Uncertainty and Auction Outcome: Evidence from Used Car Actions

Yaron Raviv 1 April 15, 2006 Abstract. I study the sequence of bidding in an open-outcry English auction to examine how uncertainty affects auction outcomes. I do this by analyzing a data set collected from a series of public auctions of used cars in New Jersey. I conjecture that the uncertainty is related to the auction’s progress and demonstrate that, empirically, an increase in the uncertainty is associated with an increase in the number of rounds required to sell an object and with a reduction in the ratio between the selling price and the presale estimate.

JEL Classification: D44 Keywords: Auction

1

Department of Economics, Claremont McKenna College, Claremont CA 91711. I am greatly indebted to Orley Ashenfelter and Han Hong. I would also like to thank Richard Burdekin, Ken Fortson, Tanjim Hossain, Gad Levanon, Andrea Podhorsky, Ryan Quillian, Jesse Rothstein, Joydeep Roy, Gabor Virag and participants in seminars at Princeton University, Tel-Aviv University, Claremont McKenna College and Ben-Gurion University. Any errors are mine.

1. Introduction Standard auction theory predicts that the level of uncertainty in an auction has a real effect on the auction outcome. In this sense, research demonstrates, both theoretically and experimentally, that an increase in the variance of bidders’ signals will reduce the expected selling price. 2 But there are no predictions as to the effect of uncertainty on the number of rounds it requires to sell an item. In this paper, I study the sequence of bidding in an open-outcry English auction to examine how the level of uncertainty affects auction outcomes. I focus on the number of bids required to sell an item and the ratio between the selling price and the presale estimate. I exploit an original data set collected during 2001 and 2002 from a series of public auctions of used cars conducted by the state of New Jersey. Although a tractable model of the strategic behavior of the parties in an openoutcry English auction is not available, I conjecture that the auction’s progress is related to the variability of valuations, and demonstrate that, empirically, an increase in that variability is associated with an increase in the number of bids required to sell an object. These findings are in line with the conjecture that, with lower variability in the unknown value of the good, the convergence to the price will be faster. I claim 2

In their seminal paper, Milgrom and Weber (1982) demonstrate that, when the seller publicly reveals some credible information about the valuation of the object, revenues increase, on average. In other words, a reduction in the variance of the true value of the object has a positive effect on the expected selling price. Wilson (1969) provides an example of a second price sealed-bid auction with two bidders who have a common prior of a diffuse Normal density and signals xi that are both normally distributed with mean V. In that case, the optimal strategy function is xi − σ π 2 where σ is the posterior marginal density of his opponent. An increase in the variance will reduce the selling price. Pai-Ling Yin (2003) conducted an empirical investigation of eBay auctions for computers. She found that the winner’s curse changes with the dispersion of information. She estimated the predicted common value of the computers using a survey, and used the survey to build a measure that is correlated with the mean and dispersion of the bidder’s signals. She found that the price declines with dispersion. McMillan and Eiichiro (2005) provide a model in which the factor that most influences the mechanism the seller will chose to conduct an auction, online or offline, is not the expected price, but the valuation uncertainty. They prove, again, that when bidders have a more precise estimator of the common value, they will bid more aggressively; hence the expected selling price is decreases in the variance. This is the case for both mechanisms. Kagel et al. (1995, 1996) also demonstrate the same thing; increases in variability reduce the expected selling prices. Goeree and Offerman (2002) provide a model combining private and common values in a first-price auction. They experiment and test the model’s predictions and find that increased competition and reduced uncertainty about the common value positively affect revenue and efficiency.

2

that, ceteris paribus, increases in uncertainty, both under the private values and the common values paradigms, are associated with an increase in the number of bids required to sell the item. Under the private values paradigm, for example, in the case of perfect certainty (where everybody knows everybody else’s values), there will be an immediate convergence to the second highest valuation. In the other extreme case, in which each player knows only his valuation and does not have any information about the distribution of the valuations, we would expect the auction to progress by the minimum bid increment required to advance the auction when the time cost is zero. In this case the expected number of bids will be the highest. In the common values case, we can model the uncertainty as the variance of the distribution from which each bidder’s signal comes. When the variance is small enough, bidders will immediately jump to the bidding option just below the object’s valuation. When there is more uncertainty about the object valuation, bidders will hesitate and bid more cautiously. The premise is that it takes bidders more time to transfer information and reveal the estimated value of the object when the variance of the signal is high. The second variable of interest is the ratio of the selling price to the presale estimate (Ratio). Many things can affect this variable - among them the order in which the item is sold, measurement errors, etc. In addition to these, I claim that the uncertainty related to the unknown valuation of the good has a negative effect on the selling price and the presale estimate ratio. A reduction in the uncertainty will, on average, increase this ratio. For two items with the same presale estimate but with different uncertainty (or as I model it, with different variances of the valuation) we would expect that the item with the lower level of uncertainty will have a higher Ratio. If, for example, an old Mercedes and a new Hyundai have the same presale estimate, but the Hyundai is new with less mileage on the odometer, we would expect that the ratio of the selling price to the presale estimate of the Hyundai will be higher. The paper is organized as follows. In the next section I describe the data I have collected and the nature of the auctions. In Section 3 I analyze the effect of uncertainty, measured as the variability of the bidders’ valuations, on the two variables

3

of interest - the ratio between the selling price and the presale estimate, and the number of rounds it take to sell an item. The final section offers concluding remarks.

2. The Data I collected the auction data in 2001-02 from the New Jersey Distribution and Support Services (DSS) in Trenton, New Jersey. 3 DSS sells surplus personal and government property through public oral English auctions and sealed-bid auctions. The open English oral auctions of cars are usually held on Saturdays once a month. Bidders can physically inspect the items the day before the auction and on the day of the auction until 9

A.M.,

when the auction begins. Each car that is auctioned is driven

through a large warehouse and stopped in front of the auctioneer, and then the bidding process begins. After the car is sold, it is driven to the parking lot, and a new car is auctioned off. The average time required to sell a car is between 1 and 2 minutes. Bids on operable vehicle units are only accepted in multiples of $25. At the time of sale, successful bidders are required to make a deposit in cash, bank money order, or certified check for $150 or 10% of the total amount of the bid, whichever is greater. If the high bidder fails to place the deposit, the vehicle is immediately resold. The DSS reveals all information available about the car’s condition such as model, year, mileage and the source of the vehicle (Turnpike Authority, criminal justice seizure, Transportation Department, taxation seizure, etc.). The state also reveals all the mechanical information known about the vehicle’s condition, for example whether it has bad transmission, bent rear axle, no vehicle identification number plate on the door, no power steering, etc. The coordinator of operations at DSS has stated that all the information known about the vehicles is made available to the bidders and that the cars are auctioned in random order (which I verified

3

For further information, see Raviv (2004, a).

4

empirically in Raviv (2004, a)), so that there is no correlation between a cars’ presale value and the sequence in which it is auctioned off. The day before each auction, I collected data on each vehicle’s condition. On the same day, I gathered the Kelly Blue Book (KBB) estimated market value of each car. KBB is a company that, among other things, provides market value estimates for cars on its Website. On the day of the auction, I collected the following data: the sequence in which the vehicles were auctioned, all the bids that each car received up to (and including) the winning bid, and data about the resold cars. During the week after each auction, I collected the official list of winning bids from DSS to compare with my notes. Table 1 gives summary statistics from the different auctions. In the first and second columns, the presale estimate (the estimated presale market value from KBB) and the price for which the item was sold are reported. The mean of the presale estimate was $2,662.19, and it was above the mean of the winning bids, which was $1,520.42. It appears that some of the cars were sold quite cheaply. Some bidders bought operable cars for as little as $50. The car with the highest presale value ($15,265) was a 1986 Porsche with 77,000 miles on its odometer. This car was eventually sold for $3,400. Although the governor of New Jersey’s car, a 1998 Buick Ultra, had a presale estimate of $11,270, it was sold for $8,650 and was the most expensive item sold in the auctions. The mean Number of bids is the average number of bids each item received before it was sold. The mean of this variable is 11.39, which indicates that it took, on average, 11.39 rounds for an item to be sold. The minimum of this variable is 1, which means that some of the cars were won by the first bidder. The Ratio variable is defined as the ratio between the selling price and the presale estimate. In order to get a better understanding of the main variables of interest of this paper, the number of rounds it takes to sell an item and the ratio between the selling price and the presale estimate, Figures 1 and 2 describe the empirical distribution of these variables. Figure 1 reports the empirical distribution of the Number of bids. Some of the cars were sold after only a few rounds, whereas it took longer to sell other cars. For

5

example, 20 cars were sold to the first bidder, but 10 other cars took 25 rounds each. I claim that, ceteris paribus, increases in the uncertainty, both under the private values and the common values paradigms, is associated with an increase in the number of bids required to sell the item. Under the private values, for example, in the case of perfect certainty (where everybody knows everybody else’s values), there will be an immediate convergence to the second highest valuation. In the other extreme case, in which each player knows only his valuation and does not have any information about the distribution, and there is no time cost, we expect the auction to progress by the minimum bid increments required to advance the auction. In this case the expected Number of bids will be the highest. In the common values case, we can model the uncertainty as the variance of the distribution each bidder’s signal came from. When the variance is small enough, bidders will immediately jump to the bidding option immediately below the object’s valuation. When there is more uncertainty about the object valuation, bidders will hesitate and bid more cautiously. The premise is that it takes bidders more time to transfer information and reveal the estimated value of the object when the variance of the signal is high. Figure 2 is a histogram with an overlay of the estimated kernel density of the ratio between the selling price and the presale estimate (Ratio). The question that arises is, “Why is this shape observed?” Why did some of the items sell for only 5% of their presale estimate, whereas some of the items sold for more than the presale estimate? The reason for the pattern of this variable is also related to the level of uncertainty. If there is a common component to a car’s value, then the variability of the common value of the object affects the selling price. When the variance of the true value of the object decreases, we would expect the ratio between the selling price and the presale estimate to increase. If the variance increases, we would expect this ratio to decrease.

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3. Variability of the Common Values and the Auction Progress In Raviv (2004, b), I concluded that my data can be characterized as coming from a common value auction.4 In this section, I will explain the pattern of the number of bids an item receives and the ratio between the selling price and the presale estimate. I will relate the outcome of these variables to the level of uncertainty. The way I measure uncertainty is by the variances of the valuation of the object and the signal each bidder received. Under the assumption of common values, the item value is the same for everyone, so bidders only differ in the unbiased signals of the item value they receive. In Figure 3, some effects of the variance on the Number of bids are demonstrated. The figure describes five different options for the pattern of jump biddings for an object that eventually sold for the same price. The item could sell in the first bid, the second bid, or even after 10 bids. For two items that sell for the same price and have the same presale estimate, we predict that, on average, the one that receives more bids has a higher variability of the unknown value. On the other hand, for two items with the same presale estimate, we cannot predict generally which will receive more bids, because the increase in the variability affects the first bid and the selling price in the same direction. 5 The premise is that, with low-enough variability, we would expect immediate convergence to the common value by placing the nearest possible bid from below. Also, as mentioned before, even with the same presale estimate, items can have different levels of uncertainty. The second conjecture is that a higher variability of the object for a given selling price and presale estimate is associated with a lower first bid. If we denote N as the Number of bids, FB as the first bid, σ 2 as the value variance, p * as the selling price and pˆ as the presale estimate, we would expect the following conditions: ∂N ∂σ 2

≥0 p*

∂N ∂σ 2

≥0 Fb , p *

∂N ∂σ 2

≥0 pˆ

∂N ∂σ 2

≥0 Fb , pˆ

∂p * ∂σ 2

≤0 pˆ

∂Fb ∂σ 2

≤0 p*

∂Fb ∂σ 2

≤0 pˆ

4

I could not reject the hypothesis that the data can be characterized as a private values data though. In the case of 0 variance, there will be an immediate convergence to the value, and the number of bids is the lowest in this case. However, it is not clear theoretically, of two items with the same presale estimate but with different variances, which will receive more bids.

5

7

We expect that the Number of bids will increase with the variance when we control for the presale estimate, the selling price, and the first bid. In general, we cannot determine theoretically the expected effect of the variance on the Number of bids because an increase in the variance leads, on average, to a reduction in the expected selling price and in the expected first bid. 6 Without any restrictions on these variables, we cannot determine how an increase in the variance affects the Number of bids. Because of this, we expect the first two conditions to hold theoretically, whereas the effect of the next two conditions cannot be precisely predicted theoretically, subsequent empirical testing reveals that these conditions hold as well, however. The next condition is the prediction by Milgrom and Weber (1982). We expect the selling price to decline with the variability of the object after we control for the presale estimate. In their seminal paper, Milgrom and Weber (1982) demonstrate that, when the seller publicly reveals some credible information about the valuation of the object, revenues increase, on average. In other words, a reduction in the variance of the true value of the object has a positive effect on the expected selling price. In addition, we would expect the first bid to decline with the variability of the valuation after controlling for the selling price and the presale estimate. The only problem left is how to estimate the variance of the valuation and examine our predictions. Although there is not yet any tractable model of the strategic behavior of the parties in an open-outcry English auction, I conjecture that the variability of valuations is related to the auction’s progress. In this sense, I predict that the winning bid is affected by the value variance, and that the agents will jump to the lower bound of the signal’s support in the first round. Then the difference between the winning bid and the first jump would be correlated with the value variability. A simple example is as follows. It is known that the equivalence between an English auction and a Second Price sealed bid auction disappears as a consequence of common values. This occurs because there is information revelation in an English auction that does not exist in a

6

When the variance of the valuation is related to the lower bound of the distribution, like in the uniform case when the mean is constant.

8

Second Price sealed-bid auction. However, to demonstrate my point, I will assume that, instead of an oral English auction, we have a Second Price sealed-bid auction. As a witness to the auctions, it seems to me that usually two bidders compete with each other over an item. 7 When there are only two bidders, the regular English and Second Price sealed-bid auctions are equivalent. If we denote V as the unknown value of the object that is drawn from a known distribution and x as a signal drawn from a known distribution X, then the symmetric equilibrium strategies in a Second Price auction are given by: B(x,x)=E[V|X=x,Y=x], where Y is the highest signal among all the other bidders (which is the opponent in two bidder cases). I will follow the specification of Kagel et al. (1995) to demonstrate my point. Assume that the real value of the object, V, is uniformly distributed over:

V ~ U [α ⋅ KBB − r ,α ⋅ KBB + r ] where 0 < α < 1 , r < α ⋅ KBB , and KBB is the Kelly Blue Book presale estimate value. In addition, assume that given V, each of the two bidders received an unbiased signal, xi, that is distributed uniformly such that: xi | V ~ U [V − ε , V + ε ] We will assume, for tractability, that ε = l ⋅ r and 0 k > V − ε ⎪ ⇒ f min {x1 , x2 }|V (min{x1 , x 2 } = k ) = ⎨ 2ε 2 ⎪⎩0 otherwise

2

And we can calculate the unconditional selling price and find: 8 ⇒ E{min (Β( xi , xi ))} = αKBB −

lr l 2 r + 3 6

It seems like a reasonable strategy to jump in the first bid to the lower bound of the support: α ⋅ KBB − r . This will be the biggest jump possible without revealing any information to the other bidders and still guaranteeing avoiding the winner’s curse. 9 Then, on average, the range between the winning bid and the first bid under our simplified assumption has an average of: lr l 2 r + − α ⋅ KBB + r 3 6 ⎛ l l2 ⎞ = r ⎜⎜1 − + ⎟⎟ = constant ⋅ r ⎝ 3 6⎠

Ε(Winning Bid − First Bid ) = α ⋅ KBB −

So Var (V ) ≡ σ 2 =

r2 ⇒ r = 3σ 2 and we can try and estimate: 3

Winning bid-First bid= constant ⋅ 3σ

2

or (Winning bid-First bid)2=constant·σ2

Three points should be noted. First, this is only an illustrative example and not supposed to capture the exact dynamics in an oral auction. It is an example that predicts that in an oral English auction the variability of the unknown value of the item is related to the difference between the First Jump and the winning bid. As mentioned above, a tractable model for this auction structure is not available yet. Second, the simple model is also valid when r is different across items. We just have to add the subscript j for the model parameters. In addition, the model is still valid when r is also

8

See derivation in appendix. Levin et al. (1996) design their laboratory experiment for an English common value auction such that it uses an ascending clock with a starting price of the lower bound of the support and increases continuously.

9

10

a function of the presale estimate, say: r = β ⋅ KBB , and β < α . In this sense, we would expect higher variability for items with higher presale estimates. It is likely that a car with a presale value of $300 has less variability in its true valuation than a car with a presale value of $3000. When we try to estimate the variance of the valuation, and test its effect on the other variables, we need to control for the size of the presale estimate. Third, the model is not identified, even though we know the relationship between the parameters. Two options have been considered. First, add another restriction (more structure to the model). Second, observe that the variance is proportionally linear 10 to the difference between the winning bid and the first bid squared and use that as a measure. I will follow the latter course and define five new variables that, under the model assumptions, should be highly positively correlated with the true variance of the item value:

sigma1 = (Winning bid - First bid) sigma2 = ( Winning bid - First bid) 2 Winning bid - First bid sigma3 = First bid Winning bid - First bid sigma4 = Winning bid Winning bid - First bid sigma5 = Estimator As a first examination I draw five graphs (Figures 4-8) of the relationship between the Number of bids variable and the suggested variance variables. On each graph, the x-axis measures the number of bids required to sell the item, and one of the suggested measures of the variance is on the y-axis. We can see from these graphs that there is a positive correlation between the Number of bids and the suggested variance variables. An increase in the estimated variance leads to an increase in the Number of bids.

10

We know that, under the model assumptions, the constant is between 219

64

and 3.

11

Tables 2 and 3 report the results of negative binomial regressions of the number of bids each item received until it was sold on several covariates. Each regression includes the following variables: intercept, the order the item was introduced in the auction divided by the total number of items, the number of years the car has been used, the presale estimate, the mileage as it appears on the odometer divided by 10,000, a dummy variable for the different auction dates, and a variable that measures the variance as defined above. For example, in the first regression, I use the difference between the winning bid and the First Jump as the variance variable. The same regression model is used in Table 3, but I add the First Jump as an explanatory variable. In both specifications, the Year and Mileage variables are negative and significant. An increase in Year or Mileage leads to a decline in the expected Number of bids. The presale estimate in all the specifications except Sigma1 is positive and significant, which means that an increase in the presale estimate will, on average, increase the expected number of bids the item receives. In the Sigma1 specification, it is negative and significant in the specification without the First Jump variable.

In all the specifications, the variance term is positive and significant,

suggesting that an increase in the variability leads to an increase in the required number of bids to sell the item. This is the case also when I control for the First Jump variable. The First Jump variable is negative and significant in most specifications, suggesting that an increase in the First Jump will, on average, reduce the expected number of bids. The reason it has positive effect in Sigma4 is because of the negative correlation between Sigma4 and First Jump. The choice of negative binomial is supported by the good deviance measure. The positive dispersion measure rejects the assumptions underlying the Poisson regression. These results support our predictions: ∂N ∂σ 2

>0 pˆ

∂N ∂σ 2

> 0 In addition, I regress the Number of bids while controlling Fb , pˆ

for the variance, winning bid, and First Jump. I do not report these results here because there might be an endogeneity problem with these procedures, although in almost all the regressions, the variance has a positive and significant effect on the expected number of bids, as predicted.

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The effect of an increase in the value’s variability on the First Jump and the winning bid is not reported here either. The variability, as defined above, affects the First Jump negatively by definition. An increase in the variability will reduce the First Jump, on average. If all the variance measures are a function of the gap between the first and winning bids when we run a regression, a negative coefficient must result for this variable. There is an endogeneity problem in running these regressions, which is why the results are not reported here. The same is true for the winning bid regression with regard to the endogeneity problem. In these regressions, however, the effect of the variance on the winning bid will be positive because the variance is a function of the gap between the winning bid and the first bid by construction. The second variable of interest is Ratio, the ratio between the presale estimate and the selling price. Many things can affect this variable - among them: the order the item is sold, measurement errors etc. In addition to these, I claim that the uncertainty related to the unknown valuation of the good has a negative effect on this variable. A reduction in the uncertainty will, on average, increase this ratio. For two items with the same presale estimate but with different uncertainty (or as we model it before, with different variances of the valuation) we would expect that the item with the low level of uncertainty will have a higher Ratio. If, for example, an old Mercedes and a new Hyundai have the same presale estimate, but the Hyundai is new with less mileage on the odometer, we would expect that the ratio between the selling price and presale estimate of the Hyundai will be higher. In table 4 I demonstrate this idea empirically. The dependent variable is Ratio and the explanatory variables are: the order the item was introduced in the auction divided by the total number of items, the number of years the car has been used, the presale estimate, the mileage as it appears on the odometer divided by 10,000, dummy variables for the different auction dates. Since I don’t have a direct measure of the variability, and since I argued earlier that an increase in the number of rounds required to sell an item is associated with higher variability of the unknown value, I will use the Number of bids, along with the Year and Mileage, as measures for the level of uncertainty. The problem is that this variable is highly correlated with the winning bid.

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In order to identify the effect of the change in the Number of bids and the effect of the presale estimate behind the sum of individual effects, I interact them. In column 1 we can see that the selling order has positive effect on Ratio. Unlike many other studies, in this data each item sold increases on average the next item’s selling price. 11 Year, Mileage and Poor condition each has a negative effect on the Ratio variable. The reason for this might be related to the unknown variability of the item as mentioned above. We expect the uncertainty to increase when the number of years the car has been used and the mileage increase. In addition a car in poor condition is associated with more uncertainty. In column 2 I add the presale estimate as a covariate. One reason for doing this is that the variance depends on the presale estimate as I mention above. We would expect a car with a $300 presale estimate to have less variability in the unknown valuation than a car with $3000 presale estimate. The statistically significant negative effect of the presale estimate on Ratio supports the claim that increases in variability are associated with a reduction in Ratio. Another reason why this variable might be negative is because of the definition of Ratio, which is the ratio between the selling price and the presale estimate. In column 3 I add the number of bids it required to sell an item as a covariate. This variable is positive and significant since an increase in this variable is associated with increase in the selling price. It is also associated with the variability but only as a second order affect and we have to control for the presale estimate to get that effect. In order to check the effect of a unit increase in the Number of bids and the presale estimate beyond the sum of individual effects I interact them in column 4. I find that the interaction between the presale estimate and Number of bids is negative. This suggests that if my previous assumptions are correct, an increase in the uncertainty is associated with a decline in the ratio between the selling price and the presale estimate. In column 5 and 6 I use an alternative measure to check the effect of uncertainty on Ratio. I construct the following variable: I take the difference between the winning bid and the first bid and divide by 25 (the minimum bid increment permitted). This is the number of rounds it would require to sell an item if the auction 11

For further information see Raviv 2004 a.

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progressed by the minimum bid increment allowed. This would be the reasonable strategy when uncertainty is high and there is no time cost. Then I subtract the number of bids it actually takes to sell the item, less one. In doing so, I create a measure that is negatively correlated with uncertainty. An increase in this variable is associated with a reduction in uncertainty and I would expect it will have a positive effect on Ratio. As we can see from column 5 and 6, it is indeed a positive and significant effect with or without the presale estimate. All the previous covariates that are presumably associated with the level of uncertainty, Mileage, Poor Condition, and the presale estimate, have significant negative signs as well, although Year is statistically significant only when the presale estimate is used as a covariate.

4. Conclusions Standard auction models describe the English oral auction as a clock auction in which an auctioneer raises the price continuously and each bidder chooses when to drop out. This description, however, does not consider jump bidding and therefore ignores the question of the length of an auction and the parameters that affect it. On the other hand, research has shown both, theoretically and experimentally, that an increase in the variance of bidders’ signals will reduce the expected selling price. In this paper, the role of uncertainty in sequential English oral auctions was empirically examined using a car auction data set I collected from 2001 to 2002 from New Jersey DSS in Trenton. I did this by studying the effect of uncertainty on the number of bids it required to sell an item and on the ratio between the selling price and the presale estimate. In this scenario, the level of uncertainty is assumed to be correlated with the variance of the distribution from which the bidders’ signals comes. I demonstrated that the variance of the unknown value of the object plays a crucial role in the auction’s progress and outcome. Increased variability is associated with an increase in the number of bids required to sell the item. These findings are in line with the conjecture that, with lower variability in the unknown value of the good, the convergence to the price will be faster. When there is more uncertainty about the object valuation, bidders will hesitate and bid more cautiously. The premise is that it

15

takes bidders more time to transfer information and reveal the estimated value of the object when the variance of the signal is high. In addition, increased variability is associated with a reduction in ratio between the selling price and the presale estimate. I claimed that uncertainty has a negative effect on this variable as well. A reduction in the uncertainty will, on average, increase this ratio. Even if two different cars have the same presale estimate I would expect that the level of uncertainty will be different if they have been used for different number of years and have different mileages on the odometer. Items with a low level of uncertainty will have a higher Ratio. A straightforward extension to this paper would be to experiment with an open English auction. This setup would allow us to check the effect of uncertainty on the ratio between the selling price and presale estimate, and whether the result that uncertainty has a positive effect on the number of rounds required to sell an item is valid, both under the private- and common-valuation paradigms.

16

References [1] Goeree, Jacob and Offerman, Theo. “Efficiency in Auctions with Private and Common Values: An Experimental Study” American Economic Review, 2002, Volume 92, 625-643. [2] Kagel, John H., Levin, Dan, and Harstad, Ronald M. “Comparative Static Effects of Number of Bidders on Behavior in a Second-Price Common Value Auction” International Journal of Game Theory, 1995, Volume 24, 293-319. [3] Kazumori, Eiichiro and McMillan, John. “Selling Online versus Live,” The Journal of Industrial Economics, 2005, Volume 53, 543-569. [4] Krishna, Vijay. “Auction Theory” Academic Press, 2002. [5] Levin, Dan, Kagel, John and Richard, Jean-Francois. “Revenue Effects and Information Processing in English Common Value Auctions” American Economic Review, 1996, Volume 86, Issue 3, 442-460. [6] Milgrom, P. and Weber, R. “A Theory of Auction and Competitive Bidding” Econometrica, 1982, Volume 50, 1089-1122. [7] Pai-Ling, Yin. “Information Dispersion and Auction Prices” Harvard Business School, 2003, Mimeo. [8] Raviv, Yaron (a). “New Evidence on Price Anomalies in Sequential Auctions: Used Cars in New Jersey,” Journal of Business and Economic Statistics, Vol.24, No.3, 2006 (forthcoming). [9] Raviv, Yaron (b). “The Role of the Bidding Process in Price Determination: Jump Biddings in Sequential English Auctions” 2004 Working Paper, Claremont McKenna College. [10] Wilson, Robert B. “Competitive Bidding with Disparate Information” Management Science, 1969, Volume 15, Issue 7, Theory Series, 446-44.

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Appendix We are interested in calculating E (min B( xi , xi )) . First, under the proposed bidding strategy, we know that V − ε ≤ xi ≤ V + ε and we can separate the support into three regions: 1.

if αKBB − r + 2ε ≤ V ≤ αKBB + r − 2ε then B( xi , xi ) = xi

if α ⋅ KBB − r ≤ V ≤ α ⋅ KBB − r + 2ε

2.

⎧ xi + α ⋅ KBB − r + ε if ⎪ then B( xi , xi ) = ⎨ 2 ⎪⎩ xi if

V − ε ≤ xi ≤ α ⋅ KBB − r + ε

α ⋅ KBB − r + ε ≤ xi ≤ V + ε

if α ⋅ KBB + r − 2ε ≤ V ≤ α ⋅ KBB + r

3.

if ⎧ xi ⎪ then B( xi , xi ) = ⎨ xi + α ⋅ KBB + r − ε if ⎪⎩ 2

V − ε ≤ xi ≤ α ⋅ KBB + r − ε

α ⋅ KBB + r − ε ≤ xi ≤ V + ε

Taking the regions into account, we can now calculate the expected value of the winning bid: E (min B (xi , xi )) = αKBB − r + 2ε

+

∫

αKBB − r αKBB + r

+

1 2r

αKBB + r

V +ε

αKBB + r − 2ε

V +ε

V + ε − xi V + ε − xi 1 1 B (xi , xi )dxi dV = x i dx i dV 2 2r V −ε 2ε 2 r V −ε 2ε 2 αKBB − r αKBB − r + 2ε

∫

∫

∫

∫

V +ε ⎡αKBB − r +ε V + ε − x i x i + αKBB − r + ε ⎤ V + ε − xi dx x dx ⋅ + ⎢ ⎥ dV i i i 2 2ε 2 2ε 2 ⎢⎣ V −ε ⎥⎦ αKBB − r + ε

∫

∫

V +ε V +ε ⎤ V + ε − xi xi + αKBB + r − ε 1 ⎡ V + ε − xi + ⋅ x dx dxi ⎥dV ⎢ i i 2 2 2r ⎢⎣V −ε 2ε 2 2ε ⎥⎦ αKBB + r − 2ε αKBB + r −ε

∫

∫

∫

1 1 = αKBB − ε + ε 2 3 6r 1 l 2r = αKBB − lr + 3 6

18

Table 1: Summary Statistics Presale Estimate

Price

Price/Estimate

Number of Bids

Mean

2662.19 (1634.84)

1520.42 (1168.18)

0.561 (0.236)

11.39 (7.13)

Minimum

318.75

50

0.051

1

Maximum

15265

8650

1.787

49

Observations

678

683

678

641

Standard errors are in parentheses.

Figure 1: Empirical Distribution of Number of Bids 60

50

Frequency

40

30

20

10

0 1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

Number of Bids

19

0

.5

D en s ity 1

1 .5

2

Figure 2: Kernel Density and Histogram of Selling Price/Estimate Ratio

0

.5

1 ratio

1.5

2

Selling Price

Figure 3: Different Jump Bidding Options for the same Selling Price

0

1

2

3

4

5

6

7

8

9

10

Number of Bids

20

49

36

31

28

25

22

19

16

13

7

10

4

7000 6000 5000 4000 3000 2000 1000 0 1

bidmfirst

Figure 4: Relation between Number of Bids and Sigma1

Figure 5: Relation between Number of Bids and Sigma2

bidmfirstsq

50000000 40000000 30000000 20000000 10000000

49

36

31

28

25

22

19

16

13

10

7

4

1

0

Figure 6: Relation between Number of Bids and Sigma3 60

40 30 20 10

49

36

31

28

25

22

19

16

13

10

7

4

0 1

bidmfof

50

21

49

36

31

28

25

22

19

16

13

7

10

4

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

bidmfow

Figure 7: Relation between Number of Bids and Sigma4

Figure 8: Relation between Number of Bids and Sigma5

0.8 0.6 0.4 0.2

49

36

31

28

25

22

19

16

13

7

10

4

0

1

bidmfokbb

1

22

Table 2: Negative Binomial Regression Results for Number of Bids Sigma1

Sigma2

Sigma3

Sigma4

Sigma5

Intercept

2.0074* (0.3320)

2.6391* (0.3394)

2.3775* (0.4166)

1.4942* (0.3525)

1.9070* (0.3482)

Order

0.0627 (0.0530)

0.1203** (0.0640)

0.1129** (0.0667)

0.0435 (0.0561)

0.0226 (0.0560)

Year

-0.0230* (0.0058)

-0.0353* (0.0069)

-0.0431* (0.0075)

-0.0625* (0.0062)

-0.0457* (0.0061)

Mileage

-0.0085** (0.0039)

-0.0163* (0.0046)

-0.0156* (0.0047)

-0.0194* (0.0040)

-0.0081* (0.0040)

Estimator

-0.0000* (0.0000)

0.0000* (0.0000)

0.0002* (0.0000)

0.0001* (0.0000)

0.0001* (0.0000)

Variance

0.0006* (0.0000)

0.0000* (0.0000)

0.0091* (0.0012)

1.6525* (0.0845)

1.7628* (0.0964)

Dispersion

0.0468 (0.0067)

0.1129 (0.0113)

0.1295 (0.0123)

0.0625 (0.0082)

0.0614 (0.0078)

Deviance

0.9592

1.0256

1.0226

0.9718

0.9665

Log Likelihood

11297.77

11158.68

11135.91

11261.13

11265.07

Observations

636

636

636

636

636

Standard errors are in parentheses. Mileage is the actual mileage divided by 10,000. Order is the order in which the object was introduced divided by total number of items. * Significant at 1% level. **Significant at 5% level. All the regressions include auction dummies fixed effects.

23

Table 3: Negative Binomial Regression Results for Number of Bids Sigma1

Sigma2

Sigma3

Sigma4

Sigma5

Intercept

1.9416* (0.3294)

2.5648* (0.3936)

2.3603* (0.4137)

1.3567* (0.3470)

1.8085* (0.3405)

Order

0.0712 (0.0528)

0.1275** (0.0632)

0.1190** (0.0665)

-0.0046 (0.0550)

0.0341 (0.0549)

Year

-0.0285* (0.0060)

-0.0432* (0.0071)

-0.0465* (0.0076)

-0.0528* (0.0062)

-0.0543* (0.0062)

Mileage

-0.0113* (0.0040)

-0.0200* (0.0047)

-0.0177* (0.0048)

-0.0128* (0.0040)

-0.0123* (0.0041)

Estimator

-0.0000 (0.0000)

0.0001* (0.0000)

0.0002* (0.0000)

0.0001* (0.0000)

0.0002* (0.0000)

First Bid

-0.0001* (0.0000)

-0.0002* (0.0000)

-0.0001*** (0.0000)

0.0004* (0.0000)

-0.0002* (0.0000)

Variance

0.0006* (0.0000)

0.0000* (0.0000)

0.0084* (0.0013)

2.1141* (0.1020)

1.7331* (0.0928)

Dispersion

0.0452 (0.0066)

0.1080 (0.0110)

0.1273 (0.0122)

0.0545 (0.0073)

0.0554 (0.0075)

0.9539

1.0266

1.0284

0.9246

0.9664

11303.56

11166.26

11137.68

11292.44

11277.63

636

636

636

636

636

Deviance Log Likelihood Observations

Standard errors are in parentheses. Mileage is the actual mileage divided by 10,000. * Significant at the 1% level. **Significant at the 5% level. ***Significant at the 10% level. All the regressions include auction dummies fixed effects.

24

25

26

Yaron Raviv 1 April 15, 2006 Abstract. I study the sequence of bidding in an open-outcry English auction to examine how uncertainty affects auction outcomes. I do this by analyzing a data set collected from a series of public auctions of used cars in New Jersey. I conjecture that the uncertainty is related to the auction’s progress and demonstrate that, empirically, an increase in the uncertainty is associated with an increase in the number of rounds required to sell an object and with a reduction in the ratio between the selling price and the presale estimate.

JEL Classification: D44 Keywords: Auction

1

Department of Economics, Claremont McKenna College, Claremont CA 91711. I am greatly indebted to Orley Ashenfelter and Han Hong. I would also like to thank Richard Burdekin, Ken Fortson, Tanjim Hossain, Gad Levanon, Andrea Podhorsky, Ryan Quillian, Jesse Rothstein, Joydeep Roy, Gabor Virag and participants in seminars at Princeton University, Tel-Aviv University, Claremont McKenna College and Ben-Gurion University. Any errors are mine.

1. Introduction Standard auction theory predicts that the level of uncertainty in an auction has a real effect on the auction outcome. In this sense, research demonstrates, both theoretically and experimentally, that an increase in the variance of bidders’ signals will reduce the expected selling price. 2 But there are no predictions as to the effect of uncertainty on the number of rounds it requires to sell an item. In this paper, I study the sequence of bidding in an open-outcry English auction to examine how the level of uncertainty affects auction outcomes. I focus on the number of bids required to sell an item and the ratio between the selling price and the presale estimate. I exploit an original data set collected during 2001 and 2002 from a series of public auctions of used cars conducted by the state of New Jersey. Although a tractable model of the strategic behavior of the parties in an openoutcry English auction is not available, I conjecture that the auction’s progress is related to the variability of valuations, and demonstrate that, empirically, an increase in that variability is associated with an increase in the number of bids required to sell an object. These findings are in line with the conjecture that, with lower variability in the unknown value of the good, the convergence to the price will be faster. I claim 2

In their seminal paper, Milgrom and Weber (1982) demonstrate that, when the seller publicly reveals some credible information about the valuation of the object, revenues increase, on average. In other words, a reduction in the variance of the true value of the object has a positive effect on the expected selling price. Wilson (1969) provides an example of a second price sealed-bid auction with two bidders who have a common prior of a diffuse Normal density and signals xi that are both normally distributed with mean V. In that case, the optimal strategy function is xi − σ π 2 where σ is the posterior marginal density of his opponent. An increase in the variance will reduce the selling price. Pai-Ling Yin (2003) conducted an empirical investigation of eBay auctions for computers. She found that the winner’s curse changes with the dispersion of information. She estimated the predicted common value of the computers using a survey, and used the survey to build a measure that is correlated with the mean and dispersion of the bidder’s signals. She found that the price declines with dispersion. McMillan and Eiichiro (2005) provide a model in which the factor that most influences the mechanism the seller will chose to conduct an auction, online or offline, is not the expected price, but the valuation uncertainty. They prove, again, that when bidders have a more precise estimator of the common value, they will bid more aggressively; hence the expected selling price is decreases in the variance. This is the case for both mechanisms. Kagel et al. (1995, 1996) also demonstrate the same thing; increases in variability reduce the expected selling prices. Goeree and Offerman (2002) provide a model combining private and common values in a first-price auction. They experiment and test the model’s predictions and find that increased competition and reduced uncertainty about the common value positively affect revenue and efficiency.

2

that, ceteris paribus, increases in uncertainty, both under the private values and the common values paradigms, are associated with an increase in the number of bids required to sell the item. Under the private values paradigm, for example, in the case of perfect certainty (where everybody knows everybody else’s values), there will be an immediate convergence to the second highest valuation. In the other extreme case, in which each player knows only his valuation and does not have any information about the distribution of the valuations, we would expect the auction to progress by the minimum bid increment required to advance the auction when the time cost is zero. In this case the expected number of bids will be the highest. In the common values case, we can model the uncertainty as the variance of the distribution from which each bidder’s signal comes. When the variance is small enough, bidders will immediately jump to the bidding option just below the object’s valuation. When there is more uncertainty about the object valuation, bidders will hesitate and bid more cautiously. The premise is that it takes bidders more time to transfer information and reveal the estimated value of the object when the variance of the signal is high. The second variable of interest is the ratio of the selling price to the presale estimate (Ratio). Many things can affect this variable - among them the order in which the item is sold, measurement errors, etc. In addition to these, I claim that the uncertainty related to the unknown valuation of the good has a negative effect on the selling price and the presale estimate ratio. A reduction in the uncertainty will, on average, increase this ratio. For two items with the same presale estimate but with different uncertainty (or as I model it, with different variances of the valuation) we would expect that the item with the lower level of uncertainty will have a higher Ratio. If, for example, an old Mercedes and a new Hyundai have the same presale estimate, but the Hyundai is new with less mileage on the odometer, we would expect that the ratio of the selling price to the presale estimate of the Hyundai will be higher. The paper is organized as follows. In the next section I describe the data I have collected and the nature of the auctions. In Section 3 I analyze the effect of uncertainty, measured as the variability of the bidders’ valuations, on the two variables

3

of interest - the ratio between the selling price and the presale estimate, and the number of rounds it take to sell an item. The final section offers concluding remarks.

2. The Data I collected the auction data in 2001-02 from the New Jersey Distribution and Support Services (DSS) in Trenton, New Jersey. 3 DSS sells surplus personal and government property through public oral English auctions and sealed-bid auctions. The open English oral auctions of cars are usually held on Saturdays once a month. Bidders can physically inspect the items the day before the auction and on the day of the auction until 9

A.M.,

when the auction begins. Each car that is auctioned is driven

through a large warehouse and stopped in front of the auctioneer, and then the bidding process begins. After the car is sold, it is driven to the parking lot, and a new car is auctioned off. The average time required to sell a car is between 1 and 2 minutes. Bids on operable vehicle units are only accepted in multiples of $25. At the time of sale, successful bidders are required to make a deposit in cash, bank money order, or certified check for $150 or 10% of the total amount of the bid, whichever is greater. If the high bidder fails to place the deposit, the vehicle is immediately resold. The DSS reveals all information available about the car’s condition such as model, year, mileage and the source of the vehicle (Turnpike Authority, criminal justice seizure, Transportation Department, taxation seizure, etc.). The state also reveals all the mechanical information known about the vehicle’s condition, for example whether it has bad transmission, bent rear axle, no vehicle identification number plate on the door, no power steering, etc. The coordinator of operations at DSS has stated that all the information known about the vehicles is made available to the bidders and that the cars are auctioned in random order (which I verified

3

For further information, see Raviv (2004, a).

4

empirically in Raviv (2004, a)), so that there is no correlation between a cars’ presale value and the sequence in which it is auctioned off. The day before each auction, I collected data on each vehicle’s condition. On the same day, I gathered the Kelly Blue Book (KBB) estimated market value of each car. KBB is a company that, among other things, provides market value estimates for cars on its Website. On the day of the auction, I collected the following data: the sequence in which the vehicles were auctioned, all the bids that each car received up to (and including) the winning bid, and data about the resold cars. During the week after each auction, I collected the official list of winning bids from DSS to compare with my notes. Table 1 gives summary statistics from the different auctions. In the first and second columns, the presale estimate (the estimated presale market value from KBB) and the price for which the item was sold are reported. The mean of the presale estimate was $2,662.19, and it was above the mean of the winning bids, which was $1,520.42. It appears that some of the cars were sold quite cheaply. Some bidders bought operable cars for as little as $50. The car with the highest presale value ($15,265) was a 1986 Porsche with 77,000 miles on its odometer. This car was eventually sold for $3,400. Although the governor of New Jersey’s car, a 1998 Buick Ultra, had a presale estimate of $11,270, it was sold for $8,650 and was the most expensive item sold in the auctions. The mean Number of bids is the average number of bids each item received before it was sold. The mean of this variable is 11.39, which indicates that it took, on average, 11.39 rounds for an item to be sold. The minimum of this variable is 1, which means that some of the cars were won by the first bidder. The Ratio variable is defined as the ratio between the selling price and the presale estimate. In order to get a better understanding of the main variables of interest of this paper, the number of rounds it takes to sell an item and the ratio between the selling price and the presale estimate, Figures 1 and 2 describe the empirical distribution of these variables. Figure 1 reports the empirical distribution of the Number of bids. Some of the cars were sold after only a few rounds, whereas it took longer to sell other cars. For

5

example, 20 cars were sold to the first bidder, but 10 other cars took 25 rounds each. I claim that, ceteris paribus, increases in the uncertainty, both under the private values and the common values paradigms, is associated with an increase in the number of bids required to sell the item. Under the private values, for example, in the case of perfect certainty (where everybody knows everybody else’s values), there will be an immediate convergence to the second highest valuation. In the other extreme case, in which each player knows only his valuation and does not have any information about the distribution, and there is no time cost, we expect the auction to progress by the minimum bid increments required to advance the auction. In this case the expected Number of bids will be the highest. In the common values case, we can model the uncertainty as the variance of the distribution each bidder’s signal came from. When the variance is small enough, bidders will immediately jump to the bidding option immediately below the object’s valuation. When there is more uncertainty about the object valuation, bidders will hesitate and bid more cautiously. The premise is that it takes bidders more time to transfer information and reveal the estimated value of the object when the variance of the signal is high. Figure 2 is a histogram with an overlay of the estimated kernel density of the ratio between the selling price and the presale estimate (Ratio). The question that arises is, “Why is this shape observed?” Why did some of the items sell for only 5% of their presale estimate, whereas some of the items sold for more than the presale estimate? The reason for the pattern of this variable is also related to the level of uncertainty. If there is a common component to a car’s value, then the variability of the common value of the object affects the selling price. When the variance of the true value of the object decreases, we would expect the ratio between the selling price and the presale estimate to increase. If the variance increases, we would expect this ratio to decrease.

6

3. Variability of the Common Values and the Auction Progress In Raviv (2004, b), I concluded that my data can be characterized as coming from a common value auction.4 In this section, I will explain the pattern of the number of bids an item receives and the ratio between the selling price and the presale estimate. I will relate the outcome of these variables to the level of uncertainty. The way I measure uncertainty is by the variances of the valuation of the object and the signal each bidder received. Under the assumption of common values, the item value is the same for everyone, so bidders only differ in the unbiased signals of the item value they receive. In Figure 3, some effects of the variance on the Number of bids are demonstrated. The figure describes five different options for the pattern of jump biddings for an object that eventually sold for the same price. The item could sell in the first bid, the second bid, or even after 10 bids. For two items that sell for the same price and have the same presale estimate, we predict that, on average, the one that receives more bids has a higher variability of the unknown value. On the other hand, for two items with the same presale estimate, we cannot predict generally which will receive more bids, because the increase in the variability affects the first bid and the selling price in the same direction. 5 The premise is that, with low-enough variability, we would expect immediate convergence to the common value by placing the nearest possible bid from below. Also, as mentioned before, even with the same presale estimate, items can have different levels of uncertainty. The second conjecture is that a higher variability of the object for a given selling price and presale estimate is associated with a lower first bid. If we denote N as the Number of bids, FB as the first bid, σ 2 as the value variance, p * as the selling price and pˆ as the presale estimate, we would expect the following conditions: ∂N ∂σ 2

≥0 p*

∂N ∂σ 2

≥0 Fb , p *

∂N ∂σ 2

≥0 pˆ

∂N ∂σ 2

≥0 Fb , pˆ

∂p * ∂σ 2

≤0 pˆ

∂Fb ∂σ 2

≤0 p*

∂Fb ∂σ 2

≤0 pˆ

4

I could not reject the hypothesis that the data can be characterized as a private values data though. In the case of 0 variance, there will be an immediate convergence to the value, and the number of bids is the lowest in this case. However, it is not clear theoretically, of two items with the same presale estimate but with different variances, which will receive more bids.

5

7

We expect that the Number of bids will increase with the variance when we control for the presale estimate, the selling price, and the first bid. In general, we cannot determine theoretically the expected effect of the variance on the Number of bids because an increase in the variance leads, on average, to a reduction in the expected selling price and in the expected first bid. 6 Without any restrictions on these variables, we cannot determine how an increase in the variance affects the Number of bids. Because of this, we expect the first two conditions to hold theoretically, whereas the effect of the next two conditions cannot be precisely predicted theoretically, subsequent empirical testing reveals that these conditions hold as well, however. The next condition is the prediction by Milgrom and Weber (1982). We expect the selling price to decline with the variability of the object after we control for the presale estimate. In their seminal paper, Milgrom and Weber (1982) demonstrate that, when the seller publicly reveals some credible information about the valuation of the object, revenues increase, on average. In other words, a reduction in the variance of the true value of the object has a positive effect on the expected selling price. In addition, we would expect the first bid to decline with the variability of the valuation after controlling for the selling price and the presale estimate. The only problem left is how to estimate the variance of the valuation and examine our predictions. Although there is not yet any tractable model of the strategic behavior of the parties in an open-outcry English auction, I conjecture that the variability of valuations is related to the auction’s progress. In this sense, I predict that the winning bid is affected by the value variance, and that the agents will jump to the lower bound of the signal’s support in the first round. Then the difference between the winning bid and the first jump would be correlated with the value variability. A simple example is as follows. It is known that the equivalence between an English auction and a Second Price sealed bid auction disappears as a consequence of common values. This occurs because there is information revelation in an English auction that does not exist in a

6

When the variance of the valuation is related to the lower bound of the distribution, like in the uniform case when the mean is constant.

8

Second Price sealed-bid auction. However, to demonstrate my point, I will assume that, instead of an oral English auction, we have a Second Price sealed-bid auction. As a witness to the auctions, it seems to me that usually two bidders compete with each other over an item. 7 When there are only two bidders, the regular English and Second Price sealed-bid auctions are equivalent. If we denote V as the unknown value of the object that is drawn from a known distribution and x as a signal drawn from a known distribution X, then the symmetric equilibrium strategies in a Second Price auction are given by: B(x,x)=E[V|X=x,Y=x], where Y is the highest signal among all the other bidders (which is the opponent in two bidder cases). I will follow the specification of Kagel et al. (1995) to demonstrate my point. Assume that the real value of the object, V, is uniformly distributed over:

V ~ U [α ⋅ KBB − r ,α ⋅ KBB + r ] where 0 < α < 1 , r < α ⋅ KBB , and KBB is the Kelly Blue Book presale estimate value. In addition, assume that given V, each of the two bidders received an unbiased signal, xi, that is distributed uniformly such that: xi | V ~ U [V − ε , V + ε ] We will assume, for tractability, that ε = l ⋅ r and 0 k > V − ε ⎪ ⇒ f min {x1 , x2 }|V (min{x1 , x 2 } = k ) = ⎨ 2ε 2 ⎪⎩0 otherwise

2

And we can calculate the unconditional selling price and find: 8 ⇒ E{min (Β( xi , xi ))} = αKBB −

lr l 2 r + 3 6

It seems like a reasonable strategy to jump in the first bid to the lower bound of the support: α ⋅ KBB − r . This will be the biggest jump possible without revealing any information to the other bidders and still guaranteeing avoiding the winner’s curse. 9 Then, on average, the range between the winning bid and the first bid under our simplified assumption has an average of: lr l 2 r + − α ⋅ KBB + r 3 6 ⎛ l l2 ⎞ = r ⎜⎜1 − + ⎟⎟ = constant ⋅ r ⎝ 3 6⎠

Ε(Winning Bid − First Bid ) = α ⋅ KBB −

So Var (V ) ≡ σ 2 =

r2 ⇒ r = 3σ 2 and we can try and estimate: 3

Winning bid-First bid= constant ⋅ 3σ

2

or (Winning bid-First bid)2=constant·σ2

Three points should be noted. First, this is only an illustrative example and not supposed to capture the exact dynamics in an oral auction. It is an example that predicts that in an oral English auction the variability of the unknown value of the item is related to the difference between the First Jump and the winning bid. As mentioned above, a tractable model for this auction structure is not available yet. Second, the simple model is also valid when r is different across items. We just have to add the subscript j for the model parameters. In addition, the model is still valid when r is also

8

See derivation in appendix. Levin et al. (1996) design their laboratory experiment for an English common value auction such that it uses an ascending clock with a starting price of the lower bound of the support and increases continuously.

9

10

a function of the presale estimate, say: r = β ⋅ KBB , and β < α . In this sense, we would expect higher variability for items with higher presale estimates. It is likely that a car with a presale value of $300 has less variability in its true valuation than a car with a presale value of $3000. When we try to estimate the variance of the valuation, and test its effect on the other variables, we need to control for the size of the presale estimate. Third, the model is not identified, even though we know the relationship between the parameters. Two options have been considered. First, add another restriction (more structure to the model). Second, observe that the variance is proportionally linear 10 to the difference between the winning bid and the first bid squared and use that as a measure. I will follow the latter course and define five new variables that, under the model assumptions, should be highly positively correlated with the true variance of the item value:

sigma1 = (Winning bid - First bid) sigma2 = ( Winning bid - First bid) 2 Winning bid - First bid sigma3 = First bid Winning bid - First bid sigma4 = Winning bid Winning bid - First bid sigma5 = Estimator As a first examination I draw five graphs (Figures 4-8) of the relationship between the Number of bids variable and the suggested variance variables. On each graph, the x-axis measures the number of bids required to sell the item, and one of the suggested measures of the variance is on the y-axis. We can see from these graphs that there is a positive correlation between the Number of bids and the suggested variance variables. An increase in the estimated variance leads to an increase in the Number of bids.

10

We know that, under the model assumptions, the constant is between 219

64

and 3.

11

Tables 2 and 3 report the results of negative binomial regressions of the number of bids each item received until it was sold on several covariates. Each regression includes the following variables: intercept, the order the item was introduced in the auction divided by the total number of items, the number of years the car has been used, the presale estimate, the mileage as it appears on the odometer divided by 10,000, a dummy variable for the different auction dates, and a variable that measures the variance as defined above. For example, in the first regression, I use the difference between the winning bid and the First Jump as the variance variable. The same regression model is used in Table 3, but I add the First Jump as an explanatory variable. In both specifications, the Year and Mileage variables are negative and significant. An increase in Year or Mileage leads to a decline in the expected Number of bids. The presale estimate in all the specifications except Sigma1 is positive and significant, which means that an increase in the presale estimate will, on average, increase the expected number of bids the item receives. In the Sigma1 specification, it is negative and significant in the specification without the First Jump variable.

In all the specifications, the variance term is positive and significant,

suggesting that an increase in the variability leads to an increase in the required number of bids to sell the item. This is the case also when I control for the First Jump variable. The First Jump variable is negative and significant in most specifications, suggesting that an increase in the First Jump will, on average, reduce the expected number of bids. The reason it has positive effect in Sigma4 is because of the negative correlation between Sigma4 and First Jump. The choice of negative binomial is supported by the good deviance measure. The positive dispersion measure rejects the assumptions underlying the Poisson regression. These results support our predictions: ∂N ∂σ 2

>0 pˆ

∂N ∂σ 2

> 0 In addition, I regress the Number of bids while controlling Fb , pˆ

for the variance, winning bid, and First Jump. I do not report these results here because there might be an endogeneity problem with these procedures, although in almost all the regressions, the variance has a positive and significant effect on the expected number of bids, as predicted.

12

The effect of an increase in the value’s variability on the First Jump and the winning bid is not reported here either. The variability, as defined above, affects the First Jump negatively by definition. An increase in the variability will reduce the First Jump, on average. If all the variance measures are a function of the gap between the first and winning bids when we run a regression, a negative coefficient must result for this variable. There is an endogeneity problem in running these regressions, which is why the results are not reported here. The same is true for the winning bid regression with regard to the endogeneity problem. In these regressions, however, the effect of the variance on the winning bid will be positive because the variance is a function of the gap between the winning bid and the first bid by construction. The second variable of interest is Ratio, the ratio between the presale estimate and the selling price. Many things can affect this variable - among them: the order the item is sold, measurement errors etc. In addition to these, I claim that the uncertainty related to the unknown valuation of the good has a negative effect on this variable. A reduction in the uncertainty will, on average, increase this ratio. For two items with the same presale estimate but with different uncertainty (or as we model it before, with different variances of the valuation) we would expect that the item with the low level of uncertainty will have a higher Ratio. If, for example, an old Mercedes and a new Hyundai have the same presale estimate, but the Hyundai is new with less mileage on the odometer, we would expect that the ratio between the selling price and presale estimate of the Hyundai will be higher. In table 4 I demonstrate this idea empirically. The dependent variable is Ratio and the explanatory variables are: the order the item was introduced in the auction divided by the total number of items, the number of years the car has been used, the presale estimate, the mileage as it appears on the odometer divided by 10,000, dummy variables for the different auction dates. Since I don’t have a direct measure of the variability, and since I argued earlier that an increase in the number of rounds required to sell an item is associated with higher variability of the unknown value, I will use the Number of bids, along with the Year and Mileage, as measures for the level of uncertainty. The problem is that this variable is highly correlated with the winning bid.

13

In order to identify the effect of the change in the Number of bids and the effect of the presale estimate behind the sum of individual effects, I interact them. In column 1 we can see that the selling order has positive effect on Ratio. Unlike many other studies, in this data each item sold increases on average the next item’s selling price. 11 Year, Mileage and Poor condition each has a negative effect on the Ratio variable. The reason for this might be related to the unknown variability of the item as mentioned above. We expect the uncertainty to increase when the number of years the car has been used and the mileage increase. In addition a car in poor condition is associated with more uncertainty. In column 2 I add the presale estimate as a covariate. One reason for doing this is that the variance depends on the presale estimate as I mention above. We would expect a car with a $300 presale estimate to have less variability in the unknown valuation than a car with $3000 presale estimate. The statistically significant negative effect of the presale estimate on Ratio supports the claim that increases in variability are associated with a reduction in Ratio. Another reason why this variable might be negative is because of the definition of Ratio, which is the ratio between the selling price and the presale estimate. In column 3 I add the number of bids it required to sell an item as a covariate. This variable is positive and significant since an increase in this variable is associated with increase in the selling price. It is also associated with the variability but only as a second order affect and we have to control for the presale estimate to get that effect. In order to check the effect of a unit increase in the Number of bids and the presale estimate beyond the sum of individual effects I interact them in column 4. I find that the interaction between the presale estimate and Number of bids is negative. This suggests that if my previous assumptions are correct, an increase in the uncertainty is associated with a decline in the ratio between the selling price and the presale estimate. In column 5 and 6 I use an alternative measure to check the effect of uncertainty on Ratio. I construct the following variable: I take the difference between the winning bid and the first bid and divide by 25 (the minimum bid increment permitted). This is the number of rounds it would require to sell an item if the auction 11

For further information see Raviv 2004 a.

14

progressed by the minimum bid increment allowed. This would be the reasonable strategy when uncertainty is high and there is no time cost. Then I subtract the number of bids it actually takes to sell the item, less one. In doing so, I create a measure that is negatively correlated with uncertainty. An increase in this variable is associated with a reduction in uncertainty and I would expect it will have a positive effect on Ratio. As we can see from column 5 and 6, it is indeed a positive and significant effect with or without the presale estimate. All the previous covariates that are presumably associated with the level of uncertainty, Mileage, Poor Condition, and the presale estimate, have significant negative signs as well, although Year is statistically significant only when the presale estimate is used as a covariate.

4. Conclusions Standard auction models describe the English oral auction as a clock auction in which an auctioneer raises the price continuously and each bidder chooses when to drop out. This description, however, does not consider jump bidding and therefore ignores the question of the length of an auction and the parameters that affect it. On the other hand, research has shown both, theoretically and experimentally, that an increase in the variance of bidders’ signals will reduce the expected selling price. In this paper, the role of uncertainty in sequential English oral auctions was empirically examined using a car auction data set I collected from 2001 to 2002 from New Jersey DSS in Trenton. I did this by studying the effect of uncertainty on the number of bids it required to sell an item and on the ratio between the selling price and the presale estimate. In this scenario, the level of uncertainty is assumed to be correlated with the variance of the distribution from which the bidders’ signals comes. I demonstrated that the variance of the unknown value of the object plays a crucial role in the auction’s progress and outcome. Increased variability is associated with an increase in the number of bids required to sell the item. These findings are in line with the conjecture that, with lower variability in the unknown value of the good, the convergence to the price will be faster. When there is more uncertainty about the object valuation, bidders will hesitate and bid more cautiously. The premise is that it

15

takes bidders more time to transfer information and reveal the estimated value of the object when the variance of the signal is high. In addition, increased variability is associated with a reduction in ratio between the selling price and the presale estimate. I claimed that uncertainty has a negative effect on this variable as well. A reduction in the uncertainty will, on average, increase this ratio. Even if two different cars have the same presale estimate I would expect that the level of uncertainty will be different if they have been used for different number of years and have different mileages on the odometer. Items with a low level of uncertainty will have a higher Ratio. A straightforward extension to this paper would be to experiment with an open English auction. This setup would allow us to check the effect of uncertainty on the ratio between the selling price and presale estimate, and whether the result that uncertainty has a positive effect on the number of rounds required to sell an item is valid, both under the private- and common-valuation paradigms.

16

References [1] Goeree, Jacob and Offerman, Theo. “Efficiency in Auctions with Private and Common Values: An Experimental Study” American Economic Review, 2002, Volume 92, 625-643. [2] Kagel, John H., Levin, Dan, and Harstad, Ronald M. “Comparative Static Effects of Number of Bidders on Behavior in a Second-Price Common Value Auction” International Journal of Game Theory, 1995, Volume 24, 293-319. [3] Kazumori, Eiichiro and McMillan, John. “Selling Online versus Live,” The Journal of Industrial Economics, 2005, Volume 53, 543-569. [4] Krishna, Vijay. “Auction Theory” Academic Press, 2002. [5] Levin, Dan, Kagel, John and Richard, Jean-Francois. “Revenue Effects and Information Processing in English Common Value Auctions” American Economic Review, 1996, Volume 86, Issue 3, 442-460. [6] Milgrom, P. and Weber, R. “A Theory of Auction and Competitive Bidding” Econometrica, 1982, Volume 50, 1089-1122. [7] Pai-Ling, Yin. “Information Dispersion and Auction Prices” Harvard Business School, 2003, Mimeo. [8] Raviv, Yaron (a). “New Evidence on Price Anomalies in Sequential Auctions: Used Cars in New Jersey,” Journal of Business and Economic Statistics, Vol.24, No.3, 2006 (forthcoming). [9] Raviv, Yaron (b). “The Role of the Bidding Process in Price Determination: Jump Biddings in Sequential English Auctions” 2004 Working Paper, Claremont McKenna College. [10] Wilson, Robert B. “Competitive Bidding with Disparate Information” Management Science, 1969, Volume 15, Issue 7, Theory Series, 446-44.

17

Appendix We are interested in calculating E (min B( xi , xi )) . First, under the proposed bidding strategy, we know that V − ε ≤ xi ≤ V + ε and we can separate the support into three regions: 1.

if αKBB − r + 2ε ≤ V ≤ αKBB + r − 2ε then B( xi , xi ) = xi

if α ⋅ KBB − r ≤ V ≤ α ⋅ KBB − r + 2ε

2.

⎧ xi + α ⋅ KBB − r + ε if ⎪ then B( xi , xi ) = ⎨ 2 ⎪⎩ xi if

V − ε ≤ xi ≤ α ⋅ KBB − r + ε

α ⋅ KBB − r + ε ≤ xi ≤ V + ε

if α ⋅ KBB + r − 2ε ≤ V ≤ α ⋅ KBB + r

3.

if ⎧ xi ⎪ then B( xi , xi ) = ⎨ xi + α ⋅ KBB + r − ε if ⎪⎩ 2

V − ε ≤ xi ≤ α ⋅ KBB + r − ε

α ⋅ KBB + r − ε ≤ xi ≤ V + ε

Taking the regions into account, we can now calculate the expected value of the winning bid: E (min B (xi , xi )) = αKBB − r + 2ε

+

∫

αKBB − r αKBB + r

+

1 2r

αKBB + r

V +ε

αKBB + r − 2ε

V +ε

V + ε − xi V + ε − xi 1 1 B (xi , xi )dxi dV = x i dx i dV 2 2r V −ε 2ε 2 r V −ε 2ε 2 αKBB − r αKBB − r + 2ε

∫

∫

∫

∫

V +ε ⎡αKBB − r +ε V + ε − x i x i + αKBB − r + ε ⎤ V + ε − xi dx x dx ⋅ + ⎢ ⎥ dV i i i 2 2ε 2 2ε 2 ⎢⎣ V −ε ⎥⎦ αKBB − r + ε

∫

∫

V +ε V +ε ⎤ V + ε − xi xi + αKBB + r − ε 1 ⎡ V + ε − xi + ⋅ x dx dxi ⎥dV ⎢ i i 2 2 2r ⎢⎣V −ε 2ε 2 2ε ⎥⎦ αKBB + r − 2ε αKBB + r −ε

∫

∫

∫

1 1 = αKBB − ε + ε 2 3 6r 1 l 2r = αKBB − lr + 3 6

18

Table 1: Summary Statistics Presale Estimate

Price

Price/Estimate

Number of Bids

Mean

2662.19 (1634.84)

1520.42 (1168.18)

0.561 (0.236)

11.39 (7.13)

Minimum

318.75

50

0.051

1

Maximum

15265

8650

1.787

49

Observations

678

683

678

641

Standard errors are in parentheses.

Figure 1: Empirical Distribution of Number of Bids 60

50

Frequency

40

30

20

10

0 1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

Number of Bids

19

0

.5

D en s ity 1

1 .5

2

Figure 2: Kernel Density and Histogram of Selling Price/Estimate Ratio

0

.5

1 ratio

1.5

2

Selling Price

Figure 3: Different Jump Bidding Options for the same Selling Price

0

1

2

3

4

5

6

7

8

9

10

Number of Bids

20

49

36

31

28

25

22

19

16

13

7

10

4

7000 6000 5000 4000 3000 2000 1000 0 1

bidmfirst

Figure 4: Relation between Number of Bids and Sigma1

Figure 5: Relation between Number of Bids and Sigma2

bidmfirstsq

50000000 40000000 30000000 20000000 10000000

49

36

31

28

25

22

19

16

13

10

7

4

1

0

Figure 6: Relation between Number of Bids and Sigma3 60

40 30 20 10

49

36

31

28

25

22

19

16

13

10

7

4

0 1

bidmfof

50

21

49

36

31

28

25

22

19

16

13

7

10

4

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

bidmfow

Figure 7: Relation between Number of Bids and Sigma4

Figure 8: Relation between Number of Bids and Sigma5

0.8 0.6 0.4 0.2

49

36

31

28

25

22

19

16

13

7

10

4

0

1

bidmfokbb

1

22

Table 2: Negative Binomial Regression Results for Number of Bids Sigma1

Sigma2

Sigma3

Sigma4

Sigma5

Intercept

2.0074* (0.3320)

2.6391* (0.3394)

2.3775* (0.4166)

1.4942* (0.3525)

1.9070* (0.3482)

Order

0.0627 (0.0530)

0.1203** (0.0640)

0.1129** (0.0667)

0.0435 (0.0561)

0.0226 (0.0560)

Year

-0.0230* (0.0058)

-0.0353* (0.0069)

-0.0431* (0.0075)

-0.0625* (0.0062)

-0.0457* (0.0061)

Mileage

-0.0085** (0.0039)

-0.0163* (0.0046)

-0.0156* (0.0047)

-0.0194* (0.0040)

-0.0081* (0.0040)

Estimator

-0.0000* (0.0000)

0.0000* (0.0000)

0.0002* (0.0000)

0.0001* (0.0000)

0.0001* (0.0000)

Variance

0.0006* (0.0000)

0.0000* (0.0000)

0.0091* (0.0012)

1.6525* (0.0845)

1.7628* (0.0964)

Dispersion

0.0468 (0.0067)

0.1129 (0.0113)

0.1295 (0.0123)

0.0625 (0.0082)

0.0614 (0.0078)

Deviance

0.9592

1.0256

1.0226

0.9718

0.9665

Log Likelihood

11297.77

11158.68

11135.91

11261.13

11265.07

Observations

636

636

636

636

636

Standard errors are in parentheses. Mileage is the actual mileage divided by 10,000. Order is the order in which the object was introduced divided by total number of items. * Significant at 1% level. **Significant at 5% level. All the regressions include auction dummies fixed effects.

23

Table 3: Negative Binomial Regression Results for Number of Bids Sigma1

Sigma2

Sigma3

Sigma4

Sigma5

Intercept

1.9416* (0.3294)

2.5648* (0.3936)

2.3603* (0.4137)

1.3567* (0.3470)

1.8085* (0.3405)

Order

0.0712 (0.0528)

0.1275** (0.0632)

0.1190** (0.0665)

-0.0046 (0.0550)

0.0341 (0.0549)

Year

-0.0285* (0.0060)

-0.0432* (0.0071)

-0.0465* (0.0076)

-0.0528* (0.0062)

-0.0543* (0.0062)

Mileage

-0.0113* (0.0040)

-0.0200* (0.0047)

-0.0177* (0.0048)

-0.0128* (0.0040)

-0.0123* (0.0041)

Estimator

-0.0000 (0.0000)

0.0001* (0.0000)

0.0002* (0.0000)

0.0001* (0.0000)

0.0002* (0.0000)

First Bid

-0.0001* (0.0000)

-0.0002* (0.0000)

-0.0001*** (0.0000)

0.0004* (0.0000)

-0.0002* (0.0000)

Variance

0.0006* (0.0000)

0.0000* (0.0000)

0.0084* (0.0013)

2.1141* (0.1020)

1.7331* (0.0928)

Dispersion

0.0452 (0.0066)

0.1080 (0.0110)

0.1273 (0.0122)

0.0545 (0.0073)

0.0554 (0.0075)

0.9539

1.0266

1.0284

0.9246

0.9664

11303.56

11166.26

11137.68

11292.44

11277.63

636

636

636

636

636

Deviance Log Likelihood Observations

Standard errors are in parentheses. Mileage is the actual mileage divided by 10,000. * Significant at the 1% level. **Significant at the 5% level. ***Significant at the 10% level. All the regressions include auction dummies fixed effects.

24

25

26