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Buy-backs, Price Wars, and Collusion
Enforcement∗
Jimmy Chan Wenzhang Zhang †
June 2009
Abstract
Recent studies of cartel operation show that in most cartels both sales
and price data were private information, and side payments between firms
were used in addition to price wars to enforce collusion. This paper pro-
poses a simple mechanism for collusion enforcement that incorporates these
features. The mechanism applies to both price and quantity competition,
and it allows for multi-product firms.
∗We thank Joe Harrington for comments and suggestions.†School of Economics, Shanghai University of Finance and Economics, Shanghai 200433,
China. E-mail: [email protected] (Chan) and [email protected] (Zhang).
1
1 Introduction
In enforcing a collusion agreement, a cartel must contend with the problem that
member firms may cut price or over sell secretly (Stigler, 1964). In a seminal paper
Green and Porter (1984) show that a cartel that could not monitor its members’
outputs may still deter cheating by using a price war as a collective punishment. In
a recent case study Harrington (2006) shows that, instead of starting a price war,
many cartels settled disputes through compensation. In the citric acid cartel, for
example, firms agreed to a set of sales quotas, and firms that sold above quota were
required to purchase product from those that sold below.1 In one case, Haarmann
& Reimer were required to purchase 7000 tons of citric acid from ADM.2
While such compensation schemes should in principle discourage over-producing,
they might not work in practice because firms could lie about their sales. Many
cartels had no safeguard against under-reporting. While some used trade asso-
ciations or other third parties to collect data, it is likely that a determined firm
would find a way to fudge the data. For any third party enforcement that would
make it impossible for a firm to lie would also likely to arouse the suspicion of
the antitrust authority. Nevertheless, Harrington (2006) concludes that cartels
appeared to take the self-reported data seriously. Suslow (2005) finds that cartels
that used some type of self-imposed penalty schemes were more stable.
In this paper we show how a cartel can enforce collusion with self-reported
data with a scheme similar to the one described above. The scheme applies to
both price and quantity competition, and it allows for multi-product firms. Under
the scheme firms agree to a collusive agreement that maximizes cartel profit and
a set of profit targets. They report their profits at the end of each period. The
reported profit shortfalls—the difference between the profit targets and reported
profits—determine the side-payments between firms and the probability that col-
lusion breaks down and a price war erupts. A firm tends to pay less when it
1Harrington (2006) reports that some form of compensation schemes were used in the car-
tels in the market for choline chloride, citric acid, lysine, organic peroxides, sodium gluconate,
sorbates, most vitamins, and zinc phosphate. Also see Levenstein and Suslow (2006).2Harrington (2006)
2
reports a lower profit, but the gain is offset by a higher probability of a price war.
In reality, collusion may break down if one or more firms suspect some firms have
been cheating about their sales.
Taking into account of both its payment to other firms and the loss of profit in
case of a price war, a firm’s expected “penalty” is independent of its own report
and increasing in the reported profit shortfalls of the other firms. If the profit
targets are set very high, then the penalty will fully internalize the externality a
firm’s action imposes on the other firms. However, in practice the profit targets
could not be set very high, because firms would refuse to pay a very high penalty.
But if the targets are set, for example, near the expected profits, a firm which
secretly cut price may get lucky and escape punishment when an exceptionally
high industry demand, due to a positive demand shock, masks the effect of the
price cut. We show that despite this “truncated distribution” problem, the firms
could be motivated to comply with the collusion agreement under a weak condition
that is satisfied either when the demand shock is multiplicative or additive, or when
its density is log-concave.
Penalties in our model must include both side payments and a price war. If
only side payments are used, firms would under-report to raise the penalties of
the other firms. On the other hand, if only a price war is used, firms would over-
report to avoid starting a price war. This explains why compensation schemes were
adopted in many cartels. Since a price war is used to ensure truth-telling, collusion
will break down occasionally due to random demand shocks. By contrast, a cartel
that monitors firms’ sales directly could enforce collusion with only side-payments
(Harrington and Skrzypacz (2007)). This suggests that arrangements that allow
firms to publicize sales information credibly may be anti-competitive.
1.1 Related Literature
Our model is inspired by the recent works of Harrington and Skrzypacz, which are
the first to analyze the role of side payments in collusion enforcement. Harrington
and Skrzypacz (2007) considers an environment where prices are private but sales
are public. They show that by requiring the firms to pay a per unit tax, a cartel
3
can induce the firms to follow the collusion agreement. Harrington and Skrzypacz
(2008) extend their 2007 model to the case where both prices and sales are private.3
The key difference between that model and ours lies in the way the firms are
induced to reveal their sales. Since in their model a firm is required to pay a tax
when its own sales is too high, it would have incentives to underreport. They show
that firms could be induced to tell the truth when the correlation between firms’
demands satisfies certain conditions. No such restriction is needed in our model
since a firm’s report is used only to determine the penalties of the other firms.
More broadly, our model is also related to the literature of repeated games
with communication under private monitoring (Kandori and Matsushima, 1998,
Compte, 1998, Fudenberg and Levine, 2007, Obara, 2008, and Zheng, 2008). In
this literature collusion is usually supported by using a statistical test that the
collusive action is the least likely to fail among all actions. In our model, that
would suggest a two-sided test where a firm is punished when other firms’ profits
are either too high or too low. On the contrary, the test we used is a one-sided
test where a firm is punished only when other firms’ profits are too low. Hence,
in our model a firm can reduce its expected penalty by raising its price, but the
gain would be smaller than the loss in its own profit that a higher price would
cause. Another difference between this paper and the repeated-game literature
is that the latter tends to focus on the question of asymptotic efficiency as the
discount factor goes to one. By contrast, the objective of the present paper is to
come up with an enforcement scheme that is both versatile and consistent with
actual cartel operation.
2 Model
There are n firms, indexed by i = 1, 2, . . . , n. The horizon is infinite and time
periods are indexed by t = 0, 1, . . . . In each period t, firms play the following
stage game: (1) each firm i simultaneously chooses an action si,t from a set Si; (2)
3Aoyagi (2002) analyzes collusion in a repeated Betrand game in which the demand shocks
are positively correlated.
4
Figure 1: Time line
the stochastic profit πi,t of each firm i is realized; both the action si,t and profit πi,t
are private information of firm i; (3) each firm i simultaneously reports a profit π̂i,t
to the other firms; (4) based on the reported profits, each firm i simultaneously
makes a side-payment τij,t to each firm j; (5) firms observe the outcome xt of a
public randomization device; xt is uniformly distributed between 0 and 1.4 See
Figure 1.
The action si can be broadly interpreted. For example, it can be price, quantity,
or quality. We assume that it is one-dimensional and the action set Si is an interval.
In section 4.1 we also allow the action si to be multi-dimensional to capture the
fact that most cartels involve firms that produce multiple goods or sell in multiple
markets.
When an action profile s ≡ (s1, s2, . . . , sn) is played, the stochastic profit πi
of firm i has a smooth distribution function Fi (·, s) with finite support [πi, πi].
Conditional on any sequence of strategies, the profits are independent over time
but possibly correlated across firms. We make two assumptions about the profit
4A note on notation: The subscript “i, t” of a variable refers to firm i in period t. Similarly,
the subscript “ij, t” refers to firms i and j in period t. The same variable with subscript t refer
to the n-vector or n× n matrix of the variable. For example τt is n× n side-payment matrix in
period t. The subscript −i refer to a n-vector minus the i-th component. We drop the subscript
t when doing so does not cause confusion.
5
distributions.
Assumption 1. For each firm i, Fi (πi, sj, s−j) is decreasing in sj for each πi and
each j.
The assumption means that a higher action of firm j shifts the distribution of
firm i’s profit to the right in the sense of first-order stochastic dominance.
Assumption 2. For each pair of distinct firms i and j, each action profile s−j of
the firms other than j, and each Kij < πi, the ratio∫ Kij
πi
∂Fi
∂sj
(πi, sj, s−j) dπi
/∫ πi
πi
∂Fi
∂sj
(πi, sj, s−j) dπi (1)
is weakly decreasing in sj.
Note that by integration by parts E [πi|s] = πi−∫ πi
πiFi (πi, sj, s−j) dπi. Roughly
speaking, Assumption 2 requires that the effect of si on the low end of the distri-
bution of πj is larger when si is smaller. Assumptions 1 and 2 are satisfied for a
large class of demand functions.
Example: Bertrand Competition. Suppose the firms are competing in price.
Each firm i has a constant marginal cost, ci, and must choose a price pi ∈ [ci, p].
Let p ≡ (p1, . . . , pn) be a price vector. Firm i’s stochastic demand is
qi (p, εi1, εi2) = h (εi1q̂i (p) + εi2) ,
where h is strictly increasing, εi1 and εi2 are random shocks, and q̂i (p) is positive
for all p, decreasing in pi and increasing in pj, j 6= i. The random demand shock,
εik, k = 1, 2, has a smooth distribution on a finite support [εik, εik] with εik ≥ 0.
Lemma 1. Assumptions 1 and 2 are satisfied in the Bertrand example if either
(1) h is linear or (2) εi1 is a constant and the density of εi2 is log-concave.
An analogous result holds for Cournot competition. The proof of Lemma 1 and
its Cournot counterpart is provided in the Appendix. Assumption 1 is satisfied
because qi (p, εi1, εi2) increases in pj for any εi1 and εi2. The conditions in Lemma
6
1 therefore are needed only for Assumption 2. If h is linear, the effect a marginal
change in pj on πj is independent of the realization of εi2, and the ratio in (1) is
equal to the probability that πj is less than Kij, an event which is less likely when pj
is high. Intuitively, the effect of pj concentrates on the high end of the distribution
of πi because πi is unlikely to be less than Kij when pj is high. The random shock
may enter the demand function in a non-linear way. For example, if h is strictly
concave, then the effect a marginal change in pj on πj will be smaller when εi2
is smaller. In this case, Assumption 2 will still hold if there is no multiplicative
shock and the density of εi2 is log-concave. Many common distributions, including
normal and logistic, are log-concave. Under either condition, there is no restriction
on the functional form of q̂i.
Let s∗ = (s∗1, s∗2, . . . , s
∗n) be an action profile that maximizes the expected
industrial profit∑n
i=1 E [πi|s]. We assume that the stage game has a pure-strategy
Nash equilibrium sN = (sN1 , sN
2 , . . . , sNn ) such that
n∑i=1
E [πi|s∗] >n∑
i=1
E[πi|sN
].
Let mpubt =
(π̂t, τt, xt
)denote the public information in period t, and mpri
i,t =(si,t, πi,t
)firm i’s private information in that period. At the beginning of period t,
all firms observe a public history hpubt = (mpub
0 , mpub1 , . . . ,mpub
t−1), and, in addition,
each firm i observes a private history hprii,t = (mpri
i,0 , mprii,1 , . . . ,mpri
i,t−1). Player i’s full
information at the beginning of period t is denoted by hi,t =(hpub
t , hprii,t
).5 An ac-
tion strategy of firm i is a function ai that maps a history hi,t to an action ai(hi,t) in
Si. A reporting strategy of firm i is a function ri that maps(hi,t, m
prii,t
)to a report
ri(hi,t, mprii,t ) in [πi, πi]. A transfer strategy of firm i is a n-vector bi = (bi1, . . . , bin),
where each bij maps(hi,t, m
prii,t , π̂t
)to a nonnegative real number bij
(hi,t, m
prii,t , π̂t
).
A strategy σi of firm i is the collection of an action, a reporting, and a profile of
transfer strategies (ai, ri, bi). Given a strategy profile σ = (σ1, σ2, . . . , σn), firm i’s
5Define each of hpub0 , hpri
i,0 and hi,0 as the empty set ∅.
7
expected average discounted payoff in the repeated game is
vi (σ) ≡ E
[(1− δ)
∞∑t=0
δt
(πi,t −
∑j 6=i
(τij,t − τji,t
)) ∣∣∣∣σ]
,
where E[ ·|σ] is the expectation with respect to the probability measure on his-
tories induced by the strategy profile σ, and δ is the common discount factor. A
strategy is a public strategy if its continuation strategy after a history hi,t de-
pends only on the public history hpubt . A profile of public strategies is a perfect
public equilibrium if the continuation strategy profile after each period constitutes
a Nash equilibrium.
3 Trigger-Strategy Equilibrium
We focus on equilibrium with a simple strategy structure. A truth-telling trigger-
strategy profile ρ is defined by three components: a collusive action s, a probability
function p : [π, π]n → [0, 1], and an n×n payment matrix β, where each component
βij : [π, π]n → [0,∞). Firms start off in the collusive state, in which they play an
action profile s and report their profits truthfully. Based on the reported profits
π̂t, each firm i pays βij(π̂t) ≥ 0 to each firm j. If all firms make the side-payments,
they switch to the non-collusive state in the next period with probability p(π̂t).
If any of them fails to pay, they switch to the non-collusive state with probability
one. The non-collusive state is absorbing. Once the firms enter the non-collusive
state, they play the stage-game Nash equilibrium forever; the reports in the non-
collusive state are irrelevant and there are no transfers.
To summarize, under a trigger-strategy profile ρ = (s, p, β), a public history
hpubt is in the collusive state if (i) t = 0, or (ii) hpub
t−1 is in the collusive state, and
τij,t−1 = βij(π̂t−1) for all i, j, and xt−1 ≥ p(π̂t−1); otherwise, t is in the non-collusive
state. Let C (s, p, β) denote the set of public histories in the collusive state. Firm
8
i’s strategy σρi = (aρ
i , rρi , b
ρi ) under ρ is given as follows:
ai(hi,t) =
{si if hpub
t ∈ C (s, p, β)
sNi otherwise
;
ri(hi,t, si,t, πi,t) = πi,t;
bij(hi,t, si,t, πi,t, π̂t) =
{βij(π̂t) if hpub
t ∈ C (s, p, β)
0 otherwise.
Firm i’s expected payoff from σρ is
vi (σρ) =
(1− δ) (E [πi|s]− E [βij(π)− βji(π) |s|]) + δE [p (π) |s] vNi
1− δ (1− E [p (π) |s]).
Define firm i’s “continuation payoff” conditional on π̂ in the collusive state—
its payoff that comes after the realization of the current period profit, including
the current period side-payments and the expected payoffs from the next period
onward—as
wρi (π̂) ≡ vi (σ
ρ)− p(π̂)(vi (σρ)− vN
i )− δ−1 (1− δ)∑j 6=i
(βij(π̂)− βji(π̂)) , (2)
where vNi = E[πi|sN ] is the expected profit of firm i in the stage-game Nash
equilibrium. The second term in (2) is the expected profit loss due to a price
war, and the third term is firm i’s net payment to other firms. Together, they
constitute a “penalty” for firm i. We will focus on trigger-strategy profiles with
the following features.
Definition 1. A trigger-strategy profile (s, p, β) is enforceable if for all i and π,
wρi (π̂) ≥ vN
i .
A unenforceable trigger-strategy profile could never be an equilibrium as firm
i would be better off not paying the side-payments it owes when wρi (π̂) < vN
i .
Definition 2. A trigger-strategy profile (s, p, β) involves no simultaneous pay-
ments if a firm never makes a strictly positive payment to one firm and receives
one from another at the same time. That is, for all i, there is no π̂ where both∑j βij (π̂) and
∑j βji (π̂) are strictly positive.
9
Let Π (s) denote all profit profiles that has a positive density when some firm
i chooses some s′i ∈ Si and all other firms j choose sj. The set Π (s) includes all
profit profiles that might occur under any choice of firm i when other firms are
choosing s−i.
Definition 3. Let K and λ be n × n matrices with λij > 0 for all i and j. A
trigger-strategy profile ρ implements a linear penalty scheme (K, λ) if for all i and
all π̂ ∈ Π (s) ,
wρi (π̂) = vi (σ
ρ)− δ−1 (1− δ)∑j 6=i
λij max (Kij − π̂j, 0) . (3)
Each Kij is a target for firm j’s profit. Under (K, λ), firm i is required to pay
a penalty linear in the profit shortfall of each firm j 6= i below Kij in the collusion
state. Its average discounted profit in the collusion state is
vK,λi (s) ≡ E[πi|s]−
∑j 6=i
λij
∫ Kij
πj
Fj (πj, s) dπj.
Proposition 1. Suppose Assumptions 1 and 2 hold. Then for any linear penalty
scheme (K, λ) such that
1. λij
∫ Kij
πj
∂Fj
∂si
∣∣∣s=s∗
dπj =∫ πj
πj
∂Fj
∂si
∣∣∣s=s∗
dπj, for each distinct i and j; and
2. (1− δ) supπ̂∈Π(s∗)
∑j 6=i λij max (Kij − π̂j, 0) ≤ δ
(vK,λ
i (s∗)− vNi
), for each
i,
then there is a perfect public equilibrium where the firms choose a trigger-strategy
profile with collusive action s∗ that implements the penalty scheme (K, λ).
Proposition 1 says that when conditions 1 and 2 are met, a cartel can use a
linear penalty scheme to enforce a collusive action that maximizes the cartel profit.
The right-hand side of the equation in condition 1 is the marginal effect of si on
firm j’s profit. Since the penalty scheme captures only the effect of si on firm j’s
profit below Kij, condition 1 requires that λij be scaled up so that the marginal
effect of si on the penalty is the same as that on firm j’s profit. This ensures
that it is optimal for each firm i to choose s∗i in the collusion state. The left-hand
10
side of the equation in condition 2 is an upper bound of firm i’s penalty under
(K, λ). Condition 2 requires that this bound be less than the value the scheme
can destroy through a price war. The condition ensures that the penalty can be
implemented by a trigger-strategy profile and that the firms have the incentive to
make the required side-payments. We prove Proposition 1 through two lemmas.
Lemma 2. Under Assumptions 1 and 2, if (K, λ) satisfies condition 1 in Propo-
sition 1, then
s∗i ∈ arg maxsi
E
[πi −
∑j 6=i
λij max (Kij − πj, 0)
∣∣∣∣si, s∗−i
]. (4)
Proof of Lemma 2. It is useful to rewrite the objective function in (4) as
E
[∑j
πj
∣∣∣∣si, s∗−i
]+∑j 6=i
E
[−λij max (Kij − πj, 0)− πj
∣∣∣∣si, s∗−i
]. (5)
The first term in (5) is the expected cartel profit, and the second is the expected
difference between firm i’s penalty associated with firm j’s profit shortfall and
firm j’s profit. By definition, s∗i maximizes the first term of (5). Hence, Lemma 2
holds if s∗i also maximizes
Hij
(si, s
∗−i; Kij, λij
)≡ E
[−λij max (Kij − πj, 0)− πj
∣∣∣∣si, s∗−i
](6)
for each j 6= i.
Notice that if Kij > πj and λij = 1 for each j 6= i, then Hij
(si, s
∗−i; Kij, λij
)=
−∑
j 6=i Kij for all si. In this case, firm i’s penalty fully captures the externality
of si on the profits of the other firms. However, as we shall see, profit targets are
bounded from above by the continuation value of collusion and might not be set
above πj.
Suppose Kij is less than πj. Through integrating Hij by parts and differenti-
ating it with respect to si, we have
∂Hij
(si, s
∗−i; Kij, λij
)∂si
= −λij
∫ K
πj
∂Fj
∂si
(πj, si, s−i) dπj +
∫ πj
πj
∂Fj
∂si
(πj, si, s−i) dπj.
(7)
11
By condition 1,∂Hij
(s∗i , s
∗−i; Kij, λij
)∂si
= 0.
It follows from Assumption 2 that
∂Hij
(si, s
∗−i; Kij, λij
)∂si
=∫ Kij
πj
−∂Fj
(πj, si, s
∗−i
)∂si
λij −
∫ πj
πj
(∂Fj
(πj, si, s
∗−i
)/∂si
)dπj∫ Kij
πj
(∂Fj
(πj, si, s∗−i
)/∂si
)dπj
≥ 0
if and only if si ≤ s∗i . Intuitively, Assumption 2 ensures that firm i’s expected
penalty decreases faster than the profit of firm j increases (as si increases) when
si is less than s∗i , and slower when si is greater than s∗i .
Lemma 3. If (K, λ) satisfies condition 2 in Proposition 1, then it can be im-
plemented by an enforceable trigger-strategy profile that requires no simultaneous
payments and in which s∗ is chosen in the collusive state.
Proof of Lemma 3. We first choose p (π̂) so that the expected value to be
destroyed through a price war is equal to the total penalty that is required. For
each π̂ ∈ Π (s), set
p (π̂) ≡
∑i
((1− δ) δ−1
∑j 6=i λij max (Kij − π̂, 0)
)∑
i
(vK,λ
i (s∗)− vNi
) . (8)
Since, by condition 2, the maximum penalty is less than the continuation value of
collusion for each firm, the total penalty for all firms must be less than the total
value. Hence, p (π̂) ∈ [0, 1] for all π̂ ∈ Π (s). To obtain the right penalty for each
firm (i.e. (3)), set the net payment of each firm i to be
βneti (π̂) ≡
∑j 6=i
λij max (Kij − π̂j, 0)− δ (1− δ)−1 p(π̂)(vK,λ
i (s∗)− vNi
). (9)
These net payments could be implemented by the side-payment scheme:
βij (π̂) ≡
βneti (π̂)
min(βnetj (π̂),0)
Pk min(βnet
k (π̂),0)if βnet
i (π̂) ≥ 0,
0 otherwise.(10)
12
According to β, each firm i with a positive net payment pays each firm j with a
negative payment an amount in proportion to firm j’s share of the total payment
changing hand. It is straightforward to see that β involves no simultaneous pay-
ments. Finally, the enforceability of ρ = (s, p, β) follows directly from condition
2.
Proof of Proposition 1. Since a firm’s report under (s∗, p, β) does not affects
its own penalty, it is always optimal for each firm to report truthfully. If firm i
refuses to make the side-payments, its continuation will be vNi . Since (s∗, p, β) is
enforceable, it is optimal for each firm i to comply with β. Finally, by (4), it is
optimal for firm i to choose s∗i in the collusion state.6
Proposition 1 says that a cartel can use a linear penalty scheme to enforce
a collusive action s∗ so long as the maximum penalty required for each firm is
smaller than the continuation value of collusion. The following proposition shows
that this condition will be met when the demand shock is small and the firms are
sufficiently patient.
Proposition 2. Suppose Assumptions 1 and 2 hold and there is some positive
constants κ1 and κ2 such that κ1 < |∂Fj (πj, s) /∂si| ≤ κ for all s, πj, i, and j.
Then there exist δ̄ < 1 and ε > 0 such that if Pr (|πi − E[πi|s]| ≥ ε | s) ≤ ε for
each i and each s, then for all δ > δ̄ there is a trigger-strategy perfect public
equilibrium where the firms choose s∗ in the collusion state. Moreover, firm i’s
average discounted equilibrium payoff converges to E [πi|s∗] as ε tends to zero.
4 Extensions
4.1 Multiple markets and multiple instruments
Suppose that there are M markets, indexed by m = 1, 2, . . . , M , and the action
space Si of each firm i is a convex subset of RL, the Euclidean space of dimension
6Strictly speaking, p and β are defined only for π̂ ∈ Π (s∗). Any profit report π̂ /∈ Π (s∗) is
not reacheable by unilateral deviation. For completeness, we can set p (π̂) = 1 and βij (π̂) = 0
for all π̂ /∈ Π (s∗).
13
L. Let πi,m denote the profit realization of firm i in market m, and si,l denote
the l-th component of an action si in Si. We assume that, for each i and each
m, there exists a function φi,m : ×nj=1Sj → R such that, conditional on a action
profile s = (sj,l)j≤n, l≤L, the cumulative distribution function of firm i’s profit πi,m
has the form Fi,m (·, φi,m(s)) with support [πi,m, πi,m].
Assumption 3. For each firm j, each market m and each level of profit realization
πj,m, we have
A. the probability Fj,m (πj,m, φj,m) is decreasing in φj,m; and
B. the partial derivative ∂Fj,m (πj,m, φj,m)/∂φj,m is integrable with respect to
πj,m, and, for each K ≤ πj,m, the ratio∫ K
πj,m
∂Fj,m
∂φj,m
(πj,m, φj,m) dπj,m
/∫ πj,m
πj,m
∂Fj,m
∂φj,m
(πj,m, φj,m) dπj,m
is weakly decreasing in φj,m.
Assumption 4. For each pair of distinct firms i and j and each market m, the
function φj,m is differentiable and quasi-concave in si.
Assumption 5. There exist positive constants κ1 and κ2 such that
κ1 ≤∂Fj,m (πj,m, s)
∂si,l
≤ κ2,
for each s, each i and j, each l, and each πj,m.
We now modify the penalty scheme in the benchmark model to encompass the
situations of multiple markets and/or multiple instruments.
We still let Π (s) denote all profit profiles that has a positive density when
some firm i chooses some s′i ∈ Si and all other firms j choose sj.
Definition 4. Let K = (Kij,m)1≤m≤M ;1≤i,j≤n;i6=j and λ = (λij,m)1≤m≤M ;1≤i,j≤n;i6=j
be such that Kij,m ≤ πj,m and λij,m > 0 for all i, j and m. A trigger-strategy
profile ρ implements a linear penalty scheme (K,λ) if for all i and all π̂ ∈ Π (s) ,
wρi (π̂) = vi (σ
ρ)− δ−1 (1− δ)∑j 6=i
M∑m=1
λij,m max (Kij,m − π̂j,m, 0) . (11)
14
Each Kij,m denotes a target for firm j’s profit in the market m, and firm i is
required to pay a penalty linear in the profit shortfall of firm j below Kij,m in the
collusion state. Its average discounted profit in the collusion state is
vK,λi (s) ≡ E
[M∑
m=1
πi,m
∣∣∣∣s]−∑j 6=i
M∑m=1
λij,m
∫ Kij,m
πj,m
Fj,m (πj,m, s) dπj,m.
As before, we let s∗ = (s∗i,l)1≤i≤n;1≤l≤L be an action profile maximizing the
cartel profit∑n
i=1
∑Mm=1 E [πi,m|s], and assume that the stage game has a pure-
strategy Nash equilibrium sN = (sNi,l)1≤i≤n;1≤l≤L such that
M∑m=1
n∑i=1
E [πi,m|s∗] >
M∑m=1
n∑i=1
E[πi,m|sN
].
Let vNi =
∑Mm=1 E
[πi,m|sN
]denote firm i’s profit given by sN .
Parallel to Propositions 1 and 2, we have the following two propositions.
Proposition 3. Suppose Assumptions 3 and 4 hold. Then for any linear penalty
scheme (K, λ) such that
1. λij,m
∫ Kij,m
πj,m
∂Fj,m
∂φj,m
∣∣∣s=s∗
dπj,m =∫ πj,m
πj,m
∂Fj,m
∂φj,m
∣∣∣s=s∗
dπj,m, for each distinct i and
j, and each m; and
2. (1− δ) supπ̂∈Π(s∗)
∑Mm=1
∑j 6=i λij,m max
(Kij,m − π̂min
j,m , 0)≤ δ
(vK,λ
i (s∗)− vNi
),
for each i,
then there is a perfect public equilibrium where the firms choose a trigger-strategy
profile with collusive action s∗ that implements the penalty scheme (K, λ).
Proposition 4. Suppose Assumptions 3, 4 and 5 hold. Then there exist δ̄ < 1
and ε > 0 such that if Pr (|πi,m − E[πi,m|s]| ≥ ε | s) ≤ ε for each i, each m and each
s, then for all δ > δ̄ there is a trigger-strategy perfect public equilibrium where
the firms choose s∗ in the collusion state. Moreover, firm i’s average discounted
equilibrium payoff converges to∑M
m=1 E [πi,m|s∗] as ε tends to zero.
The proofs for these two propositions are identical to those for Propositions 1
and 2, except that we need to replace Lemma 2 by the following lemma, which
15
generalizes the one-dimensional case and assures that the firms have incentives to
play the collusive action profile s∗.
Lemma 4. Under Assumptions 3 and 4, if (K, λ) satisfies condition 1 in Propo-
sition 3, then
s∗i ∈ arg maxsi
E
[∑m
πi,m −∑m
∑j 6=i
λij,m max (Kij,m − πj,m, 0)
∣∣∣∣si, s∗−i
]. (12)
Proof. See Appendix.
4.2 Implementing general action profiles
One distinguishing feature of our enforcement mechanism, which implements that
action profile s∗, is to exploit the fact that s∗ maximizes the cartel profit in design-
ing penalty. A natural question, then, is whether it can be extended to implement
more general action profiles that are less efficient. This is important because, when
the discount factor δ is not sufficiently close to one and the efficient profile s∗ is
not implementable, some less efficient action profiles may still be implementable.
In this subsection we address this question by first outlining in general what ac-
tion profiles could be implemented and how our mechanism should be extended to
accomplish this and then characterizing by an example the set of action profiles
that are implementable for a fixed discount factor.
Definition 5. An action profile s = (s1, s2, . . . , sn) is supportable if, for each i,
there exists a weight θi = (θi1, θ
i2, . . . , θ
in) with θi
j ≥ 0 for each j 6= i, θii > 0, and∑n
j=1 θij = 1, such that
E
[ n∑j=1
θijπj
∣∣∣∣ s] ≥ E
[ n∑j=1
θijπj
∣∣∣∣ s′i, s−i
]for each s′i.
Remark 1. By setting θij = 0 if j 6= i and θi
i = 1, for each i and each j, each Nash
equilibrium is supportable. By setting θ1 = θ2 = · · · = θn, each efficient action
profile is supportable.
16
Suppose that (θ1, θ2, . . . , θn) is a profile of weights that support an action profile
s. We still set each firm i’s continuation payoff at end of each period to be
wi(π) = vi −∑j 6=i
λij max (Kij − πj, 0) (1− δ)δ−1. (13)
Note that∫ πi
πi
πi dFi (πi, s′i, s−i)−
∑j 6=i
λij
∫ Kij
πj
(Kij − πj) dFj (πj, s′i, s−i)
=
∫ πi
πi
πi dFi (πi, s′i, s−i) +
∑j 6=i
(θii)−1θi
j
∫ πj
πj
πj dFj (πj, s′i, s−i)
−∑j 6=i
(θii)−1θi
j
∫ πj
πj
πj dFj (πj, s′i, s−i)−
∑j 6=i
λij
∫ Kij
πj
(Kij − πj) dFj (πj, s′i, s−i) .
Therefore, if we set the constants λij such that
(θii)−1θi
j
∫ πj
πj
∂Fj (πj, s)
∂s′idπj = λij
∫ Kij
πj
∂Fj (πj, s)
∂s′idπj,
then similar arguments can be used to show that, given such reward functions wi,
it is optimal for firms to follow the action profile s, and the results in Section ??
still hold.
Example: Bertrand competition with linear demand and additive
shock. For illustration we consider a special case of the Bertrand competition
discussed in Section 2. Suppose that there are two firms and demand of each firm
i takes the form
qi = q̂i(p1, p2) + εi = 1− pi + αpj + εi.
where α ∈ (0, 1) and the additive shock εi is uniformly distributed on the interval
[−1/2, 1/2]. For simplicity we set the constant marginal cost ci to be zero.
We want to characterize the sets of price pairs (p1, p2) that are supportable
and, for each supportable price pair, the pairs of weights that support it. Suppose
that θi = (θi1, θ
i2), where i = 1, 2, satisfies θi
i > 0, θij ≥ 0, and θi
i + θij = 1.
For p1 to maximize
E[θ11p1q1 + θ1
2p2q2] = θ11p1 (1− p1 + αp2) + θ1
2p2 (1− p2 + αp1) ,
17
we must have
θ11 − 2θ1
1p1 + αp2 = 0.
Similarly, for p2 to maximize E[θ21p1q1 + θ2
2p2q2], we must have
θ22 − 2θ2
2p2 + αp1 = 0.
Solving for θ11 and θ2
2 yields
θ11 =
αp2
2p1 − 1and θ2
2 =αp1
2p2 − 1.
The constraints 0 < θ11 ≤ 1 and 0 < θ2
2 ≤ 1 translate into
1− 2p1 + αp2 ≤ 0 and 1− 2p2 + αp1 ≤ 0. (14)
The nonnegativity constraints q1 ≥ 0 and q2 ≥ 0 require that
1− p1 + αp2 ≥ 0, and 1− p2 + αp1 ≥ 0. (15)
Therefore the set of supportable pairs (p1, p2) are those in the quadrangle
bounded by the four inequalities in (14) and (15), and each such (p1, p2) is sup-
portable by the pair of weight pair
θ1 =
(αp2
2p1 − 1, 1− αp2
2p1 − 1
)and θ2 =
(1− αp1
2p2 − 1,
αp1
2p2 − 1
).
A Illustration of Assumption 2
A.1 Proof of Lemma 1
Suppose that the random demand shock εi1 has a distribution Gi1 on a finite
support [εi1, εi1] with εi1 ≥ 0, and conditional on εi1, the shock εi2 has a smooth
distribution Gi2(·|εi1) on a finite support [εi2, εi2].
We use πi to denote the profit of firm i. Given the prices p and the noises εi1
and εi2, the profit πi can be calculated to be
πi = (pi − ci) qi(p, εi1, εi2).
18
Therefore πi has the distribution
Fi (πi, p) ≡∫ εi1
εi1
Gi2
(h−1
(πi
pi − ci
)− εi1q̂i (p)
∣∣∣∣ εi1
)dGi1 (εi1) .
That Fi satisfies Assumption 1 is clear as q̂i is increasing in pj for each j 6= i. To
see that Fi also satisfies Assumption 2, we fix a j different from i and differentiate
Fi with respect to pj to get
∂Fi
∂qj
= −∫ εi1
εi1
gi2
(h−1
(πi
pi − ci
)− εi1q̂i (p)
∣∣∣∣ εi1
)εi1
∂q̂i
∂pj
dGi1 (εi1) ,
where gi2 denotes the density function of εi2. Now, integrating ∂Fi/∂qj over the
truncated profits and changing the order of integrations, we obtain∫ K
πi
∂Fi
∂qj
dπi
=−∫ K
πi
∫ εi1
εi1
gi2
(h−1
(πi
pi − ci
)− εi1q̂i (p)
∣∣∣∣ εi1
)εi1
∂q̂i
∂pj
dGi1 (εi1) dπi
=− ∂q̂i
∂pj
∫ εi1
εi1
εi1
∫ K
πi
gi2
(h−1
(πi
pi − ci
)− εi1q̂i (p)
∣∣∣∣ εi1
)dπi dGi1 (εi1) .
Next we proceed separately. First suppose that the function h is linear and
h (εi1q̂i (p) + εi2) = k (εi1q̂i (p) + εi2) + l,
where k and l are constants and k > 0. Then∫ K
πi
∂Fi
∂qj
dπi
=− ∂q̂i
∂pj
∫ εi1
εi1
εi1
∫ K
πi
gi2
(1
k
(πi
pi − ci
− l
)− εi1q̂i (p)
∣∣∣∣ εi1
)dπi dGi1 (εi1)
=− ∂q̂i
∂pj
k(pi − ci)
∫ εi1
εi1
εi1Gi2
(1
k
(K
pi − ci
− l
)− εi1q̂i (p)
∣∣∣∣ εi1
)dGi1 (εi1) ,
and ∫ πi
πi
∂Fi
∂qj
dπi = − ∂q̂i
∂pj
k(pi − ci)
∫ εi1
εi1
εi1 dGi1 (εi1) .
19
So we have the following expression for the ratio of the relative effects of a marginal
change in pj,∫ K
πi
∂Fi
∂qjdπi∫ πi
πi
∂Fi
∂qjdπi
=
∫ εi1
εi1εi1Gi2
(1k
(K
pi−ci− l)− εi1q̂i (p)
∣∣∣ εi1
)dGi1 (εi1)∫ εi1
εi1εi1 dGi1 (εi1)
,
which is decreasing in pj as q̂i (p) is increasing in pj.
Now suppose, instead, that εi1 is a constant and the density gi2 is log-concave.
In this case we have∫ K
πi
∂Fi
∂qjdπi∫ πi
πi
∂Fi
∂qjdπi
=
∫ K
πigi2
(h−1
(πi
pi−ci
)− εi1q̂i (p)
)dπi∫ πi
πigi2
(h−1
(πi
pi−ci
)− εi1q̂i (p)
)dπi
.
To see that this ratio is weakly decreasing in pj, it suffices to show that
R ≡
∫ K
πigi2
(h−1
(πi
pi−ci
)− εi1q̂i (p)
)dπi∫ πi
Kgi2
(h−1
(πi
pi−ci
)− εi1q̂i (p)
)dπi
is weakly decreasing in ph. We differentiate it with respect to pj to get
dR
dpj
= −εi1∂q̂i
∂pj
∫ K
πigi2 dπi∫ πi
Kgi2 dπi
·
∫ K
πig′i2 dπi∫ K
πigi2 dπi
−∫ πi
Kg′i2 dπi∫ πi
Kgi2 dπi
,
where, to simplify notation, we omit the argument of gi2.
Note that ∂q̂i/∂pj ≤ 0 and εi1 ≥ 0, to have dR/dpj ≥ 0, it suffices that∫ K
πig′i2 dπi∫ K
πigi2 dπi
≥∫ πi
Kg′i2 dπi∫ πi
Kgi2 dπi
.
By denoting
γ1(πi) =gi2∫ K
πigi2 dπi
and γ2(πi) =gi2∫ πi
Kgi2 dπi
,
we can rewrite the above inequality as∫ K
πi
g′i2gi2
γ1(πi) dπi ≥∫ πi
K
g′i2gi2
γ2(πi) dπi,
which follows from the fact that h−1 is increasing (since h is), and that g′i2/gi2 is
decreasing (since gi2 is log-concave).
20
A.2 Cournot Competition
Suppose now that the firms are competing in quantity. Each firm i decides its
level of production qi ∈ [0, q] and has a cost function ci(·). Let q ≡ (q1, . . . , qn) be
a profile of production decisions. Firm i’s stochastic inverse demand is
pi (q, εi1, εi2) = h (εi1p̂i (q) + εi2) ,
where h is strictly increasing, εi1 and εi2 are random shocks, and p̂i (q) is positive
for all q, decreasing in qj for each j. The random demand shock εi1 has a distri-
bution Gi1 on a finite support [εi1, εi1] with εi1 ≥ 0, and conditional on εi1, the
shock εi2 has a smooth distribution Gi2(·|εi1) on a finite support [εi2, εi2].
The profit of firm i, given q, εi1, and εi2, can be calculated to be
πi = pi (q, εi1, εi2) qi − ci (qi) ,
and has the distribution
Fi (πi, q) ≡∫ εi1
εi1
Gi2
(h−1
(πi + ci (qi)
qi
)− εi1p̂i (q)
∣∣∣∣ εi1
)dGi1 (εi1) .
The following result can be proven similarly.
Lemma 5. Each distribution function Fi satisfies Assumptions 1 and 2 (with
sj = −qj for each j) if either (1) h is linear or (2) εi1 is a constant and the
density of εi2 is log-concave.
B Proof of Proposition 2
By the assumption that κ1 < |∂Fj (πj, s∗) /∂si| < κ2, we have
λij =
∫ πj
πj
∂Fj(πj ,s∗)
∂sidπj∫ Kij
πj
∂Fj(πj ,s∗)∂si
dπj
≤πj − πj
Kij − πj
κ2
κ1
.
Let ε be the smallest real number such that
Pr (|πi − E[πi|s]| ≥ ε | s) ≤ ε,
21
for each i and each s. We set
Kij = E[πj|s∗] + ε.
Since ∫ Kij
πj
Fj (πj, s∗) dπj =
∫ Kij−2ε
πj
Fj (πj, s∗) dπj +
∫ Kij
Kij−2ε
Fj (πj, s∗) dπj
≤(Kij − 2ε− πj)ε + 2ε ≤ (Kij − πj + 2)ε,
we have ∑j 6=i
λij
∫ Kij
πj
Fj (πj, s∗) dπj ≤
∑j 6=i
πj − πj
Kij − πj
κ2
κ1
(Kij − πj + 2)ε,
which tends to zero as ε tends to zero.
Therefore condition 2 holds for all δ sufficiently close to one and ε sufficiently
close to zero. This also implies that the efficiency loss
∑i
(E[πi|s∗]− vi) =∑
i
∑j 6=i
λij
∫ Kij
πj
Fj (πj, s∗) dπj
tends to zero as ε tends to zero.
C Proof of Lemma 4
Given the constants λij,m satisfying
λij,m
∫ Kij,m
πj,m
∂Fj,m (πj,m, φj,m (s∗))
∂φj,m
dπj,m =
∫ πj,m
πj,m
∂Fj,m (πj,m, φj,m (s∗))
∂φj,m
dπj,m,
(16)
we need to show that, for each firm i, the collusive action s∗i maximizes
∑m
∫ πi,m
πi,m
πi,m dFi,m
(πi,m, φj,m
(si, s
∗−i
))−∑j 6=i
∑m
λij,m
∫ Kij,m
πj,m
(Kij,m − πj,m) dFj,m
(πj,m, φj,m
(si, s
∗−i
)).
22
Again, from the assumption that s∗ maximizes the total industrial profits, it
follows that s∗i also maximizes the total industrial profits, given that the other
firms are following s∗−i. Therefore, it suffices to show that, for each si,
−∑j 6=i
∑m
λij,m
∫ Kij,m
πj,m
(Kij,m − πj,m) dFj,m (πj,m, φj,m (s∗))
+∑j 6=i
∑m
λij,m
∫ Kij,m
πj,m
(Kij,m − πj,m) dFj,m
(πj,m, φj,m
(si, s
∗−i
))≥∑j 6=i
∑m
∫ πj,m
πj,m
πj,m dFj,m (πj,m, φj,m (s∗))
−∑j 6=i
∑m
∫ πj,m
πj,m
πj,m dFj,m
(πj,m, φj,m
(si, s
∗−i
)).
(17)
Using integration by parts, we can rewrite this as
∑j 6=i
∑m
λij,m
∫ Kij,m
πj,m
(Fj,m (πj,m, φj,m (s∗))− Fj,m
(πj,m, φj,m
(si, s
∗−i
)))dπj,m
≤∑j 6=i
∑m
∫ πj,m
πj,m
(Fj,m (πj,m, φj,m (s∗))− Fj,m
(πj,m, φj,m
(si, s
∗−i
)))dπj,m.
(18)
Define sti = ts∗i + (1 − t)si and note that φj,m
(st
i, s∗−i
)is quasi-concave in t.
Consider the following two cases separately.
Case 1. Suppose that φj,m
(si, s
∗−i
)≥ φj,m (s∗). Then, since φj,m
(st
i, s∗−i
)is
quasi-concave in t, either φj,m
(st
i, s∗−i
)is decreasing in the interval [0, 1] (from
φj,m
(si, s
∗−i
)to φj,m (s∗)), or it is first increasing and then decreasing. In the
second case there must exist some t0 such that φj,m
(st0
i , s∗−i
)= φj,m
(si, s
∗−i
)and
φj,m
(st
i, s∗−i
)is decreasing in the interval [t0, 1]. Note that t0 = 0 also has these
properties in the first case.
Therefore in both cases we have
φj,m (s∗) ≤ φj,m
(st
i, s∗−i
)≤ φj,m
(st0
i , s∗−i
)= φj,m
(si, s
∗−i
)for each t ∈ [t0, 1]. Then, for each t ∈ [t0, 1], we have, by the assumption of
23
quasi-concavity,
∑m
∂φj,m
(st
i, s∗−i
)∂si,l
(s∗i,l − si,l) = −1
t∇φj,m
(st
i, s∗−i
)·(si − st
i
)≤ 0, (19)
and, by Assumption 3,
λij,m
∫ Kij,m
πj,m
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
dπj,m
−∫ πj,m
πj,m
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
dπj,m ≥ 0. (20)
Hence, from φj,m
(st0
i , s∗−i
)= φj,m
(si, s
∗−i
), we have
λij,m
∫ Kij,m
πj,m
(Fj,m (πj,m, φj,m (s∗))− Fj,m
(πj,m, φj,m
(si, s
∗−i
)))dπj,m
−∫ πj,m
πj,m
(Fj,m (πj,m, φj,m (s∗))− Fj,m
(πj,m, φj,m
(si, s
∗−i
)))dπj,m
=λij,m
∫ Kij,m
πj,m
(Fj,m (πj,m, φj,m (s∗))− Fj,m
(πj,m, φj,m
(st0
i , s∗−i
)))dπj,m
−∫ πj,m
πj,m
(Fj,m (πj,m, φj,m (s∗))− Fj,m
(πj,m, φj,m
(st0
i , s∗−i
)))dπj,m.
The right-hand side, by rewriting the integrand, is,
λij,m
∫ Kij,m
πj,m
∫ 1
t0
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
∑m
∂φj,m
(st
i, s∗−i
)∂si,l
(s∗i,l − si,l)dt dπj,m
−∫ πj,m
πj,m
∫ 1
t0
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
∑m
∂φj,m
(st
i, s∗−i
)∂si,l
(s∗i,l − si,l)dt dπj,m
or, by Fubini’s theorem, is∫ 1
t0
∑m
∂φj,m
(st
i, s∗−i
)∂si,l
(s∗i,l − si,l)
(λij,m
∫ Kij,m
πj,m
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
dπj,m
−∫ πj,m
πj,m
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
dπj,m
)dt,
which is less than or equal to zero by (19) and (20). This proves (17).
24
Case 2. Suppose that φj,m
(si, s
∗−i
)≤ φj,m (s∗). Then, since φj,m
(st
i, s∗−i
)is
quasi-concave in t, either φj,m
(st
i, s∗−i
)is increasing in the interval [0, 1] (from
φj,m
(si, s
∗−i
)to φj,m (s∗)), or it is first increasing and then decreasing. In the
second case there must exist some t0 such that φj,m
(st0
i , s∗−i
)= φj,m
(si, s
∗−i
)and
φj,m
(st
i, s∗−i
)is increasing in the interval [0, t0]. Note that t0 = 1 also has these
properties in the first case.
Therefore φj,m
(st
i, s∗−i
)≤ φj,m
(st0
i , s∗−i
)= φj,m (s∗) for each t ∈ [0, t0]. Then,
for each t ∈ [0, t0], we have, by the assumption of quasi-concavity,∑m
∂φj,m
(st
i, s∗−i
)∂si,l
(s∗i,l − si,l) =1
1− t∇φj,m
(st
i, s∗−i
)·(s∗i − st
i
)≥ 0,
and, by Assumption 3,
λij,m
∫ Kij,m
πj,m
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
dπj,m
−∫ πj,m
πj,m
∂Fj,m
(πj,m, φj,m
(st
i, s∗−i
))∂φj,m
dπj,m ≤ 0.
With these results (17) can be similarly proven as in Case 1.
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