underbidding, isthe concept of a discriminal dispersion. We have shown that it can be rational for bidders

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to underestimate their WTP if they are risk averse. The lower the WTP, the stronger thistendency towards undererstimation. Thus, incentive-compatible methods no longer yield

valid results when discriminal dispersion applies.

Future research must show if our hypothesis of underbidding due to a discriminal disper-sion of WTP can be confirmed in other empirical studies. Should our hypothesis ofa systematic downward bias be confirmed it would be important to know the size ofthe bias, its persistence and its determinants. Replications and other method compari-sons as well as investigations in other product categories seem well worthwhile. Subjectsmight exhibit a lower preference uncertainty with familiar products than with new prod-ucts, which implies that underbidding should be lower. The same might be true for highinvolvement versus low involvement products. These arguments apply to the shape of the

discriminal dispersion. Other parameters of interest that would need to be studied empir-ically include the price range and the shape of the price distribution.Finally, model extensions to other utility formulations appear as a promising area of futureresearch. As an alternative to expected utility theory, prospect theory (Kahneman andTversky (1979)) would predict an even more negative evaluation of losses, thus makingunderestimation of WTP stronger. Another possible extension would be regret theory (Bell(1982), Loomes and Sugden (1982)). Regret could arise when a bidder accidentally bidsthe wrong WTP value which leads to an outcome worse than the outcome she wouldhave achieved with the right value. In case of overestimation a monetary loss can be theresult, as demonstrated above. This leads not only to a decreasing expected value but alsoto an additional disutility accounting for the regret of having missed a better outcome.

In case of underestimation, the bidder risks not getting the product and a possibleconsumer surplus. Under expected utility theory this is not counted as a loss, since themonetary wealth of the bidder remains unchanged (see above). However, there can beregret for forgoing a possible gain due to underestimation, if the bidder knows only afterthe auction that she would have been willing to pay the resulting price or even more.Therefore utility in this case would be negative after subtracting the disutility due to regretfrom the expected value of 0.

An interesting question here is whether a bidder perceives the risk of a real monetaryloss equal to the risk of an equivalent foregone gain, and whether the foregone gain isviewed strictly in monetary terms or rather as the loss of the product. The impact ofregret when losing the product might also depend on the uniqueness of the product: ifa product is unique like, e.g., tickets for an event, then regret would probably be higher.One theory that is possibly applicable to these questions is mental accounting (Thaler(1980)). Different mental accounts might be used for products than for monetary gainsand losses, and calculation rules might differ for products in different product categoriesif they belong to different mental accounts. At this point we do not have clear sugges-tions as to how exactly this works, but we believe that mental accounting of outcomes of

bidding strategies constitutes one fertile area of future research.

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APPENDIX

CALCULATIONOFEXPECTEDSURPLUSFORBIDDERSWITHAVERAGE WTP WHOUNDERESTIMATEOR

OVERESTIMATETHEIR WTP

Expected surplus in the case of underestimation

The probability of winning at bidZequals the probability that b

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Expected surplus in the case of overestimation

In the case of overestimation, losses are possible, depending on the resulting price. Theprobability of a positive surplus is the probability that b

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