                               In the

    United States Court of Appeals
                For the Seventh Circuit
                    ____________________
Nos. 15-2385 & 15-2386
KLEEN PRODUCTS LLC, et al.,
                                                 Plaintiffs-Appellees,

                                 v.

INTERNATIONAL PAPER COMPANY, et al.,
                                             Defendants-Appellants.
                    ____________________

        Appeals from the United States District Court for the
          Northern District of Illinois, Eastern Division.
          No. 10 C 5711 — Harry D. Leinenweber, Judge.
                    ____________________

    ARGUED DECEMBER 8, 2015 — DECIDED AUGUST 4, 2016
                    ____________________

    Before WOOD, Chief Judge, and BAUER and WILLIAMS, Cir-
cuit Judges.
    WOOD, Chief Judge. The antitrust laws prohibit competing
economic actors from colluding to agree on prices, either di-
rectly or through such mechanisms as output restrictions. See
United States v. Socony-Vacuum Oil Co., 310 U.S. 150 (1940);
Palmer v. BRG of Georgia, Inc., 498 U.S. 46 (1990). That is just
what the plaintiffs in the case before us allege the producers
and sellers of containerboard did. The plaintiff-purchasers
2                                       Nos. 15-2385 & 15-2386

filed this suit under Sherman Act § 1, 15 U.S.C. § 1, seeking to
recover treble damages for the overcharges they allegedly
paid. See Clayton Act § 4, 15 U.S.C. § 15. What brings the case
before us at this time—well before the merits have been re-
solved—is the district court’s decision to certify a nationwide
class of purchasers under Federal Rule of Civil Procedure 23.
The defendants, International Paper Company, Georgia-Pa-
cific LLC, Temple-Inland Inc., RockTenn CP, LLC, and Weyer-
hauser Company (to whom we will refer collectively as De-
fendants unless the context requires otherwise), asked us to
accept this interlocutory appeal from the certification decision
pursuant to Rule 23(f). We agreed to do so. Finding no abuse
of discretion in the district court’s decision, however, we af-
firm.
                                I
    The Purchasers allege in their complaint that the defend-
ant companies agreed “to restrict the supply of container-
board by cutting capacity, slowing back production, taking
downtime, idling plants, and tightly restricting inventory.”
These actions predictably led to an increase in the price of con-
tainerboard—a price increase that caused Purchasers to pay
more for containerboard products than they would have paid
in the absence of the illegal agreement. The named plaintiff
on the complaint is Kleen Products LLC. It asked the district
court to certify the following class:
    All persons that purchased Containerboard Products
    directly from any of the Defendants or their subsidiar-
    ies or affiliates for use or delivery in the United States
    from at least as early as February 15, 2004 through No-
    vember 8, 2010.
Nos. 15-2385 & 15-2386                                          3

The proposed definition carved out the defendants them-
selves, entities or personnel related to them, and governmen-
tal entities. The Defendants opposed class certification on a
number of grounds: whether common questions predomi-
nate; whether antitrust injury can be proved using a common
method; whether the amount of damages can be proved using
a common method; and whether a class action is superior.
    As the Supreme Court emphasized in Wal-Mart Stores, Inc.
v. Dukes, 564 U.S. 338 (2011), “Rule 23 does not set forth a mere
pleading standard. A party seeking class certification must af-
firmatively demonstrate his compliance with the Rule … .” Id.
at 350. We must therefore take a careful look at the evidence
that the Purchasers presented in support of class certification
as we assess the district court’s ruling. Some of that evidence
was provided by experts, but at this stage we need say little
about them, because no defendant challenged the Purchasers’
experts under Federal Rule of Evidence 702 or the Supreme
Court’s decision in Daubert v. Merrell Dow Pharm., Inc., 509 U.S.
579 (1993). See Tyson Foods, Inc. v. Bouaphakeo, 136 S. Ct. 1036,
1049 (2016) (where there is no Daubert challenge, district court
may rely on expert evidence for class certification). The dis-
trict court also pointed out that “[f]or the most part, the par-
ties agree on the basic facts, and both parties’ experts rely
upon the same data, so there are little if any factual disputes
that the Court must resolve to decide class certification.” For
that reason, the court concluded that there was no need for a
comprehensive evidentiary hearing. This, in our view, was a
case-management decision that we have no reason to second-
guess, despite Defendants’ complaints. See American Honda
Motor Co. v. Allen, 600 F.3d 813, 815 (7th Cir. 2010) (evidentiary
hearing should be held “if necessary”); West v. Prudential Sec.,
Inc., 282 F.3d 935, 938 (7th Cir. 2002) (same).
4                                        Nos. 15-2385 & 15-2386

    Two final points are worth making before we turn to the
evidence. First, nothing in Wal-Mart changed the applicable
standard of review, which is deferential (as the cases say, only
for “abuse of discretion”). Messner v. Northshore Univ.
HealthSystem, 669 F.3d 802, 811 (7th Cir. 2012). Second, it re-
mains true that Rule 23 does not demand that every issue be
common; classes are routinely certified under Rule 23(b)(3)
where common questions exist and predominate, even
though other individual issues will remain after the class
phase. See, e.g., McMahon v. LVNV Funding, 807 F.3d 872, 875–
76 (7th Cir. 2015); Pella Corp. v. Saltzman, 606 F.3d 391, 393 (7th
Cir. 2010).
                                II
   Although the requirements for class certification under
Rule 23 are familiar, we set out the critical sections of the rule
here for ease of reference:
        (a) Prerequisites. One or more members of a class
    may sue or be sued as representative parties on behalf
    of all members only if:
       (1) the class is so numerous that joinder of all
       members is impracticable;
       (2) there are questions of law or fact common to
       the class;
       (3) the claims or defenses of the representative
       parties are typical of the claims or defenses of
       the class; and
       (4) the representative parties will fairly and ad-
       equately protect the interests of the class.
       ***
Nos. 15-2385 & 15-2386                                         5

      (b) Types of Class Actions. A class action may be
   maintained if Rule 23(a) is satisfied and if:
        ***
       (3) the court finds that the questions of law or
       fact common to class members predominate
       over any questions affecting only individual
       members, and that a class action is superior to
       other available methods for fairly and efficiently
       adjudicating the controversy. The matters perti-
       nent to these findings include:
          (A) the class members’ interests in indi-
          vidually controlling the prosecution or
          defense of separate actions;
          (B) the extent and nature of any litigation
          concerning the controversy already be-
          gun by or against class members;
          (C) the desirability or undesirability of
          concentrating the litigation of the claims
          in the particular forum; and
          (D) the likely difficulties in managing a
          class action.
Our focus is on the predominance requirement of subpart
(b)(3), since, as the district court noted, “Defendants have con-
ceded that typicality, commonality, and adequacy have been
satisfied so long as [Purchasers] have adequately proven pre-
dominance,” and no one is arguing about numbers either.
                               A
   We begin with some background facts about the contain-
erboard industry. “Containerboard” is the term for a sheet of
6                                          Nos. 15-2385 & 15-2386

heavy paper with a smooth top and bottom (the linerboard)
and a fluted layer between the two (the corrugated medium).
It is made in large, expensive mills; as of 2008, no new mills
had been built in the United States for more than 12 years. The
containerboard sheets are cut and folded into products such
as boxes of varying sizes. The industry is dominated by verti-
cally integrated producers, which means simply that the fab-
rication of the containerboard and then its processing into fi-
nal products are handled internally by a firm. This means, im-
portantly for antitrust purposes, that the Purchasers bought
directly from the alleged conspirators, not through interme-
diaries. See Illinois Brick Co. v. Illinois, 431 U.S. 720 (1977) (dis-
tinguishing between direct-purchaser suits, which are permit-
ted under Clayton Act § 4, and indirect-purchaser suits, which
are not).
    Containerboard is a commodity, sold in standardized
compositions and weights. The final products are also stand-
ardized; one trade association commented that “boxes are es-
sentially commodity items used in well established markets.”
The most common containerboard product sold in the United
States in 2010 was unbleached kraft linerboard weighing 42
pounds per thousand square feet. Pulp & Paper Week (PPW),
an industry periodical, publishes weekly price indices that in-
clude the price for the 42-pound linerboard for delivery east
of the Rocky Mountains. The PPW index, as it is called, is
widely used within the industry as a benchmark.
    A small and shrinking number of firms produce most of
the containerboard in North America. As of 1997, the five
largest firms (i.e. the current defendants or their predecessors)
were responsible for 41% of North American production. By
2007, the Defendants furnished 74% of that production. Add
Nos. 15-2385 & 15-2386                                         7

in the next two firms, and the number swelled to 84%. (This
evidence is ambiguous: a cartel’s market share will shrink
over time, to the extent that the high prices attract new entry
from fringe competitors or imports, but its market share will
grow to the extent that the cartel successfully uses exclusion-
ary devices. The existing record does not resolve that ambigu-
ity, but it does not matter for purposes of class certification.)
    There was a great deal of evidence designed to show that
the hypothesis that Defendants had organized a cartel was
one that a jury could accept. We do not need to review all of
it, but we offer some key points. During the class period,
which ran from February 15, 2004, through November 8, 2010,
Defendants attempted 15 price increases, and with one excep-
tion, all Defendants joined each one, at roughly the same time
(11 out of 15 times within the same month). Twelve times out
of 15 they increased prices by identical amounts; the remain-
ing times the increases of different firms varied by less than
2% of the average price.
     Capacity in the industry over this period was declining in
North America, though increasing elsewhere; meanwhile, de-
mand was constant or increasing. Defendants increased
prices at least once during the recession of 2008 and 2009, and
they raised prices again twice in 2010. Inventory levels were
decreasing, because of several steps they took: they closed
many mills; they indefinitely idled some; they temporarily
idled others; and they slowed down production. In 2005, they
announced mill closures representing 931,000 tons of capac-
ity, and shortly thereafter they raised prices $30/ton. They did
much the same thing in 2009. Communication among the De-
fendants was easy, thanks to trade associations.
8                                      Nos. 15-2385 & 15-2386

    These and other facts spurred the Purchasers to bring this
suit and to structure it as a class action. They filed their mo-
tion for class certification only after extensive discovery. In
that motion, they relied on the type of industry facts we have
just mentioned, as well as on reports from two experts, Mi-
chael J. Harris and Mark Joseph Dwyer. Defendants coun-
tered with reports from two other experts, Dennis Carlton and
Janusz Ordover. Harris concluded that the structure of the
containerboard industry made it likely that a conspiracy
among the Defendants could succeed in increasing prices for
all or nearly all purchasers; he also opined that Defendants’
strategy would not have made sense if it had been undertaken
unilaterally by each company. Carlton and Ordover disa-
greed, contending that Defendants’ pricing behavior could be
explained by oligopolistic interdependence (that is, by paral-
lel but independent behavior undertaken by firms in a con-
centrated market). They suggested ways in which the supply
restrictions might have been rational under the circum-
stances. Tellingly, however, they never said that there might
have been a cartel with respect to some purchasers and not
with respect to others.
    On the subject of damages, Purchasers’ expert Dwyer ex-
amined price movements. For example, he compared the ac-
tual prices paid by a sample of class members before and after
the Defendants’ price increases and found that in 92% of cases
those prices increased. He also constructed a regression
model to estimate the overcharges made possible by the con-
spiracy. That model indicated that the class paid overcharges
of approximately 3.08%, or in dollar terms, approximately
$3.8 billion too much. Defendants’ experts criticized the sam-
Nos. 15-2385 & 15-2386                                         9

ple size that Harris had used, and they asserted that Purchas-
ers’ experts had failed adequately to account for external fac-
tors influencing price and capacity.
                               B
    The district court began its analysis of predominance—the
central disputed issue in the case—by recalling the Supreme
Court’s statement in Amchem Products, Inc. v. Windsor, 521 U.S.
591 (1997), that “[t]he Rule 23(b)(3) predominance inquiry
tests whether proposed classes are sufficiently cohesive to
warrant adjudication by representation.” Id. at 623. It also
acknowledged that, as the Supreme Court puts it, “Rule
23(b)(3)’s predominance criterion is even more demanding
than Rule 23(a).” Comcast Corp. v. Behrend, 133 S. Ct. 1426, 1432
(2013). Predominance is satisfied when “common questions
represent a significant aspect of a case and … can be resolved
for all members of a class in a single adjudication.” Messner,
669 F.3d at 815 (internal quotation marks omitted); see also
Wal-Mart, 564 U.S. at 350 (a common contention for Rule
23(a)(2) purposes “must be of such a nature that it is capable
of classwide resolution—which means that determination of
its truth or falsity will resolve an issue that is central to the
validity of each one of the claims in one stroke”). With those
principles in mind, the court evaluated the two central ele-
ments of the Purchasers’ case: the alleged violation of the an-
titrust laws, and the causal link between that violation and
their alleged injury. It set the question of damages to one side,
noting that “it is well established that the presence of individ-
ualized questions regarding damages does not prevent certi-
fication under Rule 23(b)(3).” Messner, 669 F.3d at 815 (citing
Wal-Mart, 564 U.S. at 362 (“individualized monetary claims
belong in Rule 23(b)(3)”)).
10                                        Nos. 15-2385 & 15-2386

     With respect to proof of liability, the court had this to say:
     To prove each element of a conspiracy, virtually all
     class members would be relying on the same evidence
     that Plaintiffs have submitted in support of class certi-
     fication—namely, the documents, emails, phone rec-
     ords, and other indirect evidence necessary to prove
     that Defendants conspired in violation of antitrust
     laws. … [I]t is much more efficient to have a single trial
     on the alleged conspiracy rather than thousands of
     identical trials all alleging identical conspiracies based
     on identical evidence.
While acknowledging that Defendants hotly contested the
conspiracy allegation, the court found that their arguments
went to the merits, not to the suitability of the case for class
treatment.
    Turning to causation, to which it also referred as “antitrust
impact,” the court rejected the Defendants’ effort to equate
this case to Comcast, where the Supreme Court found a mis-
match between the plaintiffs’ damages theory and the evi-
dence they presented to show predominance. First, on the
question of impact, defined as whether Purchasers were
harmed, the court found that at the class-certification stage
the plaintiffs needed to demonstrate that the element of im-
pact is capable of class-wide proof at trial, through evidence
common to all class members. Looking at all the evidence, the
court found this element satisfied. For example, the Defend-
ants’ price increases were not tailored to each individual pur-
chaser; this was a commodity market with a structure condu-
cive to collusion; communications took place at a high level;
the common use of the PPW index affected all market partic-
Nos. 15-2385 & 15-2386                                        11

ipants; and the Defendants lacked any other reasonable expla-
nation for the tight correlation between the index and their
announcements of price increases.
    With respect to damages, the court found that the Purchas-
ers had the burden of producing a reliable method of measur-
ing classwide damages based on common proof. It rejected
the Defendants’ argument that the eventual need to examine
each individual purchaser’s damages was enough to defeat a
finding of predominance. Nothing in Comcast, the court said,
requires a different outcome. As this court noted in In re IKO
Roofing Shingle Products Liability Litigation, 757 F.3d 599 (7th
Cir. 2014), the plaintiffs’ damages expert in Comcast had esti-
mated harm based on the assumption that all four theories of
liability that plaintiffs offered had been established. The class
certified by the court, however, was limited to only one of
those theories. This, we explained, is what the Supreme Court
said, “made class treatment inappropriate: without a theory
of loss that matched the theory of liability, the class could not
get anywhere.” Id. at 602. With that point established, the
court assessed Dwyer’s report, concluded that both the meth-
odology and the data were reliable, and concluded that it
could be used to demonstrate class-wide damages.
   Finally, the court held that Purchasers had shown the su-
periority of proceeding under Rule 23(b)(3). The fact that
some class members had signed releases as a part of a settle-
ment of an earlier class action (dealing with an alleged liner-
board conspiracy that took place between 1993 and 1995) did
not require a contrary finding. As the court said,
   [t]he conduct at issue in the prior litigation was De-
   fendants’ allegedly collusive behavior in the mid-nine-
   ties. The actions at issue here are coordinated market
12                                      Nos. 15-2385 & 15-2386

     manipulation and price-increase announcements that
     occurred nearly a decade later. … Under Defendants’
     argument, they are free to keep colluding in violation
     of antitrust laws so long as they conspire in the same
     way as they were alleged to have behaved in a prior
     settled case. The Court is unaware of any case support-
     ing this argument; indeed, several cases are to the con-
     trary.
The court also rejected similar arguments based on particular
contractual provisions, and it decided that retaining defend-
ant Rock-Tenn in the case would not violate its bankruptcy
discharge.
                                C
                                1
    We follow the same general outline that the district court
used, looking first at predominance and then at superiority.
Within predominance, we consider two points: whether com-
mon methods of proof can be used to demonstrate the exist-
ence of the alleged collusion and its effect on prices in the con-
tainerboard market; and whether the existence and impact of
any such collusion predominate over other factors that may
affect an individual plaintiff’s damages. These inquiries apply
to all defendants. Defendant RockTenn raises some additional
arguments related to its bankruptcy; we address these at the
close of the opinion.
    In order to secure class certification, the Purchasers had to
demonstrate (not merely allege) that there is proof common
to all class members, and that this proof would show that they
suffered “injuries that reflect the anticompetitive effect of ei-
ther the violation or the anticompetitive acts made possible by
Nos. 15-2385 & 15-2386                                         13

the violation.” James Cape & Sons Co. v. PCC Const. Co., 453 F.3d
396, 399 (7th Cir. 2006); see Brunswick Corp. v. Pueblo Bowl-O-
Mat, Inc., 429 U.S. 477 (1977).
    Purchasers tendered extensive evidence that, if believed,
would be enough to prove the existence of the alleged con-
spiracy. Not surprisingly, it is largely circumstantial. But they
offered voluminous written materials of various types, which
in the aggregate pointed to the existence of both agreement
and actions to violate the antitrust laws. Indeed, Defendants
do not contest that the existence of the conspiracy could be
(perhaps had to be) proven by evidence common to the class.
    The more difficult question (though not too difficult in the
end) is whether the common evidence could show the fact of
injury on a classwide basis. See Messner, 669 F.3d at 819 (“The
ability to use such common evidence and common methodol-
ogy to prove a class’s claims is sufficient to support a finding
of predominance on the issue of antitrust impact for certifica-
tion under Rule 23(b)(3).”). At base, Defendants argue that it
is not enough for Purchasers to prove aggregate injury and
one aggregate overcharge, without allocating how much of
that overcharge was paid by each individual class member.
They urge that Purchasers have the burden of showing that
every class member must prove at least some impact from the
alleged violation. For that proposition, they rely on In re Hy-
drogen Peroxide Antitrust Litigation, 552 F.3d 305, 311 (3d Cir.
2008), and In re Rail Freight Fuel Surcharge Antitrust Litigation,
725 F.3d 244, 252 (D.C. Cir. 2013).
   While we have no quarrel with the proposition that each
and every class member would need to make such a showing
in order ultimately to recover, we have not insisted on this
level of proof at the class certification stage. To the contrary,
14                                       Nos. 15-2385 & 15-2386

we said in Suchanek v. Sturm Foods, Inc., 764 F.3d 750 (7th Cir.
2014), that “[i]f the [district] court thought that no class can be
certified until proof exists that every member has been
harmed, it was wrong.” Id. at 757; see also Parko v. Shell Oil
Co., 739 F.3d 1083, 1084–85 (7th Cir. 2014); McReynolds v. Mer-
rill Lynch, Pierce, Fenner & Smith, Inc., 672 F.3d 482, 491 (7th
Cir. 2012). There is no evidence to make us think that the class
defined by the district court either excludes too many pur-
chasers or contains troublesome internal conflicts, either of
which would indicate it should be rejected. We therefore
move on to the adequacy of the Purchasers’ showing that the
conspiracy had an effect on the prices they paid.
    The parties have jousted over the need for some kind of
“but-for” analysis, by which Defendants mean an expert con-
struction of a hypothetical market free of any anticompetitive
restraint, to which the actual market can be compared. See
Blades v. Monsanto Co., 400 F.3d 562, 569 (8th Cir. 2005). That
might be one way in which a plaintiff could satisfy its burden,
but we think that the formulation is too narrow. What is es-
sential is whether the class can point to common proof that
will establish antitrust injury (in the form of cartel pricing
here) on a classwide basis. Like the district court, we are sat-
isfied that Purchasers have done so.
    The Purchasers built up their case with several types of
evidence. First, expert Harris’s report showed that the struc-
ture of the containerboard market was conducive to success-
ful collusion. He pointed to the concentration of manufactur-
ers; the vertical integration of the market; the capital-intensive
manufacturing process (which affected the pace and likeli-
hood of new entry); weak competition from imported con-
Nos. 15-2385 & 15-2386                                          15

tainerboard; no good substitutes for the product; a low elas-
ticity of demand; and a standardized, commodity product.
These are all well accepted characteristics of a market that is
subject to cartelization. See, e.g., In re Text Messaging Antitrust
Litig., 782 F.3d 867, 872 (7th Cir. 2015); Minn-Chem, Inc. v.
Agrium, Inc., 683 F.3d 845, 859–60 (7th Cir. 2012) (en banc); In
re High Fructose Corn Syrup Litig., 295 F.3d 651, 657 (7th Cir.
2002); Jack Walters & Sons Corp. v. Morton Bldg., Inc., 737 F.2d
698, 710–11 (7th Cir. 1984).
    Defendants pooh-pooh these cases as examples of the dis-
credited structure-conduct-performance (SCP) paradigm that
ruled antitrust from the 1950s until the mid-1970s. See, e.g.,
United States v. Von’s Grocery Co., 384 U.S. 270 (1966) (striking
down a merger between a firm with 4.7% of the market and a
firm with 4.2%). We put to one side the fact that the SCP par-
adigm was used during that era primarily in merger cases and
this is a cartel case alleging hard-core price-fixing—the kind
of case in which lack of market power or reasonableness of
price is no defense. The evidence here goes well beyond the
structural. The flaw in the old SCP notion was the thought
that it was enough to know the structure of a market in order
to predict what kind of conduct would ensue, and how com-
petitively that market would perform. No such chain of as-
sumptions taints the Purchasers’ proof. They have shown ac-
tual price increases, a mechanism for those increases, the com-
munication channels the conspirators used, and factors sug-
gesting that cartel discipline can be maintained. We are not
saying that any of these points have been proven, of course,
but we are saying that this evidence is enough to support class
treatment of the merits.
16                                      Nos. 15-2385 & 15-2386

    Defendants also object to the Purchasers’ definition of the
market. The Purchasers, they say, have conflated the market
for containerboard (the material) with the market for finished
corrugated products. The district court responded that the
uniform vertical integration found in this industry makes it
appropriate to look to the finished products. That is correct:
Purchasers (and we) have no reason to dig inside the defend-
ant companies to evaluate their internal pricing of the raw
materials they use in producing the boxes and other products
they sell.
    Next, Defendants criticize Dwyer’s examination of the 15
price increases they announced and his use of the PPW index
in that connection. He analyzed “industry-wide reflections of
price and actual prices paid by class members before and af-
ter” the price-increase announcements, and he found that
nine of the 15 efforts succeeded (i.e. resulted in a durable price
increase). He also performed a regression analysis comparing
classwide aggregate prices to the PPW index and found that
“more than 97% of variation in aggregate prices is explained
by changes in the index.” He reviewed 738 contracts (those
the Defendants had produced) and found that 96% of them
“contained provisions that tied pricing to the PPW index.”
The Defendants protest that the Purchasers have shown only
correlation, not causation, but we think, taking into account
the rest of the evidence, Purchasers have not fallen into that
trap.
     Defendants also argue that Dwyer’s approach is the same
kind of “trial-by-formula” that the Supreme Court rejected in
Wal-Mart. But in that case the Court disapproved the plain-
tiff’s attempt to take a sample of the class members, who al-
Nos. 15-2385 & 15-2386                                          17

leged employment discrimination, to determine what per-
centage of that sample had actually experienced discrimina-
tion, and then to extrapolate that percentage for the whole
class. The Purchasers here are doing nothing of the sort: they
assert that every person or entity in North America paid the
overcharges that resulted from Defendants’ collusive prac-
tices. Even for transactions where prices were negotiated in-
dividually or a longer term contract existed, the district court
found, reasonably, that the “starting point for those negotia-
tions would be higher if the market price for the product was
artificially inflated.”
    We have already discussed the Purchasers’ common proof
of damages, but we add a few more words here to respond to
the Defendants’ Comcast arguments. Defendants understand
Comcast to hold that “individualized damages do foreclose
predominance if plaintiffs present no classwide method to ad-
judicate damages tethered to their theory of antitrust viola-
tions and if resolving those individualized damages issues
would ‘overwhelm questions common to the class.’” Brief for
Appellants at 36 (quoting 133 S. Ct. at 1433). We agree with
Defendants that Comcast insists that the damages theory must
correspond to the theory of liability, but that is all Comcast said
that is pertinent to our case. We must see if there is a classwide
method for proving damages, and if not, whether individual
damage determinations will overwhelm the common ques-
tions on liability and impact.
    Dwyer conducted a preliminary analysis to demonstrate
the feasibility of estimating damages on a classwide basis. He
created two categories of products, intermediate and final,
and he used two benchmark periods (before and after the
class period) to compare prices. He also accounted for many
18                                       Nos. 15-2385 & 15-2386

other variables, such as downstream demand, production and
delivery, inflation, and seasonal factors, in order to control for
other influences on price and to isolate the impact of the con-
spiracy. Using a “dummy variable,” he calculated an average
overcharge of 2.92% for the final products category and 3.81%
for the intermediate products category. He then multiplied
the average overcharges by the dollar amount of purchases by
class members, subtracting purchases that had taken place
under previously contracted prices. He found that the class
members paid $801.27 million more for intermediate prod-
ucts and $2.991 billion more for final products than they
would have absent the conspiracy.
     Defendants complain that it is wrong to calculate aggre-
gate rather than individual damages for the class. The district
court rejected that position as a matter of law, as do we. We
held in Loeb Indus., Inc. v. Sumitomo Corp., 306 F.3d 469 (7th Cir.
2002), that plaintiffs are permitted to use estimates and anal-
ysis to calculate a reasonable approximation of their damages.
Id. at 493. And we already have confirmed that at the class
certification stage, plaintiffs are not obliged to drill down and
estimate each individual class member’s damages. The deter-
mination of the aggregate classwide damages is something
that can be handled most efficiently as a class action, and the
allocation of that total sum among the class members can be
managed individually, should the case ever reach that point.
If in the end the Defendants win on the merits, this entire mat-
ter will be over in “one fell swoop.” (See WILLIAM
SHAKESPEARE, MACBETH, act 4, sc. 3, l. 220 (David Bevington
ed., Pearson Longman 6th ed. 2009.) If Purchasers prevail on
the common issues, both liability and aggregate damages will
be resolved. The district court did not commit reversible error
when it concluded that the class issues predominated.
Nos. 15-2385 & 15-2386                                           19

                                 2
    Everything we have said thus far also points to the supe-
riority of the class device for this case. We need to address
only two potential flies in the ointment that Defendants see:
first, the significance (if any) of certain releases that some class
members signed; and second, the relevance of contract de-
fenses that might apply.
    Some class members settled claims in an earlier lawsuit
against the same companies, dealing with the same industry.
Defendants represent that there are 39 relevant settlement
agreements implicating almost 10,000 of the individual claims
at issue. The Purchasers respond that these numbers are
wrong, because they involve double-counting and give an ex-
aggerated impression of the real number of affected class
members, since most people who signed releases did so with
multiple defendants. More importantly, Purchasers also note
that the Defendants’ numbers imply that most people who
signed releases did so after the events giving rise to the pre-
sent case, whereas in reality the releases are heavily distrib-
uted toward the beginning of the class period or earlier. Pur-
chasers suggest the simple expedient of limiting the recovery
period for any class member who signed a release to pur-
chases made after that release was signed. That strikes us as
an easy and effective way to handle this problem. Moreover,
as the district court observed, the fact that some plaintiffs re-
leased the defendants from further liability for their actions in
the mid-1990s is not a life-time inoculation against antitrust
liability in the same industry.
    The other contract defenses to which Defendants point in-
volve such provisions as mandatory arbitration and media-
tion clauses, forum-selection clauses, jury waivers, provisions
20                                     Nos. 15-2385 & 15-2386

shortening the statute of limitations, and clauses eliminating
remedies such as treble damages. Taking these limitations
into account, they say, will destroy the cohesion of the class.
The problem is that Defendants are relying on a case that in-
volved a class-action waiver: Lozano v. AT&T Wireless Services,
Inc., 504 F.3d 718, 728 (7th Cir. 2007), and there is no such
waiver here. The Purchasers point out that Defendants have
identified only 190 class members affected by this group of
limitations, out of over 100,000 notices that were sent out pur-
suant to Rule 23(c)(2)(B). As the record stands, this smattering
of individual contract defenses does not undermine the supe-
riority of the (b)(3) class action.
                              III
     We noted earlier that defendant RockTenn is in a different
position from the other defendants, because it filed for bank-
ruptcy and received a discharge on June 30, 2010, approxi-
mately four months before the end of the class period. The
district court refused to dismiss RockTenn on that basis be-
cause it found evidence that RockTenn re-joined the conspir-
acy after the discharge. (For example, on the very evening of
the day when the discharge order was entered, RockTenn’s
president sent an email stating “I assume we are announcing
tomorrow,” after three other defendants announced a simul-
taneous price increase.) The court also found that RockTenn
might be jointly and severally liable for actions undertaken by
its co-conspirators before the discharge, based on its post-dis-
charge participation. See Havoco of Am., Ltd. v. Shell Oil Co.,
626 F.2d 549, 554 (7th Cir. 1980) (“[A] co-conspirator who joins
a conspiracy with knowledge of what has gone on before and
Nos. 15-2385 & 15-2386                                         21

with an intent to pursue the same objectives may, in the anti-
trust context, be charged with the preceding acts of its co-con-
spirators.”).
    The district court’s reasoning was sound. RockTenn is free
to argue at trial that it did not re-join the conspiracy. There is
no conflict with bankruptcy law, however, if it did so, because
in that case its liability would be predicated on post-discharge
conduct. To the extent that these nuances need to be brought
to the jury’s attention, we are confident that the district court
can do so through proper instructions.
                               IV
    We conclude by noting again the basis of our ruling. First,
with respect to the central issue of class certification under
Federal Rule of Civil Procedure 23(a) and (b)(3), the only con-
tested points relate to predominance and superiority. Defend-
ants did not challenge Purchasers’ experts under Daubert and
Federal Rule of Evidence 702, and so we accept their reports
for what they are worth at this stage. We did not discuss the
opposing views expressed by Defendants’ experts because
they did not undermine the class certification decision. De-
fendants’ experts’ reports will be important, we assume, at the
merits stage, but the fact that class certification decisions must
be supported by evidence does not mean that certification is
possible only for a party who can demonstrate that it will win
on the merits.
   We AFFIRM the class-certification order of the district
court.
