C-1 
ATTACHMENT C 
Determining Provisionally Winning Bids and Current Price Estimates  
 
 
The licenses subject to package bidding in a Hierarchical Package Bidding auction are grouped 
into non-overlapping, successively larger packages at each level.  For purposes of this 
explanation, we are using an HPB auction as proposed in this Public Notice ? with two levels, 
with 12 individual REAG licenses in the first level and several packages of different sizes in the 
second level, as shown in Table 1 below.  After each round, all considered bids and minimum 
opening bids are used to determine the largest bid on each license and on each allowed package, 
using the random tie-breaking rule.
1
  A simple procedure is used to calculate the provisionally 
winning set of bids and a “current price estimate” (“CPE”) for each individual license.
2
  This 
appendix describes the winner determination and pricing rules, first in terms of an example and 
then in more general terms.   
 
Table 1. Two-Level Structure 
Level 2: Packages  50 States  Atlantic  Pacific  
Level 1: REAG licenses R1 R2 R3 R4 R5 R6 R7 R8 R10 R12 R9 R11
 
 
Determining Provisionally Winning Bids 
After each round of bidding, the FCC Auction System determines which combination of bids 
together provides the greatest aggregate gross amount and are therefore provisionally winning.  In 
order to determine the best combination of bids, the system takes the highest bid on each 
individual license.  It then compares the sum of the high bids on individual licenses in a package 
with the highest package bid to determine provisionally winning bid(s).   
Example 1:  Suppose that the highest bids on each license and package are as shown in Table 2 
below.  The sum of the highest bids on REAG licenses R1-R8, 80, exceeds the highest bid on the 
50-state package, 70; the sum of the highest bids on REAG licenses R10 and R12, 20, exceeds the 
highest bid on the Atlantic package, 15; and the sum of the highest bids on REAG licenses R9 
and R11, 20, exceeds the highest bid on the Pacific package.  Thus, the provisionally winning 
bids would be the bids on the individual REAG licenses.   
 
                                                 
1
 The set of considered bids in a round includes new bids for that given round and each bidder’s highest bid 
on each license or package, regardless of the round in which it was placed.  For a description of how tied 
bids are broken, see Section IV.C.6. “Provisionally Winning Bids.”   
2
 The hierarchical structure of HPB was suggested by Rothkopf, Peke?, and Harstad, “Computationally 
Manageable Combinatorial Auctions,” Management Science, 44(8), August 1998, 1131-1147.  The pricing 
mechanism for HPB was proposed by Goeree and Holt (2007) “A Simple Combinatorial Auction,” 
Working paper, May 2007.   
 C-2 
Table 2.  Example 1 with High Bids in Parentheses 
Level 2:  
Packages  
50 States  
(70) 
Atlantic  
(15)  
Pacific  
(15)  
Level 1:  
REAGs  
R1 
(10) 
R2 
(10) 
R3 
(10) 
R4 
(10) 
R5 
(10) 
R6 
(10) 
R7 
(10) 
R8 
(10) 
R10 
(10) 
R12 
(10) 
R9 
(10) 
R11 
(10) 
 
 
Example 2:  Next, suppose that the highest bid on the 50-state package changes to 120, as shown 
below in Table 3, which exceeds the sum of the high bids for the 8 individual REAGs included in 
the package.  In this case, the provisionally winning bids would be the package bid for the 50 
states and the bids for the 4 individual REAG licenses in the Atlantic and Pacific packages.   
 
Table 3.  Example 2 with High Bids in Parentheses 
Level 2:  
Packages  
50 States  
(120) 
Atlantic  
(15)  
Pacific  
(15)  
Level 1:  
REAGs  
R1 
(10) 
R2 
(10) 
R3 
(10) 
R4 
(10) 
R5 
(10) 
R6 
(10) 
R7 
(10) 
R8 
(10) 
R10 
(10) 
R12 
(10) 
R9 
(10) 
R11 
(10) 
 
 
Determining Current Price Estimates (CPEs) for Licenses 
CPEs for licenses are determined in such a way as to indicate to bidders how high they must bid 
to compete with the provisionally winning bid on the package containing those licenses in the 
event the package bid is the provisionally winning bid.  If the bids on the individual REAG 
licenses are all provisionally winning, then the CPEs are simply equal to those bids, as is the case 
in a standard simultaneous multiple round (SMR) auction.  But if the bid amount for a package is 
greater than the sum of the bid amounts for the individual licenses it contains, then the highest 
bids received so far for the licenses are scaled up to the package bid amount by adding “shares.”
3 
  
These shares are proportional to the bidding units associated with each license relative to the total 
number of bidding units in the package.   
Suppose that all REAG licenses have equal bidding units, and that the bid amounts are as shown 
as in Example 2 above.  The high bid on the 50 States package is 120, which exceeds the sum of 
the bids for the REAG licenses in the package.  Therefore, the individual bids for licenses R1-R8 
are raised from 10, to 10 + (1/8)[120 ? 80] = 15.  In other words, the “shortfall” of 40 is divided 
among the 8 licenses in the package in proportion to their bidding units (assumed to be equal).  In 
this example, a “package share” of 5 is added to the high bids on the REAG licenses in the 50-
state package, but there is no package share for the licenses in the other packages.   
                                                 
3
 If no bids have been placed on an individual license, the share will be added to the minimum opening bid 
amount. 
 C-3 
Minimum acceptable bids for licenses in the next round of the auction are calculated by adding a 
percentage to the license CPE.
4
  For packages, minimum acceptable bids are the sums of the 
minimum acceptable bids for the licenses in the package.  As a result of constructing CPEs in this 
way, the minimum acceptable bids for the next round will indicate to bidders what they must bid 
to unseat the provisionally winning bids from the last round.   
 
 
Technical Description 
To Determine Provisionally Winning Bids:  Suppose a package r is divided into individual 
licenses, i = 1,…,I
r
 where I
r
 is the number of individual licenses in package r, each with ?
ir
 
bidding units.  Bidders can submit bids on individual licenses and package bids for the entire 
package.  Let b
ir
max
 denote the highest bid for license i in package r and b
r
max
 the highest package 
bid for package r.  In this case, the revenue-maximizing allocation can be found recursively 
(Rothkopf, Peke?, and Harstad, 1998):   
 
1. Pick the larger of b
r
max
 and ?
Ir
i=1
 b
ir
max
, which determines Rev
r
, the revenue for package r.   
2. If Rev
r
 = b
r
max
, the highest package bid for package r is provisionally winning and the 
bids on individual licenses in that package are losing, otherwise the highest bids on 
individual licenses in package r are provisionally winning and the package bid for 
package r is losing.   
 
To Determine CPEs for Licenses:  A second procedure is used to calculate current price estimates 
(CPEs) for licenses (Goeree and Holt, 2007).
5
  The CPE p
ir
 of license i in package r is the 
maximum bid for the license plus a “package share” (if the sum of the high bids for individual 
licenses in the package falls short of Rev
r
).  The shares are determined by the relative sizes of the 
licenses in terms of bidding units.  Let ?
r
 = ?
Ir
i=1
?
ir
 denote the total number of bidding units in 
package r.  With this notation, the pricing formula is:   
 
'
max max
'1
 =   ( ) 
r
ir
I
ir
ir ir r
i
r
p bRevb
?
?
=
+ ?
?
 
where the difference in parentheses is non-negative by construction.  Thus,  
 
• p
ir
 = b
ir
max
 if an individual license bid wins.   
• p
ir
 = b
ir
max
 + package share if a package bid wins.   
 
Current price estimates based on this procedure address the “threshold problem” in the sense that 
they provide benchmarks for bids on individual licenses that will, when incremented, allow the 
individual license bids to displace a provisionally winning bid on the package containing those 
licenses.   
                                                 
4
 In a standard FCC SMR auction (without package bidding), the FCC Auction System adds a percentage to 
the current provisionally winning bid amount for each license to determine minimum acceptable bids for 
the next round.  This percentage may be determined by the activity-based formula, described in Attachment 
B and in Section IV.C.5. “Bid Amounts.”   
5
 The computed prices determine minimum acceptable bids for licenses by adding a percentage.  The 
minimum acceptable bid for a package is equal to the sum of the minimum acceptable bids for the licenses 
it contains.   
 C-4 
 
The procedures for determining provisionally winning bids and CPEs for licenses, set forth here 
for two levels, can be generalized for any number of levels.  With a single level these procedures 
reduce to the SMR pricing and allocation rules in which the provisionally winning bid for a 
license is the highest bid received for that license.  With multiple levels, the algorithm for 
determining provisionally winning bids follows a “bottom-up” approach and checks at every 
level, h, whether it is better to provisionally assign a level-h package to the high bidder for that 
package or to provisionally assign the smaller sub-packages that it contains.  The rule for 
determining CPEs follows from this recursive comparison of level-h and level-(h+1) packages:  if 
the larger packages provide higher revenue, then the prices of the smaller packages (at the less 
aggregated level) are incremented in proportion to bidding units to reflect this shortfall.  Once the 
algorithm finishes, the CPEs of the individual licenses add up to the sum of the associated 
provisionally winning bids.