set.seed(123456789)
M <- 1000 # number of simulated auctions.
data1 <- matrix(NA,M,12)
for (i in 1:M) {
N <- round(runif(1, min=2,max=10)) # number of bidders.
v <- runif(N) # valuations, uniform distribution.
b <- (N - 1)*v/N # bid function
p <- max(b) # auction price
x <- rep(NA,10)
x[1:N] <- b # bid data
data1[i,1] <- N
data1[i,2] <- p
data1[i,3:12] <- x
}
colnames(data1) <- c("Num","Price","Bid1",
"Bid2","Bid3","Bid4",
"Bid5","Bid6","Bid7",
"Bid8","Bid9","Bid10")
data1 <- as.data.frame(data1)Estimating Auction Models
Introduction
According to the travel presenter, Rick Steves, the Aalsmeer auction house is one the largest commercial buildings in the world. Royal Flora Holland, the owner of Aalsmeer, sold 12.5 billion plants and flowers in 2016 through its auction houses. But with $5.2 billion in auction sales, Royal Flora Holland is nowhere near the biggest auction house in the world.1 That honor goes to Google. Google sold $47.6 billion in search ads using what the economist, Hal Varian, called the biggest auction in the world (Varian 2007).2 But while that is impressive, a single auction in 2015 almost beat Google’s annual number. The US Federal Communication Commission’s auction number 97 (AWS-3) raised $44.9 billion dollars.3
Auctions are used to sell and buy a large number of products. Governments use auctions to purchase everything from paper to police body cameras. The US Federal government uses auctions to sell oil drilling rights, FCC spectrum, 10 year bonds and timber access. You can sell and buy items from eBay.com using auctions.
Economists use auction data to determine the underlying valuations for the items. We may be interested in modeling what would happen if a different auction method is used. Economists working in antitrust want to determine the effect of a merger that reduces the number of independent bidders or determine if there is evidence of collusion.
The auctions at Aalsmeer are unique. The auction runs for a short amount of time with a “clock” clicking the price down as the auction continues. As the price falls, the first bidder to hit the button, wins, at whatever price the clock is at. A spokesman for Aalsmeer stated that because the price falls, it is called a Dutch auction. But actually, she got the causality backwards. Because the Dutch popularized these types of auctions for selling flowers, we call them Dutch auctions.
The auction style you may be most familiar with is called an English auction. In this auction, there is an auctioneer who often speaks very very fast and does a lot of pointing while bidders hold up paddles or make hand gestures. In English auctions, the last bidder wins and pays the price at which the bidding stops.
Economic analysis of auctions began with William Vickrey’s seminal 1961 paper, Counterspeculation, Auctions, and Competitive Sealed Bid Tenders. Vickrey pointed out that Dutch auctions and sealed bid auctions are strategically equivalent. In a standard sealed bid auction each bidder submits a secret written bid. The auctioneer chooses the highest bid, and the bidder pays the number written down in her bid.
Vickrey characterized what a bidder should optimally bid in such an auction. He then showed that the same bidder should bid exactly the same amount in a Dutch auction. That is, in a Dutch auction, the bidder should wait until the price falls to the number written down, and then hit the button. Vickrey showed that these two auctions formats are strategically equivalent.4 However, they are not strategically equivalent to an English auction.
Vickrey invented a new auction. In a Vickrey auction, each bidder writes down a bid like in a standard sealed bid auction and the winner is the person who writes down the highest bid. However, the winner pays the amount written down by the second highest bidder. Vickrey showed that his auction was strategically equivalent to an English auction.
Econometric analysis of auctions can be split into two steps. In the first step, we use standard statistical methods to estimate the statistical parameters. In the second step, we use the game theory to determine the structural parameters. We previously used this approach to estimate demand.
This chapter discusses two of the most important auction formats, sealed bid auctions and English auctions. It presents estimators for both. The sealed bid auction estimation is based on Guerre, Perrigne, and Vuong (2000). The English auction analysis uses the order statistic approach of Athey and Haile (2002). In both cases it presents results for simulated data and analysis of timber auctions. The chapter tests whether loggers are bidding rationally in sealed bid auctions and whether loggers colluded in English auctions.
Sealed Bid Auctions
Sealed bid auctions are one of the most commonly used auction formats. These auctions are very prominent in procurement, both in government and the private sector. In a sealed bid auction, each bidder writes down her bid and submits it to the auctioneer. The auctioneer sorts the bids from highest to lowest (or lowest to highest if they are buying instead of selling). The winner is the highest bidder and she pays the amount she wrote down. This is called a first price auction, because the price is determined by the highest bid or first price.
Vickrey pointed out that sealed bid auctions are strategically complicated. To see this, assume that a bidder’s utility for an item is equal to their intrinsic value for the item less the price they pay for the item. For example, a logger bidding in a timber auction will earn profits from the logs less the price paid to the US Forestry service for access to the trees. If a logger bids an amount equal to her expected profits, then if she wins she will earn nothing from the logging. It is optimal for the logger to shade her bid down. The problem is that the more she shades down, the lower her chance of winning the auction. The bidder must calculate the trade off between the probability of winning the auction and the value of winning the auction.
Sealed Bid Model
(Independent Private Values (IPV)) Let \(v_i \stackrel{\text{iid}}{\sim} F\)
The Assumption 1 makes the exposition a lot simpler. It also seems to be a reasonable approximation for the problems considered. It states that a bidder’s value for the item is unrelated to the values of the other bidders in the auction, except that they draw their valuation from the same distribution. It is a standard simplifying assumption in the auction literature. A contrasting assumption is called “common values.” In a common values auction the item has the exact same value for everyone. Often it is assumed that while the bidders know that they all have the same value, they don’t know exactly what that value is. This leads to an over-bidding problem called the “winner’s curse.”
The bidder maximizes her expected returns from the auction. Assume that the bidder gets 0 if she loses. If she wins, assume she gets her intrinsic value for the item less her bid.
\[ \max_{b_{ij}} \Pr(\mathrm{win} | b_{ij})(v_{ij} - b_{ij}) \tag{1}\]
where \(b_{ij}\) represents the bid of bidder \(i\) in auction \(j\), and \(v_{ij}\) represents the intrinsic value of the item for bidder \(i\) in auction \(j\).
If we take first order conditions of Equation 1 then we get the following expression.
\[ g(b_i | N)(v_i - b_i) - G(b_i | N) = 0 \tag{2}\]
Let \(G(b_i | N)\) denote the probability that bidder \(i\) is the highest bidder with a bid of \(b_i\), conditional on there being \(N\) bidders in the auction, and \(g(b_i | N)\) is the derivative.
We can rearrange this formula to show how much the bidder should shade her bid.
\[ b_i = v_i - \frac{G(b_i | N)}{g(b_i | N)} \tag{3}\]
The formula states that the bidder should bid her value, less a shading factor which is determined by how much a decrease in her bid reduces her probability of winning the auction.
It will be useful for our code to write the probability of winning the auction as a function of the bid distribution. Let \(H(b)\) denote the distribution of bids in the auctions. Given Assumption 1, the probability of a particular bidder winning the auction is given by the following equation.
\[ G(b_i | N) = H(b_i)^{N-1} \tag{4}\]
If there are two bidders in the auction, then the probability of winning is simply the probability that your bid is higher than the other bidder. If there are more than two bidders, it is the probability that your bid is higher than *all} the other bidders. The independent private values assumption implies it is the probability that each of the other bidders makes a bid less than yours, all multiplied together.
We can also determine the derivative of this function in terms of the bid.
\[ g(b_i | N) = (N-1) h(b_i) H(b_i)^{N-2} \tag{5}\]
where \(h\) is the derivative of the bid distribution \(H\).
Sealed Bid Simulation
In the simulated data, each bidder draws their value from a uniform distribution. Vickrey shows that the optimal bid in this auction is calculated using the following formula.
\[ b_i = \frac{(N-1) v_i}{N} \tag{6}\]
Vickrey assumes that each bidder knows his own valuation, but only knows the distribution of valuations for other bidders in the auction. In game theory, this is called a game of incomplete information. We generally assume that the outcome of such games is a Bayes Nash equilibrium.5
Remember a game has three parts, a set of players, a set of strategies and a set of payoffs. In the case of auctions, the players are the bidders in the auction. The strategies are the bids. Actually, that is not quite correct. In games of incomplete information, the players do not observe the actions of the other players. When bidding, a bidder does not know what the other bidders are bidding. Instead, it is assumed that bidders know the function that maps from valuations to bids. In Vickrey’s game, bidders know the function represented by Equation 6. They also know their own valuation. The payoffs are the expected value of the auction accounting for the probability of winning, the intrinsic value of the item and the amount bid.
The uniform distribution simplifies the problem, which is why it is used. In each simulated auction, there are different numbers of simulated bidders.
The simulation creates a data set with 1,000 auctions. In each auction, there is between 2 and 10 bidders. Note that the bidders are not listed in order.
Sealed Bid Estimator
The estimator uses Equation 3 to back out values from observed bids. To do this, we calculate the probability of winning the auction conditional on the number of bidders. It should be straightforward to determine from this data. Once we have this function, we use the formula to determine the bidder’s valuation from their bid.
The first step is to estimate the bid distribution.
\[ \hat{H}(b) = \frac{1}{N}\sum_{i=1}^N \mathbb{1}(b_i < b) \tag{7}\]
The non-parametric estimate of the distribution function, \(H(b)\), is the fraction of bids that are below some value \(b\).6
The second step is to estimate the derivative of the bid distribution. This can be calculated numerically for some given “small” number, \(\epsilon\).7
\[ \hat{h}(b) = \frac{\hat{H}(b + \epsilon) - \hat{H}(b - \epsilon)}{2 \epsilon} \tag{8}\]
If there are two bidders, Equation 3 determines the valuation for each bidder.
\[ \hat{v}_i = b_i + \frac{\hat{H}(b_i)}{\hat{h}(b_i)} \tag{9}\]
where \(i \in \{1, 2\}\)
Sealed Bid Estimator in R
The estimator backs out the valuation distribution from the distribution of bids. It limits the data to only those auctions with two bidders. In this special case, the probability of winning is just given by the distribution of bids.8 In the code the term “eps” stands for the Greek letter, \(\epsilon\), and refers to a “small” number.
sealed_2bid <- function(bids, eps=0.5) {
# eps for "small" number for finite difference method
# of taking numerical derivatives.
values <- rep(NA,length(bids))
for (i in 1:length(bids)) {
H_hat <- mean(bids < bids[i])
# bid probability distribution
h_hat <- (mean(bids < bids[i] + eps) -
mean(bids < bids[i] - eps))/(2*eps)
# bid density
values[i] <- bids[i] + H_hat/h_hat
}
return(values)
}
bids <- data1[data1$Num==2,3:4]
bids <- as.vector(as.matrix(bids)) # all the bids
values <- sealed_2bid(bids)It is straightforward to calculate the probability of winning, as this is the probability the other bidder bids less. Given IPV, this is just the cumulative probability for a particular bid. Calculating the density is slightly more complicated. However, we can approximate this derivative numerically by looking at the change in the probability for a “small” change in the bids.9 The value is calculated using Equation 3.
The Figure 1 shows that the bids are significantly shaded from the true values, particularly for very high valuations. The figure presents the density functions for bids and derived valuations from the two-person auctions. The true density of valuations lies at 0.5 and goes from 0 to 1. Here the estimated density is a little higher and goes over its bounds. However, part of the reason may be the method we are using to represent the density in the figure.10
English Auctions
The auction format that people are most familiar with is the English auction. These auctions are used to sell cattle, antiques, collector stamps and houses (in Australia). In the 1970s they were also the standard format used by the US Forestry Service to sell timber access (Aryal et al. 2018).
Vickrey showed that English auctions are strategically very simple. Imagine a bidder hires an expert auction consultant to help them bid in an English auction.
Expert: “What is your value for the item?”
Bidder: “$2,300”
Expert: “Bid up to $2,300 and then stop.”
In sealed bid auctions there is an optimal trade-off between winning and profiting from the auction. In English auctions there is no such trade-off. In econ-speak the high bidder is pivotal in determining the price in sealed bid auctions, but is not pivotal in English auctions.
English auctions are second price auctions. The price in an English auction is determined by the highest losing bid. That is, it is determined by the second price. In English auctions the bidding continues until the second to the last bidder drops out. Once there is only one bidder, everything stops.
The optimal bid for bidder \(i\) is to bid her value.
\[ b_i = v_i \tag{10}\]
The Equation 10 suggests that empirical analysis of English auctions is a lot simpler than for sealed bid auctions. If only that were so! To be clear, the “bid” in Equation 10 means the strategy described by the expert. In the data we do not necessarily observe this strategy.
If we could observe all the bid strategies in the auction, then we would have an estimate of the value distribution. But that tends to be the problem. Depending on the context, not all active bidders in the auction may actually be observed making a bid. In addition, if the price jumps during the auction we may not have a good idea when bidders stopped bidding (P. A. Haile and Tamer 2003).
Athey and Haile (2002) provide a solution. They point out that the price in an English auction has a straightforward interpretation as the second highest bid in the auction when valuations follow Assumption 1. The price is the second highest valuation of the people who bid in the auction. Consider if the price is lower than the second highest valuation. How could that be? Why did one of the bidders exit the auction at a price lower than her valuation? Consider if the price is higher than the second highest valuation. How could that be? Why would a bidder bid more than her valuation?
If the price is equal to the second highest valuation, then it is a particular order statistic of the value distribution. Athey and Haile (2002) show how the observed distributions of an order statistic uniquely determine the value distribution.
Order Statistics
To understand how order statistics work, consider the problem of determining the distribution of heights of players in the WNBA. The obvious way to do it is to take a data set on player heights and calculate the distribution. A less obvious way is to use order statistics.
In this method, data is taken from a random sample of teams, where for each team, the height of the tallest player is measured. Assume each team has 10 players on the roster and you know the height of the tallest, say the center. This is enough information to estimate the distribution of heights in the WBNA. We can use the math of order statistics and the fact that we know both the height of the tallest and we know that 9 other players are shorter. In this case we are using the tallest, but you can do the same method with the shortest or the second tallest, etc.
The price is more or less equal to the second highest valuation of the bidders in the auction.11 The probability of the second highest of \(N\) valuations is equal to some value \(b\) which is given by the following formula:
\[ \Pr(b^{(N-1):N} = b) = N(N-1)F(b)^{N-2}f(b)(1 - F(b)) \tag{11}\]
The order statistic notation for the second highest bid of \(N\) is \(b^{(N-1):N}\). We can parse this equation from back to front. It states that the probability of seeing a price equal to \(b\) is the probability that one bidder has a value greater than \(b\). This is the winner of the auction and this probability is given by \(1 - F(b)\), where \(F(b)\) is the cumulative probability of a bidder’s valuation less than \(b\). This probability is multiplied by the probability that there is exactly one bidder with a valuation of \(b\). This is the second highest bidder who is assumed to bid her value. This is represented by the density function \(f(b)\). These two values are multiplied by the probability that the remaining bidders have valuations less than \(b\). If there are \(N\) bidders in the auction then \(N-2\) of them have valuations less than the price. The probability of this occurring is \(F(b)^{N-2}\). Lastly, the labeling of the bidders is irrelevant so there are \(\frac{N!}{1!(N-2)!} = N(N-1)\) possible combinations. If the auction has two bidders, then the probability of observing a price \(p\) is \(2 f(p)(1 - F(p))\).
The question raised by Athey and Haile (2002) is whether we can use this formula to determine \(F\). Can we use the order statistic formula of the distribution of prices to uncover the underlying distribution of valuations? Yes.
Identifying the Value Distribution
Let’s say we observe a two bidder auction with a price equal to the lowest possible valuation for the item; call that \(v_0\). Actually, it is a lot easier to think about the case where the price is slightly above the lowest possible value. Say that the price is less than \(v_1 = v_0 + \epsilon\), where \(\epsilon\) is a “small” number. What do we know? We know that one bidder has that very low valuation, which occurs with probability equal to \(F(v_1)\). What about the other bidder? The other bidder may also have a value equal to the lowest valuation or they may have a higher valuation. That is, their value for the item could be anything. The probability of value lying between the highest and lowest possible value is 1. So \(\Pr(p \le v_1) = 2 \times 1 \times F(v_1)\). Note that either bidder could be the high bidder. There are 2 possibilities, so we must multiply by 2. As the probability of a price less than \(v_1\) is observed in the data, we can rearrange things to get the initial probability, \(F(v_1) = \Pr(p \le v_1)/2\).
Now take another value, \(v_2 = v_1 + \epsilon\). The probability of observing a price between \(v_1\) and \(v_2\) is as follows.
\[ \Pr(p \in (v_1,v_2]) = 2 (F(v_2) - F(v_1))(1 - F(v_1)) \tag{12}\]
It is the probability of seeing one bidder with a value between \(v_2\) and \(v_1\) and the second bidder with a value greater than \(v_1\). Again, the two bidders can be ordered in two ways.
We can solve \(F(v_2)\) using Equation 12. We observe the quantity on the left-hand side and we previously calculated \(F(v_1)\).
For a finite subset of the valuations we can use this iterative method to calculate the whole distribution. For this to work, each bidder’s valuation is assumed to be independent of the other bidders and comes from the same distribution of valuations (Assumption 1).
English Auction Estimator
The non-parametric estimator of the distribution follows the logic above.
The initial step determines the probability at the minimum value,
\[ \hat{F}(v_1) = \frac{\sum_{j=1}^M \mathbb{1}(p_j \le v_1)}{2M} \tag{13}\]
where there are \(M\) auctions and \(p_j\) is the price in auction \(j\).
To this initial condition, we can add an iteration equation.
\[ \hat{F}(v_k) = \frac{\sum_{j=1}^M \mathbb{1}(v_k < p_j \le v_{k+1})}{2M (1 - \hat{F}(v_{k-1}))} + \hat{F}(v_{k-1}) \tag{14}\]
These equations are then used to determine the distribution of the valuations.
English Auction Estimator in R
We can estimate the distribution function non-parametrically by approximating it at \(K=100\) points evenly distributed across the range of observed values. The estimator is based on Equation 13 and Equation 14.
english_2bid <- function(price, K=100, eps=1e-8) {
# K number of finite values.
# eps small number for getting the probabilities
# calculated correctly.
min1 <- min(price)
max1 <- max(price)
diff1 <- (max1 - min1)/K
Fv <- matrix(NA,K,2)
min_temp <- min1 - eps
max_temp <- min_temp + diff1
# determines the boundaries of the cell.
Fv[1,1] <- (min_temp + max_temp)/2
gp <- mean(price > min_temp & price < max_temp)
# price probability
Fv[1,2] <- gp/2 # initial probability
for (k in 2:K) {
min_temp <- max_temp - eps
max_temp <- min_temp + diff1
Fv[k,1] <- (min_temp + max_temp)/2
gp <- mean(price > min_temp & price < max_temp)
Fv[k,2] <- gp/(2*(1 - Fv[k-1,2])) + Fv[k-1,2]
# cumulative probability
}
return(Fv)
}Consider simulated data from an English auction. The data set provides the price and the number of bidders in each auction.
M <- 10000
data2 <- matrix(NA,M,2)
for (i in 1:M) {
N <- round(runif(1, min=2,max=10))
v <- rnorm(N) # normally distributed values
b <- v # bid
p <- -sort(-b)[2] # auction price
data2[i,1] <- N
data2[i,2] <- p
}
colnames(data2) <- c("Num","Price")
data2 <- as.data.frame(data2)Given this data we can determine the value distribution for the two-bidder auctions.
price <- data2$Price[data2$Num==2]
# restrics the data to auctions w/ 2 bidders.
Fv <- english_2bid(price)
The Figure 2 shows that the method does a pretty good job at estimating the underlying distribution. The true distribution has a median of 0, which means that the curve should go through the point where the two dotted lines cross. However, the estimate of the upper tail is not good. The true distribution has a much thicker upper tail than the estimated distribution. How does the estimate change if you use auctions with a larger number of bidders? Remember to change the formula appropriately.
Are Loggers Rational?
In the 1970s, the US Forest Service conducted an interesting experiment. It introduced sealed bid auctions in 1977. Previous to that, most US Forest Service auctions had been English auctions.12 In 1977, the service mixed between auction formats. As discussed above, bidding in sealed bid auctions is strategically a lot more complicated than bidding in English auctions. In the latter, the bidder simply bids her value. In the former, she must trade off between bidding higher and increasing the likelihood of winning against paying more if she does win.
Because of the experiment, we can test whether the loggers in the sealed bid auctions bid consistently with their actions in the English auctions. Our test involves estimating the underlying value distribution using bid data from sealed bid auctions, and comparing that to an estimate of the underlying value distribution using price data from English auctions.
Timber Data
The data used here is from the US Forest Service downloaded from Phil Haile’s website.13
In order to estimate the distributions of bids and valuations it is helpful to “normalize” them so that we are comparing apples to apples. The standard method is to use a log function of the bid amount and run OLS on various characteristics of the auction including the number of acres bid on, the estimated value of the timber, access costs and characteristics of the forest and species (P. Haile, Hong, and Shum 2006).14
x <- read.csv("auctions.csv", as.is = TRUE)
lm1 <- lm(log_amount ~ as.factor(Salvage) + Acres +
Sale.Size + log_value + Haul +
Road.Construction + as.factor(Species) +
as.factor(Region) + as.factor(Forest) +
as.factor(District), data=x)
# as.factor creates a dummy variable for each entry under the
# variable name. For example, it will have a dummy for each
# species in the data.
x$norm_bid <- NA
x$norm_bid[-lm1$na.action] <- lm1$residuals
# lm object includes "residuals" term which is the difference
# between the model estimate and the observed outcome.
# na.action accounts for the fact that lm drops
# missing variables (NAs)
In general, we are looking for a normal-like distribution.
Figure 3 presents the histogram of the normalized log bids. It is not required that the distribution be normal, but if the distribution is quite different from normal, you should think about why that may be. Does this distribution look normal?15
Sealed Bid Auctions
In order to simplify things we will limit the analysis to two-bidder auctions. In the data, sealed bid auctions are denoted “S”.
y <- x[x$num_bidders==2 & x$Method.of.Sale=="S",]
bids <- y$norm_bid
values <- sealed_2bid(bids)
summary(bids) Min. 1st Qu. Median Mean 3rd Qu. Max.
-4.0262 -0.7987 -0.1918 -0.2419 0.3437 5.0647 summary(values) Min. 1st Qu. Median Mean 3rd Qu. Max.
-4.0262 -0.0218 0.9185 4.8704 2.3108 860.0647 Using the same method that we used above, it is possible to back out an estimate of the value distribution from the bids in the data. We see that comparing the valuations to the bids, the bids are significantly shaded particularly for higher valuations.
English Auctions
Again consider two-bidder auctions. The English auctions are denoted “A”.
y <- x[x$num_bidders==2 & x$Method.of.Sale=="A",]
price <- y[y$Rank==2,]$norm_bid
Fv <- english_2bid(price)We can back out the value distribution assuming that the price is the second highest bid, the second highest order statistic.
Comparing Estimates
The Figure 4 shows that there is not a whole lot of difference between the estimate of the distribution of valuations from sealed bid auctions and English auctions. The two distributions of valuations from the sealed bid auctions and English auctions lie fairly close to each other, particularly for lower values. This suggests loggers are bidding rationally. That said, at higher values, the two distributions diverge. The value distribution from the sealed bid auctions suggests that valuations are higher than the estimate from the English auctions. What else may explain this divergence?
Are Loggers Colluding?
Is there evidence that bidders in these auctions are colluding? Above, we find mixed evidence that bidders in timber auctions are behaving irrationally. Now we can ask whether they are behaving competitively.
Given the price is an order statistic, prices should be increasing in the number of bidders. Figure 5 presents a plot of the number of bidders in an English auction against the price. The figure suggests that this relationship is not clear. One explanation is that bidders are colluding. It could be that bidders in larger auctions are underbidding. Of course, there may be many other reasons for the observed relationship.
A Test of Collusion
Consider the following test of collusion. Using large English auctions we can estimate the distribution of valuations. Under the prevailing assumptions, this estimate should be the same as for two-bidder auctions. If the estimate from the large auctions suggests valuations are much lower than for two-bidder auctions, this suggests collusion.
In particular, if the inferred valuations in these larger auctions look much like auctions with fewer bidders. That is, bidders may behave “as if” there are actually fewer bidders in the auction. For example, if there is an active bid-ring, bidders may have a mechanism for determining who will win the auction and how the losers may be compensated for not bidding.16 In an English auction, it is simple to enforce a collusive agreement because members of the bid-ring can bid in the auction where their actions are observed.
We can estimate the distribution under the assumption that there are two bidders in the auction and the assumption that there are three bidders in the auction. If these estimates are consistent with the results from the actual two bidder auctions, then we have an estimate of the bid-ring size. Consider an auction with six bidders. If three of them are members of a bid ring, then those three will agree on who should bid from the ring. Only one of the members of the bid ring will bid their value. In the data, this will look like there are actually four bidders in the auction.17
“Large” English Auctions
Consider the case where there are six bidders in the auction. From above, the order statistic formula for this case is as follows.
\[ \Pr(b^{5:6} = b) = 30 F(b)^4 f(b) (1 - F(b)) \tag{15}\]
As above, order statistics are used to determine the underlying value distribution (\(F\)); however in this case it is a little more complicated to determine the starting value.
Think about the situation where the price in a 6 bidder auction is observed at the minimum valuation. What do we know? As before, one bidder may have a value equal to the minimum or a value above the minimum. That is, their value could be anything. The probability of a valuation lying between the minimum and maximum value is 1. We also know that the five other bidders had valuations at the minimum. If not, one of them would have bid more and the price would have been higher. As there are six bidders, there are six different bidders that could have had the highest valuation. This reasoning gives the following formula for the starting value.
\[ \Pr(b^{5:6} < v_1) = 6F(v_1)^5 \tag{16}\]
Rearranging, we have \(F(v_1) = \left(\frac{\Pr(p<v_1)}{6}\right)^{\frac{1}{5}}\).
Given this formula we can use the same method as above to solve for the distribution of valuations.
Large English Auction Estimator
Again we can estimate the value distribution by using an iterative process. In this case we have the following estimators.
\[ \hat{F}(v_1) = \left (\frac{\sum_{j=1}^M \mathbb{1}(p_j < v_1)}{6M} \right)^{\frac{1}{5}} \tag{17}\]
and
\[ \hat{F}(v_k) = \frac{\sum_{j=1}^M \mathbb{1}(v_{k} < p_j < v_{k+1})}{30 M \hat{F}(v_{k-1})^4 (1 - \hat{F}(v_{k-1}))} + \hat{F}(v_{k-1}) \tag{18}\]
The other functions are as defined in the previous section.
We can also solve for the implied distribution under the assumption that there are three bidders and under the assumption that there are two bidders.18 Note in each auction there are at least six bidders.19
Large English Auction Estimator in R
We can adjust the estimator above to allow any number of bidders, \(N\).
english_Nbid <- function(price, N, K=100, eps=1e-8) {
min1 <- min(price)
max1 <- max(price)
diff1 <- (max1 - min1)/K
Fv <- matrix(NA,K,2)
min_temp <- min1 - eps
max_temp <- min_temp + diff1
Fv[1,1] <- (min_temp + max_temp)/2
gp <- mean(price > min_temp & price < max_temp)
Fv[1,2] <- (gp/N)^(1/(N-1))
for (k in 2:K) {
min_temp <- max_temp - eps
max_temp <- min_temp + diff1
Fv[k,1] <- (min_temp + max_temp)/2
gp <- mean(price > min_temp & price < max_temp)
Fv[k,2] <-
gp/(N*(N-1)*(Fv[k-1,2]^(N-2))*(1 - Fv[k-1,2])) +
Fv[k-1,2]
}
return(Fv)
}
y2 <- x[x$num_bidders > 5 & x$Method.of.Sale=="A",]
# English auctions with at least 6 bidders.
price2 <- y2$norm_bid[y2$Rank==2]
Fv6 <- english_Nbid(price2, N=6)
Fv3 <- english_Nbid(price2, N=3)
Fv2 <- english_2bid(price2)Evidence of Collusion
The Figure 6 suggests that there is in fact collusion in these auctions! Assuming there are six bidders in the auction implies that valuations are much lower than we estimated for two bidder auctions from both English auctions and sealed-bid auctions. In the chart, the distribution function is shifted to the left, meaning there is greater probability of lower valuations.
If we assume the estimates from the two bidder auctions are the truth, then we can determine the size of the bid-ring. Estimates assuming there are three bidders and two bidders lie above and below the true value, respectively. This suggests that bidders are behaving as if there are between two and three bidders in the auction. This implies that the bid ring has between three and four bidders in each auction.
These results are suggestive of an active bid ring in these auctions in 1977. It turns out that this was of real concern. In 1977, the United States Senate conducted hearings into collusion in these auctions. In fact, this may be why the US Forestry Service looked into changing to sealed bid auctions. The US Department of Justice also brought cases against loggers and millers (Baldwin, Marshall, and Richard 1997). Alternative empirical approaches have also found evidence of collusion in these auctions, including Baldwin, Marshall, and Richard (1997) and Athey, Levin, and Seira (2011).
Large Sealed Bid Auction Estimator
One concern with the previous analysis is that there may be something special about auctions with a relatively large number of bidders. It may that there are unobservable characteristics of the auctions that are leading to the observed differences. To check this possibility we can compare our estimate of the value distribution from large sealed bid auctions.
For this case the probability of winning the auction is slightly different than the simple case above.
\[ \hat{G}(b | 6) = \hat{H}(b)^{5} \tag{19}\]
and
\[ \hat{g}(b | 6) = 5 \hat{h}(b) \hat{H}(b)^4 \tag{20}\]
where \(\hat{H}(b)\) and \(\hat{h}(b)\) are defined above.
Large Sealed Bid Auction Estimator in R
sealed_Nbid <- function(bids,N, eps=0.5) {
values <- rep(NA,length(bids))
for (i in 1:length(bids)) {
H_hat <- mean(bids < bids[i])
h_hat <- (mean(bids < bids[i] + eps) -
mean(bids < bids[i] - eps))/(2*eps)
G_hat <- H_hat^(N-1)
g_hat <- (N-1)*h_hat*H_hat^(N-2)
values[i] <- bids[i] + G_hat/g_hat
}
return(values)
}
y_6 <- x[x$num_bidders > 5 & x$Method.of.Sale=="S",]
bids_6 <- y_6$norm_bid
values_6 <- sealed_Nbid(bids_6, N=6,eps=3)As for two person auctions, the first step is to estimate each bidder’s valuation.
The results presented in Figure 7 provide additional evidence that there is collusion in the large English auctions. The bidding behavior in the larger sealed bid auctions is consistent with both bidding behavior in small sealed bid auctions and small English auctions.
Discussion and Further Reading
Economic analysis of auctions began with Vickrey’s 1961 paper. Vickrey used game theory to analyze sealed bid auctions, Dutch auctions and English auctions. Vickrey also derived a new auction, the sealed bid second price auction. Empirical analysis of auctions starts with the game theory.
The chapter considers two of the most important auction mechanisms, sealed bid auctions and English auctions. The sealed bid auctions are analyzed using the two-step procedure of Guerre, Perrigne, and Vuong (2000). The first step uses non-parametric methods to estimate the bid distribution. The second step uses the Nash equilibrium to back out the value distribution.
While we observe all the bids in the sealed-bid auctions, we generally only observe the high bids in English auctions. The chapter uses the order statistic approach of Athey and Haile (2002) to estimate the value distribution from these auctions.
Empirical analysis of auctions has changed dramatically over the last twenty years. This section just grazed the surface of the different issues that arise in auctions. One big issue not discussed here is unobserved auction heterogeneity. There may be systematic differences between the auctions that are observed by the bidders but not observed by the econometrician. The most important paper written on this question was written by a former office mate of mine, Elena Krasnokutskaya, (Krasnokutskaya 2011). A recent review by Phil Haile and Yuichi Kitamura is an excellent starting point (P. A. Haile and Kitamura 2019).
Baldwin, Marshall, and Richard (1997) and Athey, Levin, and Seira (2011) analyze collusion in US timber auctions. Aryal et al. (2018) uses US timber auctions to measure how decision makers account for risk and uncertainty.
References
Aryal, Gaurab, Serafin Grundl, Dong-Hyuk Kim, and Yu Zhu. 2018. “Empirical Relevance of Ambiguity in First-Price Auctions.” Journal of Econometrics 204 (2): 189–206.
Asker, John. 2010. “A Study of the Internal Organization of a Bidding Cartel.” American Economic Review 100 (3): 724–62.
Athey, Susan, and Philip A. Haile. 2002. “Identification in Standard Auction Models.” Econometrica 70 (6): 2170–40.
Athey, Susan, Jonathan Levin, and Enrique Seira. 2011. “Comparing Open and Sealed Bid Auctions: Evidence from Timber Auctions.” The Quarterly Journal of Economics 126: 207–57.
Baldwin, Laura H., Robert C. Marshall, and Jean-Francois Richard. 1997. “Bidder Collusion at Forest Service Timber Sales.” Journal of Political Economy 105 (5): 657–99.
Deltas, George. 2004. “Asymptotic and Small Sample Analysis of the Stochastic Properties and Certainty Equivalents of Winning Bids in Independent Private Values Auctions.” Economic Theory 23: 715–38.
Guerre, Emmanuel, Isabelle Perrigne, and Quang Vuong. 2000. “Optimal Nonparametric Estimation of First-Price Auctions.” Econometrica 68 (3): 525–74.
Haile, Phil, Han Hong, and Matt Shum. 2006. “Nonparametric Tests for Common Values in First-Price Sealed-Bid Auctions.”
Haile, Philip A., and Yuichi Kitamura. 2019. “Unobserved Heterogeneity in Auctions.” The Econometrics Journal 22: C1–19.
Haile, Philip A., and Elie T. Tamer. 2003. “Inference with incomplete model of English auctions.” Journal of Political Economy 111 (1).
Krasnokutskaya, Elena. 2011. “Identification and Estimation in Procurement Auctions Under Unobserved Auction Heterogeneity.” Review of Economic Studies 78 (1): 293–327.
Varian, Hal. 2007. “Position Auction.” International Journal of Industrial Organization 25 (6): 1163–78.
Footnotes
eMarketer.com, 7/26/16↩︎
Bidders choose the same bid strategy in both auction formats.}↩︎
This is a Nash equilibrium where it is assumed that players use Bayes’ rule to update their information given the equilibrium strategies. Because of the IPV assumption, there is no information provided by the other bidders. This is not the case in common values auctions.}↩︎
A non-parametric estimator makes no parametric assumptions.}↩︎
This is a finite difference estimator.↩︎
The probability of winning is the probability that your bid is higher than the other bidders in the auction.↩︎
This is an example of using finite differences to calculate numerical derivatives.↩︎
The kernel density method assumes the distribution can be approximated as a mixture of normal distributions.↩︎
Officially, the price may be a small increment above the bid of the second highest bidder. We will ignore this possibility.↩︎
You may think of this as just some academic question. But the US Senator for Iowa, Senator Church, was not happy with the decision. “In fact, there is a growing body of evidence that shows that serious economic dislocations may already be occurring as a result of the sealed bid requirement.” See Congressional Record September 14 1977, p. 29223.↩︎
http://www.econ.yale.edu/~pah29/timber/timber.htm. The version used here is available from here: https://sites.google.com/view/microeconometricswithr/table-of-contents.↩︎
Baldwin, Marshall, and Richard (1997) discuss the importance of various observable characteristics of timber auctions.}↩︎
It is approximately normal, but it is skewed somewhat to lower values. This may be due to low bids in the English auction. How does the distribution look if only sealed bids are graphed?↩︎
Asker (2010) presents a detailed account of a bid-ring in stamp auctions.↩︎
In the bid ring mechanism discussed in Asker (2010), the collusion actually leads to higher prices in the main auction.↩︎
See Equation 11 for the other cases.↩︎
For simplicity it is assumed that all of these auctions have six bidders. Once there are a large enough number of bidders in the auction, prices do not really change with more bidders. In fact, these methods may not work as the number of bidders gets large (Deltas 2004).↩︎