SUPPLEMENT TO THE PAPER
Bidding ‘as if’ risk neutral in experimental first price auctions without information feedback[*]
Tibor Neugebauer a and Javier Perote b
a Universität Hannover, Königsworther Platz 1, 30167 Hannover, Germany.
b Universidad Rey Juan Carlos, Dpto. de Economía Aplicada II y Fundamentos del Análisis Económico, Campus de Vicálvaro, 28032 Madrid, Spain.
Abstract: In this supplement we report panel data regressions for our treatments supporting the main results of the paper. In particular we document the following observations: (i) the estimates of the linear bid functions show that the risk neutral Nash bidding strategy is supported for the NoInfo1 but the functions for the other treatments exhibit a significantly steeper slope. (ii) The difference between the bid-value ratio and the RNNE decreases in the NoInfo1 treatment only. (iii) The variation of the bid-value ratio between rounds is significant for all treatments but is subject to significant changes only in the NoInfo1 treatment. (iv) The lagged winning bids are correlated to bid value ratios for Info treatments. Furthermore, we report the self assessment of subjects on the risk aversion. The statements support “risk loving” rather than “risk averse”. Finally, we append the instructions of the experiment.
S.1. Linear bid functions
Table S1 presents estimated linear bid functions of private values for both orders of treatments. Standard errors for both the slope and the intercept are recorded in parentheses and the asterisks indicate where the estimated parameters significantly differ from zero (p < 0.1). To obtain the estimates we first tested correlations between the individual effects (hk) and private values (xkit) by computing the Hausman test. Since the correlation was not significantly different from zero for any treatment,[1] we estimated the random-effects model represented in equation (S1) which is both consistent and efficient.
(S1)
In (S1), bkit and xkit denote the bid and the value of the participant i of market k in round t, (k = {1, 2, 3, 4}, i = {1, 2, …, 7}, and t = {1, 2, …, 100}),[2] bj are the parameters to be estimated, j = {0, 1, 2, 3}, and hk and vkit are zero mean, homoskedastic and mutually uncorrelated random variables.[3] Hence, the regression accounts for different individual effects for each market. Moreover model (S1) allows to test possible structural changes between Info and NoInfo treatments by using a dummy variable , DInfokit, which takes the value one for the Info treatment and zero for the NoInfo treatment. As this dummy interacts with both the intercept and the slope, the estimates displayed in Table S1 can be straightforwardly interpreted as the bid functions for each treatment.
Table S1. Random effects dummy regression: Linear Bid Function
Dependent variable: bidIndependent / NoInfo-Info order / Info-NoInfo order
Variable
Intercept / -1.3480
(.8491)
[.112] / -2.3731*
(.7151)
[.001]
Dinfo / .4442
(.4750)
[.350] / 0.6718
(.4016)
[.094]
Value / 0.8742
(.0057)
[.000] / 0.9630*
(.0050)
[.000]
DInfo ´ Value / 0.0541*
(.0082)
[.000] / 0.0122
(.0069)
[.078]
R-sq overall / 0.945 / 0.9643
Number of observations / 2800 / 2800
Number of groups / 4 / 4
Note: estimate for eq (S1) (standard errors in parenthesis); [p-values in brackets]; *significant at 1%.
As recorded in the second and fourth row of Table S1, the estimates for the intercept and the slope in the NoInfo1 treatment are -1.3480 and 0.8743, respectively. This estimated bid function is presented in the top left diagram of Figure S1 by the solid line together with the RNNE bid function which is represented by the dashed line; bids and values are scaled on the ordinate and the abscissa, respectively. Figure S1 also presents the corresponding diagrams for the Info2 (top right), the Info1 (bottom left) and the NoInfo2 (bottom right) treatments. While the estimated bid function for the NoInfo1 almost overlaps the RNNE curve, the estimates for the other three treatments all look similar and increase clearly steeper than the RNNE function.
Table S1 also reveals that while the difference of the intercepts between the two treatments in the NoInfo-Info order is insignificant, the difference in the slope is highly significant. Bids in the Info2 treatment involve a significantly higher fraction of value than the NoInfo1 treatment. In the reverse order of treatments, Info-NoInfo, intercept and slope do not change significantly from one treatment to the next. In other words, we observe a structural change in the bid function for the NoInfo-Info order of treatments but not in the reverse order of treatments. This result again shows the order effect of treatment exposure in our experiment.
S.2. Convergence to RNNE
The random effects regression of the difference between the bid-value ratio and the RNNE on a deterministic trend is significant in the NoInfo1 treatment (but not in the others); see Table S2.
Table S2. Random effects regression of bid-value ratio differences from RNNE on time
NoInfo1 / Info2 / Info1 / NoInfo2t > 0 / t > 22
Intercept / -0.0852*
(.0149)
[.000] / -0.0244
(.0243)
[.314] / 0.0026
(.0289)
[.930] / 0.0634**
(.0366)
[.083] / 0.0364**
(.0210)
[.082]
Round / 0.0021*
(.0003)
[.000] / 0.0006
(.0006)
[.325] / 0.0003
(.0003)
[.363] / -0.0006
(.0004)
[.121] / -0.0005
(.0003)
[.182]
Adj. R-squared / 0.035 / 0.001 / 0.001 / 0.002 / 0.001
Number of obs. / 1384 / 775 / 1389 / 1388 / 1386
N / 28 / 28 / 4 / 4 / 4
Model specification: DRNNE kit = (b kit - b*)/x kit = b0 + b1 t + hki + vkit for NoInfo1 and DRNNEk it = b0 + b1 t + hi + vkit for the other treatments; where t is a time trend, t Î [1;50], vkit the error term and hki and hi are the individual and market effects, respectively, according to the standard assumptions of the random effects model.
(standard errors in parenthesis); [p-values in brackets]; **significant at 10%; *significant at 1%.
Note that on the subsample which includes only the data of the last 28 rounds, both explanatory variables intercept and the deterministic trend are insignificant in the NoInfo1 treatment, too. This suggests a convergence towards equilibrium-bidding from below during the early rounds of the NoInfo1 treatment. Note that we run the random effects regression for the NoInfo1 treatment on N = 28 independent observations where hki represents the individual effects, while in the other three treatments where N = 4, hk represents the market effect.
S.3. Time pattern of bid-value differences
Given our main hypothesis that feedback information on the winning bid affects the bidding in the following round, at first sight, it may appear astonishing that precisely in the NoInfo1 treatment we observe a dependence of the bid-value ratio on a lagged variable. The data suggest that subjects learn their bid function by submitting a bid for their varying resale values.[4] To prove this claim, we run a panel data regression on the absolute deviations of individual bid-value ratios between two consecutive rounds (D(b/x)kit = | bkit / xkit - bkit-1 / xkit-1 |) on a time trend. The results of this estimation are presented for each treatment in Table S3.
Table S3. Random effects regression of absolute changes in bid-value ratio on time
NoInfo1 / Info2 / Info1 / NoInfo2t > 0 / t > 22
Intercept / 0.1503*
(.0090)
[.000] / 0.1197*
(.0267)
[.000] / 0.1056*
(.0098)
[.000] / 0.1104*
(.0287)
[.000] / 0.0982*
(.0276)
[.000]
Round / -0.0121*
(.0003)
[.000] / -0.0004
(.0007)
[.563] / 0.0026
(.0003)
[.445] / 0.0000
(.0004)
[.944] / 0.0004
(.0004)
[.245]
R-sq. overall / 0.011 / 0.001 / 0.000 / 0.000 / 0.001
Number of obs. / 1340 / 738 / 1350 / 1344 / 1348
N / 28 / 28 / 4 / 4 / 4
Model specification: D(b/x)kit = b0 + b1 t + hki + vkit for NoInfo1 and D(b/x)kit = b0 + b1 t + h k + vkit for the other
treatments; where t is a time trend, tÎ[1;50], vkit the error term and hki and hk are the individual and market effect, respectively, according to the standard assumptions of the random effects model.
(standard errors in parenthesis); [p-values in brackets]; *significant at 1%.
From these results we deduce that bidding is subject to a significant variation; the variation of the bid-value ratio between rounds is significant for all treatments as the result on the intercept shows. However, only in the NoInfo1 treatment we observe that the changes significantly decline over time. The regression on the last 28 rounds of the NoInfo1 treatment does not involve a significant trend. This result shows that the learning adjustment dynamics are particularly important in the early rounds of the NoInfo1 treatment but their importance diminishes with time.
S.4. Information feedback effect
We report the results for the random effects regression of the bid-value ratio on lagged winning bids in Table S4. For the Info treatments the lagged winning bid is significant at 10%; needless to say, the lagged winning bid does not explain bidding in the NoInfo treatments. The effect of the winning bid is only marginally significant, since there is too much noise in the data; values are randomly drawn so that subjects may not find themselves in the position of placing the winning bid in two subsequent rounds. However, the correlation of bid and lagged winning bid has been convincingly shown in Neugebauer and Selten (2006). The fact that we find a (marginally) significant effect of lagged winning bid on average bidding even in this noisy environment shows the robustness of this information feedback effect.[5] However, we plan to examine the subject with a richer data set.
Table S4. Random effects regression of bid-value ratio on lagged winning bid
NoInfo1 / Info2 / Info1 / NoInfo2Intercept / 0.8172*
(.0332)
[.000] / 0.8260*
(.0327)
[.000] / 0.9554*
(.0438)
[.000] / 0.8674*
(.0549)
[.000]
Winning bid / 0.0002
(.0004)
[.676] / 0.0007**
(.0004)
[.072] / -0.0009**
(.0005)
[.081] / 0.0001
(.0006)
[.840]
R-sq. overall / 0.000 / 0.006 / 0.004 / 0.000
Number of obs. / 1356 / 1389 / 1358 / 1388
N / 28 / 4 / 4 / 4
Model specification: (b/x)kit = b0 + b1 pkt-1 + hkt + vkit for NoInfo1 and (b/x)kit = b0 + b1 pkt-1 + hk + vkit for the other treatments; where pkt-1 is the winning bid in market k in round t-1, vkit the error term and hki and hk are the individual and market effect, according to the standard assumptions of the random effects model.
(standard errors in parenthesis); [p-values in brackets]; **significant at 10%; *significant at 1%.
S.5. Subjects’ Statements
It has been repeatedly argued in the literature that subjects overbid relatively to the RNNE under conditions of feedback information, because they are risk averse. In this section we want to briefly address this issue by looking at subjects’ statements in the debriefing. We asked subjects to self-assess their attitude toward risk on a seven point scale reaching from strongly risk seeking to strongly risk averse. In fact, no salient rewards were linked to truthful responding; therefore, we must be careful with the interpretation of the result. Still, it does make sense to report these results here, since all approaches to measuring risk aversion may involve problems. The average points assigned in this task by the 56 subjects were 3.3571, differing from a risk-neutral evaluation by -0.6429 points, involving a standard deviation of 1.4946. The statements are significantly different from risk-neutrality at the 10% level of significance, indicating risk loving rather than risk aversion.[6] It remains to be said that we do not find any significant correlation between the stated self-assessment of risk aversion and the individual average bid-value ratio. Furthermore, we find no significant correlation between self-reported risk aversion or revealed bid-value ratio and self-reported gender or age, either. The data on the statements is presented in Table S5 of this supplement.
References
Greene, W., Econometric Analysis. Prentice-Hall, NJ, 2000.
Neugebauer, T. and R. Selten, Individual Behavior of First-Price Sealed-bid Auctions: The Importance of Information feedback In Experimental Markets, Games and Economic Behavior 54 (2006), 183-204.
Weber, R.A., “Learning” without feedback in a competitive guessing game. Games and Economic Behavior 44 (2003), 134-144.
Table S5. Subjects’ stated self-assessment on risk aversion
i / Market
k / Risk
attitude / Market
k / Risk
attitude
1 / 1 / 3 / 5 / 1
2 / 0 / -2
3 / 1 / 2
4 / 1 / 0
5 / -2 / -1
6 / -1 / 1
7 / -3 / -2
1 / 2 / 0 / 6 / -2
2 / -1 / 0
3 / -2 / -2
4 / -2 / -1
5 / 1 / -1
6 / -1 / -1
7 / -1 / -2
1 / 3 / -2 / 7 / 0
2 / 1 / -3
3 / 1 / -3
4 / -1 / 2
5 / -2 / 0
6 / -1 / 1
7 / 1 / -1
1 / 4 / -2 / 8 / -3
2 / -3 / -1
3 / 1 / -2
4 / -2 / -1
5 / -1 / 2
6 / 1 / -1
7 / 1 / -1
Risk-attitude on a seven point scale; from risk loving -3 to risk averse +3; 0 indicates risk neutrality
Appendix: Instructions
General Information
1. You are about to participate in 2 x 50 rounds of an auction experiment. In each of these rounds, you will be assigned to a group of 7 bidders: yourself and 6 other participants. Your group will stay the same throughout the experiment. However, you will not receive any information about the identity of the other group members.