Delays in Simultaneous Ascending Auctions* Yuan-Chuan Lien a, ** a

Department of Economics, Stanford University, United States

June 24, 2007 Abstract This paper uses auction data from the Federal Communication Commission (FCC). It concludes that, in the FCC’s simultaneous multiple round auctions, when identical items are auctioned separately, significant delay may occur as a result of poor coordination among bidders. The progressively more restrictive activity rules which divides an auction into stages may result in bidding on only some licenses in the early stages and postponement of the others until a later stage, therefore significantly prolonging the auction. JEL classification: D44; L96 Keywords: Simultaneous ascending auction; Delay; Federal Communication Commission

1.

INTRODUCTION

Past FCC spectrum auctions have sometimes run for months in a process that is quite costly for participants.† For example, it took 112 rounds and more than three months for Auction 4, the Broadband PCS A and B Block, to reach completion. Depending on the number of licenses and the amount of money involved in the spectrum auction, bidders may be given hours or even a whole day to deliberate before placing bids in a new round. Therefore, when the number of rounds it takes to reach the end of the auction is large, the time duration is usually long as well.

*

I am indebted to Paul Milgrom for his generous support and advice. I also thank Liran Enav, Andrzej Skrzypacz, Ilya Segal, Muriel Niederle, Manuel Amador, and Anita Alves Pena for helpful comments and discussions.

**

email: [email protected]; webpage: http://www.stanford.edu/~yclien.

†

The duration or the number of rounds an auction takes to reach the end is considered as an efficiency measure. For example, see Cybernomics (2000).

1

The number of rounds an auction requires depends on the design details. Using data from Auction 4 from FCC, we can summarize the causes for elongation of this auction in three categories: 1. Nearly identical blocks of radio spectrum, block A and B in the same region, are sold separately in this simultaneous multiple round auction. 2. Relaxed activity requirements in the early stages of this auction. 3. Long pauses of bidding activities for some items during the auction. Category 1 delay arises because two identical radio spectra in each region are auctioned separately and the number of bidders is small for spectra in some regions. To illustrate this delay, suppose A and B are two identical items auctioned separately and there are three bidders, b1, b2, b3, each of whom demands exactly one item. (This means each bidder demands either A or B, but not both.) Suppose with winning bids of $10, b1 and b2 are the current winners of A and B, respectively. b3 then bids $11 on A in the new round and outbids b1. In the next round, b2 places a bid of $11 on B and outbids b2. As a result, it takes two rounds for the price of those identical items to increase by one tick ($1). This is a delay because identical items are sold at almost the same prices eventually, as shown by the data. One way to avoid this kind of delay is to pool identical items together, use a single price for those identical items and increase the price whenever total demand is greater than total supply. In this way, it will take only one round to increase the price by a tick.‡ A discussion of auctions with multiple divisible goods can be found in Ausubel and Cramton (2004).§ To measure the prevalence of this type of delay in Auction 4, we construct a measure, delay ratio, which is the ratio of the number of rounds that A or B block (or both) receives new bids, to the larger number between the number of rounds that A block ‡

A similar example can be found in Milgrom (2004), p. 279. Also, to be fair, raising prices in this way can theoretically cause the price to overshoot by on increment. This might happen, for example, if two bidders are willing to pay 10 but the one who raised the bid is willing to pay 11. The FCC’s inability to find a solution to this problem caused it to reject a proposed modification of this sort in 1998.

§

A similar type of delay may arise when goods are only closely substitutable but are not identical. This type of delay is more complicated and cannot be resolved by pooling goods together because goods are not identical. We characterize this type of delay as part of the category 3 delay in this paper.

2

receives new bids and the number of rounds that B block receives new bids. The average delay ratio is 1.74 in the data, which implies that if A and B blocks in the same market had been auctioned together and only one price had been assigned to both blocks, each market could have taken 1/1.74 = 40% fewer rounds to finish on average. Category 2 delay results from the more relaxed eligibility requirements in early stages. Auction 4, for example, is divided into three stages and has more relaxed activity requirements in the early stages. As a result, by bidding only on part of the desired licenses in the early stages, a bidder is able to maintain the eligibilities required to bid on all the bidder’s desired licenses in the final stage. Therefore, in an early stage bidders are likely to compete actively only on a fraction of the licenses, and the three stages are like three sequential auctions, each of which sells only a fraction of the licenses. In the worst case scenario, the rounds it takes to end the auction can be tripled. We show by data from Auction 4 that although at the end of each early stage the activity level was low, the total eligibilities did not decrease much. This implies that bidders were not bidding actively on all items they actually demanded and when this happened, the progress of the auction was slowed down. We also show that many licenses received most of their bidding activities at the third stage, while other licenses completed most of their bidding activities in the first two stages. Finally, we show that if progress is measured in terms of revenue increases and new high bids, the rounds near the ends of stages 1 and 2 contributed little to the progress of the auction. To reduce the category 2 delay, the FCC has already begun to experiment with tighter activity requirements. In Auction 66, for example, they employed only two stages, rather than three, and set the activity requirements for the two stages at 80% and 95%, rather than at 33%, 67% and 100%. The FCC also adopted activity related bid increments, so that the license in highest demand had their prices increase most rapidly. Category 3 delay is related to bidders’ need to switch demands between different licenses when prices change during the auction. Therefore, the delay is necessary to achieve the full flexibility provided by a simultaneous multiple round auction. Since our

3

Maximum bids (in million) for all licenses. 600 500 400 300 200 100 0 M051A

M049A

M047A

M045A

M043A

M041A

M039A

M037A

M035A

M033A

M031A

M029A

M027A

M025A

M023A

M021A

M019A

M017A

M015A

M013A

M011A

M009A

M007A

M005A

M003A

M001A

Figure 1. A and B blocks in each of the 51 region are sold at similar prices. Note that

blocks 1A, 2A and 10A are not in this auction. Block 1B, 2B, and 10B are drawn in dotted lines.

goal is to expedite an auction without sacrificing its advantage, we concentrate on the first two categories of delays in the remaining discussion. 1.1

DATA DESCRIPTION

FCC Auction 4 ran from December 1994 to March 1995, involving net bids as high as $7 billion. There were 51 markets with A and B blocks of spectrum for sale in each market, except in markets 1, 2, and 10, where only B blocks were for sale. Thus, there were 99 licenses for sale in this auction in total. Each bidder was allowed to acquire only one license between A and B blocks in each market by the FCC rule . In addition, A and B blocks in the same market had the same bandwidth, 30MHz, and can be fairly treated as identical items. However, A and B were auctioned separately in Auction 4 with two independent prices. The auction was divided into three stages with increasingly stringent activity rules from the first stage to the third stages. Since the auction rules were designed for licenses that might cover differently sized populations and different amounts of bandwidth,

4

Bidder 14 Licenses Round 80 79 78 77 76 75 74 73 72 71

Bidder 24

Bidder 26

14A

14B

14A

14B

14A

14B

0.0 79.7 0.0 0.0 0.0 0.0 0.0 0.0 68.9 0.0

0.0 0.0 0.0 0.0 75.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 75.9 0.0 0.0 0.0 0.0 0.0 0.0

82.7 0.0 0.0 0.0 0.0 0.0 71.4 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 72.3 0.0 0.0 0.0 65.6

0.0 0.0 78.7 0.0 0.0 0.0 0.0 68.0 0.0 0.0

Table 1. Bidding activities of bidder 14, 24 and 26 on licenses 14A and 14B. This table shows that the bidders bid on A and B by turns and when this happen it takes two rounds to increase the prices of both A and B blocks by a tick.

license sizes were measured in terms of the product of bandwidth and population coverage, or MHz-POPs. The eligibility rule was based on MHz-POPs of the licenses that were bid upon. Given the uniform 30MHz of bandwidth of license in this auction, this was equivalent to measuring licenses according to the populations they covered. In order to be eligible to bid on licenses with a certain sum of MHz-POPs in later rounds, a bidder is required to maintain her bidding activities on more than some specified percentage of this final sum of MHz-POPs in the early rounds. The eligibility for a bidder in each round is calculated by summing up the MHz-POPs corresponding to the licenses on which the bidder is either currently winning or actively bidding. The transition between stages are announced by FCC based on the frequencies of the total new bids being placed. In the data, the second stage starts from Round 12 and the third stage begins at Round 65. The required activity in each of the 3 stages is described more specifically as follows. Denote bidder j ’s eligibility at the start of stage i by e ( i, j ) and the end-ofstage eligibility resulting from bidder j ’s actual bidding activity by eˆ ( i, j ) . We have e ( i, j ) ≥ eˆ ( i, j ) . At the end of the first stage, if eˆ (1, j ) ≤ e (1, j ) / 3 , then at the start the

second round the eligibility is reset as, otherwise e ( 2, j ) = e (1, j ) . Similarly, at the start

5

Bidder 2

Bidder 14

Bidder 26

Bidder 30

Licenses Round

18A

18B

18A

18B

18A

18B

18A

18B

80

34.5

0.0

*34.5

0.0

0.0

0.0

0.0

0.0

79

0.0

0.0

0.0

0.0

0.0

*33.5

*32.9

0.0

78

0.0

*31.9

*31.3

0.0

0.0

0.0

0.0

0.0

77

0.0

0.0

0.0

30.4

0.0

*30.4

0.0

0.0

76

0.0

0.0

29.8

0.0

0.0

0.0

*29.8

0.0

75

0.0

*28.9

0.0

28.9

0.0

0.0

0.0

0.0

74

28.4

0.0

0.0

0.0

*28.4

0.0

0.0

0.0

73

0.0

0.0

*27.0

0.0

0.0

0.0

0.0

*27.5

72

*25.5

0.0

0.0

0.0

0.0

0.0

0.0

0.0

25.8 0.0 71 (* indicates the winning bids)

0.0

0.0

0.0

*25.8

0.0

0.0

Table 2. Bidding activities of bidder 2, 14, 26 and 30 on licenses 18A and 18B. This table shows that in a market with 4 bidders, sometimes the two losing bidders happen to bid on the same item (e.g. license A in round 74), in which case only the price of that item would increase and it takes another round to increase the price of the other item (e.g. license B in round 75.)

of the second stage, e ( 3, j ) = eˆ ( 2, j ) × 1.5 if eˆ ( 2, j ) ≤ e ( 2, j ) /1.5 , otherwise e ( 3, j ) = e ( 2, j ) . At the end of the third stage, the activity is eˆ ( 3, j ) ≤ e ( 3, j ) , which

corresponds to the licenses that bidder j ’s wins. The FCC data includes all bids placed by all bidders and the provisional winning bids at each round. Therefore, we are able to rebuild the whole bidding history. The basic unit of data can be represented by a quadruplet: bidder, round, license, bid amount, which can be conveniently sorted in the dimensions of interest. For example, to study the bidding activities, we can write the number of new bids as a function of the round number and the identification number of a license.

2. 2.1

TYPES OF DELAYS DELAYS FROM BIDDING ON TWO IDENTICAL ITEMS

When there are three bidders bidding on A and B blocks in the same market and each bidder demands only one of these two identical items, it takes about two rounds to

6

increase the prices on both items by a “tick” (the minimum required bid increment for a license in the round). The following instance excerpted from the data demonstrates this phenomenon. Example 1. The type of delay involving 3 bidders.

Table 1 shows the bidding activities from round 71 to round 80. There are 3 bidders bidding on licenses 14A and 14B. At round 73, bidder 14 and 26 are the standing high bidders of licenses A and B with prices p A = 68.9 and pB = 68.0 , respectively. At round 74, bidder 24 bids on license B and becomes the winner with price

pB + ∆pB = 71.4 . At round 75, bidder 26 wins back a license by bidding on license A with price p A + ∆p A = 72.3 , where ∆pB = 3.4 = ∆p A = 3.4 . Therefore, the number of rounds needed to reach an increment of prices for identical items is two. When there are 4 bidders bidding on two identical items, the delay still exists, which is shown in Example 2 using the data from Auction 4.

Example 2. The type of delay involving 4 bidders.

In round 73, bidders 14 and 30 are the winners of licenses A and B respectively. In round 74 bidder 2 and 26 want to win back the licenses, but they both bid on license A. As a result, only bidder 26 wins. Similarly, in round 75, bidders 2 and 14 place bids, but only bidder 14 wins. These examples show that the auction slows down because bidders do not know how to prevent bidding on the same item. If bidders are able to coordinate their biddings and bid on different items in a round, the prices for both items can increase simultaneously and the number of rounds to reach the final allocation can be reduced. However, even when there is only a slight difference between the prices of two identical items, both of the losing bidders tend to bid on the same item with the lower price. For the simplicity of analysis, in the following discussion we assume that the bidders choose items on which to bid randomly when prices are similar. Thus, the derived delay is an underestimate of the real elongation.

7

Figure 2. Histogram of number of markets with different delay ratios. Delay ratio is

defined in Definition 1. Higher delay ratio indicates a higher frequency that A and B blocks in a market receive new bids in different rounds.

As we will show shortly, a simultaneous multiround auction theoretically needs two rounds to increase the price by a tick if there are 3 bidders bidding on two identical items, and 1.5 rounds per tick if there are 4 bidders bidding on two identical items. In a round when the prices of both blocks are the same, two losing bidders may bid on the same block, which results in the need of one extra round to increase the prices of both blocks by a tick. On the other hand, two losing bidders may also bid on two different items and both prices increase in one round. We generalize this argument in the following proposition. Proposition 1. When n bidders bid on two identical items in a simultaneous multiround auction and each bidder demands only one item, the expected number of rounds it takes to end the auction is 23−n + 1 times the number of rounds it takes to end an uniform price ascending auction with multiple items, in which there is only one price for those two identical items.

When p A = p B , there are 2 / 2(

n− 2 )

chances that all n − 2 losing bidders bid on the same

item and only one of the prices increase. Another round is needed in this case to have the

8

price of the other item increased. Thus, it takes 2 × 23−n + 1 × (1 − 23−n ) = 1 + 23−n rounds

in expectation to increase both prices by a tick. When n = 3 , we get 2 . When n = 4 , we

get 1.5 .  Following the same line of thought, the above result can be extended to the case when there are more than two identical items in the market. Suppose there are n bidders bidding on m identical items. Suppose in the beginning of the auction all the items have the same starting price. If bidders choose items to bid on randomly, then it is likely that some of the items will receive no new bids. Let i be the number of items that receive new bids. Then m − i is the number of items that do not receive any new bids in round t . The prices of the m − i items will be lower in the next round t + 1 so all the losing bidders will bid only on these items in round t + 1 . Even if all the losing bidders bid on these m − i items, there is still chance that not all of the m − i items will receive new bids and we have to repeat the same argument. The expected number of rounds it takes to increase the prices of all items by a tick can be summarized as follows. Corollary 1. Let P ( i, m, n ) be the probability that n bidders bid on the same set of i items among all m items and suppose m < n . In addition, denote by f ( m, n ) the expected number of rounds it takes for the prices of all items to increase by a tick and set f ( 0, ⋅) = 1 . Recursively,

f ( m, n ) can be written as:

f ( m, n ) =

∑ P ( i, m, n ) f ( m − i, n − i )

1≤i ≤ m

In the real auction, the number of bidders bidding actively is different in different markets and the number is also changing from round to round. Using the data, we can summarize the effect of category 1 delay by comparing the maximum number rounds a market receives a new bid to the maximum number rounds a block receives a new bid. The latter represents the number of rounds to reach the end of the auction if prices of both A and B blocks in a market increase at the same time.

9

Bid difference ratio 0% − 5% 5% − 10% 10% −15% 15% − 20% 20% −

Percentage of instances 74.82% 17.13% 4.54% 1.03% 2.48%

Accumulative 74.82% 91.95% 96.49% 97.52% 100.00%

Table 3. Frequencies of bid difference ratios in all

events of new bids. Bid difference ratio is defined by the ratio of bid difference between A and B blocks to the average bid between A and B blocks in each market.

Definition 1. (Delay ratio) Let g A ( i ) denote the number of rounds with new bids placed for A block license in market i . g B ( i ) is defined similarly. g A|B ( i ) is the number of rounds in which either

an

A

or

B

license

(or

both)

receives

new

bids.

The

ratio

rD ( i )

= g A|B ( i ) / max {g A ( i ) , g B ( i )} indicates the delay ratio on average. An auction that takes

rD ( i ) T rounds to finish should be able to finish in T rounds if A and B are auctioned using the

same price.

In Figure 2, we calculate the histogram of rD ( i ) for all 48 markets with both A and B blocks available for sale. Three markets 1, 2 and 10 are excluded since each of them has only one block (block B) for sale. For studying delays, these three markets are omitted. Without these three markets, the average delay ratio is 1.7445 with a standard deviation of 0.1657. In order to show that the bids on A and B blocks increase by turns, we also need to show that most of the time during the auction, the bids on A and B blocks are with a similar value. Denoting the winning bid on a license in market i during round j by bA ( i, j ) , we calculate the bid difference ratio:

bA ( i, j ) − bB ( i, j ) / ⎡⎣( bA ( i,112 ) + bB ( i,112 ) ) / 2 ⎤⎦ , for every instance when a new bid is placed. There are 1,652 such instances. Table 3 shows the frequency of the bid difference ratios.

10

From Table 3 we know that more than 91.95% percent of the time when new bids are placed in a certain market, the bid/price difference is less than 10% of the total price. Note that the required bid increments for all licenses are around 5% to 15% of the corresponding prices in the previous round. In summary, the data shows that the new bids on A and B blocks are frequently placed in different rounds, and the bid differences among A and B blocks are small during all rounds. This implies that bidders are going back and forth bidding on both blocks during the auction, and prices on block A and B are increased by turns. Therefore, auctioning similar items separately increases significantly the rounds needed to complete the auction. Claim 1. Auctioning two identical items separately causes delay in FCC Auction 4.

Total number of rounds is 111 in Auction 4, while any of the 99 licenses receives new bids in no more than 37 rounds. This means that if bidders had placed new bids on all licenses in each round before the end of the auction, the auction would have stopped after 37 rounds. However, prices of A and B may take turns to increase. If we count the number of rounds in which either A or B license (or both) in a certain market receives new bids, then the maximum number of rounds that any such pair of A and B licenses in a market receive new bids is 61 rounds. Thus, auctioning identical items separately caused 61-37=24 rounds of delay. 2.2

DELAYS FROM MORE RELAXED ACTIVITY RULES IN THE EARLY STAGES

We want to show that because of the activity rules in Auction 4, bidders did not bid actively on all licenses in the early stages. In addition, bidders tried to prevent competition with each other in the early stages. Coordination of bidding on different licenses takes some try-and-error rounds. Therefore, the auction is prolonged significantly by dividing it into several stages, with less restrictive eligibility requirements in the early rounds. At the end of an auction, the total activity equals the sum of the MHz-POPs (or just population, since all licenses have equal bandwidth) of all licenses. FCC announces

11

transitions between stages when the bidding activity (as measured in MHz-POPs on the licenses which receive new bids) is less than a certain percentage of the total MHz-POPs of the whole market.** Only 33% and 67% of the total eligibility are needed in stages I and II to maintain the eligibility for the final stage. Since the auction has to go through stages before reaching the official end, bidding activity tends to be low at the end of each stage. To illustrate the point, assume for simplicity that activity is measured by the number of licenses a bidder bids on instead of the MHz-POPs used in FCC auction. Suppose a bidder wants to bid on all n licenses in the last stage. In the first stage, only 33% of the activity, or n / 3 licenses, need to be maintained. If at the end of the first stage there are no new bids placed by any bidder, then the total available eligibilities of all bidders are less than or equal to 3 times the number of licenses ( 3n ). One possible scenario is that there are 3 bidders all of whom want all of the licenses. If they each bid on different n / 3 licenses at the end of first stage, then they do not need to compete with each other and the prices of all licenses do no increase. All of them can still maintain the eligibility of bidding on n licenses in the second stage. However, if there are more than 3 bidders and the original total eligibilities of all bidders a0 are more than 3 times the number of licenses, a0 > 3n , then some of the bidders need to give up some of their eligibilities before the end of the first round. For some of the bidders to give up some of the licenses they demand, the prices of those licenses need to increase to a certain level. In order to prevent competition on certain licenses before the final stage, it is believed that bidders sometimes “park” their bids on other licenses in the early stages by bidding on the licenses that they do not really want. By doing so, the bidders get to maintain eligibility for the final stage.

**

“(5) Stage transition: The auction will start in Stage One. Under our general guidelines it will advance to the next stage (i.e., from Stage One to Stage Two, then from Stage Two to Stage Three) when, in each of three consecutive rounds of bidding, the high bid has increased on ten percent or less of the spectrum being auctioned (as measured in MHz-pops). However, the FCC retains the discretion to speed up this auction by announcing, at any time, that the next stage will begin in the next bidding round.“ (from Report No. AUC-96-11-A (Auction No. 11))

12

Round 1 12 (start of 2nd stage) 65 (start of 3rd stage) 112 (last round)

total eligibility 26,110,273,560 26,108,863,560 20,674,923,960 13,609,716,360

ratio to last round total eligibility 1.9185024 1.9183988 1.5191297 1

Table 4. Total eligibilities at rounds of stage transition, and

their ratios to the last round eligibility.

Now we add more details from Auction 4 into the discussion. As we will show later using the data that competition arises mainly in the second and third stages, so, we will concentrate on the transition from the second to the third stage. Recall that only 2/3 of the final stage eligibility needs to be maintained in the second stage. Since A and B blocks in a market are identical, at most n / 2 of licenses are desired by any bidder, which means n / 2 × 2 / 3 = n / 3 licenses are needed for a bidder to maintain eligibility on bidding for n / 2 licenses in the third stage. Suppose three such bidders successfully prevent

competing with each other at the end of the second stage. At the start of the third stage, at most n / 2 licenses would be bid on. This discussion can be generalized as the following proposition. Proposition 2. Let x denotes the percentage of activity required in a stage. If the total eligibility from all bidders is more than n/x at the start of this stage, then at least one license has bidding activity during this stage.

This proposition shows that when the total eligibility is large enough, then some of the licenses must have bidding activity. However, the number of licenses having activities depends on the demand structure, that is, the number of bidders in the auction and the licenses that they demand. The following examples show how the number of licenses can vary from one license to all licenses. Example 3. Bidding activity on 1 license.

Suppose there are 2n small bidders in the market and each bidder demands exactly the same license. The remaining n-1 licenses are not demanded by any bidder. Then

13

obviously there will be only one license that has bidding activity, although the total eligibilities are 2n. Example 4. Bidding activity on all licenses.

Suppose there are 4 bidders and one of them has lower valuations for all items than the rest of bidders. Also suppose all bidders demand all licenses at the start of the auction. Then all licenses will be bid on before this bidder with low values drops out completely, even when the remaining 3 bidders could avoid bidding against each other before the start of the third stage. Example 5. Bidding activity on all licenses with an one-license bidder.

Suppose the low value bidder in the above example is replaced by another low value bidder with demand of one unit of the licenses. The total eligibilities is now 3n+1. However, since any license is the same to this bidder, this low value bidder will bid on all the licenses before giving up bidding. As a result, all licenses will be bid on. Example 6. Bidding activity on part of the licenses

Suppose there are 4 identical bidders and each bidder wants to bid on all markets in the beginning of the second stage. If all 4 bidders enter the third stage, 4 × n / 2 = 2n licenses of total activity are needed in beginning of the third stage, which is n / 2 more than possible, as shown by the example with three bidders above. Therefore, some of the competition must be resolved before entering the third stage. Suppose x licenses in x / 2 markets are resolved at the end of the second stage. Two (but not all) possible scenarios are: 1. each of some two out of the four bidders gets half of x and the other two give up bidding on those x / 2 markets. The eligibility needed

for

each

losing

bidder

( n / 2 − x / 2 ) × 2 + n / 2 × 2 = 2n − x

is

(n / 2 − x / 2)

.

Total

activity

is

now

, so x = 0.5n for the eligibility to be less than

possible amount, which implies that x / 2 = 0.25n markets are given up by two of the 4 bidders. 2. there are four bidders each of whom give up bidding on a different set of x / 4

14

licenses. These licenses given up by different bidders constitute x licenses in x / 2 markets. Now each bidder only needs to bid on ( n / 2 − x / 4 ) licenses. Total activity is now ( n / 2 − x / 4 ) × 4 = 2n − x , the same as above. Now let us take a look at the eligibilities in the data. Table 4 shows the total eligibility in different stages from Auction 4. From the first stage to the start of the second stage, total eligibility almost does not change. Since the total eligibility 1.9 (ratio) is well below 3.0 (=1/33%, where 33% is the activity requirement for the first stage), it is possible for bidders not to compete on any of the licenses in the first stage and at the same time to keep the original eligibility when entering the second stage. From the activity graph (Figure 3), we can see that the number of new bids almost reaches 0% at the end of the first stage, which means bidders do postpone competition until later stages. To avoid bidding on the same items and thus increasing their prices in the first stage, bidders have to bid on different items. However, it takes rounds for bidders to coordinate bidding on different licenses. In the data, it takes 11 rounds in the first stage in Auction 4 for them to bid on different things. Bidders either become the current winners of some of the licenses they demanded, or “park” on some unwanted licenses. Since the total eligibility remains the same, which means no bidder loses her eligibility, the necessity of the first stage is questionable.

15

Table 5. Ratios of first and last round eligibilities for each bidder to the total activity in

the last round. 1st round Maximum eligibility

Company

ALAACR Communications, Inc. AT&T Wireless PCS Inc. American Portable Telecommunication Ameritech Wireless Communication BellSouth Personal Communication Boston PCS Venture CCI Data, Inc. Centennial Cellular Corp. Century Communications Corp. Cleveland PCS Venture Comcast Telephony Services II, Communications International C Continental Cablevision, Inc. Cox Cable Communications, Inc. Data Link One, Inc. GCI Communication Corp. GTE Macro Communications Corporation MICRO LITHOGRAPHY, INC. PCS America Limited Partnership PCS PRIMECO, L.P. Pacific Telesis Mobile Service PhillieCo, L.P. Poka Lambro Telephone Cooperation Powertel PCS Partners, L.P. Satellite Broadcast Network, I South Seas Satellite Communication Southwestern Bell Mobile System Western PCS Corporation Windsong Communications, Inc. WirelessCo, L.P.

1648250000 3918802150 1020000000 238896750 344226900 283581400 900000000 108715400 116419150 148372500 68248000 1410000 183634300 165464500 1410000 56468200 2500000000 100000000 289500000 2733321550 2793276810 267832450 89595750 833810300 16501300 1410000 857822900 500000000 1410000 5921893250

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Ratio of 1st Ratio of last round round eligibility to eligibility to last round last round total activity total activity 12.11% 0.00% 28.79% 23.61% 7.49% 5.84% 1.76% 1.76% 2.53% 2.53% 2.08% 0.00% 6.61% 0.00% 0.80% 0.80% 0.86% 0.00% 1.09% 0.00% 0.50% 0.00% 0.01% 0.01% 1.35% 0.00% 1.22% 0.37% 0.01% 0.00% 0.41% 0.12% 18.37% 4.27% 0.73% 0.00% 2.13% 0.00% 20.08% 12.61% 20.52% 6.84% 1.97% 1.97% 0.66% 0.45% 6.13% 1.98% 0.12% 0.00% 0.01% 0.01% 6.30% 1.46% 3.67% 3.03% 0.01% 0.00% 43.51% 32.36%

At the start of the second stage, the total bidding activity, characterized by new bids placed, is about 30%; at the end of the second stage, the bidding activity drops to about 7% (as shown in Figure 3). However, at the start of the third stage, bidding activity rises again to around but slight less than 30%. The drop and rise of activity is consistent with the conjecture that bidders try to postpone competition until the last stage. In addition, the total eligibility drops from 1.9 to 1.519 (as shown in the data) in stage 2, which means some of the bidders must have given up some of the licenses before the start of the third stage. The drop of activity within the second stage comes from two factors: 1. some bidders give up bidding on certain licenses. This results in the reduction of the total eligibility. 2. bidders find appropriate licenses and avoid competing with each other before the start of the final stage. In the data there are 13 licenses (or five markets (around 10% or slightly more)) which reach their final prices before the start of the third stage; there are only about 30% of new high bids in the start of the third stage. This 30% represents about 30 among the total of 50 markets, thus, we know there were about 15 markets that were bid on later in the third stage. These markets could be substitutes for some of those 30 markets to some bidders. These bidders bid on some licenses until the prices are too high and then switch to other licenses with lower prices. From the eligibility data of the first round (Table 5), we can tell that not all bidders want to have licenses in all markets. At the same time, no company wants more than 50% of the licenses, which is consistent with the rule that each company is allow to demand either A or B block in a market, but not both. Although bidders may not be able to avoid competition perfectly because of the demand structure and the coordination difficulty mentioned above, from the low activities in round 11 and round 64, we can see that bidders seem to try to postpone competition. As shown by the calculation above, some bidders have to give up their eligibilities, which is the same as giving up bidding on some items. There are some bidding activities and the prices of some items increase to a certain point that some bidders eventually quit bidding on them. At the same time, not all licensees need to have activity. As a result, bidders bid

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Figure 3. Source: Federal Communications Commission.

on some but not all licenses in the second stage. From data, there are only 5 markets that reach their final prices before the start of the third stage. From Figure 4, we can see that for most licenses, a great portion of bidding activities take place in the third stage, while a few of the licenses stop bidding activities before the third round. For example, market 8 has 33 rounds of bidding activities and reaches its final prices before the end of the second stage. In subsection 2.1, we know from the data that a market has at most 61 rounds of bid increments. However, different licenses have most of their bidding activities occur in different stages. The market which takes the most rounds of activities is not the same across stages. One can picture this auction as composed of two sequential auctions, with one auction ending before the other starts. In the most unfortunate case, it can take up to twice the number of necessary rounds, or about 2 × 61 = 122 rounds in our example.

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100 90 80

Percentage

70 60 50 40 30 20 10 0 M051A

M049A

M047A

M045A

M043A

M041A

M039A

M037A

M035A

M033A

M031A

M029A

M027A

M025A

M023A

M021A

M019A

M017A

M015A

M013A

M011A

M009A

M007A

M005A

M003A

M001A

License ID

Figure 4. Percentage of bidding activities in the third stage to total bidding activities.

If we count the number of new bids in stages I, II, and III separately, the largest number among all markets (regions) is 9, 35, and 37 separately. From the last section, we know this takes into account of the delay on separating identical items in auction. Therefore, if we take into account the delays from waiting to reach the end of all stages, we have a total number of 9+35+37=81, which is 20 rounds larger than the previously calculated 61 rounds, which we calculated by omitting the transition of stages. There are still 112-81=31 rounds of delay unexplained until this point. 2.3

DELAYS FROM LONG PAUSES

In order to have a closer look at the reasons of delays. The bidding activity is shown in Figure 5. Activities of A and B licenses in a market are combined in a row. From this figure, we can see that some licenses in a market have no activity for some consecutive rounds before picking up further activities. By combining activities of both licenses in a market when drawing Figure 5, we have already taken into account the kind of delay resulting from the “3-bidders-on-2-licenses” type of delay. In addition, as also shown in

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Figure 5. Bidding activities in all markets and all rounds. Each dot represents new bids placed

on Block A or B or both in a market.

the figure, the pauses occur across stages, partly caused by the activity rules mentioned in the previous subsection. The “pauses” between bidding activities in the bidding history of one pair of A and B licenses also constitute a major part of the duration in the auction. We think of this type of “delay” as necessary. In every new round, given the current high bids or prices, each bidder solves a complicated maximization problem. When the prices for some items are too high, bidders may switch their demands to other items. Sometimes bidders resume bidding on some previously bid items when the prices for others become too high. This flexibility provided in the simultaneous ascending auction is advantageous to bidders for combining licenses.

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3.

CONCLUSION

In the recent FCC Auction 66, Advanced Wireless Services (AWS-1), the number of stages was reduced to two. In the first stage, the activity requirement is to bid actively on licenses that representing 80% of the current eligibility in order to maintain the current eligibility. This requirement was much tighter compared to 33% and 66% for the first and second stages in Auction 4, in terms of number of stages and eligibility percentages. From the analysis of this paper, a tighter activity rule is supposed to reduce the delay caused by the stage transitions. However, in the second stage of Auction 66, the rule required bidders to bid on 95% of the current eligibility, instead of the 100% in the third stage of Auction 4. Therefore, further analysis is necessary to determine the total effect due to changes in the activity rules. When goods for sale are not identical but are very close substitutes, delays similar to the type I delay mentioned in this paper can also take place. One bidder with a high valuation for one unit of those close substitutes can prolong an auction for many rounds by bidding on each of those goods one round after another. (See FCC Auction 69 for an example.) Because these goods are not identical and cannot be pooled together, using a clock auction design is not a suitable solution. The physical duration of each round at the later phase of an auction could be shortened to accelerate the auction. Automating an auction by using proxy bidders to represent a bidder’s demand can also reduce the time needed. However, the effects of these solution are yet to be evaluated. References Ausubel, Lawrence M. and Peter Cramton, Auctioning Many Divisible Goods, Journal of the European Economic Association April–May 2004 2(2–3):480 – 493 Ausubel, Lawrence M., Peter Cramton, , R. Preston McAfee and John McMillan, Synergies in Wireless Telephony: Evidence from the Broadband PCS Auctions, Journal of Economics and Management Strategy, 6, 1997, 7-71. Cybernomics, “An Experimental Comparison of the Simultaneous Multi-Round Auction and the CRA Combinatorial Auction.”, Combinatorial Bidding Conference, Federal Communication Commission, March 15, 2000

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Cramton, Peter, The Efficiency of the FCC Spectrum Auctions, Journal of Law and Economics, 41, 727-736, Oct. 1998. Katok, Elena and Alvin E. Roth, Auctions of Homogeneous Goods with Increasing Returns: Experimental Comparison of Alternative “Dutch” Auctions, Management Science, Vol. 50, No. 8, August 2004, pp. 1044–1063 Milgrom, Paul, Putting Auction Theory to Work, Cambridge University Press, 2004 McAfee, R. Preston and John McMillan, Analyzing the Airwaves Auction, Journal of Economic Perspectives, 10(1):159--175, Winter 1996

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