Bidding as if risk neutral in experimental first price auctions without information feedback

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1 Exp Econ DOI /s Bidding as if risk neutral in experimental first price auctions without information feedback Tibor Neugebauer Javier Perote Received: 10 November 2006 / Revised: 21 February 2007 / Accepted: 6 March 2007 Economic Science Association 2007 Abstract Experimental research on first price sealed bid auctions has usually involved repeated settings with information feedback on winning bids and payoffs after each auction round. Relative to the risk neutral Nash equilibrium, significantly higher bidding has been reported. The present paper reports the results of experimental first price auctions with n = 7 where feedback on payoffs and winning bids is withheld. Under these conditions, average bidding is below the risk neutral Nash equilibrium prediction but converges to it with repetition. Keywords Experimental economics First-price sealed-bid auctions Independent private value model Bidding theory Risk aversion Learning JEL Classification C12 C13 C72 C92 D44 1 Introduction The present paper reports an experiment on repeated first price sealed bid auctions involving a market of seven participants in an independent values setting. The novel feature of the present experimental design (besides the market size involved) is that Electronic Supplementary Material The online version of this article ( contains supplementary material, which is available to authorized users. T. Neugebauer ( ) Institut für Finanzmarkttheorie, Universität Hannover, Königsworther Platz 1, Hannover, Germany T.Neugebauer@mbox.vwl.uni-hannover.de J. Perote Dpto. de Economía Aplicada II y Fundamentos del Análisis Económico, Universidad Rey Juan Carlos, Campus de Vicálvaro, Madrid, Spain

2 T. Neugebauer, J. Perote subjects are exposed to conditions in which they do not receive any feedback information, i.e., neither on the winning bid nor on the payoff at the end of the auction round. Under these conditions we find that average bidding is below the risk neutral Nash equilibrium and converges towards the equilibrium prediction during the early rounds of the experiment. It seems that subjects learn to bid on the basis of repeated reflection on their values. This result stands in sharp contrast to the well-documented evidence on overbidding relative to the risk neutral Nash equilibrium (for a survey see Kagel 1995). In contrast to our study, however, the literature has focused on repeated settings with feedback conditions that involve the winning bid and the payoff after each auction round. Cox et al. (1982) explained the observed overbidding in the repeated setting with a theory that allowed preferences for constant relative risk aversion, thus extending Vickrey s (1961) theory of risk neutral Nash equilibrium bidding. The relationship between overbidding in the repeated first price auctions and risk aversion received major attention by the experimental community (Harrison 1992; Cox et al. 1992; Friedman 1992; Kagel and Roth 1992; Merlo and Schotter 1992). 1 Our data suggest that the feedback information on winning bids triggers an immediate response in bidding behavior. Subjects anchor on the winning bid as if it was the price to be paid in the market. 2 Hence, when provided with feedback on the winning bid, subjects learn to bid on the basis of this winning bid rather than by reflecting on their values. A learning story in the repeated first price auction had been proposed by Selten and Buchta (1999). 3 However, only recently it has been shown (Neugebauer and Selten 2006) that bidding above risk neutrality may depend on the feedback information conditions. Neugebauer and Selten found that bidding was rather below than above the risk neutral equilibrium prediction when subjects received no quantitative feedback information. The experimental study of the present paper differs from the one presented in Neugebauer and Selten in several aspects; three essential differences follow. First, Neugebauer and Selten exposed subjects to feedback conditions which at least revealed the payoff after each round. 4 Second, the reported experiments are set in a more natural environment in the sense that they use interacting human subjects, while Neugebauer and Selten considered only markets with one human bidder and (n 1) Nash robots. 5 Third, individual values change from one round to the next, whereas 1 Friedman (1992, p. 1374) referred to this discussion as the loudest debate amongst experimentalists ever heard. Other important contributions to this debate involve Kagel et al. (1987), Harrison (1989), Kagel and Levin (1993), Cason (1995), Chen and Plott (1998), Goereeetal. (2002), Dorsey and Razzolini (2003), Armantier and Treich (2005). 2 This statement is consistent with subjects responses in the debriefings. 3 Friedman (1992) suggested such a learning story in his contribution to the debate. 4 Other experimental studies vary the feedback conditions and support a relationship between feedback information and behavior in first price auctions, too. In addition to the winning bid, Isaac and Isaac and Walker (1985) and Ockenfels and Selten (2005) offer information on all bids in the market. Additionally to this condition on information feedback, Neugebauer and Selten (2006), Dufwenberg and Gneezy (2002), Ozbay and Filiz (in press) and Engelbrecht-Wiggans and Katok (2005a) study a third feedback condition that reveals no other bid than the subject s own one. In contrast to the present paper, all these studies involve feedback on subject s payoff after each bidding round. 5 As noted in Neugebauer and Selten (p. 185), the bidding dynamics can be different when subjects interact with each other than when they interact with Nash robots. The available experimental results are not

3 Bidding as if risk neutral in experimental first price auctions in Neugebauer and Selten the value to the human subject was fixed such that the risk neutral Nash equilibrium implied degenerated bidding. The advantage of the present approach is that the data is comparable to other studies in the relevant literature. The remainder of the paper is organized as follows. Section 2 reviews the risk neutral Nash equilibrium prediction; Sect. 3 presents the details of the experimental design; the results are reported in Sect. 4; and Sect. 5 provides concluding remarks. 2 Risk Neutral Nash Equilibrium bidding RNNE Assume n>2 bidders compete in a market where a single object is auctioned. Every bidder i has a private valuation which is represented by the resale value x i 0. Resale values are independently drawn from a uniform distribution over the unit interval. Assume the first-price sealed-bid auction rule is applied such that every bidder submits a sealed-bid and the winning bidder wins the auction and pays his bid. Vickrey (1961) showed the existence of a unique symmetric Nash equilibrium for risk neutral bidders (hereafter RNNE), in which subjects bids involve a constant fraction of their value. The RNNE strategy writes as follows. b (x i ) = n 1 n x i, i = 1,...,n. (1) The RNNE bidding strategy can be interpreted as the bid which is equal to the bidder s expectation about the second highest resale value given that this value is smaller than his own. The strategy constitutes a best response only if all other market participants use the same strategy. Note that the risk neutral Nash equilibrium does not account for feedback information conditions. 3 Experimental design In the experiment, we used within-subject variation. Subjects were subsequently exposed to two feedback treatments, which we refer to as the Info and NoInfo treatment hereafter. In each treatment, subjects faced repeated play within the same group of seven participants for 50 auction rounds. In the Info treatment, subjects received information on the winning bid (i.e., the auction price) and on their own payoff after each auction round. The Info treatment must be seen as the control treatment since it involves the same feedback information conditions as most of the other available laboratory studies (see the survey by Kagel 1995). In the reported study, the novel treatment is the NoInfo treatment. In the NoInfo treatment, subjects received no information feedback on the winning bid or on the payoff for any round. The only feedback they received in the NoInfo treatment was the total payoff after the final round, i.e., t = 50. conclusive. Some studies report no differences in behavior when subjects are exposed to Nash automata than when they face human competitors (Cox et al. 1987; Engelbrecht-Wiggans and Katok 2005a), but others studies do report differences (e.g., Neugebauer 2004).

4 T. Neugebauer, J. Perote As reported below, we observe a significant order effect in bidding behavior for the NoInfo treatment. Bidding depends on the order of treatments, i.e. whether subjects are exposed to the NoInfo treatment before or after they have experienced the conditions of the Info treatment. 6 To document this effect, we have run the treatments in both orders: in the NoInfo-Info order and the reverse, Info-NoInfo, order. In the NoInfo-Info order of treatments, subjects interacted for the first fifty rounds under conditions of the NoInfo treatment and then for the second fifty rounds they were switched to the conditions of the Info treatment. In the Info-NoInfo order, subjects were exposed to the treatments in reversed order. 3.1 Procedures In total, 56 first year students (i.e., inexperienced subjects) participated in the simple computerized experiment (Urs Fischbacher s z-tree, in press); 28 were exposed to the treatments in the NoInfo-Info order, and the other 28 subjects were exposed to the Info-NoInfo order. The experiments were conducted at the Centre for Experimental Economics EXEC at the University of York in November 2001 and at the University of Hannover in November At the beginning of the experiment, subjects were given written instructions and were asked to read them carefully. 7 When they had finished, the experimenter read the instructions aloud. Any questions were answered by re-reading the corresponding sentences in the instructions. Afterwards, participants were introduced to the simple interface on their computer-screens. At the beginning of the experiment, the computer assigned participants randomly to experimental auction markets of size n = 7. Subjects competed within the same market for the entire 100 auction rounds of the experiment. In each auction round, individuals private values were independently drawn from a uniform distribution over the interval from 0 to 1, multiplied by 100 and rounded to the next integer. The subject had to submit a bid that could be a positive integer at or below her or his value. 8 According to the first-price auction rule, the winning bidder paid a price equal to her bid. In case of a tie, the winner of the auction was randomly chosen among the high bidders. After each round, all past observations, including own bids and values in both treatments and prices and payoffs in the Info treatment, were recorded in a table on the subject s screen. At the end of the experiment, subjects were asked to write a recommendation of how to bid in the auction. Then they were asked whether and how the feedback information on winning bids affected their behavior. Thereafter they had to tick one of seven boxes in a row to self-assess their inclination towards risk (from risk loving labeled at the utter left to risk averse at the utter right). Finally they filled out a questionnaire providing their personal data. Each of these queries 6 This is in line with the data of Neugebauer (2004) on elicited bid functions. In that study, subjects were exposed to different feedback conditions before they programmed a strategy for a tournament. The submitted strategies significantly differed according to the feedback conditions experienced. 7 The instructions are included in the supplement to the paper. 8 This constraint was introduced to omit the implementation of bankruptcy rules. However, eventual bidding above the resale value is of no interest in this study.

5 Bidding as if risk neutral in experimental first price auctions was successively made on a different screen. Afterwards subjects were privately paid their cumulative payoff plus a show up fee of 3 and 5 respectively. 9 4 Experimental results In this section we present our experimental data using the following notation: we denote the market, market participant and the round of the experiment by k, i and t, respectively. Since the theory predicts bids to be a constant fraction of value, much of our analysis involves the examination of the individual ratio of bid to value. Zero values, x kit = 0, are treated as missing observations. We begin the presentation by surveying the individual data in Table 1 including non-parametric tests. Thereafter, we look at the time pattern of behavior and test the average behavior for convergence to the RNNE. To preclude one possible source of confusion we index the treatments to indicate whether we refer to the first or the second run. For instance, the NoInfo treatment and the Info treatment of the NoInfo-Info order of treatments will be denoted by NoInfo 1 and Info 2 hereafter; likewise, we refer to Info 1 and NoInfo 2 in the reverse order of treatments. If a treatment is not indexed, we refer to the treatment in both orders of presentation. Most of our attention will be dedicated to the NoInfo 1 treatment, since it is the novel treatment within our experimental study. The testable hypothesis of this paper is that feedback information on winning bids and private profits affects overbidding of inexperienced subjects in the repeated game; hence, we use onetailed tests in the corresponding contexts. 4.1 Bidding without feedback According to the RNNE, subjects bid a constant fraction of their value, in particular b (x)/x = 6/ For the NoInfo-Info order of treatments, where the NoInfo 1 treatment was run first, Table 1 records the differences of the individual bid-value ratio from the RNNE ratio for the NoInfo 1 treatment (columns 1 3) and the first round of the Info 2 treatment (column 4). 10 From the two-tailed Wilcoxon signed ranks test results, recorded in the lowest row of Table 1, we deduce the following: First, average bidding is not above the RNNE in the NoInfo 1 treatment, either for the first round of the experiment (first column) or on average (second column). The negative difference between the average bidvalue ratio and the RNNE ratio along with the 95% confidence bands indicate that bidding is below the RNNE rather than above. Second, individual bids conditional on having received the highest value within the market (third column) exceed the RNNE by in the NoInfo 1 treatment; the difference of these bids from the RNNE is insignificant. This result along with the 95% confidence interval verifies the robustness of our findings for the bids which, according to the theory, should win the 9 The average payoff was 9 sterling and 13, respectively; the experiment was completed within an hour. 10 Table 3 continues to record the corresponding observations for the Info 2,Info 1 and NoInfo 2 treatments. Note that in contrast to Table 1 where the individual bid ratios within one column are always independent from another, Table 3 contains only independent data in column 7 (i.e., involving the bids submitted in the first round of Info 1 ). In the other seven columns of A1 (i.e., the columns 5, 6, 8 12), the bid-value ratios within a market are interdependent. The test results for Table 3 are neglected.

6 T. Neugebauer, J. Perote Table 1 NoInfo-Info order: bidding relative to RNNE (b kit b )/x kit Market k Participant i NoInfo 1 Info First round Average over t Average over t First round Given high value Average over t, i and k { Confidence Band 95% Z a p-value a Exact significance of the two-tailed Wilcoxon signed ranks test; H 0 : (b b )/x = 0, H 1 : (b b )/x 0. Positive values for the bid-value ratio indicate bidding above the RNNE; negative ones indicate bidding below the RNNE

7 Bidding as if risk neutral in experimental first price auctions auction. Third, the average first round bid in the Info 2 treatment (fourth column) is almost equal to the RNNE. 4.2 Between treatments comparison As recorded in the lowest row of Table 3, the average bidding in all treatments but the NoInfo 1 treatment is above the RNNE. 11 The bid-value ratio in these three treatments is statistically indistinguishable from each other, but significantly higher than in the Info 1 treatment. 12 However, the first round bids in the NoInfo 1 and Info 1 treatments are not significantly different from each other; thus suggesting homogeneous samples. The data rather suggest that the effect of feedback information on bidding behavior is immediate. The bids in the Info treatments significantly increase following the first revelation of the high bid. In all eight markets of the Info treatment, the bid value ratio increases between the first and the second round. The probability that this increase is due to chance is 0.4%. In contrast to this observation, the difference between the first and the second round bid-value ratios is not significant in the NoInfo treatments. In summary, our data indicate that bidding in the NoInfo 1 treatment is below the RNNE. Bidding is subject to an order effect as bids in the NoInfo 2 treatment are significantly higher than in the NoInfo 1 treatment. In contrast to this result, we do not observe a comparable order effect in the Info treatments, where bidding is higher relative to both the RNNE and the NoInfo 1 treatment. 4.3 Time pattern Figure 1 plots the average difference between the bid-value ratio and the RNNE over the rounds of the experiment; the top diagrams represent the time pattern for the NoInfo-Info order of treatments and the bottom diagrams represent the time pattern for the reverse order of treatments. The trajectory for the NoInfo 1 treatment (at the top left of Fig. 1) is obviously increasing while the other three trajectories are rather trend free. 13 Evidently the trajectory of the average bid-value ratio converges to some value. To examine the convergence of average bidding to the equilibrium, we consider 11 The average bid-value ratio significantly exceeds the RNNE in the Info treatments; the one-tailed randomization test rejects the null in favor of overbidding relative to the RNNE at a significance level of p = 0.027, N = In the Info 2 treatment, the average bid-value ratios, on the whole and restricted to the sample on high values only, are higher than in the NoInfo 1 treatment for each of the four markets. The probability that such an extreme event occurs by chance is The one-tailed [two-tailed] randomization test for independent samples rejects the null-hypothesis of equal bid-value ratios between the Info 1 [NoInfo 2 ] treatment and the NoInfo 1 treatment at the 10% level of significance (the corresponding p-value, denoted by p hereafter, is p = [p = 0.086]; four observations per treatment). Restricted to the data on highest values only, the corresponding p-values are p = In the NoInfo 1 treatment, the average bid is below the RNNE for the first 22 rounds. During the last 28 rounds the average bid is 14 times above and 14 times below the RNNE. For these last 28 rounds, the average difference from the RNNE is and the estimated 95% confidence bands extend from to In fact this number is a bit arbitrary, but the results are not limited to that number. Nevertheless, the sequence of under- and overbidding over the last 28 rounds of the NoInfo 1 treatment is non-systematic as a runs test indicates (p>0.5).

8 T. Neugebauer, J. Perote Fig. 1 Average deviation from the RNNE by round

9 Bidding as if risk neutral in experimental first price auctions Table 2 Unconditional mean for the difference between the bid-value ratio and the RNNE Note: standard error in parenthesis; p-value in brackets; * significantat1% NoInfo Info NoInfo-Info order of treatments * (0.013) (0.0035) [0.136] [0.000] Info-NoInfo order of treatments * * (0.0054) (0.0044) [0.001] [0.000] the average of the differences of the bid-value ratios from the RNNE for every round as a stationary time series from the family of autoregressive processes. The AR(2) process adequately fits the autocorrelation pattern of the data for the NoInfo 1 treatment, but for the other three treatments the data generating process seems to be a white noise with a significant positive drift. Table 2 shows the unconditional means of the processes, which are significantly different from zero for all but the NoInfo 1 treatment. 4.4 Supplementary results In a supplement to the paper, we present additional support for the reported findings mainly based on the panel data approach. The contents are summarized as follows. We estimate aggregate bid functions using standard econometric techniques and examine the differences that are due to the change in the information feedback conditions. We report the regression of the difference from RNNE bidding on a deterministic trend and find that the difference significantly decreases in the NoInfo 1 treatment only. We run a regression on the absolute deviations of individual bid-value ratios between two consecutive rounds on a time trend. The result suggests that subjects learn their bid function in the early rounds of the NoInfo 1 treatment by submitting a bid for their varying resale values. We estimate bids in function of lagged winning bids and observe a marginally significant dependence for the Info treatments. Finally, we report subjects self-assessment on risk aversion which they stated in the debriefings to the experiment. This assessment indicates risk loving rather than risk aversion. However, no salient rewards were offered for truthful answers in that task. 5 Concluding remarks In this paper we have tested the conjecture that subjects in a market with n = 7 subjects bid as if risk neutral in first-price sealed-bid auctions under conditions of no feedback information. On average we find that bids are below the risk neutral Nash equilibrium, but converge to the equilibrium in the first rounds of the experiment.

10 T. Neugebauer, J. Perote In contrast to the bidding behavior in the no information feedback treatment, we find evidence that the revelation of the lagged winning bid triggers an immediate response in terms of higher bidding in the information feedback treatment. In particular, we observed such an effect between the first two rounds of the feedback treatment; the average bid increased in all markets. To explain the bidding adjustments in the repeated first price auction, Neugebauer and Selten (2006) proposed a learning story to which the behavior of 92% of their subjects conformed (see also Selten 2004; Selten et al. 2005; and Ockenfels and Selten 2005); learning direction theory. According to this theory, the bidder experiences an upward impulse if she loses the opportunity to win; thus, she increases her bid in the following round. The bidder experiences a downward impulse if she makes a successful bid, because she left money on the table; thus, she decreases her bid in the following round. This logic is also in line with regret theory by Engelbrecht-Wiggans and Katok (2005b). Furthermore, Neugebauer and Selten report evidence that subjects adjust their bids more frequently when they receive clear information on the others bids than when they receive ambiguous information (i.e. feedback which reveals only the range of the others bids). Regarding the feedback conditions of our information feedback treatment, Neugebauer and Selten (2006, p. 199) thus conclude that the ambiguity of the information feedback with respect to the second highest bid, which indicates the amount of money left on the table,... apparently produced an upward drift that led to overbidding... in their Info treatment. Engelbrecht-Wiggans and Katok (2005a) make a similar observation in their conclusions as they note that fewer downward adjustments result from subjects ambiguity about the money left on the table. Hence, the feedback effect presented in our study, i.e. that the feedback treatment induces higher bids than the no feedback treatment, is entirely in line with the observations of other studies on feedback in first price auctions. The fact that there is no significant shift in bidding when subjects are switched from conditions of feedback to no feedback indicates that the information on prices may be quite abiding in a market, even if this information is turned off. When inexperienced subjects are exposed to the no feedback treatment (i.e., in our NoInfo 1 treatment), in contrast, the speed of learning and regret appears slowed down. However, with increased experience on bidding in the environment without feedback it seems that subjects learn to bid by reflecting on their changing bids and values. Such learning by introspection has also been suggested in other experimental environments (Weber 2003). Ultimately, one could speculate that subjects learn to anticipate regret. 14 Vis-à-vis the support our data give to the RNNE, we suppose that this result is not general but market size specific. In line with other experimental results on the first price auction, we are convinced that the market size n is strictly correlated to overbidding relative to the RNNE (Neugebauer and Selten 2006; Dyer et al. 1989). Neugebauer and Selten found more evidence on bidding above the RNNE for small, 14 The possibility of anticipating regret has been proposed by Ozbay and Filiz (in press). They report that subjects change their bid according to the anticipated feedback in a one-shot, contingent bid first price auction. However, this pattern seems difficult to reproduce in the first round of the repeated setting. Our results as well as the data reported in Neugebauer and Selten (2006) and Dufwenberg and Gneezy (2002) rather suggest that the first round bids are the same across feedback treatments. Nevertheless, compared to the one-shot game it might be possible that subjects take more time until they anticipate the possible impacts of their behavior in the repeated setting without feedback.

11 Bidding as if risk neutral in experimental first price auctions n ={3, 4}, and more evidence on bidding below the RNNE for greater market sizes, n ={6, 9}. Therefore, we have chosen an intermediary market size n = 7forthe purpose of inducing bidding as if risk neutral in the laboratory. Acknowledgements The authors thank Stefan Traub, Karim Sadrieh, the seminar participants at Jena, Kiel, Hannover and Pasadena, the editor and two anonymous referees for helpful comments. Financial support from the EU-TMR Research Network ENDEAR (FMRX-CT ) is gratefully acknowledged. Appendix Table 3 Bid-value ratio difference from RNNE in Info 1,Info 2 and NoInfo 2 (Table 1 continued) Participant i Market Info 2 k Market k Average over t Average over t First round Average over t Average over t (b kit b )/x kit (b kit b )/x kit (b kit b )/x kit (b kit b )/x kit (b kit b )/x kit Given high value Given high value Average

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