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Impact of valuation ranking information on bidding in first-price auctions: A laboratory study
Journal of Economic Behavior & Organization 69 (2009) 75–85
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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/econbase
Impact of valuation ranking information on bidding in first-price auctions: A laboratory study Alexander Elbittar ∗ Centro de Investigación y Docencia Económicas, Carretera Mexico-Toluca 3655, 01210 Mexico, DF, Mexico
a r t i c l e
i n f o
Article history: Received 1 December 2006 Received in revised form 25 September 2008 Accepted 28 September 2008 Available online 10 October 2008 JEL classification: C92 D44 D82 Keywords: Asymmetric auctions Laboratory experiments Bounded rationality Affiliation Economics of information
a b s t r a c t Landsberger et al. [Landsberger, M., Rubinstein, J., Wolfstetter, E., Zamir, S., 2001. First-price auctions when the ranking of valuations is common knowledge. Review of Economic Design 6, 461–480] identified optimal bidder behavior in first-price private-value auctions when the ranking of valuations is common knowledge, and they derived comparative-statics predictions regarding the auctioneer’s expected revenue and the efficiency of the allocation. The experiment reported here tests the behavioral components of these comparative-statics predictions. The results support the prediction that buyers are inclined to bid more aggressively when they learn they have the low value. Contrary to the theory, buyers are inclined to bid less when they learn they have the high value. Consistent with theory, the overall proportion of efficient allocations is lower than in the first-price auction before information is revealed. But as a result of high-value bidders decreasing their bids, the expected revenue does not increase on a regular basis, contrary to the theory’s predictions. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Traditional auction models assume that agents are unaware of how their valuations stand in relation to their rivals. In real-life situations, however, agents do have this knowledge. Such would be the case in procurement auctions, where bidders learn from experience which bidders are strongest. There is also a general perception that in privatizations, firms established in the market have a higher valuation than potential entrants. In addition, it is believed that in takeovers, firms with related activities have a higher valuation than their competing firms. For instance, in the privatization of airwaves in the United States, there was a general perception that the Pacific Telephone Company had a higher valuation for the Los Angeles area than the other bidders. Another example involves the auctioning of third generation mobile phone licenses in the United Kingdom. Here, firms already in the market were believed to have higher valuations for the new licenses being sold than new potential entrants (Klemperer, 2004). The relevance of this information in relation to bidding behavior is a common factor in all of these examples. Thus, there is a need to determine the impact of the information on bidding behavior in auction markets, on seller’s revenue, and on the efficiency of auction outcomes (Kagel, 1995; Wolfstetter, 1999). Early theoretical contributions to auction theory focused on studying a single unit auction under the assumptions of bidders’ risk neutrality, independence of bidders’ reservation values and symmetry of bidders’ beliefs (Vickrey, 1961; Myerson,
∗ Tel.: +52 55 57279800; fax: +52 55 57279878. E-mail addresses: [email protected], [email protected]. 0167-2681/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2008.09.006
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1981; Riley and Samuelson, 1981). Relaxing the last two assumptions, Landsberger et al. (2001), hereafter referred to as LRWZ, developed a theoretical model in which two bidders draw their values from the same distribution, with the ranking of these valuations being common knowledge to auction participants. This auction is ex-ante symmetric since bidders’ valuations are drawn independently from a single distribution. However, the ranking revelation makes it affiliated and asymmetric since subsequent conditional distributions differ.1 In this paper, I investigate what happens during an auction experiment where two bidders are required to bid for a single item. To do this, I follow the first-price sealed-bid allocation rule and use two different information conditions. While each bidder knows the value of the item, the valuation ranking still induces a particular asymmetry in bidding behavior. Therefore, this experiment will evaluate the comparative-statics predictions of LRWZ’s model regarding the auctioneer’s expected revenue and the efficiency of the first-price auction. This is hypothesized to occur when the ranking of valuations is common knowledge to two bidders. (i) The bidder with the lower reserve value will bid more aggressively than the highvalue bidder. Thus, the proportion of efficient allocations should be lower than when bidders lack information about the ranking of valuations. (ii) For the uniform distribution, bidders are expected to bid more aggressively than in the standard first-price auction model, where information regarding the ranking of valuations is not available. Thus, the seller’s expected revenue should be higher when rankings are known than when they are unknown. The experimental results reported here support the prediction that low-value bidders are inclined to bid more aggressively once information about the ranking of valuations has been revealed. Contrary to the theory’s predictions, high-value bidders are inclined to bid less aggressively. Thus, consistent with the theory, the overall proportion of efficient allocations is lower than in the first-price auction before information is revealed, but as a result of high-value bidders decreasing their bids, the expected revenue does not increase consistently. Although both kinds of bidders improve their average monetary payoff per auction by moving toward the asymmetric risk-neutral Nash equilibrium (RNNE) after information is revealed, high-value bidders continue to bid above it. Therefore, I consider a modified version of LRWZ’s auction model with risk-averse bidders as an alternative way to explain why high-value bidders may keep bidding above the RNNE (Cox et al., 1988). In a previous experiment, Kagel et al. (1987) tested the effects of public information about rival valuations on the seller’s expected revenue in the context of affiliated private-values (Milgrom and Weber). In that paper, bidders were required to bid for a single item, following the first-price sealed-bid allocation rule. They also followed two different information conditions. While each bidder knew the value of the item, a public signal induced a particular symmetric affiliation between valuations. As a consequence, the bidder whose valuation was below the public signal would raise his bid in an effort to win the item. In turn, this put pressure on the bidder, whose valuation was above the public signal, to raise her bid, resulting in more expected revenue for the seller. In contrast to my results, Kagel et al. showed that a public signal increases average bidding and market prices as expected. However, in the aggregate, the increase in prices is closer to the predictions of the risk-averse Nash equilibrium (RANE) model, which permits some increase in revenue lower than predictions by the RNNE model. In the context of independent private-values drawn from different distributions, Pezanis-Christou (2002) and Güth et al. (2006) tested the predictions derived respectively by Maskin and Riley (2000) and Plum (1992). In these experimental papers, one strong-bidder and one weak-bidder were required to bid for a single item following the first-price sealed-bid allocation rule where the weak-bidder was more likely to draw values lower than the strong-bidder. In this case, the strong-bidder had the incentive to low-ball, that is, to submit low bids. These authors showed that the strong-bidder was willing to low-ball as predicted. The reduction however was not as significant as expected when compared to the second-price auction. In contrast to their result, I show that contrary to expectations the high-value bidder (or strong-bidder) is willing to submit low bids. This paper is organized into five sections. The introduction is followed by Section 2, describing the experimental design. Section 3 specifies the RNNE bidding strategies for the auction models and states the comparative-statics predictions for the first-price auction when the information about the ranking of valuations is revealed. Section 4 evaluates the experimental results regarding these auction settings, and Section 5 summarizes the research results. 2. Experimental design The experimental design used directly measures the information impact of the ranking of valuations on bidding behavior in first-price private-values auctions. The particulars of the experimental design is as follows. 2.1. Structure of the auction In this experiment, two people were required to bid for a single item following the first-price sealed-bid allocation rule under two different information conditions. In the first-price sealed-bid auction, the high bidder earns a payoff equal to her value of the item less the high bid, and the second bidder earns nothing. For each auction, two private values, one for each bidder, were independently drawn from the same distribution. In one information condition (hereafter referred to as the symmetric condition), bidders had no information about the rank order of valuations. In the other information condition (hereafter referred to as the asymmetric condition), both bidders were informed
1 In auctions with affiliated private-values, values might remain private for bidders, but the higher (lower) the value is for one bidder, the more likely the value will be higher (lower) for other bidders as well (Milgrom and Weber, 1982).
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Table 1 Experimental design. Experimental treatment
Sessions
Auction periods
Markets
Information condition
Asym. condition and dual-market bidding
1 and 2
1–10 11–30
Single-auction Dual-auction
Asymmetric Sym./asym.
Sym. condition and dual-market bidding
3 and 4
1–10 11–30
Single-auction Dual-auction
Symmetric Sym./asym.
Note: In session 1, during the dual-auction markets, only 18 trading periods (out of 20) were actually run since the network system broke down at that stage.
about the rank order of their valuations before bidding, that is, whether they have the highest or lowest valuation (but not informed about the size of the differences in valuations). Each session of the experiment began with ten periods of single-auction markets, followed by 20 periods of dual-auction markets. (i) In the single-auction markets, two competing bidders submitted one bid in each period. Participants were randomly matched in each period. (ii) In the dual-auction markets, two competing bidders submitted one bid under the symmetric condition. Then, once these bids were collected, but before they were posted, information about the ranking of valuations was released, and the same two competing bidders were asked to then submit a second bid. The winner in each market was determined after the second bid was submitted. Payoffs were determined based on just one of the two markets (chosen with equal probability). Participants were also randomly matched in each period. The dual-market bidding procedure involved the same two bidders bidding for the same item under two different information conditions. It has the advantage of directly controlling the between-subject variability. The rule of flipping a fair coin to determine which auction outcome is payoff relevant ensures that under the expected utility hypothesis, the optimal strategy in the private information market should be unaffected by bids made after the ranking is revealed, and vice versa (Kagel and Levin, 1986). Each session began with bidding in single-auction markets to determine whether the dual-market bidding procedure actually affects the way bidders bid and to familiarize subjects with the auction conditions so as to make it easier for them to bid in dual-auction markets. Table 1 briefly summarizes the experimental treatments. Sessions 1 and 2 began with 10 periods of a single-auction markets under the asymmetric condition followed by 20 periods of a dual-auction markets. During these 10 initial periods, each bidder maintained his role as a high-value or low-value bidder in order to maximize the prospect that rivalrous elements will come into play. Sessions 3 and 4 began with 10 periods of single-auction markets with the symmetric condition followed by 20 periods of dual-auction markets. 2.2. Design parameters This section characterizes the basic parameters and the general procedure of the experiment. Valuation distribution All bidders were informed that their valuations were drawn randomly. The interval included all whole cent values between $0.00 and $6.00. All values in this interval had an equally likely chance of being drawn. Payoff mechanism and participation fee Each group of players received an initial cash balance of zero. At the end of each session, total earnings were distributed to participants in cash. Since bidders’ roles stayed constant during the initial 10 auction periods of sessions 1 and 2, the subjects who were assigned the role of low-value bidders received an additional $5 at the end of each session. Players were informed at the beginning about this additional payment. This was a fixed amount that was not expected to impact behavior. In addition, at the end of each day, all participants received a participation fee. Matching procedure Bidders were informed that participants would be randomly matched in every period, but that no pair of bidders would be matched twice in two consecutive periods. Bidders were also informed that in the dual-auction markets section, each of them would face the same opponent in both markets. Information feedback At the end of every period each player obtained the following information. For the single-auction markets with the asymmetric condition, each subject received feedback about his earnings and the competing bidder’s valuation and bid amount. For the single-auction markets with the symmetric condition, additional information about the ex-post ranking of valuations was presented to players. For the dual-auction markets, each subject received complete feedback about who the high bidder was in each market and how much he earns. They also knew the competing bidder’s value and bids in each market, and which market was randomly selected to pay off in. Participants’ identities were not revealed.
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Fig. 1. RNNE bid functions: bL (v) [– ·], bH (v) [—] and bS (v) [– –].
Dry runs In order to familiarize subjects with the auction procedures, two practice runs took place at the beginning of the singleauction markets, and one practice period took place prior to the dual-auction markets. Subjects For each session, subjects were recruited from a wide cross-section of students (mostly undergrads) at the University of Pittsburgh and Carnegie Mellon University. There were 18 subjects for session 1 and 20 subjects for sessions 2–4. Subjects participated in only one session. The experiment was run in the economics lab at the University of Pittsburgh using computers. 3. Theoretical predictions LRWZ considered the specific situation where a single object is auctioned between two bidders. Both bidders have riskneutral utility functions and independently drawn private valuations. These valuations are ex-ante identically distributed, and this is common knowledge. Each bidder knows whether she has the higher or the lower valuation, but does not know her opponent’s exact valuation. Under the assumption that the underlying distribution of valuations is uniform with the support [0,6], both agents maximization problems can be reduced to the following differential equation system by the first order necessary conditions: l (b)(h(b) − b) = l(b) h (b)(l(b) − b) = h(b) − l(b) h(0) = l(0) = 0 ∃b∗ ∈ [0, 6] such that h(b∗ ) = l(b∗ ) = 6 where l(·) and h(·) are, respectively, the inverse of the bid functions for the low-value bidder, bL (v), and for the high-value bidder, bH (v). The intuition behind the second boundary condition is this: if the bids functions are not equal for the upper bound of the distribution, the owner of the higher bid can lower her bid slightly and still win the object. By l’Hôpital’s Rule, the derivatives at the initial condition are h (0) = 2 and l (0) = 4/3. For a numerical approximation of the above system of equations with two boundary conditions, I follow the general method implemented by Elbittar and Ünver (2001). The bidding response for the symmetric and independent private-values first-price auction model, bS (v), can be derived analytically. For the case of n risk-neutral bidders with a uniform distribution, the unique symmetric RNNE bid function is bS (v) = ((n − 1)/n)v (Vickrey, 1961). The RNNE bidding responses for the low-value bidder and for the high-value bidder are represented in Fig. 1 by bL (v) and bH (v), respectively. In the same figure, the equilibrium bidding response for the symmetric and independent private-values model when n = 2 is represented by bS (v).2
2 The discreteness of the strategy space and of the type space introduced in the experimental design should not have an effect on the equilibrium bid functions and the resulting predictions (Athey, 2001).
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Table 2 Numerical results from the equilibrium bid functions. Auction model
Symmetricc Asymmetric
Seller’s revenue
2.000 2.219
Regular buyer
High-value
Low-value
Payoff
Pr[Win]
Payoff
Pr[Win]
Payoff
Pr[Win]
1.000 (–)
0.500 (–)
(–) 1.361
(–) 0.710
(–) 0.263
(–) 0.290
Proportion of POAa
MEb
1.000 0.710
1.000 0.957
Note: Valuations are drawn from the uniform interval [0,6]. Mean is computed by Monte Carlo using 10,000 samples of 1000 drawings. a Pareto-optimal allocations: POA. A level below 100 characterizes unrealized gains from trade. b Mean efficiency: ME = v win /vh . A level below 100 characterizes unrealized gains from trade. c Calculations are made analytically. Table 3 Observed bid proportions. Statistics
Median Mean s.e.
Bid proportiona symmetric condition
Bid proportion asymmetric condition
ıSL (v)
ıSH (v)
ıS (v)
ıAL (v)
ıAH (v)
0.697 [0.667] (0.007)
0.658 [0.644] (0.006)
0.674 [0.656] (0.005)
0.820 [0.757] (0.008)
0.607 [0.600] (0.006)
Note: ıS is the aggregation of ıSL and ıSH . a
b(v)
Bid proportion: ık = v , j = L, H and k = S, A. j
3.1. Comparative-statics predictions The comparative-statics predictions of these two auction models define the theoretical benchmark for measuring the possible impact of valuation ranking in relation to bidding behavior. These predictions also provide us with precise statements about the changes in the seller’s revenue and the proportion of optimal allocations. According to LRWZ (Corollary 1), the high-value bidder should bid less aggressively than the low-value bidder for any given value (bL (v) > bH (v), ∀v ∈ (0, 6)). Since the low-value bidder can obtain the item with positive probability, the proportion of efficient allocations should decrease. They also proved (Proposition 2) that for the uniform distribution studied here, revealing the ranking should induce the low-value bidder to bid strictly higher than when this information is not known (bL (v) > bS (v), ∀v ∈ (0, 6]). It should also induce the high-value bidder to bid the same for valuations below or equal to 4 and strictly higher for valuations above 4 (bH (v) = bS (v), whenever v ∈ (0, 4] and bH (v) > bS (v), whenever v ∈ (4, 6]). Therefore, the seller’s expected revenue should increase.3 Table 2 summarizes some theoretical predictions for both auction models using the equilibrium bid functions. The seller’s expected revenue, the proportion of pareto-optimal allocations (POAs), the proportion of the maximal payoff or mean efficiency (ME), and each type of bidder’s conditional expected payoff and probability of winning are numerically estimated, drawing valuations from the uniform distribution in the interval [0,6]. The values used in these results indicate that the theoretical impact of the valuation ranking would increase the seller’s expected revenue from $2.00 to around $2.22, with the percentages of POA and ME decreasing from 100.0 percent to 71.0 percent and 95.7 percent, respectively. 4. Experimental results This section compares the experimental results for the dual-auction markets to directly measure the ranking of valuations affecting bidding behavior.4,5 Result 1 (Bidding Behavior). While low-value bidders tend to bid higher after information about the ranking is released, high-value bidders tend to bid lower. Support for Result 1. Table 3 reports the observed bid proportion, ıkj (v), for each bidder type j and information condition k. The bid proportion tells us how much bidders bid as a proportion of their own reserve value. Fig. 2 shows all the observed bids and the RNNE bid functions as proportions of their reserve values. LRWZ predict that both types of bidders bid more
3 LRWZ proved that it will always be true that b (v) > b (v) for all values of v. LRWZ provided that there exist probability distributions for which L S bS (v) > bH (v) for some values of v. Elbittar and Ünver showed that the proposition that bH (v) ≥ bS (v) for all values of v also holds for some distributions different from the uniform. 4 The results of single-auction markets were consistent with the results for the dual-auction markets and indicate that the dual-market bidding procedure does not affect bidding behavior. A summary of these results is included in the appendix to avoid repetition. 5 The total number of bids is 1524 per market. However, some observations were excluded from the statistical analysis either because the subject’s bidding was above her/his own valuation (18 observations), or the subject’s valuation was less than two cents, so the corresponding symmetric RNNE was less than one cent (10 observations). The analysis is therefore based on 1496 bids per market.
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Fig. 2. Actual bids and RNNE bid functions as a proportion of the reserve values.
aggressively under the asymmetric condition than under the symmetric condition (ıAj (v) > ıSj (v), j = L, H). Analysis of the actual bid proportions in Table 3 indicates that, contrary to expectations, high-value bidders tend to bid less after information about the ranking of valuations was released, ıAH (v) < ıSH (v). For the pooled data, high-value bidders bid, on average, 4.4 percentage points less after the information was released, or approximately 15 cents per auction. By contrast, low-value bidders tended to bid more after information about the valuation ranking was released, ıAL (v) > ıSL (v). For the pooled data, the same bidders bid 9.0 percentage points more after the information was released, or approximately 23 cents per auction. In order to test LRWZ’s comparative static predictions, I consider a fixed-effects model for estimating the bidding proportion for each bidder type and each information condition: ıist = + ˛i + s + [ˇ1 + ˇ2 (vist − 4)dv>4 ]dH + [ˇ3 + ˇ4 (vist − 3)dv>3 ]dL In this model ıist represents the bid that each individual i submits as a proportion of her own reserve value in session s and period t; represents the mean intercept; ˛i and s represent, respectively, the deviation of individual i and of session s from the common mean ; dH is a dummy variable equal to one when a high-value bidder bids after the information was released; and dL is a dummy variable equal to one when a low-value bidder bids after the information was released. The second graph in Fig. 2 shows that for high-value bidders, the RNNE bid function as a proportion of the reserve value starts increasing measurably after valuations exceed 4. Meanwhile, the third graph in Fig. 2 shows that for low-value bidders, the RNNE bid function as a proportion of the reserve value starts decreasing measurably after valuations exceed 3. To capture these two potential effects, I consider a piecewise linear regression such that dv>4 is a dummy variable equal to one when the valuation for a high-value bidder is greater than 4, and dv>3 is a dummy variable equal to one when the valuation for a low-value bidder is greater than 3. Table 4 displays all the estimated coefficients. In contrast to the comparative-statics predictions, coefficients ˇ1 and ˇ2 are negative and significant. This means that after information is released high-value bidders tend to reduce their bids by 3.9 percentage points and 3.2 percentage points for each additional dollar for valuations greater than 4. On the other hand, while coefficient ˇ3 is positive and significant, coefficient ˇ4 is not significantly different from zero. This means that after information is released low-value bidders tend to increase their bids by 9.9 percentage points without a significant change for valuations over 3. Result 2 (Bidding Complexity). There is a larger variation in the response of high-value bidders than in that of low-value bidders. Support for Result 2. Using the pooled data, Table 5 reports how subjects alter their bids following the announcement of the valuation ranking. In the top half of this table, each entry represents the percentage of values drawn in a particular range, as in the row heading, for which high-value bidders respond as in the column heading. The same information is shown for Table 4 Fixed-effects model. Variables
Coefficient values
Common intercept dH (v − 4)dv>4 dH dL (v − 3)dv>3 dL Individual effects Session effects Number of observations Number of individuals Number of sessions
0.656*** (0.004) −0.039*** (0.009) −0.032*** (0.009) 0.099*** (0.008) 0.013 (0.011) n.r. n.r. 2992 78 4
* p < 0.05, ** p < 0.01, and *** p < 0.001. Note: The number in parentheses after each coefficient represent the coefficient standard error. n.r.: not reported.
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Table 5 Effects of valuation ranking information on bidders’ bids. Valuation range
Bids bH (v) < bS (v)
bH (v) = bS (v)
bH (v) > bS (v)
6≥v>4 4≥v>0 6≥v>0
52.5% 57.9% 54.9%
19.9% 21.2% 20.5%
27.6% 20.9% 24.6%
Valuation range
bL (v) < bS (v)
bL (v) = bS (v)
bL (v) > bS (v)
6≥v>0
13.7%
22.2%
64.1%
low-value bidders. As previously discussed, LRWZ predict that high-value bidders will raise their bids for valuations above 4 and keep them the same for valuations below or equal to 4. As shown in Table 5, a sizable proportion of bids (54.9 percent) decreased when subjects learned they had the higher valuation. Meanwhile, the other 45.1 percent of bids increased or involved no change. A similar kind of behavior is observed for valuations above and below four. The same model predicts that low-value bidders will raise their bids. As can be seen in the same table, 64.1 percent of bids increased and 22.2 percent involved no change. This result could be attributed to the greater complexity of bidding strategies for high-value bidders once information was released. For instance, when subjects were asked at the end of each session to write down their bidding strategies, most said something to the effect of “If I was a low I raised my bid, while, if I was a high I lowered my bid.” Just a few participants indicated an apparently more sophisticated reasoning: “Once I knew I was a high bidder, I usually lowered my bid. Sometimes I highered (sic) it because I knew the other person knew they were a low bidder, so they might also increase their bid.” For low-value bidders, the obvious response to such information is to increase their bids in order to increase the probability of obtaining the object. For high-value bidders, the response is less obvious. Once high-value bidders are aware of their strong position, they might either be reluctant to take on the risk of submitting lower bids because of low-value bidders’ more aggressive bidding behavior or be willing to take such a risk in hopes of increasing their average payoff. This result was also found in previous experimental studies of simple games where the iteration of dominance eliminates one, two or three strategies (see, for example, Beard and Beil, 1994; Schotter et al., 1994). These studies found little support for more than one step of deletion of dominant strategies. Although few subjects violate dominance, most of them seem unwilling to bet heavily that others will obey dominance (Camerer, 2003). Result 3 (Deviations from Predicted Behavior). Bidders under the symmetric condition and high-value bidders under the asymmetric condition keep bidding significantly above their respective RNNE. Low-value bidders under the asymmetric condition keep bidding closer to the RNNE. Support for Result 3. The baseline of the comparative-statics predictions in LRWZ is what risk-neutral, fully rational bidders bid under the symmetric condition. However, a well-known pattern of bidding above RNNE in first-price independent private value auctions are replicated here (Kagel). The overbidding factor, j = bj (v) − b∗j (v)/vj , j = L, H, S, is the relative deviation of the actual bid with respect to the RNNE bid. The mean of the overbidding factor under the symmetric condition is 15.6 percentage points more than the RNNE response. The same overbidding pattern is found for high-value bidders under the asymmetric condition. The mean of the overbidding factor for high-value bidders is around 8.0 percentage points more than the RNNE response. The proportions of bids above the RNNE are 82.6 in the symmetric condition and 68.9 percent for high-value bidders under the asymmetric condition. A significant proportion of low-value bidders under the asymmetric condition also bid above the RNNE (67.8 percent). However, the overbidding factor is 1.6 percentage points, which is not significantly different from zero (p = 0.25). Experimental results also indicate that even though bidders under the symmetric condition, and high-value bidders under the asymmetric condition, moved toward the RNNE over time, they stayed bidding well above the RNNE. In contrast, low-value bidders’ bidding behavior did not change over time. Result 4 (Risk Aversion). Changes in bidding behavior are consistent with a reduction in bidders’ aversion toward risk. Support for Result 4. Strategic considerations do not seem enough to explain the deviations from the predicted behavior. In order to reconcile the previous experimental results, I consider a modified version of LRWZ’s model, assuming a constant relative risk-aversion utility function (Cox et al., 1988). Since a ceiling in bidding is not perceived, no restriction is imposed on the upper bound of the distribution. Under the assumptions that the underlying distribution of valuations is uniform and that each agent has a constant relative risk-aversion utility function,6 both agents’ maximization problems can be expressed in terms of the following differential equation system, by the first order necessary conditions with just one boundary condition:
6 Notice that this maximization problem is equivalent to one under the assumption of risk-neutrality with transformed prior probability distributions (Marshall et al., 1994).
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Table 6 Estimated parameters of bid functions. Bid function
Estimates
bˆ S bˆ L bˆ H * +
ˆ ˇ
ˆ sˆ(ˇ)
rˆ
sˆ(ˆr )
0.601*
0.006 0.007 0.010
0.663+ 1.060 0.755+
0.017 0.051 0.030
0.779* 0.570* Reject H0 : ˇ = 0.0 for p < 0.0001. Reject H0 : r = 1.0 for p < 0.0001.
l (b)(h(b) − b) = rH l(b) h (b)(l(b) − b) = rL (h(b) − l(b)) h(0) = l(0) = 0 where rH and rL are the risk aversion coefficients (rk ∈ (0,1]). Thus, bH (v) = (1/1 + rH )v and bL (v) = ((1 + rH + rL )/(1 + rH )(1 + rL ))v. The bidding response for the symmetric model is (bS (v) = (1/1 + rS ))v (Cox et al., 1988). Therefore, the following linear specification is estimated for each kind of bidder: bist = ˇvist , where bist and vist are respectively player i’s bid and valuation in period t and session s. ˆ using the pooled data. This table also reports the riskTable 6 reports the estimated parameters of the bid functions, ˇ, aversion coefficients, rˆ , as well as the standard deviation of the estimates, sˆ(·), for the each of the bid functions. bˆ L and bˆ H correspond to the estimated bid functions under the asymmetric condition for low-value and high-value bidders, respectively. Meanwhile, bˆ S corresponds to the estimated bid function under the symmetric condition. All coefficients are significantly different from zero for a p < 0.0001. As shown in Table 6, the risk-aversion coefficients for both kinds of bidders increases after information is released. The null hypotheses that rS is equal to either rL or rH is rejected when p < 0.0001. The null hypothesis that rL is equal to rH is rejected when p < 0.01. Thus, low-value and high-value bidders become less risk-averse after information about the ranking is released. Therefore, the information in relation to the ranking seems to alleviate the strategic uncertainty associated with the first-price auction and, consequently, to induce bidders to change their attitude toward risk. Result 5 (Profitable deviation). Each group of bidders improves the average monetary payoff per auction by properly deviating their bids after the information is released. In particular, the revelation of the ranking provides incentives for high-value bidders to decrease their bids and to exploit new economic opportunities. Support for Result 5. In Table 7, the first row displays the results of a Monte Carlo experiment in which high-value bidders bid against low-value bidders, with both groups bidding based on the estimated response functions bˆ H (v) and bˆ L (v), respectively. The second row indicates results of a similar numerical experiment in which high-value bidders based their bids on the estimated response before the ranking was revealed, bˆ S (v). The third row reports results of a numerical experiment in which low-value bidders bid based on the estimated response before the revelation of the ranking, bˆ S (v), but high-value bidders based their bids on the estimated bid function after the information was revealed, bˆ H (v). Table 7 displays, for each comparison, the seller’s expected revenue, the bidder’s expected payoff conditional on winning, and the probability of winning the item. The numerical results indicate that if high-value bidders were using the bid function before the revelation of the ranking, bˆ S (v), the unconditional average payoff would drop from $1.70 to $1.48 per auction (of around 23 cents per auction); meanwhile, the probability of winning the item would increase from 0.77 to 0.85. As result, their average payoff, conditional on winning, would fall by around 7 cents per auction. Therefore, in terms of monetary payoffs, it would be better for high-value bidders to reduce their bids. On the other hand, if low-value bidders were bidding using the bid function Table 7 Estimated bid functions comparison: numerical results. Bid functions comparison
Statistics
Seller’s revenue
High-value
Low-value
Payoff
Pr[Win]
Payoff
Pr[Win]
bˆ H (v) vs. bˆ L (v)
Mean s.e.
2.366 (0.278 × 10−3 )
1.322 (0.282 × 10−3 )
0.773
0.205 (0.126 × 10−3 )
0.227
bˆ S (v) vs. bˆ L (v)
Mean s.e.
2.556 (0.293 × 10−3 )
1.251 (0.225 × 10−3 )
0.848
0.144 (0.112 × 10−3 )
0.152
bˆ H (v) vs. bˆ S (v)
Mean s.e.
2.290 (0.260 × 10−3 )
1.564 (0.237 × 10−3 )
0.908
0.130 (0.138 × 10−3 )
0.092
Note: Valuations were drawn from the common support of the bid functions. Mean and standard error were computed by Monte Carlo using 10,000 samples of 1000 drawings.
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Table 8 Effect of ranking information on revenue. Statistics
Symmetric condition pS
Asymmetric condition pA
Mean s.e. Median
2.610 (0.037) [2.655]
2.572 (0.038) [2.620]
at
−2.049 −2.378
bW *
Reject H0 against H1 at p < 0.05. a Matched pairs t-test. H : p (b) ≤ p (b) against H : p (b) ≤ p (b). 0 S 1 S A A b Matched pairs Wilcoxon test. H : p (b) ≤ p (b) against H : p (b) ≤ p (b). 0 S 1 S A A
Table 9 Effect of ranking information on efficiency. Concepts
Symmetric condition
Asymmetric condition
a ME
97.2 [87.1]
94.8 [74.8]
b POA cZ ME dZ POA
2.376* 6.134* vwinner vh × 100.
a
Mean efficiency: ME =
b
Percentage of pareto-optimal allocations: POA. Population proportion test. H0 : MES ≤ MEA against H1 : MES (b) > MEA . Population proportion test. H0 : POAS ≤ POAA against H1 : POAS (b) > POAA . Reject H0 against H1 at p < 0.05.
c d *
previous to the revelation of the ranking, bˆ S (v), they would suffer a reduction in their average payoff conditional on winning (of around 7 cents per auction). Result 6 (Expected Revenue). The experimental data do not support the prediction that the seller’s revenue increases after information about the ranking is released. Support for Result 6. With high-value bidders decreasing their bids, the enhanced revenue possibilities of the first-price auction, with the ranking of valuations revealed, does not necessarily holds; rather it becomes an empirical question of whether high-value bidders’ decreasing bids dominate low-value bidders’ increasing bids after the information is released. Table 8 reports the mean, standard error and median of the observed prices for each information condition. Each average price represents the observed seller’s expected revenue for a particular information condition j: pj , j = S, A. According to LRWZ, seller’s revenue is expected to be higher under the asymmetric condition than under the symmetric condition (pA > pS ). As noted in Table 8, however, the mean of the selling prices under the symmetric condition ($2.61) is slightly higher than under the asymmetric condition ($2.57). This observation is confirmed by a one-tailed matched pairs t-test statistic (t), the result of which is displayed at the bottom of the table. Testing for every auction period, the null hypothesis that the price under the symmetric condition is larger than or equal to the price under the asymmetric condition could not be rejected in 19 out of the 20 auction periods (p < 0.05). This shows that the seller’s expected revenue does not always increase after the information is released.7 Result 7 (Efficiency). The first-price auction is, in the aggregate, more efficient in the absence of information. Support for Result 7. This result confirms the prediction of the reduction in the proportion of optimal allocations. Table 9 reports the following two measures of optimal allocation and economic efficiency: (i) pareto-optimal allocations, reported as the percentage of objects given to the high-value bidder, and ME, reported as the percentage of the maximal payoff that was generated as a result of the auction. Since low-value bidders generally have a positive probability of obtaining the item, the percentage of efficient allocations was expected to be lower under the asymmetric condition than under the symmetric condition. As seen in the calculations displayed in Table 9, the percentage of POA and ME are consistently higher under the symmetric condition than under the asymmetric condition. This result is due to the more aggressive bidding by the low-value bidders and to the less aggressive bidding by the high-value bidders under the asymmetric condition. In the same table, a one-tailed
7 Notice that due to the significant pattern of overbidding behavior, the average revenue for each information condition was significantly higher than the expected revenue at the risk-neutral equilibrium.
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population proportion (Z) test statistic for each measure of optimal allocation is reported. When testing for every auction period, the null hypothesis was rejected in favor of the alternative hypothesis, that the efficiency under the symmetric condition is higher than under the asymmetric condition, in 19 of the 20 auction periods (p < 0.05). 5. Conclusion In this project, a first-price private-values auction experiment was conducted, where the same two bidders bid for a single item in two markets and under two different information conditions. In this study, I examine the impact of information about the ranking of valuations on bidding behavior in first-price private-values auctions. Experimental results indicate that after information about the ranking of valuations was released, the two groups of bidders responded differently. As theory predicts, low-value bidders were inclined to bid more aggressively. Contrary to the theory’s prediction, high-value bidders tended to submit lower bids. Thus, the probability of low-value bidders winning the item increased. As expected, the proportion of efficient allocations decreased with respect to the first-price auction in the absence of information. Contrary to the predictions, the seller’s expected revenue did not increase on a regular basis because high-value bidders decreased their bids once the information was released. Experimental results also indicate a larger variation in the response of high-value bidders than in that in low-value bidders. This result could be attributed to the greater complexity of bidding strategies for high-value bidders once information was released. For example, once aware of their strong position, a high-value bidder might either be reluctant to take on the risk of submitting lower bids because of the low-value bidders’ more aggressive bidding behavior or be willing to take a risk in the hopes of increasing their average payoff. Since these strategic considerations are not sufficient to explain deviations from predicted behavior, a modified version of LRWZ’s model that assume a constant relative risk-aversion utility function is considered. Changes in bidding behavior due to information about the ranking are consistent with a reduction in the bidders’ aversion toward the risk involved in the first-price private-value auction. I find my results to be consistent with those of Isaac and James (2000) and Berg et al. (2004). These authors also found that individual risk preferences elicited from the same individuals across different institutions were not stable. Based on my results, I propose that the information about the ranking seems to alleviate the strategic uncertainty associated with the first-price auction and, consequently, to induce bidders to change their attitude toward risk. To understand changes in risk-aversion degrees, I revert to Krishna (2002, p. 40), who notes in his book about auction theory that a risk-averse bidder would reduce her bid when “the effect of a slightly lower winning bid on his wealth level has a smaller utility consequence than does the possible loss if this lower bid were, in fact, to result in his losing the auction.” Following Krishna’s argument, by bidding lower after information is revealed, high-value bidders seem to be willing to “buy” less insurance against the chance of losing, while low-value bidders seem to be willing to bid higher, increasing their probability of winning the object. At the same time, they are willing to “buy” less insurance compared to a risk-averse bidder. Acknowledgements I gratefully acknowledge helpful comments from Masaki Aoyagi, Vincent Crawford, Sonia Di Giannatale, David Kaplan, Charlie Plott, Jean-Francois Richard, Tridib Sharma, Ricardo Smith, Elmer Wolfstetter, Shmuel Zamir and two anonymous referees, as well as seminar participants at CIDE, Hewlett-Packard, ITAM, ESA Summer Meeting 2000, and Econometric Society Winter Meeting 2001. I am especially grateful to John Kagel for his support and many helpful discussions and suggestions. All errors are mine. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jebo.2008.09.006. References Athey, S., 2001. Single crossing properties and the existence of pure strategy equilibria in games of incomplete information. Econometrica 69, 861–889. Beard, T.R., Beil, R., 1994. Do people rely on the self-interested maximization of others? An experimental test. Management Science 40, 252–262. Berg, J., Dickhaut, J., McCabe, K., 2004. Risk preference instability across institutions: a dilemma. Proceedings of the National Academy of Sciences 102, 4209–4214. Camerer, C.F., 2003. Behavioral Game Theory: Experiments in Strategic Interaction. Russel Sage Foundation and Princeton University, New York. Cox, J.C., Smith, V.L., Walker, J.M., 1988. Theory and individual behavior in first-price auctions. Journal of Risk and Uncertainty 1, 61–69. Elbittar, A.A., Ünver, M.U., 2001. On determination of optimal reserve price in auctions with common knowledge about ranking of valuations. In: Sertel, M., Koray, S. (Eds.), Advances in Economic Design. Springer, Berlin, pp. 79–94. Güth, W., Ivanova-Stenzel, R., Wolfstetter, E., 2006. Bidding behavior in asymmetric auctions: an experimental study. European Economic Review 49, 1891–1913. Isaac, R.M., James, D., 2000. Just who are you calling risk averse? Journal of Risk and Uncertainty 20, 177–187. Kagel, J.H., 1995. Auctions: a survey of experimental research. In: Kagel, J., Roth, A. (Eds.), The Handbook of Experimental Economics. Princeton University Press, New Jersey, pp. 501–585. Kagel, J.H., Harstad, R.M., Levin, D., 1987. Information impact and allocation rules in auctions with affiliated private values: a laboratory study. Econometrica 55, 1275–1304.
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Kagel, J.H., Levin, D., 1986. The winner’s curse and public information in common value auctions. American Economic Review 76, 894–920. Klemperer, P., 2004. Auctions: Theory and Practice. Princeton University Press, New Jersey. Krishna, V., 2002. Auction Theory. Academic Press. Landsberger, M., Rubinstein, J., Wolfstetter, E., Zamir, S., 2001. First-price auctions when the ranking of valuations is common knowledge. Review of Economic Design 6, 461–480. Marshall, R.C., Meurer, M., Richard, J.-F., Stromquist, W., 1994. Numerical analysis of asymmetric first price auctions. Games and Economic Behavior 7, 193–220. Maskin, E., Riley, J., 2000. Asymmetric auctions. Review of Economic Studies 67, 413–438. Milgrom, P.R., Weber, R.J., 1982. A theory of auctions and competitive bidding. Econometrica 50, 1089–1122. Myerson, R.B., 1981. Optimal auction design. Mathematics of Operation Research 6, 58–73. Pezanis-Christou, P., 2002. On the impact of low-balling: experimental results in asymmetric auctions. International Journal of Game Theory 31, 69–89. Plum, M., 1992. Characterization and computation of Nash-equilibria for auctions with incomplete information. International Journal of Game Theory 20, 393–418. Riley, J.G., Samuelson, W.F., 1981. Optimal auctions. American Economic Review 71, 381–392. Schotter, A., Weigelt, K., Wilson, C., 1994. A laboratory investigation of multiperson rationality and presentation effects. Games and Economic Behavior 6, 445–468. Vickrey, W., 1961. Counterspeculations, auctions, and competitive sealed tender. Journal of Finance 16, 8–37. Wolfstetter, E., 1999. Topics in Microeconomics. Cambridge University Press, Cambridge.
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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/econbase
Impact of valuation ranking information on bidding in first-price auctions: A laboratory study Alexander Elbittar ∗ Centro de Investigación y Docencia Económicas, Carretera Mexico-Toluca 3655, 01210 Mexico, DF, Mexico
a r t i c l e
i n f o
Article history: Received 1 December 2006 Received in revised form 25 September 2008 Accepted 28 September 2008 Available online 10 October 2008 JEL classification: C92 D44 D82 Keywords: Asymmetric auctions Laboratory experiments Bounded rationality Affiliation Economics of information
a b s t r a c t Landsberger et al. [Landsberger, M., Rubinstein, J., Wolfstetter, E., Zamir, S., 2001. First-price auctions when the ranking of valuations is common knowledge. Review of Economic Design 6, 461–480] identified optimal bidder behavior in first-price private-value auctions when the ranking of valuations is common knowledge, and they derived comparative-statics predictions regarding the auctioneer’s expected revenue and the efficiency of the allocation. The experiment reported here tests the behavioral components of these comparative-statics predictions. The results support the prediction that buyers are inclined to bid more aggressively when they learn they have the low value. Contrary to the theory, buyers are inclined to bid less when they learn they have the high value. Consistent with theory, the overall proportion of efficient allocations is lower than in the first-price auction before information is revealed. But as a result of high-value bidders decreasing their bids, the expected revenue does not increase on a regular basis, contrary to the theory’s predictions. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Traditional auction models assume that agents are unaware of how their valuations stand in relation to their rivals. In real-life situations, however, agents do have this knowledge. Such would be the case in procurement auctions, where bidders learn from experience which bidders are strongest. There is also a general perception that in privatizations, firms established in the market have a higher valuation than potential entrants. In addition, it is believed that in takeovers, firms with related activities have a higher valuation than their competing firms. For instance, in the privatization of airwaves in the United States, there was a general perception that the Pacific Telephone Company had a higher valuation for the Los Angeles area than the other bidders. Another example involves the auctioning of third generation mobile phone licenses in the United Kingdom. Here, firms already in the market were believed to have higher valuations for the new licenses being sold than new potential entrants (Klemperer, 2004). The relevance of this information in relation to bidding behavior is a common factor in all of these examples. Thus, there is a need to determine the impact of the information on bidding behavior in auction markets, on seller’s revenue, and on the efficiency of auction outcomes (Kagel, 1995; Wolfstetter, 1999). Early theoretical contributions to auction theory focused on studying a single unit auction under the assumptions of bidders’ risk neutrality, independence of bidders’ reservation values and symmetry of bidders’ beliefs (Vickrey, 1961; Myerson,
∗ Tel.: +52 55 57279800; fax: +52 55 57279878. E-mail addresses: [email protected], [email protected]. 0167-2681/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2008.09.006
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1981; Riley and Samuelson, 1981). Relaxing the last two assumptions, Landsberger et al. (2001), hereafter referred to as LRWZ, developed a theoretical model in which two bidders draw their values from the same distribution, with the ranking of these valuations being common knowledge to auction participants. This auction is ex-ante symmetric since bidders’ valuations are drawn independently from a single distribution. However, the ranking revelation makes it affiliated and asymmetric since subsequent conditional distributions differ.1 In this paper, I investigate what happens during an auction experiment where two bidders are required to bid for a single item. To do this, I follow the first-price sealed-bid allocation rule and use two different information conditions. While each bidder knows the value of the item, the valuation ranking still induces a particular asymmetry in bidding behavior. Therefore, this experiment will evaluate the comparative-statics predictions of LRWZ’s model regarding the auctioneer’s expected revenue and the efficiency of the first-price auction. This is hypothesized to occur when the ranking of valuations is common knowledge to two bidders. (i) The bidder with the lower reserve value will bid more aggressively than the highvalue bidder. Thus, the proportion of efficient allocations should be lower than when bidders lack information about the ranking of valuations. (ii) For the uniform distribution, bidders are expected to bid more aggressively than in the standard first-price auction model, where information regarding the ranking of valuations is not available. Thus, the seller’s expected revenue should be higher when rankings are known than when they are unknown. The experimental results reported here support the prediction that low-value bidders are inclined to bid more aggressively once information about the ranking of valuations has been revealed. Contrary to the theory’s predictions, high-value bidders are inclined to bid less aggressively. Thus, consistent with the theory, the overall proportion of efficient allocations is lower than in the first-price auction before information is revealed, but as a result of high-value bidders decreasing their bids, the expected revenue does not increase consistently. Although both kinds of bidders improve their average monetary payoff per auction by moving toward the asymmetric risk-neutral Nash equilibrium (RNNE) after information is revealed, high-value bidders continue to bid above it. Therefore, I consider a modified version of LRWZ’s auction model with risk-averse bidders as an alternative way to explain why high-value bidders may keep bidding above the RNNE (Cox et al., 1988). In a previous experiment, Kagel et al. (1987) tested the effects of public information about rival valuations on the seller’s expected revenue in the context of affiliated private-values (Milgrom and Weber). In that paper, bidders were required to bid for a single item, following the first-price sealed-bid allocation rule. They also followed two different information conditions. While each bidder knew the value of the item, a public signal induced a particular symmetric affiliation between valuations. As a consequence, the bidder whose valuation was below the public signal would raise his bid in an effort to win the item. In turn, this put pressure on the bidder, whose valuation was above the public signal, to raise her bid, resulting in more expected revenue for the seller. In contrast to my results, Kagel et al. showed that a public signal increases average bidding and market prices as expected. However, in the aggregate, the increase in prices is closer to the predictions of the risk-averse Nash equilibrium (RANE) model, which permits some increase in revenue lower than predictions by the RNNE model. In the context of independent private-values drawn from different distributions, Pezanis-Christou (2002) and Güth et al. (2006) tested the predictions derived respectively by Maskin and Riley (2000) and Plum (1992). In these experimental papers, one strong-bidder and one weak-bidder were required to bid for a single item following the first-price sealed-bid allocation rule where the weak-bidder was more likely to draw values lower than the strong-bidder. In this case, the strong-bidder had the incentive to low-ball, that is, to submit low bids. These authors showed that the strong-bidder was willing to low-ball as predicted. The reduction however was not as significant as expected when compared to the second-price auction. In contrast to their result, I show that contrary to expectations the high-value bidder (or strong-bidder) is willing to submit low bids. This paper is organized into five sections. The introduction is followed by Section 2, describing the experimental design. Section 3 specifies the RNNE bidding strategies for the auction models and states the comparative-statics predictions for the first-price auction when the information about the ranking of valuations is revealed. Section 4 evaluates the experimental results regarding these auction settings, and Section 5 summarizes the research results. 2. Experimental design The experimental design used directly measures the information impact of the ranking of valuations on bidding behavior in first-price private-values auctions. The particulars of the experimental design is as follows. 2.1. Structure of the auction In this experiment, two people were required to bid for a single item following the first-price sealed-bid allocation rule under two different information conditions. In the first-price sealed-bid auction, the high bidder earns a payoff equal to her value of the item less the high bid, and the second bidder earns nothing. For each auction, two private values, one for each bidder, were independently drawn from the same distribution. In one information condition (hereafter referred to as the symmetric condition), bidders had no information about the rank order of valuations. In the other information condition (hereafter referred to as the asymmetric condition), both bidders were informed
1 In auctions with affiliated private-values, values might remain private for bidders, but the higher (lower) the value is for one bidder, the more likely the value will be higher (lower) for other bidders as well (Milgrom and Weber, 1982).
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Table 1 Experimental design. Experimental treatment
Sessions
Auction periods
Markets
Information condition
Asym. condition and dual-market bidding
1 and 2
1–10 11–30
Single-auction Dual-auction
Asymmetric Sym./asym.
Sym. condition and dual-market bidding
3 and 4
1–10 11–30
Single-auction Dual-auction
Symmetric Sym./asym.
Note: In session 1, during the dual-auction markets, only 18 trading periods (out of 20) were actually run since the network system broke down at that stage.
about the rank order of their valuations before bidding, that is, whether they have the highest or lowest valuation (but not informed about the size of the differences in valuations). Each session of the experiment began with ten periods of single-auction markets, followed by 20 periods of dual-auction markets. (i) In the single-auction markets, two competing bidders submitted one bid in each period. Participants were randomly matched in each period. (ii) In the dual-auction markets, two competing bidders submitted one bid under the symmetric condition. Then, once these bids were collected, but before they were posted, information about the ranking of valuations was released, and the same two competing bidders were asked to then submit a second bid. The winner in each market was determined after the second bid was submitted. Payoffs were determined based on just one of the two markets (chosen with equal probability). Participants were also randomly matched in each period. The dual-market bidding procedure involved the same two bidders bidding for the same item under two different information conditions. It has the advantage of directly controlling the between-subject variability. The rule of flipping a fair coin to determine which auction outcome is payoff relevant ensures that under the expected utility hypothesis, the optimal strategy in the private information market should be unaffected by bids made after the ranking is revealed, and vice versa (Kagel and Levin, 1986). Each session began with bidding in single-auction markets to determine whether the dual-market bidding procedure actually affects the way bidders bid and to familiarize subjects with the auction conditions so as to make it easier for them to bid in dual-auction markets. Table 1 briefly summarizes the experimental treatments. Sessions 1 and 2 began with 10 periods of a single-auction markets under the asymmetric condition followed by 20 periods of a dual-auction markets. During these 10 initial periods, each bidder maintained his role as a high-value or low-value bidder in order to maximize the prospect that rivalrous elements will come into play. Sessions 3 and 4 began with 10 periods of single-auction markets with the symmetric condition followed by 20 periods of dual-auction markets. 2.2. Design parameters This section characterizes the basic parameters and the general procedure of the experiment. Valuation distribution All bidders were informed that their valuations were drawn randomly. The interval included all whole cent values between $0.00 and $6.00. All values in this interval had an equally likely chance of being drawn. Payoff mechanism and participation fee Each group of players received an initial cash balance of zero. At the end of each session, total earnings were distributed to participants in cash. Since bidders’ roles stayed constant during the initial 10 auction periods of sessions 1 and 2, the subjects who were assigned the role of low-value bidders received an additional $5 at the end of each session. Players were informed at the beginning about this additional payment. This was a fixed amount that was not expected to impact behavior. In addition, at the end of each day, all participants received a participation fee. Matching procedure Bidders were informed that participants would be randomly matched in every period, but that no pair of bidders would be matched twice in two consecutive periods. Bidders were also informed that in the dual-auction markets section, each of them would face the same opponent in both markets. Information feedback At the end of every period each player obtained the following information. For the single-auction markets with the asymmetric condition, each subject received feedback about his earnings and the competing bidder’s valuation and bid amount. For the single-auction markets with the symmetric condition, additional information about the ex-post ranking of valuations was presented to players. For the dual-auction markets, each subject received complete feedback about who the high bidder was in each market and how much he earns. They also knew the competing bidder’s value and bids in each market, and which market was randomly selected to pay off in. Participants’ identities were not revealed.
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Fig. 1. RNNE bid functions: bL (v) [– ·], bH (v) [—] and bS (v) [– –].
Dry runs In order to familiarize subjects with the auction procedures, two practice runs took place at the beginning of the singleauction markets, and one practice period took place prior to the dual-auction markets. Subjects For each session, subjects were recruited from a wide cross-section of students (mostly undergrads) at the University of Pittsburgh and Carnegie Mellon University. There were 18 subjects for session 1 and 20 subjects for sessions 2–4. Subjects participated in only one session. The experiment was run in the economics lab at the University of Pittsburgh using computers. 3. Theoretical predictions LRWZ considered the specific situation where a single object is auctioned between two bidders. Both bidders have riskneutral utility functions and independently drawn private valuations. These valuations are ex-ante identically distributed, and this is common knowledge. Each bidder knows whether she has the higher or the lower valuation, but does not know her opponent’s exact valuation. Under the assumption that the underlying distribution of valuations is uniform with the support [0,6], both agents maximization problems can be reduced to the following differential equation system by the first order necessary conditions: l (b)(h(b) − b) = l(b) h (b)(l(b) − b) = h(b) − l(b) h(0) = l(0) = 0 ∃b∗ ∈ [0, 6] such that h(b∗ ) = l(b∗ ) = 6 where l(·) and h(·) are, respectively, the inverse of the bid functions for the low-value bidder, bL (v), and for the high-value bidder, bH (v). The intuition behind the second boundary condition is this: if the bids functions are not equal for the upper bound of the distribution, the owner of the higher bid can lower her bid slightly and still win the object. By l’Hôpital’s Rule, the derivatives at the initial condition are h (0) = 2 and l (0) = 4/3. For a numerical approximation of the above system of equations with two boundary conditions, I follow the general method implemented by Elbittar and Ünver (2001). The bidding response for the symmetric and independent private-values first-price auction model, bS (v), can be derived analytically. For the case of n risk-neutral bidders with a uniform distribution, the unique symmetric RNNE bid function is bS (v) = ((n − 1)/n)v (Vickrey, 1961). The RNNE bidding responses for the low-value bidder and for the high-value bidder are represented in Fig. 1 by bL (v) and bH (v), respectively. In the same figure, the equilibrium bidding response for the symmetric and independent private-values model when n = 2 is represented by bS (v).2
2 The discreteness of the strategy space and of the type space introduced in the experimental design should not have an effect on the equilibrium bid functions and the resulting predictions (Athey, 2001).
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Table 2 Numerical results from the equilibrium bid functions. Auction model
Symmetricc Asymmetric
Seller’s revenue
2.000 2.219
Regular buyer
High-value
Low-value
Payoff
Pr[Win]
Payoff
Pr[Win]
Payoff
Pr[Win]
1.000 (–)
0.500 (–)
(–) 1.361
(–) 0.710
(–) 0.263
(–) 0.290
Proportion of POAa
MEb
1.000 0.710
1.000 0.957
Note: Valuations are drawn from the uniform interval [0,6]. Mean is computed by Monte Carlo using 10,000 samples of 1000 drawings. a Pareto-optimal allocations: POA. A level below 100 characterizes unrealized gains from trade. b Mean efficiency: ME = v win /vh . A level below 100 characterizes unrealized gains from trade. c Calculations are made analytically. Table 3 Observed bid proportions. Statistics
Median Mean s.e.
Bid proportiona symmetric condition
Bid proportion asymmetric condition
ıSL (v)
ıSH (v)
ıS (v)
ıAL (v)
ıAH (v)
0.697 [0.667] (0.007)
0.658 [0.644] (0.006)
0.674 [0.656] (0.005)
0.820 [0.757] (0.008)
0.607 [0.600] (0.006)
Note: ıS is the aggregation of ıSL and ıSH . a
b(v)
Bid proportion: ık = v , j = L, H and k = S, A. j
3.1. Comparative-statics predictions The comparative-statics predictions of these two auction models define the theoretical benchmark for measuring the possible impact of valuation ranking in relation to bidding behavior. These predictions also provide us with precise statements about the changes in the seller’s revenue and the proportion of optimal allocations. According to LRWZ (Corollary 1), the high-value bidder should bid less aggressively than the low-value bidder for any given value (bL (v) > bH (v), ∀v ∈ (0, 6)). Since the low-value bidder can obtain the item with positive probability, the proportion of efficient allocations should decrease. They also proved (Proposition 2) that for the uniform distribution studied here, revealing the ranking should induce the low-value bidder to bid strictly higher than when this information is not known (bL (v) > bS (v), ∀v ∈ (0, 6]). It should also induce the high-value bidder to bid the same for valuations below or equal to 4 and strictly higher for valuations above 4 (bH (v) = bS (v), whenever v ∈ (0, 4] and bH (v) > bS (v), whenever v ∈ (4, 6]). Therefore, the seller’s expected revenue should increase.3 Table 2 summarizes some theoretical predictions for both auction models using the equilibrium bid functions. The seller’s expected revenue, the proportion of pareto-optimal allocations (POAs), the proportion of the maximal payoff or mean efficiency (ME), and each type of bidder’s conditional expected payoff and probability of winning are numerically estimated, drawing valuations from the uniform distribution in the interval [0,6]. The values used in these results indicate that the theoretical impact of the valuation ranking would increase the seller’s expected revenue from $2.00 to around $2.22, with the percentages of POA and ME decreasing from 100.0 percent to 71.0 percent and 95.7 percent, respectively. 4. Experimental results This section compares the experimental results for the dual-auction markets to directly measure the ranking of valuations affecting bidding behavior.4,5 Result 1 (Bidding Behavior). While low-value bidders tend to bid higher after information about the ranking is released, high-value bidders tend to bid lower. Support for Result 1. Table 3 reports the observed bid proportion, ıkj (v), for each bidder type j and information condition k. The bid proportion tells us how much bidders bid as a proportion of their own reserve value. Fig. 2 shows all the observed bids and the RNNE bid functions as proportions of their reserve values. LRWZ predict that both types of bidders bid more
3 LRWZ proved that it will always be true that b (v) > b (v) for all values of v. LRWZ provided that there exist probability distributions for which L S bS (v) > bH (v) for some values of v. Elbittar and Ünver showed that the proposition that bH (v) ≥ bS (v) for all values of v also holds for some distributions different from the uniform. 4 The results of single-auction markets were consistent with the results for the dual-auction markets and indicate that the dual-market bidding procedure does not affect bidding behavior. A summary of these results is included in the appendix to avoid repetition. 5 The total number of bids is 1524 per market. However, some observations were excluded from the statistical analysis either because the subject’s bidding was above her/his own valuation (18 observations), or the subject’s valuation was less than two cents, so the corresponding symmetric RNNE was less than one cent (10 observations). The analysis is therefore based on 1496 bids per market.
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Fig. 2. Actual bids and RNNE bid functions as a proportion of the reserve values.
aggressively under the asymmetric condition than under the symmetric condition (ıAj (v) > ıSj (v), j = L, H). Analysis of the actual bid proportions in Table 3 indicates that, contrary to expectations, high-value bidders tend to bid less after information about the ranking of valuations was released, ıAH (v) < ıSH (v). For the pooled data, high-value bidders bid, on average, 4.4 percentage points less after the information was released, or approximately 15 cents per auction. By contrast, low-value bidders tended to bid more after information about the valuation ranking was released, ıAL (v) > ıSL (v). For the pooled data, the same bidders bid 9.0 percentage points more after the information was released, or approximately 23 cents per auction. In order to test LRWZ’s comparative static predictions, I consider a fixed-effects model for estimating the bidding proportion for each bidder type and each information condition: ıist = + ˛i + s + [ˇ1 + ˇ2 (vist − 4)dv>4 ]dH + [ˇ3 + ˇ4 (vist − 3)dv>3 ]dL In this model ıist represents the bid that each individual i submits as a proportion of her own reserve value in session s and period t; represents the mean intercept; ˛i and s represent, respectively, the deviation of individual i and of session s from the common mean ; dH is a dummy variable equal to one when a high-value bidder bids after the information was released; and dL is a dummy variable equal to one when a low-value bidder bids after the information was released. The second graph in Fig. 2 shows that for high-value bidders, the RNNE bid function as a proportion of the reserve value starts increasing measurably after valuations exceed 4. Meanwhile, the third graph in Fig. 2 shows that for low-value bidders, the RNNE bid function as a proportion of the reserve value starts decreasing measurably after valuations exceed 3. To capture these two potential effects, I consider a piecewise linear regression such that dv>4 is a dummy variable equal to one when the valuation for a high-value bidder is greater than 4, and dv>3 is a dummy variable equal to one when the valuation for a low-value bidder is greater than 3. Table 4 displays all the estimated coefficients. In contrast to the comparative-statics predictions, coefficients ˇ1 and ˇ2 are negative and significant. This means that after information is released high-value bidders tend to reduce their bids by 3.9 percentage points and 3.2 percentage points for each additional dollar for valuations greater than 4. On the other hand, while coefficient ˇ3 is positive and significant, coefficient ˇ4 is not significantly different from zero. This means that after information is released low-value bidders tend to increase their bids by 9.9 percentage points without a significant change for valuations over 3. Result 2 (Bidding Complexity). There is a larger variation in the response of high-value bidders than in that of low-value bidders. Support for Result 2. Using the pooled data, Table 5 reports how subjects alter their bids following the announcement of the valuation ranking. In the top half of this table, each entry represents the percentage of values drawn in a particular range, as in the row heading, for which high-value bidders respond as in the column heading. The same information is shown for Table 4 Fixed-effects model. Variables
Coefficient values
Common intercept dH (v − 4)dv>4 dH dL (v − 3)dv>3 dL Individual effects Session effects Number of observations Number of individuals Number of sessions
0.656*** (0.004) −0.039*** (0.009) −0.032*** (0.009) 0.099*** (0.008) 0.013 (0.011) n.r. n.r. 2992 78 4
* p < 0.05, ** p < 0.01, and *** p < 0.001. Note: The number in parentheses after each coefficient represent the coefficient standard error. n.r.: not reported.
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Table 5 Effects of valuation ranking information on bidders’ bids. Valuation range
Bids bH (v) < bS (v)
bH (v) = bS (v)
bH (v) > bS (v)
6≥v>4 4≥v>0 6≥v>0
52.5% 57.9% 54.9%
19.9% 21.2% 20.5%
27.6% 20.9% 24.6%
Valuation range
bL (v) < bS (v)
bL (v) = bS (v)
bL (v) > bS (v)
6≥v>0
13.7%
22.2%
64.1%
low-value bidders. As previously discussed, LRWZ predict that high-value bidders will raise their bids for valuations above 4 and keep them the same for valuations below or equal to 4. As shown in Table 5, a sizable proportion of bids (54.9 percent) decreased when subjects learned they had the higher valuation. Meanwhile, the other 45.1 percent of bids increased or involved no change. A similar kind of behavior is observed for valuations above and below four. The same model predicts that low-value bidders will raise their bids. As can be seen in the same table, 64.1 percent of bids increased and 22.2 percent involved no change. This result could be attributed to the greater complexity of bidding strategies for high-value bidders once information was released. For instance, when subjects were asked at the end of each session to write down their bidding strategies, most said something to the effect of “If I was a low I raised my bid, while, if I was a high I lowered my bid.” Just a few participants indicated an apparently more sophisticated reasoning: “Once I knew I was a high bidder, I usually lowered my bid. Sometimes I highered (sic) it because I knew the other person knew they were a low bidder, so they might also increase their bid.” For low-value bidders, the obvious response to such information is to increase their bids in order to increase the probability of obtaining the object. For high-value bidders, the response is less obvious. Once high-value bidders are aware of their strong position, they might either be reluctant to take on the risk of submitting lower bids because of low-value bidders’ more aggressive bidding behavior or be willing to take such a risk in hopes of increasing their average payoff. This result was also found in previous experimental studies of simple games where the iteration of dominance eliminates one, two or three strategies (see, for example, Beard and Beil, 1994; Schotter et al., 1994). These studies found little support for more than one step of deletion of dominant strategies. Although few subjects violate dominance, most of them seem unwilling to bet heavily that others will obey dominance (Camerer, 2003). Result 3 (Deviations from Predicted Behavior). Bidders under the symmetric condition and high-value bidders under the asymmetric condition keep bidding significantly above their respective RNNE. Low-value bidders under the asymmetric condition keep bidding closer to the RNNE. Support for Result 3. The baseline of the comparative-statics predictions in LRWZ is what risk-neutral, fully rational bidders bid under the symmetric condition. However, a well-known pattern of bidding above RNNE in first-price independent private value auctions are replicated here (Kagel). The overbidding factor, j = bj (v) − b∗j (v)/vj , j = L, H, S, is the relative deviation of the actual bid with respect to the RNNE bid. The mean of the overbidding factor under the symmetric condition is 15.6 percentage points more than the RNNE response. The same overbidding pattern is found for high-value bidders under the asymmetric condition. The mean of the overbidding factor for high-value bidders is around 8.0 percentage points more than the RNNE response. The proportions of bids above the RNNE are 82.6 in the symmetric condition and 68.9 percent for high-value bidders under the asymmetric condition. A significant proportion of low-value bidders under the asymmetric condition also bid above the RNNE (67.8 percent). However, the overbidding factor is 1.6 percentage points, which is not significantly different from zero (p = 0.25). Experimental results also indicate that even though bidders under the symmetric condition, and high-value bidders under the asymmetric condition, moved toward the RNNE over time, they stayed bidding well above the RNNE. In contrast, low-value bidders’ bidding behavior did not change over time. Result 4 (Risk Aversion). Changes in bidding behavior are consistent with a reduction in bidders’ aversion toward risk. Support for Result 4. Strategic considerations do not seem enough to explain the deviations from the predicted behavior. In order to reconcile the previous experimental results, I consider a modified version of LRWZ’s model, assuming a constant relative risk-aversion utility function (Cox et al., 1988). Since a ceiling in bidding is not perceived, no restriction is imposed on the upper bound of the distribution. Under the assumptions that the underlying distribution of valuations is uniform and that each agent has a constant relative risk-aversion utility function,6 both agents’ maximization problems can be expressed in terms of the following differential equation system, by the first order necessary conditions with just one boundary condition:
6 Notice that this maximization problem is equivalent to one under the assumption of risk-neutrality with transformed prior probability distributions (Marshall et al., 1994).
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Table 6 Estimated parameters of bid functions. Bid function
Estimates
bˆ S bˆ L bˆ H * +
ˆ ˇ
ˆ sˆ(ˇ)
rˆ
sˆ(ˆr )
0.601*
0.006 0.007 0.010
0.663+ 1.060 0.755+
0.017 0.051 0.030
0.779* 0.570* Reject H0 : ˇ = 0.0 for p < 0.0001. Reject H0 : r = 1.0 for p < 0.0001.
l (b)(h(b) − b) = rH l(b) h (b)(l(b) − b) = rL (h(b) − l(b)) h(0) = l(0) = 0 where rH and rL are the risk aversion coefficients (rk ∈ (0,1]). Thus, bH (v) = (1/1 + rH )v and bL (v) = ((1 + rH + rL )/(1 + rH )(1 + rL ))v. The bidding response for the symmetric model is (bS (v) = (1/1 + rS ))v (Cox et al., 1988). Therefore, the following linear specification is estimated for each kind of bidder: bist = ˇvist , where bist and vist are respectively player i’s bid and valuation in period t and session s. ˆ using the pooled data. This table also reports the riskTable 6 reports the estimated parameters of the bid functions, ˇ, aversion coefficients, rˆ , as well as the standard deviation of the estimates, sˆ(·), for the each of the bid functions. bˆ L and bˆ H correspond to the estimated bid functions under the asymmetric condition for low-value and high-value bidders, respectively. Meanwhile, bˆ S corresponds to the estimated bid function under the symmetric condition. All coefficients are significantly different from zero for a p < 0.0001. As shown in Table 6, the risk-aversion coefficients for both kinds of bidders increases after information is released. The null hypotheses that rS is equal to either rL or rH is rejected when p < 0.0001. The null hypothesis that rL is equal to rH is rejected when p < 0.01. Thus, low-value and high-value bidders become less risk-averse after information about the ranking is released. Therefore, the information in relation to the ranking seems to alleviate the strategic uncertainty associated with the first-price auction and, consequently, to induce bidders to change their attitude toward risk. Result 5 (Profitable deviation). Each group of bidders improves the average monetary payoff per auction by properly deviating their bids after the information is released. In particular, the revelation of the ranking provides incentives for high-value bidders to decrease their bids and to exploit new economic opportunities. Support for Result 5. In Table 7, the first row displays the results of a Monte Carlo experiment in which high-value bidders bid against low-value bidders, with both groups bidding based on the estimated response functions bˆ H (v) and bˆ L (v), respectively. The second row indicates results of a similar numerical experiment in which high-value bidders based their bids on the estimated response before the ranking was revealed, bˆ S (v). The third row reports results of a numerical experiment in which low-value bidders bid based on the estimated response before the revelation of the ranking, bˆ S (v), but high-value bidders based their bids on the estimated bid function after the information was revealed, bˆ H (v). Table 7 displays, for each comparison, the seller’s expected revenue, the bidder’s expected payoff conditional on winning, and the probability of winning the item. The numerical results indicate that if high-value bidders were using the bid function before the revelation of the ranking, bˆ S (v), the unconditional average payoff would drop from $1.70 to $1.48 per auction (of around 23 cents per auction); meanwhile, the probability of winning the item would increase from 0.77 to 0.85. As result, their average payoff, conditional on winning, would fall by around 7 cents per auction. Therefore, in terms of monetary payoffs, it would be better for high-value bidders to reduce their bids. On the other hand, if low-value bidders were bidding using the bid function Table 7 Estimated bid functions comparison: numerical results. Bid functions comparison
Statistics
Seller’s revenue
High-value
Low-value
Payoff
Pr[Win]
Payoff
Pr[Win]
bˆ H (v) vs. bˆ L (v)
Mean s.e.
2.366 (0.278 × 10−3 )
1.322 (0.282 × 10−3 )
0.773
0.205 (0.126 × 10−3 )
0.227
bˆ S (v) vs. bˆ L (v)
Mean s.e.
2.556 (0.293 × 10−3 )
1.251 (0.225 × 10−3 )
0.848
0.144 (0.112 × 10−3 )
0.152
bˆ H (v) vs. bˆ S (v)
Mean s.e.
2.290 (0.260 × 10−3 )
1.564 (0.237 × 10−3 )
0.908
0.130 (0.138 × 10−3 )
0.092
Note: Valuations were drawn from the common support of the bid functions. Mean and standard error were computed by Monte Carlo using 10,000 samples of 1000 drawings.
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Table 8 Effect of ranking information on revenue. Statistics
Symmetric condition pS
Asymmetric condition pA
Mean s.e. Median
2.610 (0.037) [2.655]
2.572 (0.038) [2.620]
at
−2.049 −2.378
bW *
Reject H0 against H1 at p < 0.05. a Matched pairs t-test. H : p (b) ≤ p (b) against H : p (b) ≤ p (b). 0 S 1 S A A b Matched pairs Wilcoxon test. H : p (b) ≤ p (b) against H : p (b) ≤ p (b). 0 S 1 S A A
Table 9 Effect of ranking information on efficiency. Concepts
Symmetric condition
Asymmetric condition
a ME
97.2 [87.1]
94.8 [74.8]
b POA cZ ME dZ POA
2.376* 6.134* vwinner vh × 100.
a
Mean efficiency: ME =
b
Percentage of pareto-optimal allocations: POA. Population proportion test. H0 : MES ≤ MEA against H1 : MES (b) > MEA . Population proportion test. H0 : POAS ≤ POAA against H1 : POAS (b) > POAA . Reject H0 against H1 at p < 0.05.
c d *
previous to the revelation of the ranking, bˆ S (v), they would suffer a reduction in their average payoff conditional on winning (of around 7 cents per auction). Result 6 (Expected Revenue). The experimental data do not support the prediction that the seller’s revenue increases after information about the ranking is released. Support for Result 6. With high-value bidders decreasing their bids, the enhanced revenue possibilities of the first-price auction, with the ranking of valuations revealed, does not necessarily holds; rather it becomes an empirical question of whether high-value bidders’ decreasing bids dominate low-value bidders’ increasing bids after the information is released. Table 8 reports the mean, standard error and median of the observed prices for each information condition. Each average price represents the observed seller’s expected revenue for a particular information condition j: pj , j = S, A. According to LRWZ, seller’s revenue is expected to be higher under the asymmetric condition than under the symmetric condition (pA > pS ). As noted in Table 8, however, the mean of the selling prices under the symmetric condition ($2.61) is slightly higher than under the asymmetric condition ($2.57). This observation is confirmed by a one-tailed matched pairs t-test statistic (t), the result of which is displayed at the bottom of the table. Testing for every auction period, the null hypothesis that the price under the symmetric condition is larger than or equal to the price under the asymmetric condition could not be rejected in 19 out of the 20 auction periods (p < 0.05). This shows that the seller’s expected revenue does not always increase after the information is released.7 Result 7 (Efficiency). The first-price auction is, in the aggregate, more efficient in the absence of information. Support for Result 7. This result confirms the prediction of the reduction in the proportion of optimal allocations. Table 9 reports the following two measures of optimal allocation and economic efficiency: (i) pareto-optimal allocations, reported as the percentage of objects given to the high-value bidder, and ME, reported as the percentage of the maximal payoff that was generated as a result of the auction. Since low-value bidders generally have a positive probability of obtaining the item, the percentage of efficient allocations was expected to be lower under the asymmetric condition than under the symmetric condition. As seen in the calculations displayed in Table 9, the percentage of POA and ME are consistently higher under the symmetric condition than under the asymmetric condition. This result is due to the more aggressive bidding by the low-value bidders and to the less aggressive bidding by the high-value bidders under the asymmetric condition. In the same table, a one-tailed
7 Notice that due to the significant pattern of overbidding behavior, the average revenue for each information condition was significantly higher than the expected revenue at the risk-neutral equilibrium.
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population proportion (Z) test statistic for each measure of optimal allocation is reported. When testing for every auction period, the null hypothesis was rejected in favor of the alternative hypothesis, that the efficiency under the symmetric condition is higher than under the asymmetric condition, in 19 of the 20 auction periods (p < 0.05). 5. Conclusion In this project, a first-price private-values auction experiment was conducted, where the same two bidders bid for a single item in two markets and under two different information conditions. In this study, I examine the impact of information about the ranking of valuations on bidding behavior in first-price private-values auctions. Experimental results indicate that after information about the ranking of valuations was released, the two groups of bidders responded differently. As theory predicts, low-value bidders were inclined to bid more aggressively. Contrary to the theory’s prediction, high-value bidders tended to submit lower bids. Thus, the probability of low-value bidders winning the item increased. As expected, the proportion of efficient allocations decreased with respect to the first-price auction in the absence of information. Contrary to the predictions, the seller’s expected revenue did not increase on a regular basis because high-value bidders decreased their bids once the information was released. Experimental results also indicate a larger variation in the response of high-value bidders than in that in low-value bidders. This result could be attributed to the greater complexity of bidding strategies for high-value bidders once information was released. For example, once aware of their strong position, a high-value bidder might either be reluctant to take on the risk of submitting lower bids because of the low-value bidders’ more aggressive bidding behavior or be willing to take a risk in the hopes of increasing their average payoff. Since these strategic considerations are not sufficient to explain deviations from predicted behavior, a modified version of LRWZ’s model that assume a constant relative risk-aversion utility function is considered. Changes in bidding behavior due to information about the ranking are consistent with a reduction in the bidders’ aversion toward the risk involved in the first-price private-value auction. I find my results to be consistent with those of Isaac and James (2000) and Berg et al. (2004). These authors also found that individual risk preferences elicited from the same individuals across different institutions were not stable. Based on my results, I propose that the information about the ranking seems to alleviate the strategic uncertainty associated with the first-price auction and, consequently, to induce bidders to change their attitude toward risk. To understand changes in risk-aversion degrees, I revert to Krishna (2002, p. 40), who notes in his book about auction theory that a risk-averse bidder would reduce her bid when “the effect of a slightly lower winning bid on his wealth level has a smaller utility consequence than does the possible loss if this lower bid were, in fact, to result in his losing the auction.” Following Krishna’s argument, by bidding lower after information is revealed, high-value bidders seem to be willing to “buy” less insurance against the chance of losing, while low-value bidders seem to be willing to bid higher, increasing their probability of winning the object. At the same time, they are willing to “buy” less insurance compared to a risk-averse bidder. Acknowledgements I gratefully acknowledge helpful comments from Masaki Aoyagi, Vincent Crawford, Sonia Di Giannatale, David Kaplan, Charlie Plott, Jean-Francois Richard, Tridib Sharma, Ricardo Smith, Elmer Wolfstetter, Shmuel Zamir and two anonymous referees, as well as seminar participants at CIDE, Hewlett-Packard, ITAM, ESA Summer Meeting 2000, and Econometric Society Winter Meeting 2001. I am especially grateful to John Kagel for his support and many helpful discussions and suggestions. All errors are mine. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jebo.2008.09.006. References Athey, S., 2001. Single crossing properties and the existence of pure strategy equilibria in games of incomplete information. Econometrica 69, 861–889. Beard, T.R., Beil, R., 1994. Do people rely on the self-interested maximization of others? An experimental test. Management Science 40, 252–262. 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