Comparing the FCC's Combinatorial and Non-Combinatorial

Simultaneous Multiple Round Auctions:

Experimental Design Report

Prepared for the Federal Communications Commission

Jacob K. Goeree, Caltech

Charles A. Holt, University of Virginia

This Version: April 27, 2005

The authors wish to acknowledge FCC staff Mark Bykowsky, William
Sharkey, and Martha Stancill for input and helpful comments. Any views
expressed in this paper are those of the authors and are not purported
to reflect those of Caltech, the University of Virginia, the Federal
Communications Commission, Wireless Telecommunications Bureau, or other
members of the Commission's staff.

**I. Introduction**

The Federal Communications Commission (FCC) has decided to use
laboratory experiments to compare the performance properties of its
simultaneous multiple round (SMR) auction design with its newly
developed simultaneous multiple round auction that permits
combinatorial, or package, bidding (SMRPB). The purpose of this
comparison is to shed light on the conditions under which the FCC should
employ one auction form rather than the other for the assignment of
spectrum licenses.

Experiments are a natural way to compare the different auction formats,
since many aspects of package bidding have proven to be extremely
difficult to analyze using economic theory. Experiments may also be able
to capture behavioral elements (risk aversion, winner's curse effects,
bounded rationality) that are hard to incorporate in standard
theoretical models. These behavioral elements may have important effects
on auction performance, and these effects may be mitigated or
exacerbated by the auction form used. Furthermore, controlled laboratory
experiments allow the researcher to examine the extent to which the
tested auction design assigns licenses to the bidders which value them
most highly. Because the researcher assigns bidders' values for licenses
and packages, at the close of the auction it is possible to evaluate the
economic efficiency of the allocation. In contrast, field data obtained
from actual auctions do not permit one to make a direct inference about
the efficiency of the final allocation since real bidders' values are
unknown. Finally, experimentation offers the opportunity to run numerous
auctions under controlled conditions while changing a single design
feature. This replication and control allows the researcher to spot
systematic differences across auction formats that are not due to random
variations from one auction to another.

The FCC has only limited experience with package bidding. Therefore, the
experiments may help the Commission to avoid any procedural problems and
unintended side effects that become apparent, and will facilitate the
effective use of package bidding.

An important first step in this process is the development of an
"experimental design," which specifies the economic environment, the
experimental procedures, and the criteria used to measure performance.
The purpose of this paper is to propose an experimental design. In
addition to permitting a comparison between the SMR and SMRPB auctions,
the experimental design can also be used as a benchmark for possible
future evaluations of the performance properties of other auctions
(e.g., price-driven "clock" auctions).

While the FCC has sponsored an earlier experimental study of auctions
that allow for package bidding, the currently proposed SMRPB design has
not yet been evaluated in the laboratory.[^1] Furthermore, the proposed
experimental design includes several new features that supply additional
realism (e.g., bidders may face budget constraints and their valuations
consist of both private and common-value components). Finally, besides
revenue and efficiency, the performance measures used to evaluate the
different designs now include other factors, such as the effects on
small bidders, and the extent to which collusion is suppressed.

Section II of this report discusses a method for generating bidder
valuations for licenses based on common and private-value elements,
population covered by the licenses, and economies of scale. Several
treatment conditions are outlined in Section III based upon structural
conditions that might make one auction form more attractive than
another. Section IV describes experimental procedures such as the number
of experimental sessions, and the number and experience level of
subjects. Section V details the performance measures by which the
auction formats are evaluated and compared. Appendix A reviews the
definitions of competitive equilibrium and the core. Appendix B suggests
some specific parameter values for different treatments. The auction
rules are specified in detail in Appendices C and D.

**II. Modeling Bidder Valuations**

1.  *Private and Common Values*

Auction models typically fall into one of two categories. In *private
value auctions*, bidders know their own value for the object but are
unsure about others\' valuations, as in the purchase of a painting for
personal enjoyment. *Common value auctions* pertain to situations where
the object is worth the same to everyone but bidders have different
information about its true value, as in the auctioning of oil drilling
rights. While this dichotomy is useful from a theoretical viewpoint,
real world auctions, including spectrum auctions, are likely to exhibit
both characteristics. For instance, when a painting is auctioned, it may
be resold in the future and the resale price will be the same for all
bidders, which adds a common value element. And in the oil drilling
example, private value differences may arise because of differences in
production technologies.

One way to capture both private and common value elements is provided by
Milgrom and Weber's (1982) "affiliated-values" approach.[^2] Roughly
speaking, bidders' assessments of their values, or "signals", are
affiliated when one bidder's high (low) signal makes it more likely that
others' signals are high (low) as well. This approach is able to
reproduce the independent private values model and the pure common-value
model as limit cases. However, in this setup, each bidder receives a
single signal (and in equilibrium, the bidder with the highest signal
wins), which precludes an analysis of the possible inefficiencies that
arise when separate private and common values are present.

To illustrate, consider the case of spectrum licenses where bidders'
valuations involve private value elements (e.g., geographic preferences)
and common value elements (e.g., future consumer demand for wireless
services). Here, an inefficient allocation results when a bidder with a
relatively high private value for a particular license but who is
relatively pessimistic about future demand is outbid by an overly
optimistic bidder with a relatively low private value. This inefficiency
can be modeled only by letting each bidder receive a private value
signal and a separate independent common value signal.

For purposes of the experiments, we propose that the common and private
value elements enter a bidder's valuation in a simple multiplicative
manner. For example, consider the value of bidder *i* for a single
license *j* that pertains to a particular geographic area with a
population of *q*~j~. Bidder *i*'s valuation for license *j* is modeled
as the product of a "per pop" valuation and the total population covered
by the license. The "per pop" valuation, in turn, is the product of the
private value component for license *j* (i.e. *U*~ij~) and the common
value component (i.e. *CV*~j~). That is:

(1.1) ![](media/image1.wmf){width="1.3465277777777778in" height="0.5in"}

The private value component of the "per pop" valuation, *U*~ij~,
reflects bidder *i*'s per capita profit in region *j* in the "baseline"
state of the market characterized by no consumer demand uncertainty
(i.e., *CV*~j~ = 1). Bidder *i* may believe that future consumer demand
in region *j* may be higher than expected (i.e., *CV*~j~ \> 1) or lower
than expected (i.e., *CV*~j~ \< 1) due, for example, to differences in
expected population growth.[^3] Bidders do not know the common value
component *CV*~j~ at the time of bidding but, instead, receive prior to
bidding an unbiased signal about its value: *s*~ij~ = *CV*~j~ ∙ *e*~ij~,
where the "error" *e*~ij~ has a mean of one (1) and is independent
across bidders.[^4] In the presence of this type of common value
uncertainty, bidders face an *inference problem* in the sense that an
optimistic common value signal may be due to favorable market
circumstances (high common value *CV*~j~ ) or due to overly optimistic
expectations (high error *e*~ij~).

However, this formulation may provide subjects with too much
information: for each license they receive a private value and a common
value signal. A simplification that we recommend involves giving each
bidder receives a single common value signal, *s*~i~, that applies to
all the licenses of interest.[^5] Equation (1.1) then becomes:

(1.2) ![](media/image2.wmf){width="1.2083333333333333in"
height="0.2638888888888889in"}

The introduction of a single common value for each bidder may change the
extent to which bidders are affected by the "winner's curse," which is
caused by the tendency for the bidder with the highest bid to be one
with the most optimistic value estimate. Separate license-specific
common value elements would tend to mitigate the winner's curse effect
since the effects of the error terms would tend to "average out" when
bidders acquire packages of multiple licenses. In contrast, a single
common value for each bidder may exacerbate winner's curse effects since
a bidder's high estimate for one license would imply high estimates for
all other licenses. For example, with a single common value, a bidder
who is optimistic about the demand for wireless services in one
geographic area is equally optimistic about the demand in all areas.

2.  *Packages: Substitutes and Complements*

The value a bidder places on a package of licenses need not be equal to
the sum of the values for the individual licenses. For instance, if
licenses A and B are considered substitutes, the value of the package AB
may be less than the values of A and B added together. Alternatively, if
licenses A and B are considered complements, the value of the package AB
may exceed the values of A and B added together.

The degree of complementarity or substitutability for licenses or
packages can be generated from assumptions about the distributions of
the private and common values within the bidder population. As an
example, consider the case of pure private values (i.e. *CV* = 1) and
suppose bidders' values for licenses A and B are uniformly distributed
on \[100, 200\]. The relationship between license values may indicate
either substitutes or complements if the AB package value is distributed
on \[200, 500\]. When the value draw for the AB package is towards the
lower end (close to 200) the package value will generally be less than
the sum of the license values (although higher than each of the license
values individually). Conversely, when the value draw for the AB package
is towards the upper end (close to 500), the package value exceeds the
sum of the individual license values, indicating synergies between the
two licenses.

Through the use of such distributional assumptions, together with
assumptions regarding license preference overlap, a wide variety of
efficient license assignments can be created. For example, assume there
are two bidders with strong local preferences, one for A and the other
for B. Each bidder's value for its "own" location is drawn from \[100,
200\], with no value for the other location. Assume, further, that there
is a single national bidder whose single item values for licenses A and
B are drawn from \[100, 200\], and whose value for the AB package is
drawn from \[200, 500\]. Depending on the draws, the efficient
allocation may involve assigning both licenses to one bidder while, in
other instances, it would involve assigning each license to different
bidders.

3.  *A Parametric Approach to Defining Valuations*

Another consideration is how to scale up the experiment (in terms of the
number of bidders and packages) without overloading subjects with
information about distributional assumptions on private values (for each
license and package), the common value, and common-value signals. One
approach is to take a setup like the one above for two licenses and a
single package and replicate it -- increasing the numbers of licenses
and bidders, with values drawn from the same distributions as before.

We prefer instead to use a formula-based approach that includes
parametric terms to capture market size, geographic location, and
complementarities. In this manner, the experimenter specifies a large
array of combinatorial valuations with particular properties by
specifying values for a small number of parameters. More importantly,
this formula-based approach may be presented to subjects in a way that
makes it easier for them to understand the nature of value
complementarities on an intuitive level.

For example, consider the value for bidder *i* of a package *X*
consisting of several licenses, where each license
*j*![](media/image3.wmf){width="0.1388888888888889in"
height="0.1388888888888889in"}*X* pertains to a particular geographic
area with a market size of *q*~j~. The package value is the sum of the
individual valuations, modeled as the product of the market size and a
valuation "per pop" that is now the product of three elements: a
bidder-specific private value *U*~ij~ times the common value *s*~j~, and
a *scale component*, *f*~i~(*n*~i~) that is a function of the number of
licenses bidder *i* acquires,[^6] which affects the profitability of the
overall operation.

(1.3) ![](media/image4.wmf){width="2.5416666666666665in"
height="0.4166666666666667in"}

Comparing (1.1) and (1.3) shows that a package consisting of only a
single license has a scale component *f*~i~(1) = 0. There are
complementarities when *f*~i~(*n*~i~) is positive. This kind of function
could be specified parametrically, e.g. a quadratic form where convexity
for some bidders would provide strong incentives for those bidders to
bid on packages, budget constraints permitting. In any case, the
function should be presented to subjects in tabular form for integer
values of *n*~i~, which would enable subjects to obtain a quick
intuitive understanding of the incentives for package bidding, an
understanding that would be supplemented by providing package value
calculations for specific packages on request. These calculations,
however, are contingent on the uncertainty over common values, so a
table show would be especially useful.

The approach where *n~i~* corresponds to the total number of licenses
acquired by bidder *i* (corresponding roughly to the total area served)
is probably simplest to explain to subjects in an experiment.
Alternatively, one could introduce "region-dependent" synergies by
having a scale component *f*~i~(*n*~ij~), where *n~ij~* is the number of
licenses bidder *i* wins that are located "close" to the area for
license *j*, according to some distance metric (e.g. those adjacent or
within a specified distance).[^7] The complementarities being modeled
here might correspond to increased value to subscribers of being able to
obtain service in adjacent areas without paying roaming fees, or to
reductions in average costs due to economies of providing service to
adjacent areas (marketing, inventory, repair, etc.). These cost economy
and roaming factors, of course, could be modeled separately, but the
goal here is to provide for a flexible form of complementarities without
introducing too much complexity into the setup from the subjects' point
of view.

4.  *Budget Constraints*

Along with a set of valuations for individual licenses and packages of
such licenses, each bidder will be assigned prior to bidding a budget
that limits the total payment the bidder can make at the close of the
auction for the licenses it has won. The bidder's objective is to
maximize its net profit (i.e., license or package valuation minus
payment) subject to this budget constraint.

The experiments should, at a minimum, include some situations in which
budget constraints are binding for winning bidders. However, budget
constraints should not be so tight as to eliminate opportunities to
observe bidding behavior related to the financial exposure issue. For
example, we are interested in whether bidders will risk exposure by
bidding above the stand-alone values for a group of licenses with
superadditive values in the SMR auction, but they will not take the risk
if their budget is inadequate.

If bidders default (i.e. when their winning bids total more than their
allotted budgets), they must default on *all* winning bids, forfeiting
the licenses and incurring the penalties specified in the auction rules.
In the actual FCC auctions, a bidder who defaults is required to pay the
difference between the withdrawn winning bids and the amount that the
licenses sell for subsequently as well as an additional amount based on
a percentage of the winning bids. This percentage is higher if the
default occurs after a package bidding auction. In the experiment, this
process can be mimicked by having the subsequent sale prices be
determined as a random draw from a known distribution.

5.  *Additional Considerations*

An important question is whether aids should be provided to experimental
subjects to help them make thoughtful decisions in complex environments.
This issue typically gets clarified after some pilot experiments. At
present, we envision providing subjects with an interface that contains
a "Package information" table, a "Bidding basket," and "Bidding History"
plus a separate "Auction Result" page, see Figure 1. In addition, it
would be useful to incorporate a message area, which the experimenter
can use to send general messages (e.g. "the experiment will start in a
few seconds"). In the subject's "Package Information" table there would
be a row for each license/package for which the subject has positive
values. For example, in Figure 1, bidder 1 has positive values for
packages A, B, C, AB, AC, BC, and ABC. The columns of the "Package
information" table would identify the license or package, the current
price, the private value component, and the common value signal.[^8]
Subjects can place bids on licenses/packages they are interested in by
moving them into the "Bidding basket," which they can submit after all
relevant licenses/packages have been entered.

In each round, bidders are informed if they are the provisional winner
on a license or package through the use of colors: in Figure 1, for
instance, the grey cells indicate that bidder 1 is the provisional
winner on package BC. When licenses/packages are added to the bidding
basket, bidders will see the minimum bids that need to be submitted in
the next round: for non-provisional winners, it is the current price
plus a pre-specified increment.[^9] In the right bottom panel, the
"Bidding history" shows the bidding activity during the period on the
packages relevant to the subject. The experimenter should be able to
choose different information feedback conditions, e.g. show all bids and
identities, show all bids (but not identities), or show only own
bids.[^10] Finally, by clicking on the "Auction Result" tab on the
top-left panel, subjects can view previous auction results. The
corresponding screen will appear automatically after an auction has
ended, and subjects will get some time to interpret the results before a
new auction starts and the screen changes back to the one shown in
Figure 1. Of course, other experimenters are likely to have different
ideas about how to best represent all the information to subjects, so
the interface presented in Figure 1 should be seen as a suggestion not a
requirement.

![](media/image5.png){width="5.985416666666667in"
height="4.679166666666666in"}

**Figure 1**. Example of Proposed Client Interface

In any case, the experiment software would have to be extremely simple
from the subject's point of view, without many of the extra details
required in the actual FCC auctions. In contrast, the software must
provide the experimenter with all of the options needed for the
treatments discussed below. In particular, the setup pages should allow
the experimenter to specify the auction format, the protocol for
matching subjects in auction groups (random or fixed), the common and/or
private value distributions, the information feedback conditions, the
enforcement of bidding budget and activity rule constraints, etc.
Moreover, the software must provide for instructions and easy to
understand earnings calculations at the end of each auction period.

**III. Proposed Treatments**

Before we discuss the treatments that are most relevant to test the SMR
auction vis-à-vis the SMRPB auction, we first review the attributes of
package bidding. We also discuss elements likely to affect both
institutions.

1.  *Attributes of Package Bidding*

The principal reason to consider the SMRPB auction is to allow bidders
to express their preference for a combination of licenses without
risking financial exposure. The "*exposure problem"* arises when a
bidder obtains part, but not all, of the preferred package but spends
more for the obtained pieces than they are worth to him. For example,
suppose a bidder values the package AB at 10 but has no value for either
A or B separately. In an SMR auction with no package bidding, a bidder
who attempts to acquire both licenses but is outbid for one will incur a
financial loss. To avoid the loss, the bidder may decide not to bid
aggressively on either item, resulting in an inefficient allocation when
the total value of licenses A and B to others is less than 10. In
contrast, in the SMRPB auction the bidder would be willing to bid up to
10 on the AB package, and this bidder will efficiently be assigned the
package if the total value of licenses A and B to others is less than
10. Intuitively, the exposure problem of the SMR auction will become
more severe as the degree of complementarity between the licenses rises
and bidder license preference overlap increases.

The ability to express preferences for packages may thus enhance
efficiency and revenues. However, the SMRPB auction may create problems
for "small bidders," which we define here to be those bidders interested
only in small packages or even a single license. This is known as the
"*threshold problem*." Suppose in the above example that there are two
other small bidders who have values of 8 for license A or license B, but
no value for the package AB. Furthermore, suppose the current prices in
the SMRPB auction are 2 for A, 2 for B, and the package bidder is the
provisional winner by bidding 9 for AB. The small bidders could
overthrow the package bid by both increasing their bids to 5, but both
would prefer a smaller increase in their own bid hoping the other will
make up for the difference. In other words, the positive effect of an
increase in one small bidder's bid for other small bidders creates
incentives for small bidders to "free ride," i.e. to let others bear the
cost of outbidding the package bidder. Moreover, small bidders face a
coordination problem in that their combined total bid needs to rise to
overthrow the package bid, but an increase that overshoots the minimum
required amount is wasteful from their point of view. (The situation the
small bidders face is therefore akin to a step-level public goods
game.[^11]) Intuitively, this coordination problem may get worse as more
small bidders are needed to overthrow the package bid. Note that the
threshold problem is a problem in the sense that free riding and/or
failure to coordinate bid increases may result in an inefficient
allocation (i.e., the licenses included in the package are worth more to
two or more non-winning bidders than to the winning bidder).

In addition to the exposure and threshold issues which are likely to
vary according to whether a package or non-package format is used, the
winner's curse and tacit collusion may also affect the outcomes of the
alternative auction designs differentially. The winner's curse effect is
due to a failure of a bidder to recognize that winning is more likely
when one's common value signal is relatively high. In other words, even
though the value signal is unbiased *ex ante*, there is an upward bias
conditional on winning. The failure to realize that winning is an
informative event may cause bidders to submit bids that are too high for
given signals, and as a consequence, the winner may end up paying more
than the license turns out to be worth to him. When there is a single
common value that applies to all licenses, winner's curse effects may be
stronger for larger packages since the adjacency factor in equation
(1.2) exaggerates any misjudgments that bidders may make regarding the
level of the common value. Experimentation provides an ideal vehicle to
test how these winner's curse effects interact with the auction format.

Tacitly collusive outcomes may result from coordinated attempts to
divide the market at low prices. By *tacit collusion*, we mean mutual
pricing restraint that is a result of a "meeting of the minds" without
any explicit communication or bid agreements, which would have to be
reported under current FCC rules. Intuitively, tacit collusion is more
likely to be observed if each bidder has a local advantage for one
license or, more generally, when bidders' preferred packages show little
overlap, and if complementarities are low.[^12] The lower the
complementarities, the smaller the economic value a cooperative bidder
surrenders when agreeing to a collusive strategy that splits the market.
In addition to the valuation environment, auction rules may have an
important effect on the ability of bidders to identify a collusive
outcome. For example, in order to divide up the market, bidders must
signal to other bidders their desired collection of licenses. This is
facilitated if bidders have information regarding the identities of the
submitters of the provisionally winning and non-winning bids. Moreover,
such information is needed in order for cooperative bidders to impose,
in the form of a retaliatory bid, a cost upon bidders that behave
non-cooperatively. Finally, the ability to make publicly announced bids
on packages may facilitate bidders' abilities to lay claims on
non-overlapping segments of the market.[^13] An important auction design
question is whether bidders can successfully divide up a market and
whether changing the information environment and allowing for package
bids would enhance their ability to do so. The suggested treatment
structure in Appendix B is designed to focus on the effects of auction
rules on tacit collusion, in environments where collusive outcomes are
more likely to appear.

Finally, both the SMR auction and the SMRPB auction described in
Appendices C and D provide price feedback on individual licenses that
may help bidders decide where and how to adjust their bids in the course
of an auction. When the auction closes and no bidder wishes to further
raise their bids at the current prices, the resulting set of final
prices essentially corresponds to a competitive equilibrium in which all
licenses are allocated. This process may be unstable when
complementarities in valuations preclude the existence of a set of
individual-license prices that clear the market. Experiments can show if
in these circumstances the extra pricing flexibility provided by package
bidding results in improved performance of the auction. Therefore, it is
essential that the experimental environment include some cases where no
competitive equilibrium exists. See Appendix A for a detailed discussion
of the non-existence of a set of competitive equilibrium prices in the
presence of license value complementarities (i.e., non-convexities) and
the related concept of the core.

2.  *Experimental Treatments*

To evaluate all four effects (i.e., financial exposure, threshold
problem, winner's curse, and tacit collusion) we propose to allow for
both common and private value elements, and for bidders of different
sizes and license preferences (i.e., small bidders interested in a
single license and large bidders interested in sets of licenses). Within
this framework, we need to define the variables under experimental
control. These are:

(i) The degree of complementarities among licenses.

(ii) The amount of overlap in bidders' preferences.

(iii) The "strength" of small bidders vis-à-vis large bidders (i.e., the
      difference between the sum of the valuations small bidders place
      on a set of licenses and the value the highest value large bidder
      places on those same licenses as a package).

(iv) The amount of information bidders receive during the course of the
     auction regarding the identities of the provisionally winning and
     non-winning bids.

We propose to include two cases for each variable: low/high degree of
complementarity, low/high overlap in bidder preferences, "weak/strong"
small bidders, and full/incomplete bidder information disclosure. The
combinations of variables, or "treatments," are categorized in the table
below. Each would be tested in both the SMR and SMRPB formats. If
desired, additional treatments, including, for example, varying degrees
of common-value uncertainty, could be considered. Also, depending upon
results, it may be unnecessary to include all of the treatments listed.

+-------------+-------+------+----+----+-----+-------+------+-------+
| Treatments  | **D   |      | ** |    | *   |       | *    |       |
|             | egree |      | Pr |    | *Sm |       | *Inf |       |
|             | of**  |      | ef |    | all |       | orma |       |
|             |       |      | er |    | Bi  |       | tion |       |
|             | **Co  |      | en |    | dde |       | Disc |       |
|             | mplem |      | ce |    | r** |       | losu |       |
|             | entar |      | ** |    |     |       | re** |       |
|             | ity** |      |    |    | **S |       |      |       |
|             |       |      | *  |    | tre |       |      |       |
|             |       |      | *O |    | ngt |       |      |       |
|             |       |      | ve |    | h** |       |      |       |
|             |       |      | rl |    |     |       |      |       |
|             |       |      | ap |    |     |       |      |       |
|             |       |      | ** |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
|             | **    | **Hi | *  | ** | **  | **Str | **Fu | *     |
|             | Low** | gh** | *L | Hi | Wea | ong** | ll** | *Part |
|             |       |      | ow | gh | k** |       |      | ial** |
|             |       |      | ** | ** |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| LC,OL,SW,IF | X     |      | X  |    | X   |       | X    |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| LC,OL,SS,   | X     |      | X  |    |     | X     | X    |       |
| IF          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| LC,OH,SW,   | X     |      |    | X  | X   |       | X    |       |
| IF          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| LC,OH,SS,   | X     |      |    | X  |     | X     |      |       |
| IF          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| HC,OH,SS,   |       | X    |    | X  |     | X     |      |       |
| IF          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| HC,OH,SW,   |       | X    |    | X  | X   |       |      |       |
| IF          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| HC,OL,SS,   |       | X    | X  |    |     | X     |      |       |
| IF          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| HC,OL,SW,   |       | X    | X  |    | X   |       |      |       |
| IF          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| LC,OL,SW,IP | X     |      | X  |    | X   |       |      | X     |
+-------------+-------+------+----+----+-----+-------+------+-------+
| LC,OL,SS,   | X     |      | X  |    |     | X     |      | X     |
| IP          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+
| LC,OH,SW,   | X     |      |    | X  | X   |       |      | X     |
| IP          |       |      |    |    |     |       |      |       |
+-------------+-------+------+----+----+-----+-------+------+-------+

Each of the eleven treatments will require six groups (six independent
data points) for a total of 66 groups for the 2 basic auction formats,
which results in a grand total of 2\*11\*6=132 groups. Appendix B gives
a specific proposal for the parameterizations of these sessions.

**IV. Experimental Procedures**

It is anticipated that adequate data collection will involve about 6-12
subjects[^14] for each of the 132 groups, and another 150 subjects for
debugging and pilot studies, for a total between 942 and 1,734 subjects.
These numbers are only an approximation, and may be negotiated with FCC
staff as the project proceeds. The sessions will probably be somewhat
long, perhaps 2 hours or more, due to the need to explain complex
procedures and obtain enough replications. Financial motivation should
be high enough to merit serious attention, and therefore, it is
anticipated that earnings will be about \$50 per person, including
show-up fee.

The auction rules should be based closely on the description of the two
alternative auctions, contained in Appendix C, which was provided by FCC
staff. The actual FCC auction framework will be done on-line, and
therefore, the experiments will also be computerized. The FCC will
provide the needed pricing and assignment algorithms (see Appendix D),
but it is the responsibility of the experimenter to develop the software
that would be used for the experiments (not for the actual FCC auction).
The software developed by the FCC for the actual auction may be of
limited use to run the experiments, although it may be useful to do some
comparability testing to assure that the experimental and FCC interfaces
induce similar results. In particular, it is necessary to provide the
experimenter with the flexibility to run alternative treatments, induce
a desired array of private and common values, enforce budget and
activity constraints, and allow for a variety of information feedback
conditions. Furthermore, the software should provide subjects with an
easy-to-understand interface for instructions, valuations and signals,
bidding decisions and earnings.

Of course, all information collected in the experiment (current and past
bids, values, signals, earnings, etc.) should be recorded and reported
to the FCC. In addition, it is desirable to report and control any prior
experience that particular bidders have had in prior auction
experiments, along with some demographic information. This information
can be collected with a questionnaire that should be done *ex post*. The
structure of the survey is up to the experimenter.

**V. Performance Measures**

Economic efficiency, the degree to which the auction assigns the
licenses to the bidders with the highest valuations, is the most
commonly used performance measure in economic experiments. Efficiency is
measured as a ratio of the sum of the valuations winning bidders place
on the obtained items to the maximum valuation placed by bidders under
the efficient allocation of items, and is independent of prices paid.
For example, suppose two bidders compete for a single license; one of
them values the license at 10 while the other bidder values it at 6.
Awarding the license to the lower-valuing bidder would result in an
efficiency level of 6/10, or 60%. One of the main benefits of laboratory
experiments is that bidders' valuations are induced and known to the
researcher, which makes it possible to examine efficiency and to compare
average efficiency levels across different auction formats.

The FCC may be interested in other measures as well. The experimental
data should also include sales revenue comparisons between auctions run
with and without package bidding. The reports should distinguish between
revenue resulting from sales which were profitable to bidders, and
revenue from sales in which bidders "overbid" due to exposure problems
or a "winner's curse."

Since the length of FCC auctions is important to actual FCC bidders, the
experiments should report the number of rounds for each auction, and
compare treatments. In addition, there should be some measure of the
extent to which the time patterns of bidding activity vary. One
simplistic measure would be in terms of *cumulative* numbers of bid
changes, by round, which would give a picture of the extent to which
bidding is concentrated towards the beginning or end of the auction. A
more economically relevant measure would take bid amounts into account
and would attempt to ascertain the extent to which bids begin to reveal
final valuations during the auction process. To this end, the
provisional winning bids in each round would be calculated, summed, and
expressed as a fraction of the maximum-value allocation or as a fraction
of the final sales revenue.

It may also be useful to try to evaluate the extent to which the various
auction formats and treatments produce stable prices and outcomes.
Loosely speaking, final auction prices can be thought of as
market-clearing prices, since no new bids are being made. However, the
final prices and allocation may not comprise a competitive equilibrium;
indeed, a competitive equilibrium may not exist in some of the more
complex fitting environments that these experiments are designed to
test. In this context, it is important to consider whether or not such
equilibria are possible, and if so, whether the final experimental
allocations and prices correspond to competitive equilibrium
outcomes.[^15] Given that in theory there are situations in which
competitive equilibria do not exist in non-package bidding auctions but
are possible in package bidding auctions,[^16] we are interested in the
extent to which the auction format contributes to the potential for
attaining a competitive equilibrium outcome. The researcher should note
whether competitive equilibria exist, and whether the experimental
outcomes correspond to those equilibria.

A weaker measure of the stability and desirability of an outcome is to
examine whether an experimental auction results in prices and
allocations which are in the "core."[^17] The researcher should report
the percentage of times that the final allocations are in the core. Of
course, as with competitive equilibria, noise and other variations in
bidding behavior are likely to result in inefficiencies which may
preclude core outcomes, even if such outcomes are predicted in theory.
However, differences which vary systematically by auction format are of
interest.

Finally, the FCC may be interested in assessing whether the success of
small bidders in winning licenses, relative to the success of large
bidders, varies according to auction format, taking into account
valuations and budgets. One way to examine this effect is to consider
the efficient allocation of licenses (with some provision for ties), and
to calculate the sum of the license valuations that all small bidders
obtain in the efficient allocation, V\*. Then let V represent the sum of
all prize valuations obtained by small bidders in the actual allocation
that results from the auction. These valuations would include the
complementary effects but would not include prices paid. Then the ratio,
V/V\*, provides a license-based measure of bidding success, which can be
compared across auction formats. This ratio, however, does not
incorporate profit information based on prices paid. An alternative is
to compare ratios of small bidders' earnings to large bidders' earnings
across auction formats. These and possibly other measures may be used to
help the FCC assess performance in this dimension.

**Appendix A: Competitive Equilibrium and the "Core"**

The draws from the various distributions determine whether there exists
a set of prices that supports the efficient assignment of licenses.
Recall that a *feasible allocation* is an assignment of licenses such
that no license is assigned to more than one bidder. A *competitive
equilibrium* is a feasible allocation and a set of prices, one for each
license, such that: 1) all licenses are assigned to a bidder (market
clearing), and; 2) no bidder can earn a higher profit by adding licenses
or deleting licenses from the assigned allocation, and altering the
payment based on the prices for those licenses (profit maximization
given prices). More formally:

[Definition]{.underline}: A competitive equilibrium is a set of prices
![](media/image6.wmf){width="0.8333333333333334in" height="0.25in"} and
allocations ![](media/image7.wmf){width="0.9166666666666666in"
height="0.3055555555555556in"} where the *X ^k\ \*^* span the entire set
of licenses *X*, ![](media/image8.wmf){width="1.0416666666666667in"
height="0.2222222222222222in"} for![](media/image9.wmf){width="0.5in"
height="0.2222222222222222in"}, and
![](media/image10.wmf){width="3.3208333333333333in"
height="0.3055555555555556in"}for all *k*,
![](media/image11.wmf){width="0.2222222222222222in"
height="0.20833333333333334in"}.

We are interested in considering competitive equilibria since their
existence or non-existence may affect the stability of auction outcomes,
even though an alternative equilibrium concept, the Bayesian-Nash
equilibrium, is more typically used to analyze auctions. In determining
experimental parameters, we recognize that value complementarities will
tend to introduce precisely the kinds of non-convexities that may
preclude the existence of a competitive equilibrium, even in setups that
may seem reasonable in terms of their economic structure. Consequently,
in evaluating whether an auction results in a competitive equilibrium
outcome, we will need to consider whether such a result is possible.

Consider the following example, with three licenses (A, B, and C) and
three bidders (S1, S2, and L1).[^18] Table 1A below presents the set of
valuations for individual licenses and combinations of such licenses for
each bidder. Each bidder has a positive value of 10 for two of the
licenses. For the small bidders, S1 and S2, the value of the package
consisting of those two licenses is double the sum of the individual
values. For the large bidder, L1, the value of the AB package is 2.5
times the sum of the individual values for A and B.

## Table A1: License and Package Valuations

  --------- -------- -------- -------- -------- -------- -------- --------
  Bidders   A        B        C        AB       AC       BC       ABC

  S~1~      10       0        10       10       40       10       40

  S~2~      0        10       10       10       10       40       40

  L~1~      10       10       0        50       10       10       50
  --------- -------- -------- -------- -------- -------- -------- --------

The efficient allocation is to assign the package AB to bidder L1 and
assign license C to either of the small bidders. The set of competitive
equilibrium prices is determined by the requirement that the "market
clears," i.e. demand equals supply. If the price of license C, p~C~, is
less than 10 both small bidders would want to buy license C and there
would be excess demand. In contrast, if p~C~ exceeds 10 then neither
small bidder would be interested in license C and there would be excess
supply. Hence, p~C~=10 in equilibrium. Next, p~A~+ p~B~ ≤ 50 since
otherwise bidder L1 would not be willing to buy the package AB. Finally,
there are two conditions that keep the small bidders from demanding
packages AC and BC: p~A~+ p~C~ ≥ 40 and p~B~+ p~C~ ≥ 40. Together with
the constraint that p~C~=10 this yields p~A~ ≥ 30 and p~B~ ≥ 30, which
contradicts the condition p~A~+ p~B~ ≤ 50. Therefore, for the example in
Table 1A no competitive equilibrium prices exist.[^19] Note that
equilibrium prices would exist if the large bidder's value for the AB
Package were raised to 60. Appendix B describes how the example in Table
1A can be generalized to accommodate the treatments discussed in section
3 while maintaining the flexibility of parameterizing auctions with and
without competitive equilibrium prices.

As a relaxation of the competitive equilibrium notion we next consider
the set of *core payoffs*. Let N denote the set of players including the
seller, N = {0, 1, ..., n} where the 0-label corresponds to the seller,
and let v~k~ (x~k~) denote bidder *k*'s value for package x~k~ (with
v~0~ (x~k~) = 0 for all possible packages x~k~). Define the coalitional
value function, w(S), for S  N as:

w(S) = max *~x\ \ X~* { ~k\ \ S~ v~k~ (x~k~ )},

if 0  S, and w(S) = 0 otherwise (since the bidders have nothing to
trade among themselves). Let π~k~ = v~k~ (x~k~) -- p~k~ denote bidder
*k*'s profit when obtaining package x~k~ at price p~k~ and let π~0~ =
^n^ ~k=1~ p~k~ denote the seller's revenue. The set of core payoffs is
defined as:

Core = {π \|  ~k\ \ N~ π~k~ = w(N) and w(S)   ~k\ \ S~ π~k~ for all
S  N}.

Note that for payoffs to be in the core, the corresponding allocation
has to be efficient. In the example, this implies that the large bidder
obtains the AB package and one of the small bidders obtains license C.
Each of the inequalities w(S)   ~k\ \ S~ π~k~ for all S  N requires
that the sum of the payoffs for the members of coalition S be no less
than the total surplus generated by the coalition S in isolation of the
rest. In other words, if one of the inequalities was not satisfied then
the coalition S could form a "blocking coalition" that provides its
members with a higher payoff than they would receive otherwise. For the
example above, some of the relevant inequalities are:

![](media/image12.wmf){width="1.6111111111111112in"
height="1.1111111111111112in"}

By adding the first three inequalities and using the final equality, we
get π~0~ ≥ 50. Moreover, by subtracting the second or third inequality
from the final equality yields ![](media/image13.wmf){width="0.5in"
height="0.2638888888888889in"}and
![](media/image14.wmf){width="0.5138888888888888in"
height="0.2638888888888889in"}. So the profits of the small bidders are
zero while the large bidder's profit can be anything between 0 and 10
and the seller's revenue is anything between 50 and 60. These payoffs
correspond to a price equal to 10 for license C and a price between 40
and 50 for the AB package.

The notion of core payoffs is less restrictive than the notion of a
competitive equilibrium in the sense that it will always exist in the
framework being considered and it does not require the calculation of
prices for individual licenses. In the presence of complementarities,
individual-license prices that clear the market often do not exist and
the notion of the core thus seems the relevant generalization to deal
with such situations. It will be easy to verify whether the final
outcomes of the auction are in the core, since the experimenter has
complete knowledge of the bidders' valuations, their payoffs and the
seller's revenue.[^20]

**Appendix B: Parameter Choices for the Different Treatments**

As an illustration of the proposed parametric approach to defining
valuations, below we take the "three-bidder ABC design" of the previous
appendix and scale it up by having two A licenses (A1 and A2), two B
licenses, and two C licenses. With these labels, the valuation structure
matches that of Table A1 with the deterministic, commonly known values
replaced by privately known, uniformly distributed random variables. In
Table B1, each V~i~ is a realization of a uniformly distributed variable
(e.g. between 0 and 100). Therefore, each bidder obtains a *single*
private value that pertains to each of the licenses in which it is
interested. This example assumes a pure private value structure, so that
*CV*~i~ = 1 for each bidder *i* in equation (1.3). Table B1 displays the
values corresponding to *U*~ij~ for each bidder *i* and individual
license *j*. Values for packages of two or more items are represented
using Tables B2 and B3 as explained below.

To capture synergies between licenses in a simple manner (see section
3), we assume that a small bidder's value for a package consisting of N
licenses is simply the sum of values of the individual licenses
multiplied by a scale factor 1+ (N-1)α (see Table B2). For instance, if
V~1~ = 10 and α=1, bidder 1's value for the package A~1~C~1~ is
(10+10)\*(1+1) = 40, as in the example of Appendix A. (Note that our
setup also allows for substitutes by letting α be less than 0.)
Likewise, for a large bidder the value of a package that consists of N
licenses is the sum of values of the individual licenses in the package
multiplied by a scale factor 1+ (N-1)β, see Table B3. For example, if
V~5~ = 10 and β =1.5, the value of the package A~1~B~1~ to bidder 5 is
(10+10)\*(1+1.5) = 50, as in the example of Appendix A.

Because we want to study the effects of overlapping preferences among
bidders, we replicate the basic market by introducing two additional
small bidders, one additional large bidder, and additional licenses D~1~
and D~2~, E~1~ and E~2~, and F~1~ and F~2~. This second market is
completely separate if the "cross-over" values labeled by W~5~ and W~6~
are constrained to be zero. In contrast, there is full overlap if W~5~ =
V~5~ and W~6~ = V~6~.

## Table B1: Private Value Structure

  ------------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
  Bidders      A~1~   A~2~   B~1~   B~2~   D~1~   D~2~   E~1~   E~2~   C~1~   C~2~   F~1~   F~2~

  Small Bidder V~1~   V~1~                                             V~1~   V~1~          
  1                                                                                         

  Small Bidder               V~2~   V~2~                               V~2~   V~2~          
  2                                                                                         

  Small Bidder                             V~3~   V~3~                               V~3~   V~3~
  3                                                                                         

  Small Bidder                                           V~4~   V~4~                 V~4~   V~4~
  4                                                                                         

  Large Bidder V~5~   V~5~   V~5~   V~5~   W~5~   W~5~   W~5~   W~5~                        
  5                                                                                         

  Large Bidder W~6~   W~6~   W~6~   W~6~   V~6~   V~6~   V~6~   V~6~                        
  6                                                                                         
  ------------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------

## Table B2. Scale Factors for Small Bidders

(α \> 0 for complements, and α \< 0 for substitutes)

## 

## 

## 

## 

## Table B3. Scale Values for Large Bidders

(The magnitude of β determines the strength of complementarities, with β
\> 0)

+-------------------+-----+-----+------+------+-----+-----+-----+-----+
| Number of         | N=1 | N=2 | N=3  | N=4  | N=5 | N=6 | N=7 | N=8 |
| Licenses          |     |     |      |      |     |     |     |     |
|                   |     |     |      |      |     |     |     |     |
| Won by Large      |     |     |      |      |     |     |     |     |
| Bidder            |     |     |      |      |     |     |     |     |
+-------------------+-----+-----+------+------+-----+-----+-----+-----+
| Scale Factor      | 1   | 1+β | 1+2β | 1+3β | 1   | 1   | 1   | 1   |
|                   |     |     |      |      | +4β | +5β | +6β | +7β |
+-------------------+-----+-----+------+------+-----+-----+-----+-----+

There are two main advantages to the value structure in Table 1B. First,
the possible non-existence of competitive equilibrium prices for the
example in Appendix A carries over directly to the present setup when
the markets do not overlap. Depending on the actual realizations of the
V~i~, a competitive equilibrium may or may not exist for certain values
of α and β. The second advantage is that the licenses of exclusive
interest to the small bidders (C1, C2 and F1, F2) provide them with an
opportunity to compete actively and make some earnings even when the
large bidders have a scale advantage.

Using this setup, we can address the treatment variations called for in
Section III. For example, by choosing high or low values of α and β, we
can vary the degree of complementarity between licenses. Likewise, by
restricting the realizations of W~i~ to be zero, the degree of overlap
is small while high overlap occurs for W~i~ = V~i~. The strength of
large vis-à-vis small bidders is determined by the scale factor α and β
and the extent to which the large bidders' random values are drawn from
a higher or lower range of values. The remaining treatment change is
implemented by altering the amount of information bidders receive
concerning the identities of bids submitted by others. Other variations
that might be of interest to the FCC include (i) changing the degree of
common-value uncertainty, (ii) allowing bidders to submit bids in
separate OR-groups thereby allowing the possibility that a bidder could
have multiple winning bids, with at most one in each OR-group,[^21]
(iii) changing the activity and stopping rules, (iv) varying the pricing
rules, and finally (v) allowing for alternative auction designs (e.g.
clock auctions or clock-proxy auctions).

**Appendix C: Auction Rules**

The following describes the rules of the SMR and SMRPB auctions.

**1. SMR Auction**

a.  [Simultaneity and Bid Structure]{.underline}

> All licenses are put up for bid simultaneously. Participants have the
> opportunity to submit bids on individual licenses, for as many
> licenses as they wish.

b.  [Iterative]{.underline}

> The auction consists of discrete, successive rounds in which buyers
> have the opportunity to place bids on their desired licenses.
> Following each round, the high bid for each license is posted. These
> high bids then become the standing bids for the next round of bidding.

c.  [Minimum Opening Bid]{.underline}

> The minimum opening bid is defined as the minimum price the Commission
> demands in exchange for selling the license.

d.  [Minimum Acceptable Bid]{.underline}

> In the first round and until a bid has been placed on a license, an
> acceptable bid must be equal to or exceed the minimum opening bid.
> Subsequently, in order to be acceptable, a bid must exceed, by a
> specified percentage, the provisionally winning bid for the license.
> In order to move the auction along more quickly, the increment
> percentage can be calculated on a per-license basis, as an increasing
> function of the number of bids placed on the license in previous
> rounds. If speed of the auction is a concern in these experiments, we
> may want to use this feature -- known as the exponential smoothing
> formula.

e.  [Bid Increments]{.underline}

> Bidders are given the choice of making the minimum acceptable bid, or
> of making one of eight incrementally higher bids.

f.  [Bid Withdrawal]{.underline}

> In a limited number of rounds (usually two), participants are
> permitted to withdraw any of their provisionally winning bids. After
> the withdrawal, the Commission becomes the provisionally winning
> bidder for the withdrawn license and the minimum acceptable bid in the
> following round equals the second highest bid received on the license
> (which may be less than or equal to, in the case of tied bids, the
> amount of the withdrawn bid). A withdrawing bidder pays a penalty
> equal to the maximum of zero or the difference between the price at
> which the bidder withdrew its bid and the final sale price in the
> current auction or in a subsequent auction in which the license is
> sold. If the license remains unsold in the current auction, the
> withdrawing bidder pays an interim payment of 3% of its withdrawn bid.

g.  [Bidding Eligibility and Activity]{.underline}

> Each license is assigned bidding units typically based on its
> bandwidth and the size of the population "covered" by the geographic
> area of the license. A bidder's upfront payment determines the total
> number of bidding units a bidder can "bid on" at any one time. The
> total number of bidding units available to the bidder establishes the
> bidder's maximum "eligibility" to bid in the auction. To encourage
> active bidding, the auction has rules that impose a cost upon bidders
> if they fail to display a minimum amount of bidding activity. In each
> round, bidders must be active on at least a fixed percentage of the
> total bidding units available to the bidder. High standing bids for
> licenses from the previous round and new bids in the current round are
> considered "active" bids for purposes of measuring bidding activity.
> The failure of a bidder to satisfy the activity rule leads to a
> reduction in his bidding eligibility sufficient to bring him into
> compliance with the rule (unless the bidder uses an activity rule
> waiver, as described below). Therefore, in subsequent rounds, the
> maximum number of bidding units on which the bidder may be active is
> below the original number. The bidder may also actively reduce his
> eligibility, if he does not wish to maintain the minimum bidding
> activity required by the activity rule. While a bidder's eligibility
> determines the maximum number of bidding units on which a bidder can
> bid on and the activity rule defines the minimum level of bidding
> activity the bidder must exhibit in order to maintain its current
> bidding eligibility, the activity and eligibility rules do not
> restrict the amount of the bids.

h.  [Activity Rule Waiver]{.underline}

> Each bidder is granted a limited number of opportunities (usually 3-5)
> to avoid being subject to the activity rule. If the bidder does not
> meet his minimum activity requirement for a round, the system will
> automatically apply a waiver, if the bidder has one remaining, or the
> bidder may actively submit a waiver. An activity rule waiver applies
> to an entire round, not to particular licenses (i.e., by using an
> activity rule waiver, a bidder is essentially taking a "time out" for
> the round).

i.  [Closing Rule]{.underline}

> The auction closes after the first round in which no new bids were
> placed, no bids were withdrawn, and no waivers were actively
> submitted.

j.  [Information Environment]{.underline}

> Prior to entering the auction, bidders have information showing the
> number of bidding units associated with each license and the number of
> bidding units each bidder is able to "bid on." Following the end of
> each round, participants receive information on provisionally winning
> bids, the identity of each high bidder, the name of each non-winning
> bidder and their bids, the identity of any bidders using activity
> waivers, the eligibility requirement for each bidder in the subsequent
> round, and the minimum acceptable bid for each license for the
> subsequent round. After each round, bidders also know the identity of
> the bidders that have withdrawn bids and the license(s) on which they
> have withdrawn bids.

k.  [Payment Default Rule]{.underline}

> If a winning bidder defaults on payment for the licenses he won, he is
> liable for a deficiency payment, equal to the difference between the
> amount of the bidder's bid and the amount of the winning bid the next
> time the licenses are won in an auction, [plus]{.underline} an
> additional payment equal to a percentage of the defaulter's bid or of
> the subsequent winning bid, whichever is less. In instances in which
> the amount of a default payment cannot yet be determined, the
> Commission assesses an initial default deposit of a percentage of the
> defaulted bid amount. (If a bidder defaults, he must default on all of
> his winning bids.)

2.  **SMRPB Auction**

```{=html}
<!-- -->
```
a.  [Simultaneity and Bid Structure]{.underline}

> All licenses are put up for bid simultaneously. Participants have the
> opportunity to submit sets of bids, each of which may be for one or
> more licenses. Multiple bids submitted by each bidder will have an
> "exclusive OR" (i.e., "XOR") relationship. For example, bids of the
> form "AB XOR C" would be interpreted as the bidder's desire to acquire
> A and B or C, but not A or B separately or the set A, B, and C.
> Implementing this structure requires that only one of a bidder's bids
> can be included in the provisionally winning set of bids.

b.  [Iterative]{.underline}

> The auction consists of discrete, successive rounds in which buyers
> have the opportunity to place bids on their desired licenses.
> Following the submission of bids, an algorithm identifies the
> provisionally winning assignment of licenses through revenue
> maximization of gross bids. In addition, an algorithm estimates a set
> of individual item prices (i.e., "current price estimates" ("CPEs"))
> that is consistent with this assignment.[^22] The CPEs then become the
> basis for establishing minimum bids for the next round of bidding.

c.  [Minimum Opening Bid]{.underline}

> The minimum opening bid is defined as the minimum price the Commission
> demands in exchange for selling the license. It serves as the minimum
> acceptable bid on a license in the first round and in subsequent
> rounds if no bids have been placed covering that license. For a
> package, the minimum opening bid is the sum of the minimum opening
> bids for its component licenses.

d.  [Minimum Acceptable Bid]{.underline}

> In all rounds subsequent to the first round in which a bid is placed
> on a license or on a package containing the license, the minimum
> acceptable bid for a license must exceed the CPE by a specified
> percentage. This percentage is either a flat rate or a
> license-by-license percentage based on bidding activity. The minimum
> acceptable bid for a package of licenses is the sum of the minimum
> acceptable bids for its component licenses.

e.  [Bid Increment]{.underline}

> Bids can also be made in one of eight incrementally higher bid
> amounts. The increment is based on a fixed percentage of the minimum
> acceptable bid amount for the license or package.

f.  [Bidding Activity]{.underline}

> Each license is assigned bidding units typically based on its
> bandwidth and the size of the population "covered" by the geographic
> area of the license. A bidder's upfront payment determines the total
> number of bidding units a bidder can "bid on" at any one time. The
> total number of bidding units available to the bidder establishes the
> bidder's maximum "eligibility" to bid in the auction. To encourage
> active bidding, the auction has rules that impose a cost upon bidders
> if they fail to display a minimum amount of bidding activity. In each
> round, bidders must be active on at least a fixed percentage of the
> total bidding units available to the bidder. A bidder\'s activity in a
> round is equal to the number of bidding units in the bidder's largest
> (in terms of bidding units) active bid. Active bids include current
> provisionally winning bids, new bids and any bids from previous rounds
> which are at or above the current minimum acceptable bid. While a
> bidder's eligibility determines the maximum number of bidding units on
> which a bidder can be active and the activity rule requires a minimum
> level of activity, the activity and eligibility rules do not restrict
> which licenses the bidder can bid on or the amounts of the bids. The
> failure of a bidder to satisfy the activity rule leads to a reduction
> in his bidding eligibility sufficient to bring him into compliance
> with the rule (unless the bidder uses an activity rule waiver, as
> described below). Therefore, in subsequent rounds, the maximum number
> of bidding units on which he may be active is below the original
> number corresponding to his upfront payment. He may also actively
> reduce his eligibility, if he does not wish to maintain the minimum
> bidding activity required by the activity rule.

g.  [Activity Rule Waiver]{.underline}

> Each bidder is granted a limited number of opportunities (usually 3-5)
> to avoid being subject to the activity rule. If the bidder does not
> meet his minimum activity requirement for a round, the system will
> automatically apply a waiver, if the bidder has one remaining, or the
> bidder may actively submit a waiver.

h.  [Closing Rule]{.underline}

> The auction closes when no new bids have been submitted and no
> proactive waivers applied in one or two consecutive rounds. Whether
> the auction closes after one round or two consecutive rounds with no
> new bids and no proactive waivers will be established prior to the
> auction.

i.  [Information Environment]{.underline}

> Prior to entering the auction, bidders have information showing the
> number of bidding units associated with each license and the number of
> bidding units each bidder has available to assign. Following the end
> of each round, participants receive information on the provisionally
> winning set of bids, the identity of each provisionally winning
> bidder, the name of each non-winning bidder and their bids, the
> identity of any bidders using activity waivers, the eligibility
> requirement for each bidder, and the minimum acceptable bid for the
> next round for each license and for each package that has already
> received a bid. In addition, bidders are given the per-license CPEs.

j.  [Payment Default Rule]{.underline}

> If a bidder defaults on payment for a winning bid, he is liable for a
> deficiency payment, equal to the difference between the amount of the
> bidder's bid and the amount of the winning bid the next time licenses
> covering the same spectrum are won in an auction, [plus]{.underline}
> an additional payment equal to 25 percent of the defaulter's bid or of
> the subsequent winning bid, whichever is less. In instances in which
> the amount of a default payment cannot yet be determined, the
> Commission assesses an initial default deposit of between 3 percent
> and 20 percent of the defaulted bid amount.

**Appendix D: Pricing Rules**[^23]

# This appendix describes the method by which bid information on packages and licenses is used to approximate a "price" associated with each license at the close of every round. These "current prices," as they are called,[^24] are then used in the next round when calculating minimum acceptable bid amounts. Specifically, for a license, this value is the current price of the license plus a percentage. For a package, the minimum acceptable bid amount is the sum of the minimum acceptable bid amounts of its component licenses.

# 

# The current item prices of the licenses are based on the concept that every linear optimization problem has a dual problem that provides pricing information. We begin by discussing a simplified representation of the FCC winner determination problem and then discuss its linear programming relaxation before explaining the dual problem of interest. The winner determination problem is shown in (P1):

(P1): ![](media/image15.wmf){width="2.879861111111111in" height="1.2in"}

# 

# where *B^t^* is the set of considered bids in round *t,*

> *b~j~* is the bid amount of bid *j,*
>
> *L* is the set of licenses being auctioned,
>
> ![](media/image16.wmf){width="2.2222222222222223in" height="0.5in"}
> and,
>
> ![](media/image17.wmf){width="2.611111111111111in" height="0.5in"}

In this formulation, *x~j~* is an indicator variable that equals one if
bid *j* is in the provisionally winning set and zero otherwise. Thus,
the sum of the bid amounts of all provisionally winning bids produces
the maximum obtainable revenue for round *t*. Constraints (1) ensure
that each license is awarded exactly once. The constraints that ensure
that a bidder's bids between rounds are mutually exclusive are not
represented in (P1) since they will be ignored in the linear
representation of the problem.[^25]

The linear program of (P1) relaxes the restriction on the variables
*x~j~*, for all *j∈B^t^*, allowing these variables to take on any value
between zero and one. The linear programming representation of (P1) is
shown in (P2):

(P2): ![](media/image18.wmf){width="2.7895833333333333in"
height="1.2in"}

The dual formulation of (P2) can be used to identify a price, *π~i~*,
for each license *i*, and is shown in the following linear program (P3):

(P3): ![](media/image19.wmf){width="5.4743055555555555in"
height="1.3694444444444445in"}

where *F⊂* *B^t^* is the set of FCC bids on each license[^26] and,

![](media/image20.wmf){width="2.5833333333333335in" height="0.5in"}.

The optimal value of each variable, *π~i~*, in (P3) corresponds to a
dual price[^27] -- often called a "shadow price" -- for each constraint,
i.e., each license, in (P2). The dual price of each license measures the
monetary cost of not awarding the license to whom it has been
provisionally assigned under the solution to (P2). Thus, this monetary
cost has a clear and natural use in estimating the current price of a
license given the bids considered in the current round.

# 

# Constraints (2) in (P3) ensure that the dual price of a license must be at least as large as the greatest bid made on that license. For a package, these constraints ensure that the sum of the dual prices of the licenses that make up a particular package must be at least as large as the greatest bid made on that package. Constraints (3) in (P3) ensure that if a license has not been bid on, the dual price of that license is at least as large as the FCC bid amount.

# 

# Ideally, the solution to (P2) is identical to the solution of (P1). When this occurs, the sum of the dual prices of the licenses comprising any provisionally winning bid equals the winning bid amount. However, (P2) is only an approximation to the integer problem[^28] and often *overestimates* the maximum revenue of (P1). When this occurs, the sum of the dual prices of the licenses in at least one provisionally winning bid will be greater than the respective bid amount. Thus, using the dual prices of (P3) can result in minimum acceptable bid amounts that are too high.

We propose to resolve this issue by using *pseudo-dual prices,*[^29]
rather than the dual prices of (P3)*.* These pseudo-dual prices are
obtained by forcing the sum of the dual prices of the licenses
comprising a provisionally winning bid to equal its respective bid
amount. For example, suppose there are two bids in the provisionally
winning set in round *t*: a bid on license A for \$10 and a bid on
package BC for \$25. The pseudo-dual price of A would exactly equal \$10
and the sum of the pseudo-dual prices of B and C would exactly equal
\$25. These restrictions ensure that the sum of the pseudo-dual prices
equals the maximum revenue for the round (e.g. \$35) and that minimum
acceptable bid amounts reflect the bid amounts of bids in the
provisionally winning set.

# Pseudo-dual prices for each license *i*, denoted *π~i~*, satisfy the following constraints:

![](media/image21.wmf){width="5.447916666666667in"
height="1.7284722222222222in"}

where *W^t^⊂ B^t^* is the provisionally winning bid set in round *t*
and*,*

> *δ~j~* is a slack variable that represents the difference between the
> bid amounts of
>
> non-winning bid *j* and the sum of pseudo-dual prices of the licenses
> contained
>
> in non-winning bid *j*

Constraints (5) ensure that for each provisionally winning bid, the sum
of the dual prices of the licenses comprising that bid equal its
respective bid amount. This new restriction requires that we ease
restriction (2) in (P3) for non-winning bids in order to ensure that a
feasible solution exists. Constraints (4) provide this needed slack.
Constraints (6) are equivalent to constraints (3) in (P3) and
constraints (7) force the slack variables to be non-negative.

Satisfying constraints (5) implies that the sum of the pseudo-dual
prices always yields the maximum revenue for the round. There are likely
to be many sets of pseudo-dual prices that satisfy this constraint set.
For instance, in the example provided earlier, the pseudo-dual prices of
B and C might be any two numbers that together sum to \$25.

By keeping constraints (4)-(7), we have the flexibility to choose an
objective function that will help in selecting among multiple solutions
while still ensuring that the sum of the pseudo-dual prices yields the
maximum revenue of the round. We would like an objective function that
minimizes the values of the slack variables *δ~j~,* for all *j∈ B^t^ \\
(W^t^ ∪ F)* in order to obtain pseudo-dual prices that are close to the
dual prices of (P3). We have tested a number of alternative objective
functions:

1.  Minimization of the maximum *δ~j~* for all *j∈B^t^ \\ (W^t^ ∪ F)*
    followed by maximization of the minimum *π~i~* for all *i* in
    license set *L*, in an iterative manner. (DeMartini, Kwasnica,
    Ledyard and Porter, 1999)

2.  Minimization of the sum of the squares of *δ~j~* for all *j∈B^t^ \\
    (W^t^ ∪ F)*. (also DeMartini, Kwasnica, Ledyard and Porter, 1999)

3.  Minimization of the sum of the *δ~j~* for all *j∈B^t^ \\ (W^t^ ∪ F)*
    using a "centering" algorithm[^30] to solve, essentially finding an
    average among all sets of optimal pseudo-dual prices.

In testing the above alternatives, we frequently observed instances
where the pseudo-dual price of a license significantly changed from
round to round. We acknowledge that prices of licenses should be allowed
to reflect real changes, both increases and decreases, in the way
bidders value the licenses over time. However, we believe that large
oscillations in minimum acceptable bid amounts for the same bid that are
due to irrelevant factors such as multiple optimal solutions, can be
confusing to bidders. We have therefore chosen a method that attempts to
balance minimizing the slack variables and reducing the fluctuations in
pseudo-dual prices from round to round. This method requires solving two
optimization problems, the first of which is alternative 3 above, which
we present as (P4):

(P4): ![](media/image22.wmf){width="5.264583333333333in"
height="2.1958333333333333in"}

Since multiple optimal solutions can exist to (P4) we solve a second
optimization problem that chooses a solution in a way that reduces the
magnitude of price fluctuations between rounds. Specifically, we use an
objective function that applies the concepts of exponential
smoothing[^31] to choose among alternative pseudo-dual prices with the
additional constraint on the problem that the sum of the slack variables
equals Ω^\*^ (the optimal value of (P4)). This objective function
minimizes the sum of the squared deviations of the resulting pseudo-dual
prices in round *t,* from their respective smoothed prices in round
*t-1.*[^32] At the start of the auction, we use the minimum opening bid
prices as the prior smoothed prices. Since these opening prices are
based on bandwidth and population, the pricing algorithm begins with *a
priori* information about the differences among licenses.

Let ![](media/image23.wmf){width="0.20833333333333334in"
height="0.2638888888888889in"} be the pseudo-dual price of license *i*
in round *t.* The smoothed price for license *i* in round *t* is
calculated using the following exponential smoothing formula:

> ![](media/image24.wmf){width="1.5138888888888888in"
> height="0.2638888888888889in"}

where ![](media/image25.wmf){width="0.3055555555555556in"
height="0.2638888888888889in"}is the smoothed price in round *t*-1,

0 ≤ α ≤ 1, and

![](media/image26.wmf){width="0.2222222222222222in"
height="0.2638888888888889in"} = the minimum opening bid amount for
license *i.*

Consistent with prior practice of the Commission, a weighting factor of
![](media/image27.wmf){width="0.5416666666666666in" height="0.19375in"}
has been chosen but can change, as the Commission requires*.*

The following quadratic program (QP) will find the pseudo-dual price,
![](media/image28.wmf){width="0.20833333333333334in"
height="0.2638888888888889in"}, for each license *i* in round *t* that
minimizes the sum of the squared deviations from the respective smoothed
prices in round *t-1* while ensuring that the pseudo-dual prices sum up
to the provisionally winning bid amounts and that the sum of the slack
variables is minimized.

(QP): ![](media/image29.wmf){width="5.445138888888889in"
height="2.6465277777777776in"}

where ![](media/image30.wmf){width="0.2916666666666667in"
height="0.2638888888888889in"} is known and treated as a constant within
the optimization.[^33]

Among alternative prices that satisfy all constraints, the objective
function of this optimization problem chooses one that forces the
pseudo-dual prices to be as close as possible to the previous round's
smoothed price. Thus, we call this the *Smoothed Anchoring Method* since
we "anchor" on the smoothed prices when solving for the pseudo-dual
prices. We define the "current price" for license *i* in round *t* as
the pseudo-dual
price,![](media/image23.wmf){width="0.20833333333333334in"
height="0.2638888888888889in"}, obtained by solving (QP).

The minimum acceptable bid amount for a license in round *t+1* will be
the current price of the license, as calculated above, plus a
percentage. For a package, the minimum acceptable bid amount will be the
sum of the minimum acceptable bid amounts of the licenses that make up
the package.

[^1]: Among previous studies, the most relevant is Cybernomics (2000),
    "An Experimental Comparison of the Simultaneous Multi-Round Auction
    and the CRA Combinatorial Auction." Available at
    http://wireless.fcc.gov/auctions/conferences/combin2000/releases/98540191.pdf.

[^2]: Milgrom and Weber (1982) "A Theory of Auctions and Competitive
    Bidding," *Econometrica*, 50, 1089-1122.

[^3]: There may be other sources of uncertainty. For example, per capita
    costs may be lower or higher than expected.

[^4]: Bidders' private values, *U*~ij~, the common values, *CV*~j~, and
    bidders' common value signals, *e*~ij~, are draws from pre-specified
    distributions, assumed to be common knowledge among the bidders.

[^5]: In the oil drilling rights case, for example, this could
    correspond to uncertainty about the future price of oil, an effect
    that is independent of tract-specific considerations.

[^6]: Here, *n*~i~ includes only the licenses of interest to bidder *i*,
    i.e. for which *U*~ij~ \> 0 (since otherwise bidders could increase
    the value of a package by buying unrelated licenses that have no
    value to them).

[^7]: An alternative way to define the notion of geographic closeness is
    to locate licenses on a circle, as in the Cybernomics (2000)
    experiments, in which case *n*~j~ would take on the values of 0, 1,
    or 2. Another approach is to locate licenses on a rectangular grid,
    where adjacency is determined by shared boundaries. In this case,
    each location in the interior of the grid would have 8 neighbors, so
    *n*~j~ would take on integer values from 0 to 8. Our approach (see
    also Appendix B) is even simpler, i.e. to have scale effects be a
    function of the total number of licenses acquired by a bidder.

[^8]: Figure 1 does not show a column for the common value signal.

[^9]: Alternatively the "Current Price" column could be shown in the
    "Bidding Basket" table next to the "Bid" column.

[^10]: The experimenter should also be able to set budget constraints
    and activity rules if desired. In the example given in Figure 1 no
    such constraints were imposed (see the top panel).

[^11]: See, for instance, Isaac, R.M., D. Schmidt, and J. Walker (1989)
    "The Assurance Problem in a Laboratory Market," *Public Choice*,
    62(3), 216-236; Ledyard, J. O. (1995): \"Public Goods: A Survey of
    Experimental Research,\" in *A Handbook of Experimental Economics*,
    ed. by A. Roth, and J. Kagel. Princeton: Princeton University Press,
    111-194.

[^12]: There is some evidence that in an ascending auction without
    package bidding, collusive outcomes are less likely when there are
    strong complementarities between licenses. See Anthony Kwasnica and
    Katerina Sherstyuk, "Collusion and equilibrium selection in
    auctions". Presented at the FCC, September 2004.

[^13]: Of course, revealing information about all bids and bidder
    identities may have benefits, as well, such as making the auction
    more "transparent" and informing bidders about demand for the
    licenses on which they are bidding.

[^14]: Larger groups can result from scaling up the basic six-bidder
    design in Appendix B or from randomly reconfiguring a group of
    twelve subjects among two smaller groups of six.

[^15]: See Appendix A.

[^16]: See, for example, Bykowsky, Cull and Ledyard.   "Mutually
    Destructive Bidding:  The FCC Auction Design Problem," *Journal of
    Regulatory Economics*, 17(3), May 2000.

[^17]: The FCC will provide an algorithm to determine whether or not a
    final allocation is in the Core.

[^18]: In the remainder of this section we assume a pure private values
    environment.

[^19]: Bykowsky, Cull, and Ledyard (2000) note that bidders in the SMR
    auction are likely to lose money when a set of prices consistent
    with a competitive equilibrium fails to exist.

[^20]: One way to determine whether a competitive equilibrium exists is
    to apply the RAD algorithm (DeMartini, Kwasnica, Ledyard and Porter,
    1999) described in Appendix D and verify whether the slack variables
    δ~j~ are zero for all *j*.

[^21]: The current FCC SMRPB auction format specifies an XOR bidding
    language in which each bidder can have at most one winning bid. This
    makes it easier for bidders to avoid spending more than their
    budgets but may require a large number of bids to be specified.

[^22]: Broadly speaking, the price estimate of a license corresponds to
    the monetary cost of not awarding that license to the bidder to whom
    it is provisionally assigned. Appendix B contains a detailed
    description of the pricing algorithm.

[^23]: This Appendix was written by contractors to the FCC from Decisive
    Analytics Corporation, under contract to Computech, Inc.

[^24]: The "current prices" referred to in this Appendix are called
    "current price estimates" or "CPEs" by the FCC, and are referred to
    as such in the rest of this document.

[^25]: These constraints will be ignored in the linear program
    representation since they are rarely binding in the relaxation of
    the integer-programming problem and because adding such constraints
    to the dual problem creates "degeneracy" in the solution thereby
    causing multiple alternative solutions.

[^26]: The bid amount for a FCC bid is some small amount less than the
    minimum-opening bid for that license. *See* Auction Of Licenses in
    the 747-762 and 777-792 MHz Bands Scheduled for March 6, 2001;
    Modifications to the Calculation for Determining Minimum Acceptable
    Bids and the Provisions Concerning "Last and Best Bids" and Other
    Procedural Issues, DA 01-12, *Public Notice*, 16 FCC Rcd 217
    (2001)(Section III discusses the reasons for this approach).

[^27]: We note that for non-linear problems, these dual prices are also
    known as *Lagrange multipliers.*

[^28]: When the problem is a convex optimization problem, the primal and
    dual problems yield the same objective function values. This is
    called strong-duality. These conditions do not hold for integer
    programming problems, often resulting in a gap between the linear
    programming and integer programming solution values.

[^29]: In our research we found this term first applied to auction
    pricing in the paper by Rassenti, Smith and Bulfin (1982), "A
    combinatorial auction mechanism for airport slot allocation," *Bell
    Journal of Economics*, vol. 13, pp. 402-417.

[^30]: The centering algorithm used in this testing was the barrier
    method available in CPLEX, a commercial optimization package.

[^31]: Exponential smoothing often is used in determining minimum
    acceptable bids in FCC auctions. *See, e.g.*, Auction of Licenses in
    the 747-762 and 777-792 MHz Bands; Auction Notice and Filing
    Requirements for 12 Licenses in the 700 MHz Bands Auction Scheduled
    for May 10, 2000; Minimum Opening Bids and Other Procedural Issues,
    DA 00-292, *Public Notice*, 15 FCC Rcd 2921, Attachment G (2000).

[^32]: This objective function is a convex, quadratic function. This
    quadratic optimization problem is solved using the barrier method.

[^33]: Once the pseudo-dual prices,
    ![](media/image28.wmf){width="0.20833333333333334in"
    height="0.2638888888888889in"}, have been determined, the smoothed
    prices, ![](media/image31.wmf){width="0.20833333333333334in"
    height="0.2638888888888889in"}, can be calculated and used for
    solving (QP) in round *t+1*.