# Assessing the Role of Ordering in Sequential English Auctions

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Assessing the Role of Ordering in Sequential English Auctions
– Evidence from the Western Video Market Auction

## Abstract

In a sequential English auction, where multiple homogenous objects are offered, it is often found that the price tends to decrease over the course of the auction, which is referred to as the “declining price anomaly”. This article studies the price trend in a sequential live cattle auction. Motivated by the fact that in an English first-price auction, the winning price is the maximum among all the bids offered, we examined the existence of the declining price anomaly by introducing the Generalized Extreme Value (GEV) theory framework, and nesting it with a hedonic regression, where the price of cattle relates to the market values for cattle attributes. Empirical results indicate that the unit price difference between the first and the last lot sold ranges from \$7.82 to \$2.31, depending on the average animal weight in that lot. On average, the unit price of a stocker cattle lot sold in the last position of a typical sale is \$5/cwt (or 3%) less than if it was sold in the first place, regardless of the scale of the sale and keeping all other factors constant. Based on the theoretical framework developed by Brendstrup and Paarsch (2006), we found that the buyer’s maximum valuation on the subsequent object actually declines as he wins more objects, which  provides a plausible explanation for the declining prices anomaly in sequential auctions.

Key Words: Sequential Auction, Cattle Auction, Declining prices.

Assessing the Role of Ordering in Sequential Auctions
– Evidence from the Western Video Market Auction

## 1Introduction

Cattle prices in the U.S. are established through various market channels, such as direct sales and auction markets. As one of the major price discovery mechanisms, the livestock auction often takes the form of a sequential and multi-object English auction, which is known to be an efficient mechanism in terms of consumer surplus and seller revenue under various assumptions. One often observed phenomenon is that in sequential auctions, the wining prices of homogenous goods tend to decline as the sale proceeds. This has been referred as the ‘Declining Price Anomaly’, which has been of research interest for decades.

Empirical findings of declining prices in sequential auctions have been reported in multiple studies. In sequential livestock auctions, Buccola (1982) found evidence that the quality-corrected prices of individual cattle lots trend downwards as the sale proceeds. Pitchik and Schotter (1988) experimentally show that prices typically decline due to budget constraints in the course of sequential auctions, given independent private values. Ashenfelter (1989) also found that price declined slightly in sequential auctions of identical units of wine. Empirical studies by McAfee and Vincent (1993) show that the price of the second unit of wine is 1.4 percent lower than that of the first unit on average. Ashenfelter and Genesove (1992) and Beggs and Graddy (1997) took the heterogeneity of objects into account and price decline is still detected. Van Den Berg, et al. (2001) also found the declining price trend in sequential auctions with multi-unit demand in a sequential Dutch auction of roses. The declining price anomaly in sequential auctions was also noted for other objects, for example, Picasso prints (Pesando and Shum, 1982), commercial real estate (Lusht, 1994), satellite transponder leases (Milgrom and Weber, 2000), antiques (Ginsburgh and van Ours, 2007), fish (Gallegati et al., 2011), and lobsters (Salladarre et al., 2017), etc.

A number of theoretical studies explain this declining price anomaly from various perspectives. From the perspective of bidder preferences, McAfee and Vincent (1993) attribute the declining price trend to the non-decreasing absolute risk aversion of bidders. From the perspective of auction structures, Milgrom and Weber (2000) suggest that the use of agents in auctions may explain the declining prices; Black and De Meza (1993) explain this price trend with a buyer’s option, which is that the winner of the first auction has the opportunity to buy the remaining objects at the winning price; Von der Fehr (1994) and Menezes and Monteiro (1997) relate the declining price trend to auction participation costs. Engelbrecht-Wiggans (1994), Bernhardt and Scoones (1994), and Gale and Hausch (1994) explain the declining prices from the nature of heterogeneity of the objects in auction. Gale and Stegeman (2001) have theoretically shown that in sequential auctions with two completely informed bidders, prices weakly decline as the auctions progress.

However, opposite empirical results have also been observed. In the analysis of Donald et al. (1997), auction price increases in timber auctions where bidders are interested in more than one object. Jones, et al, (2004) also find prices increase through some sales and decrease through others. Because of the mixed findings from previous auction literature about ‘Declining Price Anomaly’ phenomenon and limited literature on sequential cattle auctions, in this article, we examine the price trend in the live cattle sequential auction, held by the Western Video Market, Inc. (WVM), based on sales dated from August 2017 to May 2019.

The sequential, asymmetric bidding model of Brendstrup and Paarsch (2006) provides a theoretical platform for both the choice of estimator and the validation of the declining price anomaly in this study.  Their model applies to multi-unit, sequential English auctions under the assumption that bidders may value characteristics differently, i. e., the presence of asymmetries.  They recognize that the winning bidder will have the highest valuation of that lot and that this valuation can be represented by an order statistic.  They specify the distribution of a bidder’s highest remaining valuation of a subsequent lot given the number of lots in the auction and the number of lots the bidder has already won.  Thus bidders’ valuations are asymmetric depending on the number of lots each has won. In our study, we followed their work to show a bidder’s expected maximum valuations decreases with the number of the objects he has won. Our empirical model differs from theirs in one important aspect, however.  They assume that each potential bidder demands each of the lots in the auction.  Clearly this is an unrealistic assumption for the auctions we study—yet with some mild assumptions it can be finessed.

The contribution of this study to existing empirical literature on the declining price anomaly is three-fold: 1) To the best of our knowledge, this is the first research to apply Generalized Extreme Value (GEV) theory to examine English auction data; 2) Instead of evaluating the nominal order number of a lot, we propose a more reasonable representation of relative lot position by using the standardized lot order, which also makes it comparable across sales of various scales; 3) Under the GEV framework, we provide credible empirical findings using the theoretical model of Brendstrup and Paarsch (2006) for the declining price anomaly. This result has important implications for all participants’ decisions in a sequential auction including the seller, buyer, and auction organizer, and possibly, for the design of optimal sequential auctions.

This article is outlined as follows: In Section 2, we describe the auction data applied in this study and the characteristics of the video auction. In Section 3, we introduce the Extremal Types Theorem and set up the empirical model. Estimation results are reported and discussed in Section 4. We draw conclusions and provide further discussion in Section 5.

## 2Video Auctions and Data Description

Increasingly, large numbers of cattle are being priced through online video auctions. As a major satellite auction operator, Western Video Market (WVM) hosts a live cattle sale in the form of a sequential English (ascending price) auction about every month. In 2016, WVM marketed more than 290,000 head of cattle for more than 850 consignors. As the sale is also broadcast nationwide via satellite and internet, buyers can either attend the sale in person and bid on site or place bids on the telephone or internet. In this way, time and travel costs have been largely reduced and it likely generates a large pool of potential buyers. Previous research has found that satellite video auctions have increased the efficiency of cattle markets, especially in pricing mechanisms. Bailey and Peterson (1991) find that video auction prices were equal to or greater than regional market prices.

Most animals sold on WVM are cattle, including calves, stocker cattle, and bred cows, etc. In very rare cases, lambs are sold as well. A few days earlier than a sale, for each lot detailed information in the form of a sale catalog is available on the WVM website, and it is also distributed electronically and mailed to buyers throughout the country so potential buyers will have the opportunity to study the lots, as well as be informed of the scale of the sale up front. For each lot in a sale, such information includes: the lot number (which determines the position of the lot to be sold during a sale), the lot size (measured in head), average animal weight (measured in cwt/head), sex (steer/heifer/mixed), name of consignor, breed, state of origin, delivery dates, etc. Animals are sold in the unit of a tractor-trailer sized lot and have roughly homogenous characteristics within each lot. Lots are numbered consecutively and the sale proceeds in the order of lot number. In a typical sale, one to two hundred cattle lots are available for sale, with the lot size ranging from 50 head to as many as 1000 head. Some of the lots to be sold are ready for delivery in the next day while some need windows as long as 6 months before they can be shipped.

We collected and pooled the data of historical sales dated from August 2017 to May 2019, during which over 3,000 cattle lots were sold. Sales normally last one day, while big sales may last for two or three days when supplies are abundant. In our analysis, we treated sales that lasted more than one day as several independent sales. For example, the Aug 2017 sale was split into two sales that were held on 7th and 8th. Sales on November 29th, 2017 and March 2nd, 2018 are omitted due to the low trading volumes on these two sales, resulting in 22 sales in the dataset. Given the heterogeneity across cattle lots, we only focus on stocker cattle (500-800lbs) sales, which is a majority type of cattle listed. Moreover, we dropped lots with delivery windows exceeding four months (120 days) from the day of sale, as longer delivery windows are more likely to introduce significant uncertainty. Variables in the dataset include wining prices (in USD/cwt), average weights (in cwt/head), head counts (in hundreds), order of the lot, gender (steer/heifer), state of origin, delivery window (in days). Note that in our analysis, the lot order variable is re-defined as the lot position offered within the subsample of the sale (500-800lbs calves), not within the entire day’s sale.

Table 1 summarizes the mean of each variable (except the state of origin) in the sampled data by sale. The overall means of each variable across all sales are reported in the last row. Of all the 1,827 lots sold in the 22 sales, 83 stocker cattle lots were sold per sale on average, with an average price of \$157.05 per hundredweight. The mean size of sampled lots ranges from 32 to 158 head with an overall average of 113 head per lot. 61% of the total animals sold are steers. The mean delivery window is 42 days, which is nearly one and a half month. 33 percent of the lots were contract for delivery in over two months (60 days). We also listed the price of the first lot sold in each sale and find that in 19 out of 22 sales, the lots sold in the first place are above the average selling price in that sale. Compared with the overall mean price of \$157.05/cwt, the mean price of first lot sold is \$174.32/cwt, which is 11% higher. Although not reported, most of the cattle lots offered are from western states – California, Oregon, and Nevada are the top three states that offered 34%, 21%, and 12%, respectively, of the sampled 1,827 lots.

Based on the characteristics of the video auction held by WVM, we define it is a sequential multi-object English (ascending) first-price auction, where the auctioneer announces the current highest bids to all participants until no higher bid is placed and the winner pays the last price he offered. Since buyers can buy more than one cattle lot, they are defined as multiunit-demand buyers instead of the often-assumed unit-demand buyers. We also assume the buyers’ valuations of an object are independent and identically distributed, as the various forms (in-person, online, and on-phone) of participation create an obstacle to forming affiliation among buyers.

Another key element in auction analysis is the number of bidders, which determines the level of competition among the bidders thus playing an important role in buyer’s bidding strategy (i.e. how high to bid) as well as the selling price, thus the seller’s revenue. In the most naïve auction model setting, each bidder is assumed to know the number of bidders, and to know everyone else knows that he knows this. When analyzing video auction, however, it is not appropriate to assume the number of bidders is constant and known to all, especially when the auction is broadcast via internet and satellite so the bidder can enter or exit at any time point of the auction. In our study, this would be less of an issue because the winning price tends to be the highest possible valuation among buyers as the numbers of bidders approaches infinity, regardless of the form of auction (Holt, 1980). Moreover, if information is sufficiently dispersed among the bidders, then the selling price converges to the item’s true value as the number of bidders becomes arbitrarily large (Milgrom 1979; Wilson 1977). That is, with perfect competition, the winning price is equal to the true value even though the number of bidders is unknown to all and no individual in the economy knows the true value. In our study, given the scale of the sale and since the sale is broadcast nationwide via internet and satellite, we assume that the number of bidders is sufficiently large although unknown, and as such there is perfect competition among bidders. We also assume each bidder is aware of the large number of bidders and the existence perfect competition.

## 3The Extremal Types Theorem and Model Setting

Analogous to the Central Limit Theorem that indicates the mean of a random sample drawn from an arbitrary distribution is asymptotically normally distributed, the Extremal Types Theorem focuses on the asymptotic distribution of the largest/smallest order statistic, or the sample extremes (minima/maxima). Suppose that

is a sequence of independent and identically distributed (i.i.d) random variables with a common arbitrary distribution, one way of characterizing extremes is by considering the distribution of the maximum order statistic

${X}^{\left(N\right)}=\mathrm{max}\left\{{X}_{1},{X}_{2},\dots ,{X}_{N}\right\}$

Surprisingly, such distribution of

${X}^{\left(N\right)}$

only falls into one of the three types of distributions: Gumbel (Type I), Fréchet (Type II), and Weibull (Type III) distributions, known collectively as the extreme value distributions (EVD). This fundamental result is known as the Extremal Types Theorem, first discovered by Fisher and Tippett (1928) and then proved in general by Gnedenko (1943). These three distributions can be linked to form an expression which is referred to as the Generalized Extreme Value (GEV) distribution (Von Mises, 1954, and Jenkinson, 1955). The Extreme Value Theory has been widely applied in engineering, environment, especially in disaster studies.

The cumulative distribution function (CDF) and the probability density function (PDF) of the GEV distribution are given by

CDF:

PDF:

Where

. The parameters

$\mu ,\sigma >0,$

and

$\xi$

are the location, scale, and shape parameters respectively. The three EVDs differ in the sign of

$\xi$

, which controls the behavior in the tails. When

$\xi <0,$

it suggests a GEV distributed random variable has an upper bound, which is a finite value that the maximum cannot exceed, and it gives the Weibull distribution. In contrast, the Fréchet distribution is for

$\xi >0$

, which indicates the maximum has a lower finite bound.

$\xi =0$

corresponds to the unbounded Gumbel distribution (Type I), which is also referred to as the log-Weibull distribution, double exponential distribution, and sometimes the Laplace distribution.

Given the fact that the winning price for an object in an English (or ascending price) auction is the maximum bid among all bidders, it naturally motivates us to apply the Extremal Types Theorem to the live cattle auction data. In an auction and among all the bids offered (observable or not), the winning bid always attracts extra research attention. “Since the winning bidder’s estimate is the maximum among all the estimates, the winning bid conveys a bound on all the loser’s estimates.  When there are many bidders, the price conveys a bound on many estimates, and so can be very informative.” (Milgrom and Weber, 1982). Now we formally define the settings: in a cattle sale, the winning price for a cattle lot,

${p}^{\left(N\right)}$

, is the highest among all the prices,

${p}_{i}‘s$

, offered by N bidders:

${p}^{\left(N\right)}=\mathrm{max}\left\{{p}_{1},{p}_{2},\dots ,{p}_{N}\right\}$

${p}_{i}‘s$

are assumed to be independent and identically distributed, which implies independent valuations among buyers. N is unknown but is assumed to be sufficiently large enough to result in perfect competition among bidders as discussed in Section 2. For each cattle lot at each sale, the CDF of the winning price

follows the GEV distribution:

where

${p}_{\mathit{it}}$

is the winning price (measured in USD dollars per cwt, \$/cwt) for the

${i}^{\mathrm{th}}$

cattle lot of a date-specific sale, denoted by

$t$

. All other parameters are defined the same as in Equations (1) and (2). When we estimate the shape parameter

$\xi$

, the standard error for

$\xi$

accounts for our uncertainty in choosing between the three types of GEV distribution as it is inconvenient to determine the specific type of distribution upfront in practice. Moreover, the GEV model can incorporate covariates. Specifically, the covariate(s) enter the model through the location parameter,

$\mu$

,

(4)

where

$X$

is the covariate matrix and

$\beta$

is the vector of coefficients (including a constant) to be estimated. Note that for GEV distribution:

The model is estimated by the maximum likelihood method.

## 4Empirical Model and Estimation Results

To examine how the winning price responds to the attributes of the cattle being sold, we nest the likelihood function with a hedonic regression. The hedonic price analysis has been used in empirical studies with a long history. Since the paper of Waugh (1928), who studied the influence of quality factors on vegetable price, numerous applications of hedonic price models have been carried out considering very different agricultural and food products. Focusing on cattle, Schroeder et al. (1988) and Bailey and Peterson (1991) used hedonic price models to measure the implicit value of the most important cattle attributes. The basic assumption of the model is the consideration that buyer preferences are based on the sum of the values for cattle, lot, and market characteristics of the good rather than on market goods. In our study, cattle buyers are buying a bundle of separate attributes such as: weight, sex, lot size, days to delivery, state of origin, etc., as they define the value of the whole cattle lot. Specifically, we introduce the following set of attributes in the model for a sale held at date

$t$

:

the relative position of the lot in the sale in lot

$i$

. Specifically, this variable is obtained in two steps: First, as discussed in Section 2, we applied specific criteria to the full sample and re-arranged the lot order number to reflect the lot position offered within the subsample of the sale, not within the entire day’s sale; Second, we standardized the re-arranged order numbers by subtracting the date-specific mean and dividing it by the date-specific standard deviation. Therefore, the

${\mathit{ORDER}}_{i}$

variable reflects the relative position of the lot in a sale and is comparable across sales of various scales. After standardization, the order variable for each sale is within

${\mathit{WT}}_{i}–$

the average animal weight in cwt in lot

$i$

${\mathit{HEAD}}_{i}–$

the number of head in lot

$i$

divided by 100

${\mathit{STR}}_{i}–$

coded one if lot

$i$

is composed of steers (otherwise zero)

${\mathit{WINDOW}}_{i}–$

coded one if the cattle lot

$i$

will be delivered in over 60 days, otherwise zero

${\phi }_{j}–$

state of origin fixed effects

${\lambda }_{t}–$

auction date-specific fixed effects

Following Equations (3) and (4), we specify the location parameter,

$\mu$

, as a linear function of the covariates above:

Model I:

(6)

${\mathit{HEAD}}_{\mathit{it}}^{2},$

to account for the possibility of nonlinear effects of lot size. The price effect of a lot’s relative position is represented by

${\beta }_{1}$

, which is the parameter of interest. Given the various forms of participation (i.e. internet, telephone, or on-site), it is plausible to assume that the number of bidders for a sale is sufficiently large enough, although unknown, to draw reliable inferences.

### 4.1 Empirical Results

Estimated coefficients are obtained by maximum likelihood method and are summarized in Table 2 (estimated fixed effects are not reported, but are available from the authors). All the estimates are statistically significant at 1% level except the window effect and some of the estimated date- and state-specific fixed effects.

As discussed in Section 3, we use standardized order number in the regression as we believe the relative position (or percentile) of a lot in a sale matters more to buyers than its nominal order number, thus making it comparable across sales of different scales (in terms of the number of lots to be sold). In this way, for example, the 40th lot in a sale with 80 lots to be sold is in the same position relative to the 80th lot in a larger-scale sale with 160 lots to be sold. Therefore, the order effect, which is of the interest in this study, measures the mean change in unit price (\$/cwt) when the position of a lot of stocker cattle is advanced by one standard deviation, with the other factors remaining constant. Our estimates show that the sale order effect is -1.46 and statistically significant at 1% level. After standardization, the order variable for any sale is within

sale-specific standard deviations about zero, regardless of the scale of a sale. That is, the last lot is about 3.44 sale-specific standard deviations away from the first lot. Therefore, keeping the other factors constant, the winning price of the cattle lot offered in the last place is estimated to be 1.46*3.44 = 5.02 USD/cwt lower than that of the one offered in the first place. As summarized in Table 1, a typical sale may have an average lot size of 113 head of animals with a mean weight of 6.42 cwt (or 642 pounds) and the first lot price is 174 USD/cwt on average. In such a typical sale, a lot offered in the end of the sale is

, or

$\frac{5}{174}*100%\approx 2.9%$

, lower than if it is offered in the very first position.

Average animal weight of a lot has a significant and negative effect on prices: the mean value of estimated mean weight effects indicated that mean price drops \$8.29/cwt on average when the average weight increases by one hundredweight. Effects of lot size (HEAD) and quadratic term of head (HEAD2) show a negative relationship between price and lot size and that the speed of price decreasing is faster as the size grows. Steers would be sold at a higher price than heifers by \$13.55/cwt on average. As far as delivery window, it shows no significant impact on prices although has negative estimates.

Both the scale and shape parameters are statistically significant and the negative value of estimated shape parameter (

$\stackrel{̂}{{\xi }}{=}{–}{0}{.}{235}{\right)}$

suggests the underlying GEV distribution of the winning price is Weibull (Type III) distribution.

### 4.2 Association between Lot Order and Animal Weight

As noted by Buccola (1982), a problem with livestock auction data is that, in most cattle sales, the lot’s position is, to some degree, purposely determined by animal weight and/or some other attributes. Thus, measures of the latter factors are often found to be highly correlated with lot position. For example, lot position may be offered in order of ascending animal weight, which is often associated with decreasing unit price (\$/cwt). In cases when there exists multicollinearity between lot position and other factors, the magnitude of the position effect on price will be biased down or up, depending on the direction of the correlation. Moreover, the precision of the estimated lot position effect will be decreased, compared with results might be obtained with uncorrelated data. In our sample, we also noticed that in 9 out of 22 sales, the lot position has a highly positive correlation (Pearson’s correlation coefficient greater than 0.8) with animal weight (i.e. heavier animals are offered in latter positions). Although in Model I, the precision of position effect is not severely affected by multicollinearity given the large sample size, the association between the lot position and the average animal weight motivated us to explore the possible varying order effect due to different animal weight. Therefore, we propose another specification below

Model II:

where we include an interaction term,

which allows the order effect to vary as the animal weight changes. Maximum likelihood estimation results (except sale- and state-specific fixed effects) are reported in the last column of Table 2. The estimated order effect is now a function of weight,

$–4.944+0.534*\mathit{Weight}$

, suggesting the magnitude of order effect decreases with average animal weight. Since in our dataset, the weight is restricted between 5 and 8 cwt with an average of 6.42 cwt and the last lot is 3.44 standard deviation away from the first on average, thus the maximum price difference due to order effect ranges from

to

$\left(–4.944+0.534*8\right)*3.44=–2.31,$

keeping other factors constant. We also plot the estimated maximum price difference against weight along with the 90% confidence interval in Figure 1. It turns out the estimated mean price effect of Model II given an average animal weight is very close to that estimated by Model I:  the estimated mean price difference is

$–4.944+0.534*\stackrel{̅}{\mathit{Weight}}=–1.52$

if the lot order is advanced by one standard deviation and therefore, the mean price difference between the first and the last lots sold is

$–1.52*3.44=–5.21$

, which is -3.0% considering the average price of first lot sold is \$174.32. Additionally, all other parameter estimates are very close to those in Model I, except that the coefficient on

$\mathit{Window}$

turns out to be negative and statistically significant., indicating on average the price of cattle lots to be delivered in over two months is 29.8 cents lower than that of lots delivered earlier. Likelihood ratio test comparing two models showed results in favor of the model with the interaction term (P-value = 0.027).

### 4.3Declining Price Anomaly Validation

Brendstrup and Paarsch (2006) developed a theoretical platform to investigate how an individual buyer behaves in a multi-unit, sequential English auction. They recognize that the winning bidder will have the highest valuation of that lot and that this valuation can be represented by an order statistic.  They specify the distribution of a bidder’s highest remaining valuation of a subsequent lot given the number of lots in the auction and the number of lots the bidder has already won.  Thus bidders’ valuations are asymmetric depending on the number of lots each has won. Our empirical model differs from theirs in one important aspect, however.  They assume that each potential bidder demands each of the lots in the auction.  Clearly this is an unrealistic assumption for the auctions we study—yet with some mild assumptions it can be finessed.

Following their work, we are able to derive a buyer

$i$

’s expected maximum valuation toward the subsequent lot, given that there are m lots on sale and he already won

${w}_{i}$

lots (m

$>>{w}_{i}$

),

where f(y) is defined from Equation (2), and

$F\left(y\right)=\mathrm{exp}\left\{–{{\left[1+\xi \left(\frac{y–\mu }{\sigma }\right)\right]}_{+}}^{–\frac{1}{\xi }}\right\},$

$i$

’s cumulative valuation distribution. When

$F\left(y\right)=1,$

his valuation reaches the highest value, which is

Using Model II, we now have enough information to construct a hypothetical buyer’s maximum valuation of a lot. Consider an average expected sale price of

$\mathrm{E}\left(Y\right)=157.$

Using the estimated values of

$\sigma =7.569$

and

$\xi =–.239$

, and following Equation (5), we calculate

$\mu =\mathrm{E}\left(Y\right)–\frac{\sigma \left[\mathrm{\Gamma }\left(1–\xi \right)–1\right]}{\xi }=154.11$

. And then we plug

$\mu$

=154 into Equation (9) and can calculate the expected maximum of the distribution is

, which is the upper bound of the integral in Equation (8).

Brendstrup and Paarsch (2006) specify that each bidder is a potential buyer of each of the

$m$

lots in the auction.  It is likely that buyers have already established their requirements that lots must satisfy due to their production practices and the physical endowments of their operations.  In this case we can substitute some assumed value,

${m}^{*}$

, for

$m$

such that we expect

${m}^{*}$

to be considerably less than

$m$

. To make this concrete, suppose there are

$m=80$

lots in an auction, and assume that a bidder has a potential interest in 10 lots; hence

${m}^{*}=10$

.  Given the values expressed above and the GEV distribution assumed, we can evaluate the integral (Equation 8) to observe how a bidder’s expected maximum valuations may change in the course of an auction depending on the number of lots won, as shown in the

${m}^{*}=10$

column of Table 3, where we also assessed a buyer’s expected maximum valuation given different numbers (

${m}^{*}$

) of lots he demanded. The paths of the expected maximum valuation given a buyer’s demand are plotted in Figure 2. As a bidder wins more lots, it is clear his expected maximum valuation declines and hence he is less likely to win additional lots.  This acts to drive bids down.  Our finding that sale prices decline with order is consistent with the empirical findings of Brendstrup and Paarsch who find that the price of the last unit of fish sold declines as the number of units sold increases

## 5Conclusion and Discussion

This article has examined the role of ordering in sequential cattle auctions. It should be expected that at a sequential English auction, the quality-corrected prices of goods tend to decline during the course of the sale. This hypothesis has been tested by examining the winning prices for stocker cattle (500-800lbs) at 22 video cattle auctions dated from August 2017 to May 2019. We proposed two models to fit the auction data based on the Extremal Types Theorem. Both models showed empirical evidence for declining quality-corrected prices at stocker cattle sales. The estimated mean effects of the relative order of a cattle lot in two models are quite close: in the sampled sales, the unit price is estimated to be about \$1.5/cwt lower if the relative position of the lot is advanced by one sale-specific standard deviation within that sale. Comparing two identical lots offered in the last and the first position in a sale, it results in about \$-5/cwt (or -3%) difference in the total cost, which is a clearly noteworthy drop. The second model with an interaction term between lot order and animal weight is favored because it takes into account the association between these two factors and allows the order effect to vary given changing animal weight.

A mix of factors could result in the price downtrends in a sequential auction. Brendstrup and Paarsch (2006) provided us with a theoretical platform to examine a possible explanation. They developed a framework, which applies to a multi-unit, sequential English auction, to investigate the buyer’s expected maximum valuation toward the subsequent object, given the number of objects he demanded and the number of objects he already won. Based on that and assuming a bidder’s maximum valuation is from a GEV distribution, we found the bidder’s expected maximum valuation actually decreases as the bidder wins more lots, which drives the winning price downward during the course of a sequential auction.

As a consequence, the downward trend in prices indicates that sellers as a group extracted economic rent from buyers, especially during the early phase of a sale. However, the rent is unevenly distributed among sellers: owners of livestock offered early in a sale tend to receive additional gains associated with lot order, while owners whose livestock were offered late in an auction would even suffer losses. In the meanwhile, the sale organizer may take advantage of the downward price trend and could extract the part of gain associated with position from the seller by imposing charges on early positions. In this way, sellers need to pay the organizer higher premium for their items to be sold earlier in a sale. On the buyers’ side, the valuation as well as the bidding strategy may be adjusted with the expectation of downward quality-corrected price trend.

Overall, this study made three contributions to the existing empirical literature on sequential auctions: 1) To the best of our knowledge, this is the first research to apply Generalized Extreme Value (GEV) theory to examine the sequential auction data; 2) Instead of evaluating the nominal order number of a lot. we proposed a more reasonable representation of relative lot position by using the standardized lot order, which also made it comparable across sales of various scales; 3) Under GEV framework, we provided justification for one of the possible explanations of declining price anomaly. This result has important implications for all participants’ decisions in a sequential auction including the sellers, buyers, and auction organizer, and possibly, for the design of optimal sequential auctions.

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## 6Tables and Figures

Table 1 Summary Statistics

 Date of Sale (22 in total) Number of Lots (1827 in total) Mean Price (\$/cwt) 1st Lot Price Mean Weight (cwt/head) Proportion of Steer Mean Head Window (days) Proportion of Window > 60 8/7/2017 92 147.44 175.50 6.63 70% 98 62 58% 8/8/2017 158 161.12 175.00 5.71 73% 119 73 89% 9/11/2017 111 162.42 150.00 6.38 66% 120 37 15% 10/26/2017 56 159.88 176.25 6.42 53% 116 8 0% 1/4/2018 47 154.59 181.00 6.47 55% 75 7 0% 1/25/2018 46 151.95 170.00 6.53 35% 93 6 0% 4/11/2018 59 154.52 184.00 6.42 57% 113 16 0% 5/3/2018 73 147.96 184.50 7.17 54% 171 16 0% 6/7/2018 41 148.64 172.00 7.00 68% 101 9 0% 7/9/2018 59 154.62 155.00 7.54 66% 128 32 12% 7/10/2018 165 160.56 178.00 6.46 64% 104 79 76% 7/11/2018 83 173.27 167.00 5.58 62% 138 110 99% 8/6/2018 91 153.92 177.50 7.06 62% 106 55 46% 8/7/2018 136 170.67 202.50 5.71 65% 122 77 91% 9/10/2018 133 167.15 160.50 6.39 68% 119 37 14% 10/25/2018 74 154.41 161.50 6.33 55% 93 13 0% 11/28/2018 154 153.05 180.50 6.35 59% 98 14 0% 1/3/2019 64 149.16 186.75 6.22 56% 86 6 0% 1/24/2019 45 147.92 170.50 6.29 51% 104 8 0% 4/10/2019 57 153.88 186.00 6.70 64% 116 27 4% 5/2/2019 51 144.35 165.00 7.23 61% 133 18 0% 5/30/2019 32 133.02 176.00 7.24 66% 116 9 0% Mean 83 157.05 174.32 6.42 61% 113 42 33%

Table 2 Parameter Estimates

 Variable Model I Model II Intercept 185.516* (2.217) 185.815* (2.220) Order -1.459* (0.208) -4.944* (1.597) Weight -8.285* (0.271) -8.281* (0.271) Weight*Order 0.534* (0.243) Head 5.250* (0.413) 5.181* (0.411) Head2 -0.579* (0.064) -0.572* (0.063) Steer 13.551* (0.378) 13.580* (0.378) Window -0.162 (0.734) -0.298* (0.736) Scale – ${\sigma }$ 7.555* (0.120) 7.569* (0.120) Shape – ${\xi }$ -0.235* (0.008) -0.239* (0.009) -2*Log-likelihood 12541 12537 AIC 12617 12615 BIC 12826 12829

Note: 1) the standard error is in parenthesis

2) * denotes significant at 5%

Table 3 A buyer’s Expected Maximum Valuation (\$/cwt) Given Different Demand (m* lots) and Number of Lots Won (w)

 Lots Won (w) Buyer’s Demand (m* lots) 10 9 8 7 6 5 4 3 2 0 \$ 169.18 \$ 168.76 \$ 168.27 \$ 167.70 \$ 167.03 \$ 166.19 \$ 165.12 \$ 163.65 \$ 161.39 1 \$ 164.95 \$ 164.38 \$ 163.73 \$ 162.95 \$ 162.01 \$ 160.82 \$ 159.23 \$ 156.89 \$ 152.61 2 \$ 162.13 \$ 161.44 \$ 160.63 \$ 159.66 \$ 158.45 \$ 156.85 \$ 154.54 \$ 150.47 3 \$ 159.84 \$ 159.02 \$ 158.04 \$ 156.83 \$ 155.25 \$ 153.01 \$ 149.11 4 \$ 157.79 \$ 156.82 \$ 155.62 \$ 154.07 \$ 151.89 \$ 148.13 5 \$ 155.84 \$ 154.66 \$ 153.14 \$ 151.02 \$ 147.38 6 \$ 153.87 \$ 152.38 \$ 150.31 \$ 146.77 7 \$ 151.74 \$ 149.71 \$ 146.27 8 \$ 149.21 \$ 145.84 9 \$ 145.46

Figure 1 Mean Unit Price Difference (with 90% Confidence Interval) between the First and the Last Lot Sold in a Sale.

Figure 2 A buyer’s Expected Maximum Valuation Given Different Demand (m lots) and Number of Lots Won.

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