4 Multimarket Contact With Uncertainty

This section extends the analysis of Section 3 by allowing both markets to be subject to demand
uncertainty, which arguably is the most realistic case to consider. Assume the model is identical
to the one presented in Section 3, except now market B has low demand with probability ag,
where 0 < ag < aa. Furthermore, assume that the realizations of demand are independent across
markets and across time. If the firms treat the markets in isolation, then the maximum profits for
the firms in the non-coordinated equilibrium are those found in Section 2.

Recalling the notation in Section 3, the firms maximize

Tate

Vy = (1—aa)(1 an) | 7

+ w'| +(1-oa)ag [Fs + ov + (12)

ag(1— ap) [2 + vy + agp [5V25]

subject to
(1 an)8 [V°— V4] + an6 [VE - Vin] > “A (18)
rons 5
(1 a.4)6 [V° — VB] + 48 [VA - Vag) 2 F (14)
and

(1 ~ a4)(1 — ap )6[V° — VEq] + (1 - alan d[V§ - Vip] + @a(l ~ o18)6[V A — Vip! (15)

Tr Tr,
> Q-aa)+0-an)y

I wish to maximize Vy with respect to 74, 7B, Vi, VB, Vip, and V°, subject to the incentive
compatibility constraints (13)-(15) and the feasibility constraints. As in the single market model,
Ve = Vj; successful collusion is followed by again setting the collusive prices in cach market.
Multimarket contact is exploited by raising V2, say, while simultaneously decreasing Vg or Vip, to
keep the incentive constraints satisfied.

One difference between this approach with imperfect information and BW’s approach with
perfect information is that with the latter approach one need only be concerned with defection in

both markets simultaneously. Given that any deviation can be observed perfectly and dealt with

probability that the market permanently collapses, then it is more plausible to assume discount factors differ across
markets. In any event, this change in structure, while not qualitatively changing the method of examination, may
greatly complicate the analytics.

14


Figure 1

(l-a,)4
A B
| (i-a.)8=5
1 E| \p al.
2

<+#—_— (1-a 8 = 1-8

Nile


occur even though no firm defects.

Using V¢ = V and the result that (2) holds with equality, I can rewrite the problem to maximize

(1—aa)% +. a46V?

Va Gene ®@)
subject to
p — (2(L—aa)6—1]%
Wwe 26(1 - 6) 7

and the remaining feasibility constraints. From (7) it is clear that if (1 — a,)é < 4, then 7 = 0,
vP = 0, and hence V = 0. This condition nicely generalizes the standard Bertrand result that
5 > 4 (with a4 = 0) is necessary to sustain supracompetitive pricing.

If (1—a@a)5 > 4, then substitute 6V? from (7) into (6) to yield

a _ (L-2a,4)t
v= "30 -8) ~

The right hand side of this expression is maximized, while still satisfying (7) and the feasibility
constraints, at 7 = 7, so
(1 - 2a,

2(1 - 6)

Vea
The following proposition summarizes the results.

Proposition 1 Supracompetitive prices cannot be sustained if either the probability of low demand
is too high, or firms care too little about the future, and conversely. Formally, if (1 —a,)6 < b
then 7 = 0, VP =0, and hence V = 0. If (1—aa)5 > 4, then

= (L—2a4)nTM
V= 21 ay

and
_ [2(1 — a4)é — aa"

Pp
Vv 35(1 — 6)

Not surprisingly, the maximum payoff is increasing in 6 and is decreasing in a4, the degree of
uncertainty. The intuition for the condition (1 — a4)6 > 4 has to do with the feasibility of

punishment.” Suppose that (1 — a,4)5 < $ and that V? = 0, but that firms attempt to set a price

The feasibility of punishment also lies at the heart of the incentive constraint in a market with perfect information,
though that interpretation often is overlooked. If one solves the perfect information problem and considers the largest


as a, and ag increase even further, beyond the point at which no collusion can be sustained in

the non-coordinated equilibrium, multimarket contact does not increase profits.

5 Conclusion
This paper examines the effect on tacit collusion of the interaction between imperfect information
and multimarket contact. When firms have imperfect information about the state of demand, I
show that reliance on multimarket contact theory confined to markets with perfect information
leads one to misstate the effect of multimarket contact on the degree of sustainable collusion. This
misstatement is not simply a question of the magnitude of increased profits, but is a question of
the existence of profit increases. The central theme of the results is that a market must generate
sufficiently noisy signals for it to benefit another market through strategic linkage. This outcome is
surprising, given one’s intuition that linking a high noise market to a low noise market might lead
to shocks from the high noise market spreading to and adversely affecting the low noise one. Such
destabilization does occur, but not in all instances in which it seems plausible. This insight is useful
to parties involved in antitrust enforcement and litigation. It also is useful to firms considering either
strategic linkage of existing markets or expansion into new markets with the hope of sustaining
higher degrees of collusion.

I introduce imperfect information into the multimarket contact framework in two stages. First,
I examine the strategic linkage of two markets, one that is subject to demand uncertainty and
one that is not. A firm presumably would consider such a link more valuable than a linkage of
two markets that both are subject to unseen demand fluctuations. However, such a link may not
increase profits, despite the slack in incentive constraints exploited in perfect information models.
The results indicate that the benefits from multimarket contact do not arise simply from the
presence of excess punishment power, but also from the existence of an outlet through which any
excess punishment power can be applied. Second, I examine the arguably more realistic setting
in which demand uncertainty is present in and asymmetric across both markets. If both markets
have too little uncertainty, then multimarket contact does not increase profits. The reason for
this is that it becomes impossible to punish defection in both markets severely enough to make
defection in only one market undesirable. If firms attempt to secure additional industry profits,
then an individual firm’s incentive is to cheat in one market, bank on the low likelihood of the other
market having a demand shock that will trigger severe punishment, then simply appeal to bad luck
or the law of large numbers while professing its innocence. While it seems this problem could

be eliminated by even more severe punishment when both markets experience a negative demand

18


Abstract

With perfect information about relevant strategic variables, economic theory predicts that firms
engaged in competition across several markets sometimes can use their multimarket contact to
blunt competitive forces. In practice, perfect information likely is not available, and it is well
known that the existence of imperfect information can impede firms’ collusive efforts. I extend a
standard oligopoly supergame to examine simultaneously the effects of imperfect information and
multimarket contact on the degree of cooperation that firms can sustain, and I reach the following
conclusions. First, linkage of onc market with perfect information and another with imperfect
information may not increase profits, despite the slack in incentive constraints exploited in perfect
information models. Second, multimarket contact does not increase profits if each market has too
little uncertainty. Third, profits can be increased in a market in which some collusion initially is
sustainable by linking it with a market in which no collusion initially is sustainable. The central
theme of these results is that a market must generate sufficiently noisy signals for it to benefit
another market through strategic linkage. This contrasts with the finding in individual markets

that collusion decreases as the level of noise increases.


{8] Gertner, R. and McCutcheon, B. “Multimarket Contact and Tacit Collusion with Imperfect

Monitoring.” mimeo (1994).

[9] Green, E., and Porter, R. “Noncooperative Collusion Under Imperfect Price Information.”

Econometrica, Vol. 52 (1984), pp. 87-100.

(10] Parker, P. and Roller, L. “Collusive Conduct in Duopolies: Multimarket Contact and Cross-
Ownership in the Mobile Telephone Industry.” CEPR Discussion Paper 989 (1994).

[11] Phillips, O. and Mason, C. “Mutual Forbearance in Experimental Conglomerate Markets.”
RAND Journal of Economics, Vol. 23, No. 3 (1992), pp. 395-414.

{12] Singal, V. “Airline Mergers and Multimarket Contact.” Managerial and Decision Economics,

Vol. 17, No. 6 (1996), pp. 559-574.

(13) Tirole, J. The Theory of Industrial Organization. Cambridge, Mass.: M.I.T. Press (1988).


simply by setting p = ¢ forever, all the V? terms are set to zero, in which case (15) implies both
(13) and (14). With imperfect information, one must account for firms’ incentives to defect only
in one market at a time, and so the incentive constraints in the original problem cannot simply be
added together when considering the use of multimarket contact to enhance coordination.

As the first step in the solution process, it can be shown (see Lemma 1 in the Appendix) that
the incentive constraints preventing cheating in A or B separately, (13) and (14), both cannot be
slack. If both were slack, then it would be the case that V? = VB = V°. If so, then with no
reversion to punishment in the event only one market had a bad outcome, the sum of the expected
loss from defecting in A only and B only is less than the expected loss from defecting in A and B
simultaneously. This is the case because there exists an additional source of loss when defecting
in both markets at one time: with such a defection, a firm can go from a good outcome in both
markets to a bad outcome in both markets. Such an outcome, induced by cheating, cannot occur
when defecting in only one market. However, the sum of the expected gains from defecting in A
only and B only is identical to the expected gain from defecting in both markcts sirnultaneously,
because the demand shocks in the two markets are independent. Consequently, if (13) and (14) both
are slack, then (15) must also be slack. However, with demand uncertainty, all three constraints
cannot be slack at the joint profit maximizing level.

The finding that at least one of (13) and (14) must bind permits characterization of four
situations in which using mu!timarket contact affects profits. Given the complex interaction between
the degree of impatience, 5, and the degrees of uncertainty, a4 and ag, a complete characterization
does not necessarily generate especially useful insights. However, such a characterization can be

obtained using results presented in the Appendix.

Proposition 5 Suppose some collusion can be sustained in each market in the non-coordinated
equilibrium. If there is a sufficiently small degree of uncertainty in both markets, then multimarket

contact does not increase profits.

This result is consistent with Proposition 3, in which market B always has high demand, the
smallest degree of uncertainty. The intuition for this surprising result is as follows. Recall that
multimarket contact is exploited by making incentive compatible punishing less harshly when one
bad outcome is observed by punishing more harshly when two bad outcomes are observed. There
are limits to how much one can punish deviations, however, and the problem that occurs when a4

and ag get too small is that it becomes very unlikely that even one market will experience low

15


Proof of Proposition 2: If 6 < 3 then (27) is satisfied only if 74 = 7_ = 0. This implies
Vz=0.0

Proof of Proposition 3: If (1 - a4)6 > 3, then 6 > 3 and both 74 and 7%, can be as large as
possible and still satisfy (27). As 7, and 7g enter the objective function positively, taking 74 = 1”
and 7, = 7" yields the maximum payoff. 0

Proof of Proposition 4: If 6 > } and (l-a4)é < 3, then letting 7 = nTM increases the objective
function and relaxes (27) by as much as possible. However, (1 — a@4)d < } implies that the m
term in (27) enters negatively. If a, > } then ig also enters the objective function negatively.
Therefore, V; is maximized by letting 74 = 0. If a4 < bs then tm! enters the objective function
positively. Therefore, Vy is maximized by letting 7, be as large as possible, subject to (27). This

occurs when
(26-1)0r"
7-20 aaa” (28)

7,4 =min
Analyzing Section 4
I wish to maximize Vy with respect to 74, 7B, V}. VE, Vig, and V°, subject to the feasibility
constraints and the following incentive constraints:
+7B
2

(1 a4)(1 - ag) [As + wv +(1-aa)ap [# +ov5] (29)

A
+a,(1— ap) [2 + ov + agag [6VE5|

Iv

(1-@4)(1 - a8) [rs +2 +0v4] +(1-aa)ap [a+ 6V25]

+a4(1— ap) [# + ov + agen [6V2,]

and

(1—a4)(1— ap) bees + wv] 4+ (1 ag)og [- + wv} (30)

+a,(1— ap) [2 + vy + aap [6Vinl

IV

(1-a4)(1 — ag) [7 +%p+ vs] +(1-ag)ap [i + evs]

t+a,(1 — ap) [Fp + 6V25] + aoe [5Vig]

21


associated punishment payoffs when using multimarket contact are
Vy = 2.20" VE =180" VE =2.2nTM Vig = 0.0330"

By successfully using multimarket contact, the payoff following defection in only market B increases
by 136%, in only market A increases by 13%, and in both markets A and B decreases by 96%.
Surprisingly, by linking a low noise market with a high noise market, the largest payoff gain is in the
low noise market. Note that if a4 decreases to 0.2, then multimarket contact no longer increases

profits. Of course, total profits in that case are
Vy = 280" VE = 2.1330" VE = 2.1330" VE, = 1.4660"

and they exceed the profits from using multimarket contact with the higher degree of uncertainty
in market A. That is, the ability to profitably employ multimarket contact does not make more

uncertainty preferable to less.

Proposition 8 If each market has a moderate degree of uncertainty, then multimarket contact
increases profits. This can be true whether or not some collusion initially was sustainable in either

market in the non-coordinated equilibrium.

This result links the preceding three. A market can gain from linkage with another only if it does
not have too much or too little uncertainty. Consequently, medium ranges of uncertainty in each
market are more conducive to gaining from multimarket contact than from having too much or too
little.

The preceding four propositions illustrate how the imperfect information setting relates to the
perfect information setting examined by BW. In BW, the authors present an “irrelevance result,”
in which they give conditions under which multimarket contact does not increase profits. The basic
reasoning is that in the non-coordinated equilibrium, either collusion can be sustained in both
markets, or collusion cannot be sustained in either market. Thus, there is no benefit to linking
the markets. Propositions 5, 6, and 8 offer continuity between the all-or-nothing outcomes in the
non-coordinated equilibrium of the perfect information setting. If a4 = ag = 0, then full collusion
can be sustained in the non-coordinated equilibrium, and multimarket contact does not increase
profits. Multimarket contact does not increase profits until a4 and ag increase to moderate levels,

including levels such that no collusion is sustainable in the non-coordinated equilibrium. Finally,

17




never will be invoked, because market B always has high demand. Reducing the payoff following an
apparent defection in A, say by switching to less profitable behavior in market B, does not reduce
defection in market A, as there already was no defection in equilibrium. Reducing that payoff
merely reduces the overall payoff, because the punishment phase is entered only when there is a
negative demand shock. If firms already are not cheating, and only enter punishment phases to
ensure that they do not cheat, then B’s excess power has no place to be applied.

Proposition 3’s result suggests that markets in which collusion is possible in isolation do not
benefit from multimarket contact, but this is at odds with GM’s Proposition 1 that shows that in
markets with symmetric uncertainty (and no demand shock correlation) there always are gains to
multimarket contact. It seems unlikely that making one market’s chance of high demand much more
certain reduces the effectiveness of multimarket contact. However, as described in the preceding
paragraph, the problem when market B always has high demand is that it is impossible to reduce
the payoff following defection in both markets in a way that permits increasing the payoff following a
low demand realization in the uncertain market. This reasoning suggests that the problem lies with
the certainty associated with market B, and therefore suggests that multimarket contact increases
profits even with the smallest degree of uncertainty in market B. I show in the next section, however,

that this argument is incorrect.

Proposition 4 [f, in the non-coordinated equilibrium, collusion is sustainable in the certain market

but not in the uncertain market, then multimarket contact may increase profits. If uncertainty is

the primary hindrance to collusion, then profits do not increase. Formally, suppose 6 > $ and

(l-ag)6< 3. Tfaa< 3, then

oz _ (L— 2a) aTM
Y= "sa-8) *30G>8
and
Ve = mm [i= 20.- CA) RA
2(1 - 6) 26(1 — 6)
where
7,4 = min 6-10” nm
as 1—2(1—aa)o’

However, if a4 > 3 then Pa =c, 74 = 0, and

7”

Vi = 35"

12


first instance, the result follows from the inability to exploit the excess punishment power that exists
in the market with perfect information. Multimarket contact is exploited by decreasing the payoff
following negative shocks in both markets, but there never will be such shocks in the market with
perfect information. Hence, the threatened lower punishment never will be invoked, and therefore
has no deterrent effect on the incentive to deviate in the market with imperfect information. In
the second instance, the result follows if the market with imperfect information exhibits too much
demand uncertainty. In this case, though firms could link the markets while satisfying all necessary
constraints, doing so would be less profitable than keeping the markets separate. The frequently
occurring shocks in the market with imperfect information have a destabilizing effect on the market
with perfect information.

The intuition for the second result is straightforward. The benefit associated with multimarket
contact is the coordination of punishments it permits. In equilibrium no firm actually defects
from the collusive price, and price wars occur even though it is known that no firm defected.
Multimarket contact is exploited by increasing the payoff following bad outcomes in only one
market. while decreasing the payoff following bad outcomes in both markets. However, if there is
too little uncertainty, then there is little chance of even one market having a bad outcome, let
alone both. In this instance, a firm’s incentive is to defect in one market, rely on the low likelihood
that the other market has a low demand realization, and thus avoid the increased punishment.
The defecting firm then enjoys the increased payoff following a bad outcome in only one market,
relative to the non-coordinated equilibrium, all the while professing its innocence about defection
and appealing to bad luck or the law of large numbers to explain the ever increasing string of low
demand realizations in one of the markets. While it seems this problem could be eliminated by even
more severe punishment when both markets experience a negative demand shock, there is a limit to
how strongly one can punish apparent deviations. This limitation on the severity of punishment,
coupled with the low likelihood of the punishment being induced, prevents multimarket contact
from increasing profits.

In contrast, the third result is that if some collusion initially can be sustained in one market,
but not in another, then multimarket contact may increase profits in the initially collusive market.
This is orchestrated by increasing the payoff following a bad outcome only in the initially collusive
market, while decreasing the payoff following a bad outcome in both markets. Knowing that they
will receive a larger payoff even if the outcome in the initially collusive market is bad, it appears

that a firm’s incentive is to defect in that market. However, the high likelihood that the other


the solution in Lemma 3 to apply, it must be the case that

apie <aaTa

wy < ran) [A +04) 6 ~ Fp
a@a(l-2(—aa)d])

Combining the preceding two constraints, and recalling that a4 > h it must be the case that

(1ta4)6-1

OBS Baa —1)6

That is, wg must be sufficiently small for multimarket contact to increase profits. For example, if
a, = 0.4 and 6 = 0.75, then ag must be less than 4. Suppose ag = 0.1. Then the constraints all

can be satisfied with #4 = 7, =m”, and the resulting payoffs are
Vy =2.2n" VR = 180" VE= 227" VE, = 0.0380

The maximal profit when employing multimarket contact exceeds the maximal profit in the non-
coordinated equilibrium, which is 1.67TM.

Proof of Proposition 8: First, suppose that some collusion can be sustained in each market in
the non-coordinated equilibrium. That is, 2(1—a4)6-1> 0 and 2(1-—ag)6-12>0. If aa, ap,
and 6 are such that (1+ a@4)6-1>0 and (1+ ag) 6-1 > 0, then Case 1 of Lemma 3 applies. In
this instance, multimarket contact increases profits.

Second, suppose that no collusion can be sustained in either market in the non-coordinated
equilibrium. That is, 2(1—a4)6-—1<0 and 2(1—ag)6-—1<0. If a4, ag, and 6 are such that
(1+ a,4)6-12> 0 and (1+ ag)5—1 > 0, then Case 4 of Lemma 3 applies. As a first step, suppose
that (33) binds and (34) is slack. For the solution in Lemma 3 to apply, it must be the case that

QpTB <Q@aTa

and
ry (629s) (A+ ea) 6 = te
A @a (1 —2(1— aa) 6]

27


satisfies (83)-(35) is
_ (l= 2a4)ta | (1—-op)ita

Y= "Sa-8 * 90-5”
with
ve =Vy,
subject to
yey 2ilsoa)6= Uta Te ap ((Ltaa)6—-1]%p
A 26(1 — 6) 2(1 - 6) 2a6(1 — 6) ’
V2, < (2(1-—a4) 6-17 4 (1 — ag) (1 + a) 6 - 1] 7B

26(1 — 6) 2a46(1 — 6) ,
and the remaining feasibility constraints. The case of (33) being slack and (34) binding is analyzed

in symmetric fashion, with a change of subscripts.

Proof of Lemma 3: First note that if (33) binds and (34) is slack, then VZ = V°. Therefore,
(35) holds strictly and is superfluous. Eliminate V4 using (33). Multiply (33) by a4 and (34) by
(1 —ag). Substitute a4(1 - ag)6V4 from (33) into V, and (34). From these manipulations one

determines that
Te (L-2a,)%4 | (1-2)
7~ 91-6) 2(1 — 6)

and
[2(1—a4)6-1] 7 4 (1= ap) (+e) 6 = 17a |

>
VaB < 9501 — 8) 2aad(l — 6)

Now work backwards to determine V3. From (33),

(2(1-—a4) 5-1] Tp ap [(1t+a4) 6-1] 7B

Pp _
Va> 35-8) SO OE -) aad(1 — 6)

One also must ensure that V? and Vi, satisfy the feasibility constraints. For Vi < Vy, it must

be the case that

[2(1-aa)6—174 7p ap [(1+ea)6~17%_  (L—20a)Fa , (l~on)te
26(1 — 4) 30-4) Qa6(1 — 4) 21 — 4) 2-6)

This is equivalent to

apt BR < QATA-

If the preceding relationship holds, then there exists V2 satisfying the incentive and feasibility

constraints.

24


1 Introduction

Firms often interact with their rivals in several distinct geographic or product markets. For example,
airlines compete in different city-pair markets, pharmaceutical manufacturers compete in treatment
markets for different ailments, while conglomerates and multinationals compete across products and
national borders. This multimarket contact may reduce the vigor with which such firms compete, as
they realize that aggressive behavior in one market may be met by severe reactions in other markets,
not only the one in which the disruptive behavior occurred. The possibility of reduced competition
associated with multimarket contact creates profit incentives for firms considering either strategic
linkage of existing markets or expansion into new markets with the hope of sustaining higher degrees
of collusion. In turn, the possibility of these incentives may cause concern to antitrust agencies
trying to prevent anticompetitive behavior.

From the perspective of applications, one weakness of existing multimarket contact theory is
that it assumes that firms have perfect information about all relevant strategic variables. In prac-
tice, this assumption is unlikely to hold, and it is well known that such imperfect information can
impede firms’ collusive efforts.! Given the benefits associated with multimarket contact in perfect
information settings, it is important to consider multimarket contact in a way that incorporates
uncertainty. To that end, this paper examines the incentive for joint decision making and the
potential for anticompetitive harm arising from multimarket contact in markets characterized by
imperfect information regarding demand fluctuations. It is conceivable that the possibility of “mis-
takes” caused by noise in the marketplace may make strategically linking markets an unprofitable
course of action. If so, then multimarket contact should be less of a concern to antitrust agencies
and should be less of a strategic consideration for firms.

The potential effect of multimarket contact in perfect information settings previously has been
explored by economists. In their 1990 paper, Bernheim and Whinston (hereinafter BW) formalized
both conventional wisdom and anecdotal evidence regarding the ability of firms engaged in com-
petition across several markets to use that contact to blunt competitive forces. In the context of
tepeated Bertrand competition with homogenous products, the authors illustrate how firms may
use slack enforcement power in one market to sustain collusion in another market that otherwise
would be unsustainable. The slack enforcement power mathematically is embodied in the looseness
of the incentive constraints that must be satisfied for joint profit maximization, and that power

is “used” in another market by pooling the incentive constraints together. For example, it is well

1For example, see Green and Porter [1984].




Multimarket contact is exploited in this instance by decreasing the payoff in market B if there
is a bad outcome in market A. This is evident by noting that V{ is less than its value in the
non-coordinated equilibrium. In that equilibrium, Vf = aes which is the payoff associated
with perpetual collusion in market B. This optimal strategy for exploiting multimarket contact is
effective if there is not too much uncertainty. However, as a, increases beyond 3, then the increased
uncertainty lowers profits if firms insist on strictly linking the two markets. For example, suppose
that the pooled incentive constraint, 6 + (1 — a4)6 > 1, from the two incentive constraints in the
non-coordinated equilibrium, is satisfied. In this instance, Proposition 4 indicates that 74 = 1".
If a4 increases beyond }, then firms still can prevent defection in market A by threatening lower
profits in market B, but it is not worth doing so. The amount firms gain by setting p4 > c in
market A cannot compensate for the loss so frequently incurred in market B. In this instance, the
destabilizing nature of the shocks in market A so adversely affects market B that linking the two
markets is unprofitable.

Figure 1 illustrates the parameter dependent results of Propositions 2 through 4. In region ABC,
to the northeast. of (l—a4)6 = 3, collusion is sustainable in the two markets without coordination.
In that case, multimarket contact does not increase industry profits. In region ACD, in the non-
coordinated equilibrium collusion is sustainable in the certain market. but is not sustainable in the
uncertain market, yet the multimarket contact allows the firms to set the monopoly price in that
market. This accords with the intuition from BW, as the line ADF is the locus above which the
pooled incentive constraint (1—a,4)6+6 > 1 is satisfied. Given the previous result, behavior in the
following two regions is of interest. First, though the pooled incentive constraint is satisfied in region
CDF while the single incentive constraint for the uncertain market is not, multimarke: contact does
not increase industry profits. The occurs because of the previously described destabilizing nature
of the shocks in market A. Thus, simple application of the notion of pooling incentive constraints
leads one to overstate the degree to which additional collusion may be sustained using multimarket
contact. Second, in the region ADE, though it is below the pooled incentive constraint embodied by
AD, multimarket contact does permit higher industry profits. These last two observations suggest
that the degree of uncertainty plays a larger role in sustaining additional collusion via multimarket

contact than does the degree of firms’ impatience.!!

“One way to examine this more rigorously is to suppose that the two markets have different discount factors.
However, this assumption is difficult to justify if the discount factor simply embodies a firm’s rate of time preference.
If the markets meet with different frequencies or grow at different rates, or if the discount factor also includes some

13


provides an initial examination of this issue. In addition to a number of interesting applications, the
authors demonstrate that the superiority of joint decision making proven in BW can be extended
to markets that are identical ex ante but that are subject to unobservable and possibly correlated
demand shocks. This conclusion contrasts with the result in BW for identical markets under
perfect information, which BW refers to as an “irrelevance result,” and it illustrates the role that
uncertainty can play in enhancing the effectiveness of joint decision making in multimarket contact
settings. As GM’s main interest with respect to multimarket contact theory is the preceding point,
they restrict their model to a symmetric environment with symmetric treatment of each market.
The necessity of symmetry stems from their use of a punishment scheme that differs from the one
I employ.

The present paper’s model contrasts with GM’s model in two primary respects. First, it permits
differences in the degree of demand uncertainty across markets, while GM’s model assumes the
markets are symmetric. This asymmetry leads to richer results regarding the effect of multimarket
contact. In particular, I find in several situations that firms have no incentive to strategically link
markets, even though naively pooling incentive constraints suggests that multimarket contact is
profitable. Second, my model assumes the demand shocks are uncorrelated, while GM’s model
allows correlation of the demand shocks across markets. Given that GM examines the effect of
correlation in the symmetric model, I feel that the computational cost of including correlation in
the asymmetric model outweighs the benefits.

My analysis has three primary conclusions. First, linkage of one market with perfect information
and another with imperfect information may not increase profits, despite the slack in incentive
constraints exploited in perfect information models. Second, multimarket contact does not increase
profits if each market has too little uncertainty. Third, profits in a market in which some collusion
initially can be sustained can increase following linkage with a market in which no collusion initially
can be sustained. The central theme of these results is that a market must generate sufficiently noisy
signals for it to benefit another market through strategic linkage. This contrasts with the finding in
individual markets that collusion decreases as the level of noise increases. This insight is useful to
parties involved in antitrust enforcement and litigation. It also is useful to firms considering either
strategic linkage of existing markets or expansion into new markets with the hope of sustaining
higher degrees of collusion.

The first result holds if any collusion can be sustained in the market with imperfect information,

and in some cases if no collusion can be sustained in the market with imperfect information. In the


shock, there is a limit to how strongly one can punish apparent deviations. For moderate levels of
uncertainty in each market, multimarket contact increases profits. Finally, I show that a market
with a great deal of uncertainty, in which collusion cannot be sustained in isolation, sometimes can
be used to bolster collusion in a market subject to more stable demand.

These results are convincing in light of earlier work for two reasons. First, I use a simple
extension of the perfect information model employed by Bernheim and Whinston [1990], so their
work provides a good benchmark for comparison. Second, I solve for the highest sustainable payoffs
for the firms, so the results showing that multimarket contact has no role do not rely on my inability
to find a suitably profitable collusive scheme. In this setting, the solution concept I employ from
Abreu, Pearce, and Stacchetti {1986, 1990] is both powerful and simple.

The model offers testable predictions that likely can be evaluated within the experimental
literature examining the effect of multirnarket contact on firms’ prices and profits.!? Because it
is difficult to quantify the degree of uncertainty in actual markets, it is likely that experiments
designed to measure demand uncertainty’s impact on the effectiveness of multimarket contact as a
tool for increasing profits will be more useful than trying to discern the effects in field data.18

There are at least four ways one may consider extending the theoretical analysis. First, onc may
increase the number of firms or markets. Second, one may permit different numbers of firms in the
different markets. as is done in Section 4 of Bernheim and Whinston [1990]. These two extensions
should have no qualitative effect on the results. Third, one may add correlation of demand shocks
across markets, as is done in Gertner and McCutcheon [1994]. Their analysis shows that correlation
affects the nature of the results in the symmetric model. Fourth, one could consider a homogenous
product Cournot model or a differentiated product Bertrand model to check the sensitivity of the

results to the discontinuous nature of the profit function used in this paper’s model.

Appendix
Analyzing Section 3
I wish to maximize Vy with respect to 74, 7, V4. VB, Vag, and V°, subject to the feasibility

constraints and the following incentive constraints:

(=a, [7 F%8

+ ov +e [% + ev] > (1-04) [rs +32 +ov4| toa [? + v4] (18)

See Feinberg and Sherman [1984, 1988] and Phillips and Mason [1992].
13See Evans and Kessides [1994], Barla [1994], and Singal [1996] (airlines); Jans and Rosenbaum [1997], Parker and
Roller [1994} (cement and cellular phones).

19


Multimarket Contact and Imperfect Information

Charles J. Thomas'

This revision: November 29, 1998

Federal Trade Commission. I am grateful to Dan Hosken, Nancy Lutz, Dave Reiffen, and Bart Wilson
for their helpful comments. Send correspondence to Federal Trade Commission, 600 Pennsylvania Ave., NW,
Washington, DC 20580, or to cjthomas@ftc.gov. This work represents the opinions of the author and is not
intended to represent the position of the Federal Trade Commission or any individual Commissioner.


highest payoff associated with a symmetric sequential equilibrium (SSE).*> This solution follows
the method demonstrated by Abreu, Pearce, and Stacchetti [1986,1990] (hereinafter APS). The
reason for using such a tool is clear: In order to make convincing statements about the effect of
multimarket contact on the ability to collude, it is necessary to find the highest possible payoffs
for the firms. This is particularly true regarding claims that multimarket contact does not increase
profits.

A brief description of the APS methodology is in order. Of primary importance for the present
analysis is the insight that a repeated game, much like a dynamic programming problem, may be
decomposed into a family of static games. Consider play following the first period of an SSE. The
SSE specifies successor SSEs to be followed in each state of the world after the first period. These
successor SSEs have associated payoffs. If the truncated game for each state of the world following
the first period is simply replaced by the payoff to the associated successor SSE, then this new
game’s equilibrium is exactly the first. period behavior specified by the original SSE. This approach
is analogous to using backward induction to determine the subgame perfect Nash equilibrium in a
finite extensive form game. The APS methodology permits one to concentrate on payoffs rather
than strategies, which is useful because the strategies potentially are quite complicated.

Let V € ® denote the set of payoffs of all SSEs for the game under consideration. V is nonempty
(setting p = ¢ forever is an SSE) and is compact.® Thus, there exists V, the largest element of
Y, which is associated with an SSE I denote &. In the first period of play, specifies that firms
set price p, receiving z with probability (1 — a4) and receiving 0 with probability a4. Given the
structure of uncertainty, in equilibrium there are only two common knowledge events following
the price-setting in period one: Either both firms received a profit or at least one firm did not
receive a profit. That is, the finest partition of the information sets on which the firms agree is
{{W},{X,Y,Z}} in Table 2. For example, if market demand is high and firm 2 undercuts firm 1,
then the firms both could not say that they are in state Y. Firm 1 is unsure whether it received no

profit because firm 2 undercut, state Y, or because demand was low, state Z.

“The restriction to symmetric equilibria is natural given the symmetric nature of the problem. See Abreu, Pearce,
and Stacchetti [1990] for a discussion of asymmetric equilibria.

5Note that this differs from the treatment in Tirole [1988], which examines an equilibrium consisting of a collusive
phase and a punishment phase. In the collusive phase, both firms charge pTM until one firm earns zero profit. Seeing
zero profit triggers a punishment phase lasting I’ periods in which both firms charge c. At the end of T'4 periods,
play reverts to the collusive phase, This method does not guarantee the highest SSE payoff, though it can be shown
that it actually does generate the highest payoff. The problem with using this method in a multimarket contact
setting is discussed in the next section.

Y clearly is bounded, and a limit argument shows that it is closed.


market will experience a bad demand shock is enough to dissuade the firm from defecting and
increasing the likelihood that both markets will have negative outcomes. In this instance, there
exist severe enough punishments such that one can increase profits yet prevent deviation in the
initially collusive market.

The next section presents an oligopoly supergame in which firms have imperfect information
about the level of demand and their rival’s actions. Section 3 extends this model to consider
multimarket contact between a market with perfect information and one with imperfect information.
This simple method shows how the strategic linkage of markets does not necessarily enhance the
ability of firms to collude, despite the slack in incentive constraints that was exploited in BW.
Section 4 extends the model further, by assuming imperfect information exists in both markets.
There I find greater support for the initial findings in BW and discuss the difference in predictions

of the results in sections 3 and 4. Section 5 briefly concludes.

2 Collusion in the Face of Uncertainty
This section presents a one-market model of a repeated game with uncertainty taken from Chapter
6 of Tirole [1988}.3 Two firms 1 and 2, each with marginal cost c, choose prices every period for
a product sold in market A. A firm’s price offer is not seen by its rival. The firms’ products are
perfect substitutes, so consumers always purchase from the low price firm. Assume that demand
is split evenly between the two firms in the event of a tie. In each period, there are two possible
states of demand. With probability a4 there is no demand in that period, while with probability
1—a, demand is given by D(p). Denote the unique monopoly price and profit in the “high demand
state” by pTM and nTM, respectively. The realizations of demand are assumed to be independently
and identically distributed over time. This is a Bertrand version of the model employed in Green
and Porter [1984]. Examples of markets with similar informational characteristics include wholesale
markets for cement and lumber.

A note on terminology. I refer to an increase in @4 as an increase in uncertainty, in the sense
that for a higher a4 and a given strategy for a firm’s rival, a firm assigns lower probability to a
zero-profit outcome’s being the result. of a deviation by its rival as opposed to a low demand shock.

In the finitely repeated game, both firms charge c each period. In the supergame, I solve for the

3] describe the model only briefly. Readers wishing for a fuller treatment should see Section 6.7.1 in Tirole [1988].
As this model is a prelude to those used in succeeding sections, I carefully show how to solve for the highest sustainable
payoffs. In the later sections most of the mathematical arguments are relegated to the Appendix.


which simplifies to
Ve (1 —3a,)nTM
T= ofl—= (+ a4) 5}

For a and ag sufficiently small, the preceding value for Vz is strictly less than the profits obtained
in the non-coordinated equilibrium, in which each market is treated separately.

A symmetric argument holds if (33) is slack and (34) binds.
Proof of Proposition 6: Suppose that no collusion can be sustained in either market in the
non-coordinated equilibrium. That is, 2(1 —a@4)6—1<0 and 2(1—ap)6—-1<0.

First, if a4, ag, and 6 are such that (1+a4)6—1 <0 and (1+ag)6-1<0, then Case
2 of Lemma 3 applies. In this instance, no collusion can be sustained using multimarket contact.
Hence, multimarket contact does not increase profit.

Second, if a4, ap, and 6 are such that (1+a@4)6—1> 0 and (1+ag)6—12 0, then Case 4
of Lemma 3 applies. As a first step, suppose that (33) binds and (34) is slack. For the solution in

Lemma 3 to apply, it must be the case that
Opie <aaTa

and
— | (l~ag) [(1+ a4) 6-17
TA < ~~, [l 20 — aa) 6]

The preceding two constraints can be combined to yield a new constraint,

(1 ap) [(1 +04) 6-1 7a

oBTB <0 aal)

that also must be satisfied for the solution in Lemma 3 to apply. This combined constraint may be
simplified to

0<[(1+aa)(1—ag) +2(1—@,) ag] 6—1.

For a4 and ay sufficiently large, this constraint cannot hold. In this instance, no collusion can be
sustained using multimarket contact. A symmetric argument holds if (33) is slack and (34) binds.
Hence, a4 and ag sufficiently large, multimarket contact does not increase profit.

Proof of Proposition 7: Suppose that a4 and 6 are such that 2(1— a4) 6-1 <Oand(1+ aa) o—
1> 0. In this instance it must be the case that a4 > } Further, suppose that ag < 2. In this

instance, Case 4 of Lemma 3 applies. As a first step, suppose that (33) binds and (34) is slack. For

26


For Viz = 0, it must be the case that

[2(1~aa)6—- ta (l= op) (+ oa) $= Ui

0 HEL =H Yaad(t — 6)

There are four cases to consider.

1) If [2(1 — a) 5 — 1] > 0 and [(1 + a4) 5 — 1] > 0, then the preceding relationship holds easily
and there exists V2, satisfying the incentive and feasibility constraints.

2) If [2(1 — a4) 5 — 1] < 0 and [(1 + a4) 6 — 1] <0, then the preceding relationship holds only
for 74 = 7p = 0.

3) If (2(1 — a4) 6 — 1] > 0 and [(1 + a4) 6 — 1] <0, then the preceding relationship holds only

if
wg < OAR U = ea) b= UR
°° (= an) [t= (Fea) 6)
4) If [2(1 - a4) 6 — 1] <0 and [(1 + a4) 6 — 1] > 0, then the preceding relationship holds only
if

(ag) (lta) 6-7
TAS aa =20l = aa) 8) .

Proof of Proposition 5: Suppose that some collusion can be sustained in each market in the
non-coordinated equilibrium. That is, 2(1—a4)6-—1 > 0 and 2(1—ag)6—12> 0. If a4, ap,
and 6 are such that (1+ a,)6—1 <0 and (1+ag)6 —1 <0, then Case 3 of Lemma 3 applies.
As a first step, suppose that (33) binds and (34) is slack. For the solution in Lemma 3 to apply, it
must be the case that

QptB < a4,
and

_ ag [2(1— aa) 6-1 Fa
TMB <Q —ap)l- (1 tea) 5)

Setting 74 = 7” relaxes the preceding constraints. Because ag < a4, the first constraint holds for
all feasible 7g. If a, is sufficiently small, then the second constraint requires Tg < 1”. In fact,

for a4 sufficiently small, the second constraint will just bind. Hence, I can write

(1-2a,4)nTM  (1—ag) [ aa[2(1-a4)6-l]7a

Va= 2 — 4) 201-6) |G —apg) [1-G +a) 4]]’



known that, in an infinitely repeated Bertrand duopoly with homogenous goods and constant mar-
ginal costs, the discount factor, 6, must weakly exceed 3 for supracompetitive prices to arise in
equilibrium. Similarly, in a market with a per period discount factor of 6?, 6 must weakly exceed
vi = 0.707 to sustain supracompetitive prices. If the incentive constraints in the two markets
are added together, then the pooled incentive constraint is 6 + 6? > 1. In this instance, 6 need
only exceed 0.618 for the constraint to be satisfied. Thus, multimarket contact, as embodied by
the pooling of incentive constraints, enlarges the set of discount factors for which collusion in both
markets is sustainable.

To illustrate how a stochastic element might reduce the effectiveness of multimarket contact,
suppose firms use grim strategies to enforce collusive behavior. That is, each firm responds to
a low firm-specific demand realization, whose cause is unknown but is either a negative demand
shock or a rival’s price cutting, by setting price at the competitive level thereafter. In equilibrium
there are no deviations from the collusive price, so price warfare is induced solely by the negative
demand shocks.? Suppose there are two markets, cach of which receives a negative demand shock
with probability }. If the markets are considered in isolation, then the probabilities of being in any
particular configuration of collusive and non-collusive phases are shown in the first row of Table
1. If the two markets are strategically linked in a multimarket contact sense, then the strategies
employed may be such that collusion breaks down in both markets if there is a negative shock in
at least one market. In that instance, the probabilities of being in any particular configuration of
collusive and non-collusive phases are shown in the second row of Table 1. As is evident, the middle
ground is lost; that is, the multimarket linkage eliminates the possibility of low-grade competition

as opposed to either full competition or full collusion.

Table 1

Collusion in Both | Collusion in One | Collusion in None
9/16 6/16 1/16
9/16 0 7/16

Though the candidate strategies in the preceding example obviously are too simple, they do
raise the possibility that strategic linkage of markets may not increase profits when firms have

imperfect information. Gertner and McCutcheon [1994] (hereinafter GM) is a recent paper that

2Note that the price wars still must occur, else there will be incentives for firms secretly to cut price.


demand. Hence, a firm’s defection strategy will be to defect in one market, be fairly certain that
the other market still will have high demand, and thus avoid the harsh punishment while professing
its innocence. Such a deviation would be profitable, because the punishment necessary to prevent
the deviation entails a negative stream of payoffs in the continuation equilibrium, which violates
individual rationality. Thus, the limitation on the severity of punishment, coupled with the low

likelihood of the punishment being induced, prevents multimarket contact from increasing profits.

Proposition 6 Suppose that collusion cannot be sustained in either market in the non-coordinated
equilibrium. If each market has a sufficiently high degree of uncertainty, then multimarket contact

does not increase profits.

This result is not too surprising. Even though, due to the other market’s high likelihood of having
a negative demand shock, cheating in one market is quite likely to induce severe punishment, the
low value of future payoffs makes cheating today profitable. Therefore, supracompetitive profits

cannot be sustained.

Proposition 7 Suppose that collusion cannot be sustained in at least one of the markets in the
non-coordinated equilibrium. If that market has a sufficiently large degree of uncertainty and the

other market has a sufficiently small degree of uncertainty, then multimarket contact increases

profits.

This result is consistent with Proposition 6. The market with a very small degree of uncertainty
cannot help increase profits in the market with a large degree of uncertainty, because the probability
of having a bad outcome in the more certain market is so low. However, the reverse is true. If a
firm cheats in the more certain market, then it is quite likely the less certain market will have a
negative shock, thus inducing the severe punishment. Therefore, a firm will not cheat in the more
certain market. This permits a much higher payoff following a negative demand shock in market
B, as the following example illustrates.

Suppose that a4 = 0.4, ag = 0.1, and 6 = 0.75. Using the solution obtained in the non-
coordinated equilibria, the payoffs are

Vy = 1.60" VR = 1.60" V2 = 0.9330" V2_ = 0.9337"

Using Case 3 of Lemma 4 in the Appendix, one can determine that the maximal profits and the

16


and

(1—a4)(1— a) [Pay + ove] +(1-aa)an [# +6v5| (31)

+a,4(1— ag) [2 + ov + aap [6Vi5|
> (1—aa4)(1 - ag) [Fa t+ 7p + 6VE_] + (1 -aa)ap [a + 6VE 5]

+aa(1 — ag) [Fe + 6VE_| + oace [6Vi 5]
Vj; and (29)-(31) may be written more compactly as maximizing

Va = (ag) +(1- an) +(1-aa)(1—ap)6V°+(—aa)andVp+ (32)

aa(1 — ap)dV9 + agapoVin

subject to
(1—ap)8[V°~ V3) + an8(VB - Van) > (33)
(1 aa)6V* ~ VB] + andl — Vip] > “2 (34)

(1 —a4)(1 — @g)6[VS ~ V2n) + (1 — aa)apé[VE — Vig] + a4(1 — ap) 6[VE — Vip} (35)
> ( ~ aa) +(1 -ap)o

These constraints simply state that the expected loss from a given defection must outweigh the
expected gain from that defection. Consider (33), for example. Defecting in market A only changes
payoffs if A is in the high demand state. If so, then a player gains by defecting. If market B is
in the high demand state, then play moves from ° to 0%, inducing loss (V° — V3) one period in
the future. If market B is in the low demand state, then play moves from o%, to 74 ,, inducing loss
(VE —Vjp) one period in the future. If the expected loss exceeds the expected gain, then defection

will not occur. The same analysis applies to (34) and (35).
Lemma 1 In equilibrium, at least one of (33) and (34) must bind.

Proof of Lemma 1: Suppose (33) and (34) both are slack. If (33) is slack, then V? = V°. This
is the case because increasing Vj increases the objective function and relaxes both (34) and (35).

Similarly, if (34) is slack, then VB = V°. Substitute the new values of V{ and VB into (33)-(35),

22




greater than marginal cost. From (2), it must be the case that 6V° > %. If 6 is too low, then a firm
cares too little about the future to be deterred from cheating on the tacit agreement. The gain from
cheating earned today, 5, outweighs the loss from cheating incurred tomorrow, 6V°. If a4 is too
high, then a firm expects future collusive profits to be low. That is, as a4 increases, V increasingly
is determined by V?, which equals 0. Therefore, the loss from cheating incurred tomorrow, 6V°, is
outweighed by the gain from cheating earned today, $. Thus, even though a firm knows that if it
defects it will get a payoff of zero for the rest of the game, the punishment already is so likely to

be induced following tomorrow’s play that today the firm is willing to induce punishment for sure

and take the chance of getting an increased payoff in the current period.

3 Multimarket Contact With Partial Uncertainty

The previous section illustrates how imperfect: information about demand can impede firms’ collu-
sive efforts. Having learned from the BW analysis that multimarket contact increases profitability,
a firm might consider linking a market that has imperfect information with a market that has
perfect information. For example, price and quantity information could be easy to monitor, while
advertising or R&D outlays could be very difficult to monitor. Firms might sce a strategic linkage
of prices and advertising expenditures as their best opportunity to increase profits, as opposed to
a linkage of two markets that both have imperfect information. I show that in some instances this
intuition is incorrect.

Suppose firms 1 and 2 now compete in two markets, A and B, and choose prices in each market
in each period. A firm’s prices are not seen by its rival. For simplicity, marginal cost is c in each
market and for each firm. With probability a4, market A is in the “low demand state,” facing no
demand for the product. With probability 1-4, market A is in the “high demand state” with
demand D(p). Market B has no demand fluctuations, with firms always facing demand D(p).®
Assume that the realizations of demand in market A are independently and identically distributed
across time. If the firms treat the markets separately, then the maximum profits for the firms in
this non-coordinated equilibrium are those found in Section 2.

This model adds a small amount of uncertainty to offer an extremely simple way to study the
interaction between imperfect information and multimarket contact. In fact, by having uncertainty

in only one market, the analysis looks much like that in Section 2. Denote by Vy the non-empty and

payoff following defection consistent with maximal collusion, then it becomes clear that the maximal payoff following
defection approaches zero as 6 | 3.

The assumption that the markets are identical except that one is more likely to have low demand than is the
other serves to highlight the effect that uncertainty has on the decision to link markets strategically.

9


B is followed by setting p = c in both markets forever. Therefore, V3 = Vi, = 0.
There are three possibilities when the two markets are considered in isolation: collusion may

be sustainable in zero, one, or both markets. I consider each possibility separately.

Proposition 2 Jf collusion is not sustainable in either market in the non-coordinated equilibrium,

then multimarket contact does not increase profits. Formally, if 6 < 3, then Vy =0.

This result is not too surprising, given the result from a standard Bertrand supergame. In this
case, neither market has excess enforcement power that can be used to increase collusion in the

other market.

Proposition 3 If collusion is sustainable in both markets in the non-coordinated equilibrium, then

multimarket contact does not increase profits. Formally, if (l—aa)é > 4, then

_ m. m
1 Gay a

The previous two results appear to be similar to the “irrelevance result” in BW that shows that
jdentical markets do not benefit from multimarket contact. The basic intuition of their irrelevance
result is that in isolation either each market already can sustain full collusion (as in Proposition 3),
in which case there does not exist a need for multimarket contact, or neither market can sustain
any collusion (as in Proposition 2), in which case neither market has excess punishment power to
use to increase profits. Though in the present case markets A and B are not identical, under the
conditions of Proposition 3 some collusion is possible in both in the non-coordinated equilibrium.
What is surprising is that, in isolation, the certain market, with its slack incentive constraint, has
excess punishment power,!© while the uncertain market’s incentive constraint is binding and full
collusion cannot be sustained. Multimarket contact raises profits by coordinating punishments, in
that if a firm cheats in A, then the payoff in B also is lowered. By pooling the incentive constraints
it seemingly should be possible to increase the overall level of profits. The flaw in this reasoning
is that the probability of centering the punishment phase associated with defection in market A is
unchanged following the multimarket link. The payoff following an apparent defection in A cannot
be increased with a concomitant decrease in the payoff following low demand realizations in both A

and B, as this will induce players to defect in A only in every period. The more severe punishment

19“Fxcess punishment power” in a world of perfect information formally means that the payoff following defection
does not have to be zero.

11


Abstract

With perfect information about relevant strategic variables, economic theory predicts that firms
engaged in competition across several markets sometimes can use their multimarket contact to
blunt competitive forces. In practice, perfect information likely is not available, and it is well
known that the existence of imperfect information can impede firms’ collusive efforts. I extend a
standard oligopoly supergame to examine simultaneously the effects of imperfect information and
multimarket contact on the degree of cooperation that firms can sustain, and I reach the following
conclusions. First, linkage of one market with perfect information and another with imperfect
information may not increase profits, despite the slack in incentive constraints exploited in perfect
information models. Second, multimarket contact does not increase profits if each market has too
little uncertainty. Third, profits can be increased in a market in which some collusion initially is
sustainable by linking it with a market in which no collusion initially is sustainable. The central
theme of these results is that a market must generate sufficiently noisy signals for it to benefit
another market through strategic linkage. This contrasts with the finding in individual markets

that collusion decreases as the level of noise increases.




(1-aa) eee + wv] +04 [# +ov4] > (1-aa) [ +7p +65] +a, [tp + 6VR_] (17)

(tay) [FAST 4 ove] tag [RP 4 avg] 2 aa) [Rate + Rol +04 Re + ORs) (08)

(16) through (18) force equilibrium payoffs to exceed those from defecting in A only, B only, or A
and B together, respectively. They may be simplified to obtain

TA

5[ve-Va] 2 > (19)

(1 = 04)6 [V°— VB] + aad [VP - VR,] > 2 (20)

B A AB. 2 7

(1 = a4)6 [V* = Vig] +a.48 [VE —VEp] > (1 an) St + 2 (21)

Because V2 and Vi, do not enter the objective function, they create the most slack in the incentive
constraints by being as small as possible. Thus, VB = Viz = 0, so that any defection in B is followed
by setting p = c in both markets forever. Given the preceding condition, (20) holds whenever (21)

holds, so (20) is superfluous, and the constraints are
é[vo—va] > “4 (22)

7B

3 (23)

(1—arg)5Vo + a 5V2 > (A on) +

If a4 = 0, then this collapses to the perfect information problem. If a4 > 0, then (22) binds. If
it did not, then one could increase V2, increasing V, without violating the two constraints. Using

(22) in (23) and the objective function yields

Tp

Vy = (1 204)%4 + 6V" + 5 (24)
and
coy TA, TB 5
6ve> gt" (25)
Now V¢ = Vy, so
v,-(—.) la-2yTMM+TM
Vj= (5) [a 2a) 2 +3 (26)
and
TA Tp
(201 — @4)6 -— Yy + 25 — vy >0. (27)

20




Combining the preceding two constraints, it must be the case that

(= ap) (1 +04)6-1)

OBST 8 —aa)

This combined constraint may be simplified to
0 < [(1+ a4) (1 — ag) + 2(1 — a4) og] 6-1.

This constraint holds if both a4 and ag are close to a for 6 sufficiently large. For example, if
aaq = ag = hy then 6 must exceed $. As both a, and ag get sufficiently large, then the com-
bined constraint cannot hold. Therefore, for moderate degrees of uncertainty, multimarket contact

increases profits even though no collusion could be sustained in the non-coordinated equilibrium.

References

{1] Abreu, D., Pearce, D., and Stacchetti, E. “Optimal Cartel Equilibria with Imperfect Monitor-
ing.” Journal of Economic Theory, Vol. 39 (1986), pp. 251-269.

[2

Abreu, D., Pearce, D., and Stacchetti, E. “Toward a Theory of Discounted Repeated Games
with Imperfect Monitoring.” Econometrica, Vol. 58, No. 5 (1990), pp. 1041-1063.

(3

Barla, P. “Multimarket Contact and Pricing Strategy in the U.S. Domestic Airline Industry.”
Cornell University Ph.D. Dissertation (1994).

[4] Bernheim, B. Douglas and Whinston, Michael D. “Multimarket Contact and Collusive Behav-
ior.” RAND Journal of Economics, Vol. 21, No. 1 (1990), pp. 1-26.

[5] Evans, W. and Kessides, I. “Living by the ‘Golden Rule’: Multimarket Contact in the U.S.
Airline Industry.” Quarterly Journal of Economics, Vol. 109, No. 2 (1994), pp. 341-366.

(6

Feinberg, R. and Sherman, R. “An Experimental Investigation of Mutual Forbearance by
Conglomerate Firms,” in Industry Structure and Performance, Schwalbach, J. (ed.). Berlin:
Bohn, Sigma (1985), pp. 139-166.

(7

Feinberg, R. and Sherman, R. “Mutual Forbearance Under Experimental Conditions.” South-
ern Economic Journal, Vol. 54, No. 4 (1988), pp. 985-993.

28


then add (33) and (34) after premultiplying them by (1 — a4) and (1 — ag), respectively. This

summation yields

TB

(1- an)apélV* — Ven] + aa(t — ap)61V*— Vig] > (1 a4) + (1 — an),

whereas (35) is

(1 — a4)(1 — a) 8[V° — V2'g] + (1 — aa) d[V* — Vip] + oa(l — op)6[V° — Vig]

> (a- an)A +(1- an)

Consequently, if (33) and (34) both are slack, then (35) also must be slack. In equilibrium, all three

incentive constraints cannot be slack.

Lemma 2 Suppose (33) and (33) both bind. Then the solution that maximizes V and that satisfies

(33)-(35) is

=o _ (l—2a,4)%4 , (1-2ag)%B
TV, = “Vasa

Ma — 6) ya 4)”

with
p _ (2(L-a4) 6-74 | (1—20n)FB
“a= 5-8 + 20-8)”
VP = (1-2a4)%4 , (2(1—ag)6-1) 7B
B~ 3-5) 21-6)”

and

Pp [(2(1-a4)6—-1)7%4 , [(2(1-ag)6—1]7p
VaB =

w-6) | was  ”

subject to the feasibility constraints.

Proof of Lemma 2: Multiply (33) by (1— aa) and (34) by ag. Substitute (1— a4) ap6VE
from (33) into Vy, (34), and (35). The sign of the coefficient on V? in (34) is ambiguous, so
substitute V2, from (34) into Vy and (35). Increase V° in (35) until (35) just binds. This yields
6VE = 6Ve— x, so substitute V7 from (35) into Vy. Using V* = Vy yields

vy, _ (l= 2a), , (L-20n)t a
Vi="SG-6 + 30-8)

Moving backward through the proof yields the values for V?, VZ, and Vig.
Lemma 3 Suppose (33) binds and (34) is slack. Then the solution that maximizes Vz and that

23


Table 2

2 received a profit | 2 received no profit
1 received a profit Ww xX
1 received no profit Y Z

Given the two common knowledge events following first period play, & specifies two SSEs truncated
to the remaining game. Call these truncated SSEs o? and o°, as mnemonics for punishment and
collusion, respectively. These truncated SSEs have associated payoffs V? and V°, which both are
elements of V. In what follows I am unconcerned with the strategies specified in o? and 0°, just

the associated payoffs. The payoff to & is
V=(1-a4) B +ov4 + a4 [6V?). (1)

I wish to maximize V with respect to 7, V?, and V°, subject to the following constraints:

(1 aa) EB + wv] + 4 [6V?] > (1 — axa) [i + 6V?] + cx [6V?] (2)
O0<Ve<v (3)
O<VP<V (4)
O<m@<nTM (5)

Equation (2) is an incentive compatibility constraint, according to which the expected profit from
colluding must. weakly exceed the expected profit from defecting. Equations (3) through (5) are
constraints that embody both individual rationality (each firm always can achieve a payoff of 0 by
setting p = c) and feasibility (e.g. 7 must be weakly less than the monopoly profit, 7”). To save
space, later in the analysis I simply will refer to equations such as (3) through (5) as feasibility
constraints, and usually will not show them in the text.

There are two immediate conclusions from the form of the maximization problem: First, because
V° increases the objective function and relaxes (2) when increased, V° = V. Intuitively, if the firms
receive their share of the collusive profits in the current period, then they continue with the same
behavior in the next period. Second, (2) must bind. If it did not, then one could increase V?, still
satisfy (2), yet increase V. That is, the firms should not punish poor market outcomes any more

than is necessary to prevent deviation, because the demand variability ensures that price wars will


compact set of SSE payoffs for this game permitting decisionmaking jointly across markets. Let 7)
be the SSE which yields the largest element of ¥;, Vy. In the first period of play, 7; specifies that
firms set prices B4 and Dp, receiving payoff =p with probability one and payoff % with probability
(1-4). As in Section 2, in equilibrium there are only two common knowledge events following
the price-setting in period one: Either both firms received a profit in market A, or at least one firm
did not.® 7, specifies two SSEs truncated to the remaining game, one for each common knowledge
contingency. Call these 0% and o°, with associated payoffs V4 and V°, both elements of Vy. Other
relevant SSEs, which come into play only out of equilibrium, are of and oF gp» which are the SSEs
specified by a7 following a defection in B or in both A and B. These SSEs have associated payoffs
Vg and Vi',. The payoff to 7, is

Tatts, wv +a [2 + vy! . (8)

I wish to maximize Vy with respect to 74, Fp, Vi, VB; Vip, and V°, subject to the incentive

compatibility constraints below and the feasibility constraints.

(l-a4) [st +5v4] +04 [# +ov4| > (1-aa) [ra = +6v4| +04 [2 +6v3| (9)

) [Pz + w +a [7 + ov] > (1-a,) [5 +ia+ wvs| +aa [Fp + 6Vi5] (10)
Tate

(1a) [

+ ov tag [2 + ov] > (I-aa) [Ra -+ p+ 6V2,] +04 [Fe + 6V25] (11)

ic)

The incentive constraints (9)-(11) specify that it must not be profitable for a firm to deviate in
market A only, market B only, or both markets A and B, respectively.

As in the single market model, V° = Vy; successful collusion is followed by again setting
the collusive prices in each market. Also, certainty regarding demand in market B allows strong
punishment without fear of mistakenly invoking punishment. Because Vg and Vi, do not affect

the objective function and relax the incentive constraints when decreased, any deviation in market

°At this point, one sees the utility of using the APS methodology versus that used in Tirole [1988] and GM.
Specifically, it is not clear how one would specify the punishments with the alternative approach. It is not obvious
whether only one length of punishment is used in both markets simultaneously, or whether each market has its own
punishment length. Using the latter method, one must specify how the punishment regimes interact. For example,
suppose market A is in the punishment regime, while market B has finished punishment and has returned to the
collusive equilibrium for that market. Does the market A punishment begin again if there is a defection in B? Using
the APS methodology, one ignores the specifics of the punishment profiles and concentrates simply on the payoffs,

10