 United States Court of Appeals
         FOR THE DISTRICT OF COLUMBIA CIRCUIT



Argued March 27, 2015              Decided August 14, 2015

                        No. 13-1276

            SETTLING DEVOTIONAL CLAIMANTS,
                      APPELLANT

                             v.

 COPYRIGHT ROYALTY BOARD AND LIBRARY OF CONGRESS,
                    APPELLEES

     WORLDWIDE SUBSIDY GROUP, DOING BUSINESS AS
          INDEPENDENT PRODUCERS GROUP,
                   INTERVENOR



       On Appeal from The Copyright Royalty Board



     Matthew J. MacLean argued the cause for appellant
Settling Devotional Claimants. With him on the briefs were
Clifford M. Harrington and Victoria N. Lynch.

     Sonia K. McNeil, Attorney, U.S. Department of Justice,
argued the cause for appellees. With her on the brief were
Stuart F. Delery, Assistant Attorney General at the time the
brief was filed, and Mark R. Freeman, Attorney.
                               2
    Brian D. Boydston was on the brief for intervenor
Independent Producers Group in support of appellees.

    Before: BROWN, KAVANAUGH and MILLETT, Circuit
Judges.

    Opinion for the Court filed by Circuit Judge MILLETT.

     MILLETT, Circuit Judge: Cable operators’ retransmission
of religious and devotional programming from 2000 to 2003
produced a pool of royalties that Congress charged the
Copyright Royalty Judges with distributing to the copyright
owners.      The Appellant Settling Devotional Claimants
(“Devotional Claimants”) and Intervenor Independent
Producers Group (“IPG”) vigorously contested their
respective shares of that pool before the Royalty Judges. The
Devotional Claimants now appeal, arguing that the Royalty
Judges wrongly calculated their share of the pie by allowing
IPG to press claims without proper authority and refusing to
accept the Devotional Claimants’ evidence regarding how the
relative value of claims should be calculated. They also argue
that, after the Royalty Judges rejected both their and IPG’s
proposed methodologies, the Royalty Judges’ final allocation
simply split the difference between the two parties, and that
decision was arbitrary and capricious and unsupported by
substantial evidence.

    We agree with the Devotional Claimants’ latter claim.
King Solomon was not subject to the Administrative
Procedure Act; the Royalty Judges are. Congress thus
required that the Royalty Judges’ determinations rest on a
focused analysis of the record, not an arbitrary splitting of the
baby. We affirm the Royalty Judges’ procedural rulings
resolving which IPG claims could go forward and whether the
                              3
Devotional Claimants’ methodological evidence could be
properly considered.

                              I

                   Statutory Background

     The statutory framework governing the distribution of
royalties for the retransmission of copyrighted material by
cable system operators is discussed in detail in Independent
Producers Group v. Librarian of Congress (“IPG II”), ---
F.3d ---, 2015 WL 3952243, at *1–*2 (D.C. Cir. June 30,
2015); and Independent Producers Group v. Library of
Congress (“IPG I”), 759 F.3d 100, 101–103 (D.C. Cir. 2014).
As relevant here, the Copyright Act, 17 U.S.C. §§ 101 et seq.,
is designed to further two potentially competing statutory
policies—protecting intellectual property while also ensuring
that information flows freely. One way the Copyright Act
balances those interests is by providing in appropriate
circumstances for compulsory licensing accompanied by
royalty payments to the copyright holder. See IPG I, 759 F.3d
at 101; see also 17 U.S.C. §§ 107–122.

     Cable retransmission is an area for which Congress
designed such a compulsory licensing scheme. Under 17
U.S.C. § 111(c), a cable system operator may retransmit to its
viewers copyrighted material initially aired on a broadcast
station without obtaining the permission of the relevant
copyright owners for that second transmission. The cable
system operators, however, must deposit a statutorily
prescribed royalty fee with the Register of Copyrights. IPG
II, 2015 WL 3952243, at *1. Congress charged the Copyright
Royalty Judges with annually distributing the pool of funds
that accrues to the copyright holders. Id. (citing 17 U.S.C.
§ 801(b)(3)).
                              4
     Copyright owners and their agents who assert entitlement
to those royalties must file a claim with the Royalty Judges in
July of the year following the retransmission of their
programming. See 17 U.S.C. § 111(d)(4)(A); 37 C.F.R.
§ 360.2. Once the claims have been filed, the Royalty Judges
“determine whether there exists a controversy concerning the
distribution of royalty fees.” 17 U.S.C. § 111(d)(4)(B). If all
claimants have agreed on the proper distribution, the Royalty
Judges may conclude that no controversy exists and distribute
the fees consistent with the claimants’ agreement. See IPG II,
2015 WL 3952243, at *1 (citing 17 U.S.C. §§ 111(d)(4)(B)–
(d)(4)(C), 801(b)(7)). In the absence of such agreement, the
Royalty Judges must “conduct a proceeding to determine the
distribution of royalty fees.” 17 U.S.C. § 111(d)(4)(B).

    That proceeding has two phases. In Phase I, the Royalty
Judges apportion the total pool of royalties collected for a
year across broad categories of retransmitted programming—
such as sports, public television, or devotional (religious)
shows—and assign a percentage of the overall fund to each
category based on its comparative value. See IPG II, 2015
WL 3952243, at *1. In Phase II, the Royalty Judges divide up
the amount allotted to each category among the individual
claimants within that category. Id.

     At the outset of each phase, the Royalty Judges announce
the commencement of dispute proceedings in the Federal
Register, which puts interested claimants on notice to file
petitions to participate. See IPG II, 2015 WL 3952243, at *2;
see also 17 U.S.C. § 803(b)(1)(A). The statute then provides
for a three-month “voluntary negotiation period,” during
which the parties attempt to reach agreement. See IPG II,
2015 WL 3952243, at *2; see also 17 U.S.C. § 803(b)(3). For
claimants who have not resolved their disputes during this
period, the Royalty Judges accept written submissions,
                               5
oversee a period for discovery, and provide for a post-
discovery settlement conference period. See IPG II, 2015 WL
3952243, at *2; IPG I, 759 F.3d at 102.

     If the claimants remain at loggerheads, the Royalty
Judges conduct a hearing and issue a final determination
allocating payments. See IPG II, 2015 WL 3952243, at *2.
That decision is subject to a 60-day period of review by the
Register of Copyrights and then published in the Federal
Register. IPG I, 759 F.3d at 102–103 (citing 17 U.S.C.
§§ 802(f)(1)(D), 803(c)(6)). A disappointed claimant may
seek judicial review of the final decision in this court by
appealing within 30 days of the Federal Register publication.
IPG I, 759 F.3d at 103 (citing 17 U.S.C. § 803(d)(1)).

           Factual and Procedural Background

     As in IPG II, this appeal challenges the Phase II
determination of the Copyright Royalty Judges for the years
2000 to 2003. IPG II involved the sports programming and
program suppliers categories, see 2015 WL 3952243, at *2,
while this opinion addresses the allocation of royalties within
the devotional programming category.

     The Devotional Claimants are twenty-three religious
ministries that own copyrights for devotional television
programming. The group participated in the Royalty Judges’
Phase I proceeding, which resulted in a partial settlement
(including as to the devotional programming category) and,
for the non-settling claimants, a final determination allocating
royalties among the other programming categories. See
Distribution of the 2000, 2001, 2002, and 2003 Cable Royalty
Funds, 78 Fed. Reg. 64,984, 64,988 (Oct. 30, 2013).

    The Phase II proceeding to allocate the devotional
programming royalties began in February 2011, with only the
                               6
Devotional Claimants and IPG filing petitions to participate.
Both of them subsequently filed written direct statements
summarizing their claims. IPG, employing a methodology of
its own devising, asserted entitlement to a share of the
royalties ranging from 37.30% to 53.10% for each year from
2000 to 2003. The Devotional Claimants claimed they were
due 100% of the royalties because they knew of no other
“valid, compensable claims” within the devotional category.
J.A. 7638. Seeing no need to share the royalties, the
Devotional Claimants’ direct statement set forth no specific
methodology or calculations to govern apportionment of the
royalties between them and IPG.

     The closest the Devotional Claimants came to an
allocation methodology was the proposed testimony of Dr.
William Brown, who discussed “potential quantifiable
criteria” that the Royalty Judges should consider in making
any allocation that may be required. J.A. 7650. While Dr.
Brown endorsed a survey of cable system operators called the
Bortz survey for making allocations among Phase I
categories, he acknowledged that the Bortz survey did not
include data needed to make the pending Phase II allocation.
In the absence of such Phase II-relevant data, Dr. Brown
suggested that viewership ratings could be “[a] valuable tool”
in determining the “relative marketplace value of particular
programs.” J.A. 7651. He noted that a different group had
previously shown how such data could be used to help project
relevant viewership levels. But Dr. Brown attempted no such
projections or calculations for this case. Rather, his testimony
ended with the general observation that, in a Phase II
determination, “[t]he most useful quantifiable data is Nielsen
viewing data, projected to distant cable households,
supplemented, where applicable with Bortz study data.” J.A.
7652.
                               7
    Both the Devotional Claimants and IPG challenged each
other’s authority to represent the claims of certain copyright
holders. As relevant here, the Devotional Claimants objected
in particular to IPG’s asserted representation of seven of its
eight claimants. 1

     After conducting hearings in November and December
2012, the Royalty Judges issued a Memorandum Opinion and
Order on March 21, 2013, addressing the preliminary
evidentiary and procedural objections of the claimants. The
decision dismissed IPG’s claims as to three claimants, but
rejected the Devotional Claimants’ argument that IPG did not
properly represent Benny Hinn Ministries, Creflo A. Dollar
Ministries, and Life Outreach International. The Royalty
Judges concluded that IPG had come forward with sufficient
evidence of representational authority for each claimant “for
purposes of this preliminary stage of the proceeding.”
Memorandum Opinion and Order Following Preliminary
Hearing on Validity of Claims 6 (March 21, 2013) (“March
21 Op.”), J.A. 3177.

     With respect to IPG’s seventh claimant, Billy Graham
Evangelistic Association (“Billy Graham”), the Royalty
Judges found that IPG had been authorized to file claims on
Billy Graham’s behalf for the years 2001 to 2003, but that the
agreements for 2002 and 2003 had been terminated by Billy
Graham in June 2005. Other correspondence, however,
supported the conclusion that, following a remonstrative letter
from IPG, Billy Graham later revoked this termination of
IPG’s authority. Taking a “dim view” of what they concluded
were IPG’s “mischaracterization of [Billy Graham’s] rights
1
  Other motions challenged IPG’s claims in the program suppliers
and sports programming categories. Our decision in IPG II
addressed those disputes. See 2015 WL 3952243, at *4–*7.
                                8
under the Copyright Act” and “strong-arm tactics [IPG] used
to seek to prevent [Billy Graham] from severing the
principal/agency relationship that [Billy Graham] had clearly
revoked,” March 21 Op. at 8, J.A. 3179, the Royalty Judges
nonetheless declined to dismiss the claims IPG had filed on
behalf of Billy Graham, since doing so would unfairly punish
Billy Graham. The Royalty Judges ordered IPG to obtain
written confirmation of Billy Graham’s continuing interest in
the royalty dispute and agreement to IPG’s representation.

      At the close of their opinion, the Royalty Judges stated
that they would not hear any further testimony or review any
further exhibits on the denied claims and “d[id] not request
and w[ould] not accept revised Written Direct Statements
from the parties, relying instead on their respective abilities to
filter admissible from inadmissible material.” March 21 Op.
at 18, J.A. 3189.

     Royalty Judge Roberts filed a dissenting opinion. In his
view, the March 21 Opinion had improperly left open the
status of certain claims and contemplated additional
challenges, while at the same time “expressly forbid[ding] the
parties from amending their Written Direct Statements prior
to hearings.” J.A. 6220. He further disagreed with the
majority opinion’s analysis of which claims could go forward
and, in particular, with the suggestion that Billy Graham
could pursue its own claims.

     Following those two opinions, Billy Graham submitted a
letter on April 19 acknowledging IPG’s authority to represent
it for the 2002 and 2003 royalty years. With the last
outstanding representation issues thus resolved, the parties
exchanged rebuttal statements in the lead-up to a hearing on
the Phase II royalty distribution. The Devotional Claimants’
response included proposed testimony from a new witness,
                              9
Alan Whitt, who had previously worked with the Motion
Picture Association of America (“MPAA”) in developing a
viewership-based model for assigning relative values to
individual programs. His proposed testimony offered data
derived from the database he had helped develop reflecting
the projected viewership of devotional programming in
distant markets for the years 2000 to 2003.

     The Devotional Claimants also presented additional
testimony from Dr. Brown, who attested that IPG’s
methodology was unreliable, “premised on faulty and
unsupported contentions for valuation of devotional
programming,” and “riddled with calculation errors.” J.A.
6445. Using the viewership projections compiled in Whitt’s
testimony, Dr. Brown also calculated the proportion of the
total viewership of devotional programming attributable to the
Devotional Claimants. He proposed that the Devotional
Claimants should receive between 65 and 75 percent of the
royalties in the devotional category for each year from 2000
to 2003, with IPG receiving the remainder.

     At the allocation hearing, the Royalty Judges excluded
Whitt’s testimony, concluding that it constituted “testimony
regarding the development of information or data” that should
have been disclosed in the Devotional Claimants’ direct case.
J.A. 3839. Dr. Brown was allowed to testify without
objection, and his proposed rebuttal testimony was introduced
as an exhibit.

     On July 10, 2013, the Royalty Judges issued their initial
allocation decision. 78 Fed. Reg. at 64,985. In assigning
royalty amounts for the separate program suppliers category,
the Royalty Judges “ultimately relied heavily” on the
viewership-based methodology put forward by the MPAA.
IPG II, 2015 WL 3952243, at *7; see also id. at *8. In doing
                             10
so, the Royalty Judges extensively criticized IPG’s proposed
methodology. They noted that IPG’s methodology was
advanced not by an econometric or statistical expert, but by
the “imperfect messenger” Raul Galaz, an IPG employee who
had no econometric or statistical expertise, but did have a
fraud conviction arising out of a previous copyright royalty
proceeding. 78 Fed. Reg. at 65,000. They also noted that the
methodology produced numerous results that could not be
justified. Id. at 65,000–65,001.

     For similar reasons, the Royalty Judges dismissed the
validity of IPG’s model for the devotional category, finding
that “IPG’s formula produced absurd results in the Devotional
category, as it did in the Program Suppliers category.” 78
Fed. Reg. at 65,003. The Royalty Judges also dismissed the
Devotional Claimants’ argument that they should use the
same viewership-based model employed in the program
suppliers category as untimely raised and thus procedurally
barred. Id. In particular, the Royalty Judges noted that the
Devotional Claimants waited until their rebuttal case to
present Whitt’s testimony and the data on which Dr. Brown
relied to apply the methodology in question. Id. at 65,004.
The Royalty Judges stressed that, by providing Whitt’s
testimony just three weeks before the hearing, the Devotional
Claimants had “prejudiced IPG and, in essence, engaged in
trial by ambush, in violation of the letter and spirit of the
Judges’ procedural rules * * * [and] deprived IPG of the
opportunity to review the work undertaken by Mr. Whitt.” Id.

     The Royalty Judges’ rejection of both parties’ proposed
methodologies left them empty-handed in allocating the
devotional-programming royalties. See 78 Fed. Reg. at
65,004. Explaining that they were “nevertheless[] obligated
to reach a determination based on the existing record,” the
Royalty Judges observed that, for the year 2000, the
                              11
allocations proposed by IPG happened to fall within a range
proposed by the Devotional Claimants. Id. They ruled that
this “some degree of agreement” provided a basis for
awarding to IPG the 37.14% of royalties it proposed. Id. For
the year 2002, the Royalty Judges similarly found that IPG’s
proposed allocations were “almost equal to the lower bound”
of the range proposed by the Devotional Claimants. Id. They
thus concluded that it would be within the “zone of
reasonableness” to give effect to IPG’s number, holding that
IPG would receive 41.02% of the royalties. Id.

     For the years 2001 and 2003, the Royalty Judges
recognized that the parties’ proposals were not even in the
same ballpark, 78 Fed. Reg. at 65,004, but stated that “there is
no record evidence explaining why the percentage allocations
for 2001 and 2003 should be so markedly different in those
years compared to 2000 and 2002.” Id. at 65,004–65,005.
The Royalty Judges accordingly allocated royalties for 2001
and 2003 by simply averaging the allocations for the two
other years. Under that approach, the Devotional Claimants
received 60.92% of the royalties for each year, and IPG
received 39.08%. Id. at 65,005.

     The Devotional Claimants subsequently filed a petition
for rehearing, which the Royalty Judges denied. With respect
to the Devotional Claimants’ objection to the exclusion of
Whitt’s and Dr. Brown’s rebuttal testimony, the Royalty
Judges reiterated that 37 C.F.R. § 351.4(b) required that
Whitt’s testimony be included in the written direct statement.
They concluded that the Devotional Claimants had been
afforded “ample time” to submit an amended written direct
statement with this additional information and simply did not
do so. J.A. 6877. Finally, the Judges rejected a challenge to
the final allocation of royalties, emphasizing the overlap or
near-overlap in the parties’ proposals for two years and
                              12
insisting that, for the other two years, the Royalty Judges had
not relied on IPG’s proposed allocations.

     The Royalty Judges issued their final determination on
August 13, 2013. It was approved by the Librarian of
Congress and published in the Federal Register on October
30, 2013. This appeal followed.

                              II

                          Analysis

     We have jurisdiction over the Devotional Claimants’
timely filed appeal pursuant to 17 U.S.C. § 803(d)(1). That
jurisdiction includes the Devotional Claimants’ objections to
the Royalty Judges’ March 21 procedural and evidentiary
order, because that earlier interlocutory order merges into and
is reviewable as part of the Royalty Judges’ final
determination. See IPG II, 2015 WL 3952243, at *4.

    We review decisions of the Copyright Royalty Judges
under the familiar standards of the Administrative Procedure
Act, reversing only if their decision is arbitrary, capricious,
contrary to law, or not based on substantial evidence. See 17
U.S.C. § 803(d)(3) (incorporating by reference 5 U.S.C.
§ 706). Our review is “highly deferential,” Intercollegiate
Broadcast Sys., Inc. v. Copyright Royalty Board, 571 F.3d 69,
79 (D.C. Cir. 2009), and, in reviewing royalty distribution
decisions specifically, we ask only whether the Royalty
Judges’ assigned allocation percentages are “within a zone of
reasonableness,” Christian Broadcasting Network, Inc. v.
Copyright Royalty Tribunal, 720 F.2d 1295, 1304 (D.C. Cir.
1983) (internal quotation marks omitted).
                              13
    IPG’s Authority to Represent Four Claimants

     The Devotional Claimants repeat on appeal their
objections to IPG’s representation of Benny Hinn Ministries,
Creflo A. Dollar Ministries, Life Outreach International, and
the Billy Graham Evangelistic Association. Those largely
factual inquiries implicating the Royalty Judges’ own
proceedings, however, fall squarely within the Royalty
Judges’ area of expertise. The only question before us is
whether the Royalty Judges’ rulings were supported by
substantial evidence and were not arbitrary or capricious.
Their decision crosses that deferential threshold.

     The Devotional Claimants do not dispute that
representational authority turns on a factual inquiry into
“whether the claimant intended for its claim to be filed on its
behalf by another.” March 21 Op. at 4, J.A. 3175. Such
intent must be expressed prior to the filing of the relevant
claim. See Distribution of 1993, 1994, 1995, 1996 and 1997
Cable Royalty Funds, 66 Fed. Reg. 66,433–66,435 (Dec. 26,
2001) (noting Library of Congress’s determination that “what
the law requires[] is a factual determination as to which of the
owners and distributors identified [in a claim] were in fact
represented by [the filer] at the close of the filing period for
1997 cable claims”) (quoting Order in Docket No. 2002-2
CARP CD 93-97 at 7 (June 22, 2000)); see also 37 C.F.R.
§ 360.3(b)(1)(vi), (2)(ii), (2)(vii) (requiring declaration or
statement of authorization to be filed with claims for
compulsory license royalties).

     For the Benny Hinn and Creflo Dollar Ministries, the
Royalty Judges rested their judgment on IPG’s evidence of
agreements in which each ministry authorized IPG “to apply
for and collect any and all monies distributed by audiovisual
copyright collection societies throughout the world (e.g.,
                              14
monies derived from rights set forth on Exhibit ‘A’ hereto)”
for works that the ministry owned or distributed. J.A. 6195,
6198. Exhibit A, in turn, included among the list of
recoverable monies “Cable and Satellite Retransmission
Royalties,” defined as “[r]oyalties and charges imposed by
law with respect to the retransmission by cable or satellite of
terrestrial broadcast signals.” J.A. 6197, 6201. IPG also
produced email correspondence in which the ministry
identified for IPG programs for which claims could be made.

     The Devotional Claimants argue that there was
insufficient evidence that the agreements were signed before
the claims were filed and that IPG needed to present
confirmatory testimony from the representatives of the
ministries. Looking through the highly deferential lens of
substantial evidence review, however, we hold that the
documentary agreements themselves provided sufficient
evidence that IPG was authorized to file claims on behalf of
the two ministries. The Devotional Claimants suggest that the
Royalty Judges’ decision contravened earlier precedent
requiring a heftier evidentiary showing in the distribution of
the 1997 cable royalty funds. But that distribution involved a
case in which there was actual evidence of backdating the
dates on which the contracts were signed. See 66 Fed. Reg. at
66,438–66,439. There was no analogous evidence here, and
thus it was not unreasonable for the Royalty Judges to rely on
the written representation agreements themselves, as
corroborated by subsequent email correspondence.

     Finally, invoking our decision in National Broadcasting
Co. v. Copyright Royalty Tribunal (“NBC”), 848 F.2d 1289
(D.C. Cir. 1988), the Devotional Claimants argue that the
Royalty Judges’ reading of the phrase “audiovisual copyright
collection societ[y]” in the two agreements was the product of
impermissible contract interpretation that went beyond the
                               15
“face of these [contracts]” to consider their exhibits.
(Devotional Claimants’ Br. 9).                  That argument
misunderstands NBC. Disputes regarding who may claim
royalties generally raise two issues:           “(1) the proper
distributee of the royalties allocated by the [agency] to [a
particular program]; and (2) the ownership of the copyright to
which the royalties attach.” NBC, 848 F.2d at 1293. NBC
made clear that the Royalty Judges have no authority to
decide the second question—the proper ownership of the
copyright. Id. Accordingly, when the Royalty Judges assign
a claim to a particular claimant as part of their allocation
process (the first question), the ruling “should not be seen at
all as adjudicating the contractual entitlement rights of [the
parties], but rather as setting forth a general rule for the
distribution of cable royalties in these cases,” and the
disposition “leaves the parties free to litigate their contractual
claims in an appropriate forum.” Id. at 1296.

     That holding does not preclude the Royalty Judges from
looking at a contract in resolving the first question or other
antecedent questions concerning whether to allocate royalties
to a particular program in the first place. Instead, the statute’s
silence on how to resolve a dispute over an agency
relationship for purposes of royalty-allocation proceedings
leaves the Royalty Judges discretion to which we must defer.
See NBC, 848 F.2d at 1296 (deferring to the Copyright
Royalty Tribunal’s “presumption * * * in the face of
congressional silence” regarding allocation as “permissible
interpretation of the statute”).

     In short, then, the Copyright Act does not confine the
Royalty Judges to a face-of-the-contract-only analysis of
representational authority in proceedings before them, and
nothing in the Royalty Judges’ consideration of the exhibits
to the Benny Hinn and Creflo Dollar agreements contravened
                                 16
NBC. While the Royalty Judges’ decision would not bind the
parties in any future contractual dispute, see NBC, 848 F.2d at
1293, 1296, it was an entirely appropriate step for the Royalty
Judges to take here.

     For largely the same reasons, the Royalty Judges’
decision to allow IPG to continue to represent Life Outreach
also passes muster. Indeed, the agreement with Life Outreach
specifically identified monies associated with cable and
retransmission royalties under 17 U.S.C. §§ 111 and 119 as
within IPG’s authority to collect. While the Devotional
Claimants object to the absence of a date associated with
IPG’s signature on the agreement, we cannot say that the
Royalty Judges’ reliance on the date that Life Outreach’s
representative signed was unreasonable or based on
insubstantial evidence.

     The Devotional Claimant’s objection to IPG’s
representation of Billy Graham focuses less on the evidence
of initial authorization than on Billy Graham’s subsequent
effort to terminate IPG’s representation. While the Royalty
Judges’ decision could have been clearer, it was clear enough
to be sustained on this point, too. 2

    Billy Graham terminated IPG as its agent in June 2005,
which was after the filing date for claims for each of the

2
   With respect to the question of initial authorization, the
Devotional Claimants highlight that two of the Billy Graham
agreements were not signed by IPG. But there was testimony
before the Royalty Judges that IPG had lost some copies of the
agreements. And the termination letter sent by Billy Graham is
itself evidence that there was a prior agreement to be terminated for
at least one of the years that the Devotional Claimants contest.
                                  17
royalty years at issue in this proceeding. Accordingly, for
purposes of this argument, IPG’s authority to have filed the
claims in the first instance is not at issue. The question,
instead, is whether IPG had authority to petition to participate
in the Phase II proceeding on behalf of Billy Graham. To do
so, IPG would again have had to be authorized to represent
Billy Graham at least by the latest point at which Billy
Graham could have filed its own petition to participate. See
37 C.F.R. § 351.1(b)(2)(ii)(E) (joint petition to participate
filed by copyright owners’ representative must include
certification that, “as of the date of submission of the joint
petition, such * * * representative has the authority and
consent of the participants to represent them in the royalty
distribution proceeding.”). 3

     In finding such authority, the Royalty Judges relied in
part on correspondence between Denise Vernon, an IPG
executive, and Billy Graham reflecting that, at some
unspecified date, Vernon employed what the Royalty Judges
termed “strong-arm tactics” to induce Billy Graham to allow
continued representation by IPG. March 21 Op. at 8, J.A.
3179. Subsequent emails, which at least extended to after the
date by which Billy Graham would have had to file a petition
to participate, indicate that Billy Graham acceded to IPG’s
efforts to collect royalties for the years at issue. A witness for
IPG also testified that Billy Graham had “recanted” its initial
termination of IPG. J.A. 2907.


3
    See also 37 C.F.R. § 351.1(d) (Royalty Judges “may, for
substantial good cause shown, and if there is no prejudice to the
participants that have already filed petitions, accept late petitions to
participate at any time up to the date that is 90 days before the date
on which participants in the proceeding are to file their written
direct statements”); 17 U.S.C. § 803(b)(1)(A)(ii) (same).
                             18
     The record before the Royalty Judges certainly does not
demand the conclusion that Billy Graham’s “recantation” was
timely. But all that matters is that we cannot say that the
Royalty Judges lacked substantial evidence in finding that
Billy Graham’s reauthorization of IPG was timely made. See,
e.g., Christian Broadcasting Network, 720 F.2d at 1313
(concluding substantial evidence existed for the Copyright
Royalty Tribunal’s conclusion that it had made awards only to
bona fide copyright owners).

     The Royalty Judges also obtained contemporaneous
corroboration from Billy Graham of IPG’s authority, and we
see nothing in the Copyright Act or the Royalty Judges’ rules
that precluded that measure. Indeed, given the questionable
representations that IPG made in obtaining that
reauthorization, the reconfirmation made sense. Accordingly,
given the very narrow scope of our review, we uphold the
Royalty Judges’ determinations concerning the scope of
IPG’s representation authority.

    Exclusion of the Devotional Claimants’ Allocation
    Methodology

     The Devotional Claimants next object to the Royalty
Judges’ exclusion of testimony and evidence they presented in
their rebuttal case suggesting a possible viewership-based
valuation methodology for devotional programming royalties.
We conclude that the Royalty Judges’ application of their
own procedural regulations was reasonable.

     The regulations governing proceedings before the
Royalty Judges provide that all parties who have filed a
petition to participate in a proceeding “must file a written
direct statement.” 37 C.F.R. § 351.4(a). A written direct
statement is defined by statute to mean “witness statements,
testimony, and exhibits to be presented in the proceedings,
                              19
and such other information that is necessary to establish * * *
the distribution of royalty payments * * * as set forth in
regulations issued by the Copyright Royalty Judges.”
17 U.S.C. § 803(b)(6)(C)(ii)(II). The Judges’ regulations, in
turn, provide that “[t]he written direct statement shall include
all testimony, including each witness’s background and
qualifications, along with all the exhibits.” 37 C.F.R.
§ 351.4(b)(1). In a royalty distribution proceeding, the parties
must further include in their direct statement their “percentage
or dollar claim to the fund,” but “[n]o party will be precluded
from revising its claim * * * at any time during the
proceeding up to, and including, the filing of the proposed
findings of fact and conclusions of law.” Id. § 351.4(b)(3).

     Focusing on the requirement that the written direct
statement include “all testimony” a party intends to present,
the Royalty Judges concluded that the Devotional Claimants’
direct statement should have included analysis of how the
viewership-based methodology that they endorsed in general
terms would apply in a relative valuation of the claims
brought by the Devotional Claimants and IPG.             The
Devotional Claimants challenge both that interpretation of
Section 351.4 and the reasonableness of its application.

     We generally accord deference to an agency’s
interpretation of its own regulation “unless that interpretation
is plainly erroneous or inconsistent with the regulation.”
Decker v. Northwest Envtl. Defense Ctr., 133 S. Ct. 1326,
1337 (2013) (internal quotation marks omitted). Such
deference is perhaps particularly appropriate here because the
rule concerns the Copyright Royalty Judges’ own procedures.
Cf. National Ass’n of Broadcasters v. Copyright Royalty
Tribunal, 675 F.2d 367, 375 n.8 (D.C. Cir. 1982) (noting that
the Copyright Act “gives the [Copyright Royalty Judges’
predecessor] considerable freedom to determine its own
                               20
procedures”). Such deference may not even be needed,
though, because the Royalty Judges’ interpretation wholly
comports with the plain text of the regulation. It should be no
surprise that a requirement that a party present in its initial
direct statement “all testimony” necessary to establish its
claimed entitlement to royalties would obligate the party to
include in that statement the testimony applying its proposed
methodology to the case at hand. Indeed, linking the
proposed methodology to the facts of the case is the
testimonial heart of the matter. Abstract discussion of
potential methodologies, without any application, would do
nothing to support the party’s desired outcome.

     Nor do the rules provide any reason to think that this
weighty evidence may be saved for the rebuttal case. Indeed,
in civil litigation generally, courts have recognized that a trial
court “has the discretion to ‘limit the scope of rebuttal
testimony to that which is directed to rebut new evidence or
new theories proffered in the defendant’s case-in-chief.” Toth
v. Grand Trunk R.R., 306 F.3d 335, 345 (6th Cir. 2002)
(quoting Martin v. Weaver, 666 F.2d 1013, 1020 (6th Cir.
1981)). When, as here, the proposed rebuttal testimony
“could have been included in the same witness’ direct
examination,” it may be excluded. Waterview Mgmt. Co. v.
FDIC, 203 F.3d 54, 1999 WL 503921, at *2 (D.C. Cir. May
20, 1990) (unpublished) (citing Geders v. United States, 425
U.S. 80, 86 (1976)). Nothing in the text of the regulations
suggests a different operation here. The Royalty Judges thus
acted within the bounds of reasonable discretion in affording
their rule an interpretation grounded in text and practice.

     The Devotional Claimants point out that a predecessor to
the Royalty Judges once allocated royalties relying in part on
a study presented only in the rebuttal phase of the proceeding.
See Distribution of 1998 and 1999 Cable Royalty Funds,
                              21
69 Fed. Reg. 3,606, 3,619 (Jan. 26, 2004). In that case,
however, the Librarian turned aside an objection to the
consideration of the evidence in part by noting the very
limited use that was made of it. See id. Indeed, the Librarian
specifically noted that, had the Copyright Arbitration Royalty
Panel “fully credited [the study in question] and used it as the
basis for determining [the objecting party’s] award,” the
outcome may have been different. Id. That one-time
dispensation hardly evidences a binding precedential blessing
of parties lobbing in critical evidence, like the determinative
application of a decisional methodology, only at the rebuttal
phase.

     The Devotional Claimants’ reliance on National
Association of Broadcasters as another case in which
evidence introduced during rebuttal was used is no help
either. No one even raised the admissibility issue in the case.
See 675 F.2d at 376.

     We also hold that the Royalty Judges’ application of the
rule in this case was neither arbitrary nor capricious. The
Devotional Claimants assert that they did not know until after
the March 21 order that IPG in fact had valid claims to assert,
but by that point, the Royalty Judges had closed the door on
amended direct statements. See March 21 Op. at 18, J.A.
3189; see also 37 C.F.R. § 351.4(c) (“A participant in a
proceeding may amend a written direct statement based on
new information received during the discovery process,
within 15 days after the end of the discovery period.”). The
Devotional Claimants’ bad strategy call does not make
enforcement of written rules of which they had fair notice
arbitrary or capricious.

    First, the Devotional Claimants had full notice prior to
the time an amended written direct statement would have
                              22
been due that IPG claimed a share of the royalties as well.
Any uncertainty about which claims IPG might be able to
represent—or hope that they would all be dismissed—was no
excuse for failing to prepare for the chance of a disputed
allocation. Indeed, the Devotional Claimants did not even
specifically object to one of IPG’s representations, so it was
more than likely that a contested allocation would go forward.
In any event, there was certainly nothing to prevent them
from timely identifying testimony by the August 2012
deadline, rather than making a tactical choice to wait until
May 2013—just 20 days before the hearing—to disclose their
anticipated testimony.

     Tellingly, the MPAA was faced with similar uncertainties
in the program suppliers category. But when the MPAA
received IPG’s direct statement claiming a share of the
royalties, the MPAA timely filed an amended written direct
statement specifying how a relative valuation should be
calculated under its proposed methodology. The Devotional
Claimants could have done the same.

     Second, the Devotional Claimants point out that, on the
eve of the hearing, IPG was permitted to provide a great deal
of additional data, reflecting recalculations performed in
response to an error identified in its initial methodology. See
78 Fed. Reg. at 65,001–65,002. They argue that the disparate
treatment of their late evidence and IPG’s was arbitrary and
capricious. We disagree. IPG’s submission was a correction
to data that had been presented along with IPG’s applied
methodology in IPG’s direct statement, affording the
Devotional Claimants fair notice. The Royalty Judges could
reasonably conclude that such amendments were far less
prejudicial than interjecting the application of an allocation
methodology for the very first time in rebuttal just weeks
before the hearing was to begin.
                               23
     Third, the Devotional Claimants are wrong to argue that,
because Dr. Brown’s written rebuttal testimony was accepted
into evidence without objection, the Royalty Judges were also
obligated to consider the portions of this testimony applying a
viewership-based      methodology       to     the   devotional
programming category. “In for a penny, in for a pound” is not
an evidentiary rule binding the Royalty Judges. That is
especially true when, as here, the foundation for Dr. Brown’s
rebuttal testimony was the excluded testimony of Whitt. With
Dr. Brown’s testimony stripped of its proffered factual basis,
the Royalty Judges reasonably concluded that they could not
give it any weight.

     To be sure, as the Devotional Claimants note, experts
may base their opinion on facts that may otherwise be
inadmissible. See FED. R. EVID. 703 (“If experts in the
particular field would reasonably rely on those kinds of facts
or data in forming an opinion on the subject, they need not be
admissible for the opinion to be admitted.”). But the
Devotional Claimants do not claim to have made any showing
before the Royalty Judges that Dr. Brown’s testimony had
been based on the kinds of facts on which an expert would
reasonably rely.

    The Ultimate Allocation Decision

    The Devotional Claimants’ final objection is to the
Royalty Judges’ decision to go forward with the allocation of
royalties despite having rejected every proposed methodology
making that decision for the devotional programming
royalties. As to this final challenge, we agree that, despite its
Solomonic pedigree, the Royalty Judges’ approach was
quintessentially arbitrary and capricious.

   With respect to the royalties from the year 2000, the
Royalty Judges ruled that there was a functional agreement—
                              24
or at least “some degree” of agreement—between the parties
to the extent that IPG’s proposed allocation percentages
happened to fall within the range proposed by the Devotional
Claimants. 78 Fed. Reg. at 65,004. We are told on appeal
that the parties “effectively” agreed on this point. Gov’t Br.
50.

     We cannot agree. Settling royalty distributions by
agreement reflects a separate avenue for resolving royalty
distributions under the Copyright Act, subject to its own
requirements. See 17 U.S.C. § 801(b)(7)(A)(i) (providing that
Royalty Judges may use agreement as the basis for a royalty
distribution provided that the parties that would be bound by
the determination have an opportunity to comment). In this
case, any intersection of the two parties’ numbers was the
product of accident, not agreement, as evidenced by the fact
that the Devotional Claimants were awarded almost 5% less
of the total fund than they had requested. Indeed, even to find
overlap in the parties’ numbers at all, the Royalty Judges
adopted a “two wrongs make a right” methodology,
(i) crediting IPG percentages that were derived from a
discredited methodology and partially based on claims that
the Royalty Judges had ruled IPG could not bring, and
(ii) giving effect to allocations derived from a methodology
that the Royalty Judges had refused to consider. In the face of
that actual non-agreement, simply picking a number out of
IPG’s flawed and otherwise-rejected proposal just because it
happened to roughly coincide with the lowest bound proposed
by the Devotional Claimants falls beyond the bounds of
reasoned decisionmaking.

    It gets still worse for subsequent royalty years, in which
even the fig leaf of “some degree” of agreement fell away.
For 2002, the Royalty Judges chose IPG’s (otherwise infirm)
proposed allocation solely because it was deemed to be close
                              25
enough to the lower bound proposed by the Devotional
Claimants. The decision gives no hint as to how close is close
enough, nor any explanation for why they picked IPG’s
proposal in lieu of the supposedly close-enough lowest bound
of the range proposed by the Devotional Claimants. And for
2001 and 2003, the Judges, with a blank slate of an
evidentiary record, simply split the difference of the
allocations from the two other years.

     That cannot be sustained.        The Royalty Judges’
obligation is to make reasoned decisions supported by the
written record before them. See 17 U.S.C. § 803(c)(3). They
do not satisfy that burden by bridging over a lacuna in the
record with a purported agreement that does not actually exist.

    That is not to say that royalty distributions must be
perfect. “[A]n agency’s choice of a particular number or
percentage is not reviewable for exact precision, but simply
for broad reasonableness.” National Ass’n of Broadcasters,
675 F.2d at 374. Indeed, “mathematical exactitude in these
matters appears well-nigh impossible; rough justice in
dividing up the royalty pie seems to be the inevitable result of
the process that Congress ordained.” National Ass’n of
Broadcasters v. Copyright Royalty Tribunal, 772 F.2d 922,
926 (D.C. Cir. 1985) (citation omitted). And we have on rare
occasion sustained a superficially similar rough-justice
approach. But in those cases, the administrative body relied
on some relevant and creditable methodological evidence,
even if it was “far from perfect,” National Cable Television
Ass’n, Inc. v. Copyright Royalty Tribunal, 724 F.2d 176, 184
(D.C. Cir. 1983), or “fairly but not wholly satisfactory,”
Association of American Publishers, Inc. v. Governors of U.S.
Postal Service, 485 F.2d 768, 773 (D.C. Cir. 1973). In this
case, even that minimal foundation is lacking.
                             26
     We thus face here on a larger scale the type of situation
we confronted in Intercollegiate Broadcast System, Inc. v.
Copyright Royalty Board, 571 F.3d 69 (D.C. Cir. 2009). In
that case, the Royalty Judges’ rate-setting decision adopted a
certain minimum fee to be charged to each noncommercial
webcasting channel or station to cover administrative costs.
See id. at 86–87. There the Royalty Judges relied on a single
party’s proposed fee amount, notwithstanding the lack of any
demonstrated link between the proposal and the expenses it
was designed to compensate. Id. at 87. We rejected that
approach, holding that the Royalty Judges’ decision was
“inconsistent with rational decisionmaking.” Id.

     A reasoned justification “requires more than an absence
of contrary evidence; it requires substantial evidence to
support a decision.” Intercollegiate Broadcast, 571 F.3d at
87. That is missing here. And making matters still worse, the
Royalty Judges’ approach to allocation was “first presented in
the Judges’ determination and not advanced by any
participant.” Id.

     Perhaps the Royalty Judges’ decision was driven by the
statutory mandate to decide the case within eleven months of
the end of the settlement conference period. See 17 U.S.C.
§ 803(c)(1). In an appropriate case, the conditions imposed
on the Royalty Judges and the resource constraints they face
may be relevant to our consideration of the reasoned nature of
their decisionmaking. See National Cable Television Ass’n,
724 F.2d at 187. But the bottom-line obligation to produce a
reasoned decision remains. Moreover, the Royalty Judges
were not limited to the deficient methodological evidence the
parties put before them. The Copyright Act permits the
Royalty Judges to request information even from a
nonparticipant if relevant to a material issue of fact. 17
U.S.C. § 803(b)(6)(C)(ix). Indeed, the Royalty Judges have
                              27
made use of similar initiatives before to obtain additional
methodological evidence. See 66 Fed. Reg. at 66,441.

     For those reasons, we vacate the Royalty Judges’
allocation of royalties in the devotional programming
category for 2000 to 2003. We leave to the Royalty Judges
on remand how to balance their legitimate interest in
preventing parties before them from engaging in trial by
ambush with the need to have a sufficient factual basis to
make a reasoned decision.

                              III

                         Conclusion

     We hold that the Royalty Judges reasonably determined
that IPG had the authority to represent the four claimants
challenged by the Devotional Claimants on appeal. The
Royalty Judges also reasonably declined to consider the
Devotional Claimants’ methodological evidence given its
untimely presentation. We conclude, however, that the
Royalty Judges’ ultimate royalty allocation, in the wake of the
evidentiary gap left by their rejection of all proffered
methodologies, was arbitrary and capricious.               We
consequently vacate that portion of the determination and
remand for further proceedings consistent with this opinion.

                                                   So ordered.
