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working paper
department
of
economics
Optimal Collusion With Private Information
Susan Athey
Kyle Bagwell
No. 99-17
October 1999
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
•_-_..-
WORKING PAPER
DEPARTMENT
OF ECONOMICS
Optimal Collusion With Private Information
Susan Athey
Kyle Bagwell
No. 99-17
October 1999
MASSACHUSETTS
INSTITUTE OF
TECHNOLOGY
50
MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
99-17
ABSTRACT
We consider an
infinitely-repeated Bertrand game, in which prices are perfectly
observed and each firm receives a privately-observed, i.i.d. cost shock in each period. In
our base model, side-payments are prohibited. In order for a cartel to achieve
productive efficiency, firms that receive a high cost draw in the current period must be
induced to give up market share. To provide this incentive, firms may need to reduce
current prices, make compensating adjustments in future market shares or allow for future
price wars. In the most profitable collusive schemes, firms implement productive
efficiency, but they do not resort to low prices or future price wars; instead, firms with bad
cost draws may be favored with higher expected market share in future periods. If types
are discrete, there exists a discount factor strictly less than one above which first-best
profits can be attained purely through history-dependent reallocation of market share
between
equally-efficient firms.
We provide further characterizations of equilibrium play
as well as several computational examples.
We next consider the role of three characteristics of the institutional environment.
First,
we
examine the costs and benefits to the cartel of explicit communication about cost types
in a given period. We show that firms may find it beneficial to communicate after some
histories but not others. Second, we show that if firm identities cannot be tracked over
time (so that firm-specific future market-share favors are unavailable), the best collusive
scheme sacrifices all productive efficiency. Third, we examine the role of explicit sidepayments, which may entail inefficiencies if they are illegal and bear the risk of
detection. Unless side-payments are perfectly efficient, optimal collusive equilibria are
non-stationary and thus involve the use of future market-share favors.
_-—trrepffHMsTffUTE
Optimal Collusion with Private Information
Susan Athey and Kyle Bagwell
May, 1998. This Draft: September, 1999.
First Draft:
Abstract:
We
consider an infinitely-repeated Bertrand game, in which prices are
perfectly observed and each firm receives a privately-observed,
i.i.d.
cost shock in
each period. In our base model, side-payments are prohibited. In order for a cartel to
achieve productive efficiency, firms that receive a high cost draw in the current period
must be induced to give up market
to reduce current prices,
share.
To provide
this incentive, firms
make compensating adjustments
in future
may need
market shares or
allow for future price wars. In the most profitable collusive schemes, firms implement
productive efficiency, but they do not resort to low prices or future price wars; instead,
bad cost draws may be favored with higher expected market share
firms with
in future
periods. If types are discrete, there exists a discount factor strictly less than one above
which
first-best profits
can be attained purely through history-dependent reallocation
of market share between equally-efficient firms.
We
provide further characterizations
of equilibrium play as well as several computational examples.
We
next consider the role of three characteristics of the institutional environment.
First,
we examine the
costs
cost types in a given period.
after
some
histories
and benefits to the
communication about
cartel of explicit
We show that firms may find it beneficial to communicate
but not others. Second, we show that
if
firm identities cannot be
tracked over time (so that firm-specific future market-share favors are unavailable),
the best collusive scheme sacrifices
role of explicit side-payments,
all
productive efficiency. Third, we examine the
which may
entail inefficiencies
if
bear the risk of detection. Unless side-payments are perfectly
lusive equilibria are non-stationary
illegal
and
optimal
col-
they are
efficient,
and thus involve the use of future market-share
favors.
JEL
Classification
Numbers: C73, L13, L41.
Keywords: Collusion, cartel design, repeated games, private information, mechanism
design, auctions, information-sharing, communication.
*M.I.T. and
NBER
(Athey) and Columbia University and
NBER
providing detailed comments on an earlier draft.
Mullin, David Pearce, Lars Stole, Jean Tirole
Fabra), Berkeley, Federal Reserve
We would like
Bank Board
We
1997-1998,
We
are espe-
thank the
NSF
Ellison,
at Barcelona
UC-San
Game Theory
Wally
(Pompeu
Diego, the
Conference
Summer
for helpful
(Athey: SBR-9631760; Bagwell: SES-9905460) for generous
Athey thanks the Cowles Foundation
when much
thank Glenn
of Governors, Harvard, M.I.T., Minneapolis
Conference on I.O. in Toronto, and the Stony Brook
financial support.
to
and seminar participants
Federal Reserve Bank, S.I.T.E., Stanford, Toulouse (I.D.E.I.),
discussions.
(Bagwell).
Pavel Grigoriev for exceptional research assistance, and to Eric Maskin for
cially grateful to
of this research
was conducted.
at Yale University for hospitality in
Introduction
1.
we
In this paper,
consider an infinitely-repeated Bertrand game, in which each firm
privately informed of
and time.
across firms
i.i.d.
unit cost level in each period,
its
We
and the unit-cost
realization
is
is
thus represent collusion in the context of a repeated
hidden-information model with publicly-observed actions (prices). 1
Building on work by
Abreu, Pearce and Stacchetti (1990) and Fudenberg, Levine and Maskin (1994), we use
the tools of dynamic programming to recast the characterization of optimal Perfect Public
Equilibria
firms are
(PPE)
in terms of a static
drawn from a
restricted set (namely, the set of
who
we show how
firms
self-enforcing
scheme where,
another, the collusive
mechanism design problem,
are prohibited from
in place of
mechanism
an
PPE
in
which transfers across
continuation values). Thus,
making side-payments can
explicit
specifies that
still
implement a
monetary transfer from one firm to
one firm
is
favored over another in future
play.
Our
analysis
ent market
and
is
motivated by the question of how collusive conduct varies across
legal environments. In particular, the design
policies vary widely across countries
and
industries,
and enforcement of anti-trust
and the
different manifestations of
anti-trust policies naturally affect the organization of collusive activity.
environment
is
permissive, then firms
may
set
differ-
up industry associations
2
in
If
the anti-trust
which they con-
firm the terms of their agreement, communicate about current conditions, keep records of
past experiences and perhaps even exchange side-payments.
environments, firms
may
still
may communicate
3
In less permissive anti-trust
as to current conditions, but the conversations
occur surreptitiously, in "smoke-filled rooms," and explicit monetary transfers
avoided. Finally, a vigorous anti-trust policy might even deter
among
tion
firms.
This latter possibility
holds that communication
While anti-trust
1
Our model
among
is
some forms
of
may be
communica-
the evident aim of current U.S. policy, which
firms as to current prices
is
4
a per se violation.
policies clearly influence the organizational structure of the collusive
of hidden information can be contrasted with the hidden-action
model of Green and Porter
(1984) and Abreu, Pearce and Stacchetti (1986), as well as the model of observable
demand
fluctuations
pioneered by Rotemberg and Saloner (1986).
2
Levinsohn (1995) describes the significant variation in anti-trust law and enforcement that is found
Significant variation also occurs within countries and across industries; for example, in
across countries.
many
3
countries, the legal stance toward cartels
is
more
tolerant in export industries.
is often permissive in Japan, and
McMillan (1985) discusses that colluding firms there sometimes form "dangos" within which communication
and even side-payments are found. As Stocking and Watkins (1946) detail, U.S. anti-trust policy was also
relatively permissive in the first part of the 20th century, and industry associations in which firms shared
information and records, and exchanged side-payments, were then common. Sophisticated cartels of this
nature were found in the steel, aluminum, incandescent electric lamp, and sugar (as Genesove and Mullin
For example, anti-trust policy with respect to collusive conduct
(1998, 1999) describe) industries.
4
However,
it
is
not illegal to share production plans; see, for example, Doyle and Snyder's (1998)
analysis of information sharing in the
motor vehicle industry.
endeavor, the implications for market conduct are more subtle.
all,
For example, how,
does the ability to communicate influence prices, market shares and profits?
might these variables be affected,
if
one another (perhaps at some cost,
firms were able as well to
if
side-payments are
make
illegal
at
And how
direct side-payments to
but detection occurs with
some intermediate probability)?
Practically, a theory that addresses such questions
be useful in two main respects.
First,
it
if
might
could provide a lens through which to interpret
observed (historic or current) collusive conduct in terms of the surrounding market and
environment. Second,
legal
it
might provide a framework with which to better anticipate
the possible consequences of a change in anti-trust policies for the conduct of collusive
firms.
5
Motivated by these considerations, we analyze in this paper optimal collusion in a
range of possible environments. To understand our findings,
tradeoffs that confront the legalized cartel,
which uses
payments) that are enforced by binding contracts.
cartel
that production
is
Cramton and Palfrey
is
(1990)
it is
explicit
An
helpful first to recall the
monetary transfers
important consideration
allocated efficiently over cartel members.
As Roberts
(side-
for
the
(1985),
and Kihlstrom and Vives (1992) have shown, when firms are
privately informed as to their respective costs of production, productive efficiency requires
communication and transfers between members. Communication enables firms to establish
before production the identity of the lowest-cost firm, while transfers (from this firm to the
other cartel members) ensure that firms have the incentive to communicate truthfully.
Outside of a legalized
cartel,
however, the collusive relationship must be self-enforcing.
In addition, the collusive environment
may
effect transfers
collusive
may
from one to another.
restrict the set of
instruments with which firms
Thus, we characterize optimal self-enforcing
conduct among privately-informed firms operating
in collusive
environments that
are distinguished on the basis of restrictions on the instruments available to the firms. In
our base model, we make the following assumptions:
regard to current cost conditions;
(ii).
(i).
firms can "communicate" with
firms can use "market-share favors," which requires
that the identities of individual firms can be followed over time, and whereby individual
firms are treated asymmetrically as a reward or punishment for past behavior;
(iii).
firms
can not make explicit monetary transfers (use "bribes" ). After studying this model in some
detail,
we then analyze
restrictions
Our
is
the
way
in
which optimal collusion changes as each of the three
imposed or removed.
^
characterization of the optimal collusive
mechanism
in the base
model highlights
the central role played by market-share favors: the prospect of a future favor can provide
a high-cost firm with the incentive to truthfully acknowledge
low market share
5
in the present.
This consideration
is
We show
that
when such
its
current state
and accept a
favors are possible, the scope for
perhaps of some special significance in the current environment, as new instiEuropean Union and the former communist countries, and further as
tutions are being developed in the
countries increasingly raise competition-policy concerns in their trade-policy negotiations.
self- enforcing
collusion
can be induced as
is
significant. Indeed,
follows: if
critical
firms are sufficiently patient, truth-telling
a firm announces low cost in the present,
market share only in the event of
we compute a
if
forgoes
it
where both firms are equally
"ties,"
some future
efficient.
Formally,
discount factor (strictly less than one) above which the cartel can
In essence, the cartel uses asymmetric future
achieve first-best profits in every period.
market-share assignments to duplicate the transfer patterns that would be employed by a
legalized cartel, without
When
efficiency loss.
firms are less patient, however,
favors. In particular,
when
the disadvantaged firm
of ties.
any
When
(in particular,
its
implement market-share
costly to
the firms attempt to implement a scheme that favors one firm,
may
not be willing to give up enough market share in the event
assigned market share
when both
may be
it
is
too low following a particular cost realization
may be tempted
firms realize low cost), the disadvantaged firm
to undercut the collusive price
and thereby capture the
war may subsequently ensue).
In order to mitigate this temptation, firms
entire
market (though a price
may
find
it
necessary to lower prices, in order to diminish the gain from undercutting the collusive
price. Alternatively, the
scheme may require the disadvantaged firm to give up some market
share in the state in which
efficient
it is
most
Intuitively,
efficient.
under a scheme that
calls for
production across firms, the disadvantaged firm has no incentive to undercut the
collusive price in the state of the world
when
serving the entire market. Thus,
relatively easy to induce the disadvantaged firm to
give
up a
At the
little
market share
start of the
game,
it is
it is
most
efficient, since it
already anticipates
in that state.
less patient firms anticipate
the costs associated with using
future market-share favors to provide incentives for truthful revelation.
They then weigh
the disadvantages of future market-share favors against the productive efficiency gains
from truth-telling
up productive
in the present.
efficiency rather
One theme
of our analysis
is
that such firms
may
give
than lowering prices in the present or bearing large future
inefficiency.
In addition to our theoretical characterizations,
several examples that illustrates the
The computation
manner
in
we
offer
a numerical computation of
which collusive conduct unfolds over time.
highlights an empirical implication of our model: optimal collusion gen-
erates negative correlation over time in the market shares of firms.
referring to our theoretical
and computational
findings,
we
Later in the paper,
discuss further the extent to
which the predictions of our model can be linked to real-world case studies and empirical
applications.
We next turn to evaluate each of our three assumptions about
We
begin with the role of communication (assumption
(i)).
the collusive environment.
In our model,
allows firms to smoothly divide the market on a state-contingent basis.
nication, firms
must allocate market share
in
communication
Without commu-
a decentralized fashion, thereby limiting the
range of market-sharing plans available. Thus, we show that
to choose plans that require communication.
At the same time, we
moderate patience may choose to avoid communication
Accordingly,
its price.
if
in
know
firms do not communicate, a given firm does not
chooses
often profitable for firms
it is
some
its
also find that firms of
periods. Intuitively,
when
opponent's cost type when
it
the opponent's price varies with cost, then the firm also
does not know the exact price that
its
opponent
will choose.
Uncertainty about the op-
ponent's price can, in turn, diminish the incentive that the firm has to "cheat" and select
a slightly lower price. Therefore, while communication can generate productive efficiency
gains along the equilibrium path,
it
may
also heighten the incentive of
an impatient firm to
deviate from the collusive agreement. Firms thus choose to avoid communication in
periods and use
Our
it
to allocate market share in others.
finding that firms sometimes, but not always, choose to
rally to the question of
We
communication.
how
collusive behavior changes
show that even
count factor strictly
less
in the absence of
than one above which
In particular,
we
by reducing productive
mal rationale
Next,
we
assumption
efficiency
7
may have
further,
it
anti-trust enforcement prohibits
communication, there exists a
first-best profits
may respond
without lowering
for the "smoked-filled
with collusive ventures;
conversations
find that such firms
if
communicate leads natu-
can be achieved.
on communication can diminish
firms of moderate patience, however, restrictions
profits.
some
prices.
6
dis-
For
collusive
to communication restrictions
Thus, our theory
room" conversations that traditional
offers
a
for-
I.O. associates
suggests that anti-trust enforcement that limits such
perverse welfare implications.
consider the ability of firms to track firm-level behavior over time, modifying
(ii)
to suppose that firms face informational or coordination limitations which
preclude rewarding an individual firm's current behavior with more favorable treatment in
the future. Such limitations arise, for example,
when
where only the amounts of the bids (but not the
Individual behavior
ent identities, or
6
also difficult to chronicle,
if
identities of the bidders) are publicized.
firms can exit
and
participants use agents to conduct business (as
is
re-enter under differ-
common
in auctions).
This finding hinges on the assumptions that goods are perfect substitutes and firms compete in prices.
More
7
if
is
firms interact in an auction market
generally, restricting
communication lowers both prices and productive
efficiency.
We may contrast the role of communication in our model with alternative roles identified in the previous
literature.
First,
Compte
(1998)
and Kandori and Matsushima (1998) consider repeated games
in
which
players have private information about the history of play, arguing that public communication can then
constitute a state variable that facilitates the application of recursive techniques. In our model, by contrast,
over time and firms' choices (prices) are public, and so the game already has
McCutcheon (1997) provides an alternative interpretation of communication,
the relationship between communication and the renegotiation of the equilibrium. Finally, in the
information sharing literature (e.g., Vives (1984) and Shapiro (1986)), firms commit to share or
the private information
is i.i.d.
a recursive structure. Second,
involving
oligopoly
not share information prior to the play of a static oligopoly game. Incentive compatibility issues do not
arise in this literature, as any shared information is, by assumption, directly received or honestly relayed.
See
Kuhn and
Vives (1994) for a general survey of research that explores the competitive implications of
communication among
firms.
Collusion in such environments can be characterized using the solution concept of
metric
PPE, that
PPE
is,
basis, so that all firms
in
which collusion among firms proceeds on an industry-wide
move through
collusive
and war phases together. Athey, Bagwell
and Sanchirico (1998) provide a more complete analysis of symmetric collusion
with a continuum of types and downward-sloping demand.
type model) two of their findings.
symmetric collusion among
sacrifices
We
productive
off of
same
equilibria, firms charge the
efficiency.
8
We
confirm here
make
in a
model
(for
a two-
a wide range of parameter values, optimal
is
achieved using stationary equilibria,
the equilibrium path. Second, in the optimal symmetric
(high) price irrespective of their cost types,
Clearly,
communication
is
transfers
and the
cartel
not necessary for this scheme.
communication has no value among patient
therefore conclude that
firms are also able to
First, for
sufficiently patient firms
where price wars occur only
Sym-
firms, unless the
from one to another as through asymmetric future
market-share assignments.
Finally,
make
we examine assumption
bribes,
(iii)
by considering environments in which firms can
though these must be self-enforcing and may incur
inefficiencies, as
through
a risk of detection. Then, firms can potentially substitute current-period bribes for future
market-share favors. In practice, monetary transfers
may
occur directly, with one firm pay-
ing other firms for the right to produce. Bribes can be associated with
processes, too; for example, colluding suppliers
determines the firm that
is
to
may
9
a concern, so that bribes are not
is
hold a "knockout" auction that
win the procurement contract, and then
to ensure that this firm wins at a profitable price.
trust officials
first
more sophisticated
future market-share favors as a
means
We
rig the actual bids
show that when detection by
fully efficient, bribes
of transferring utility.
Put
anti-
never fully replace
differently, unless bribes
are perfectly efficient, firms strictly prefer to keep track of history, using non-stationary
equilibria that specify a future advantage to firms
who admit
high cost in the present.
Before proceeding, we pause to place this analysis in the context of recent methodological
We
advances in repeated game theory.
build directly on tools developed by Abreu,
Pearce and Stacchetti (1990) and Fudenberg, Levine and Maskin (1994). The latter paper discusses optimal
an
efficient
are impatient.
Sec also
in a repeated
game with
private information, establishing that
outcome can be achieved when firms are
of our analysis
8
PPE
is
10
that
we
Within
characterize optimal
this context,
Damania and Yang
(1998),
who
we
PPE
further
infinitely patient.
A
novel feature
with hidden information when firms
show that the optimal scheme can be
consider a class of symmetric collusion schemes for duopolists
with private information as to demand parameters.
9
Side-payments may also be implemented through a common-fund arrangement, whereby a cartel
member that exceeds its production quota contributes a penalty payment to the common fund, while
a member that operates below its quota receives a compensatory payment from the common fund. As
Stocking and Watkins (1946) describe, arrangements of this general nature appeared in the
and incandescent
10
electric
lamp
steel,
aluminum
cartels of the early 1900's.
Independently, Aoyagi (1998) analyzes an auction model where bidders
who win
in the current
period
characterized using tools developed for static
must
fers
mechanism design
theory, but
continuation values. Thus, in addition to computational analysis,
and comparative
ing the connections to the static literature.
PPE
motivated by the shape of the set of
satisfy particular kinds of restrictions
theoretical characterizations
where trans-
we provide a
rich set of
statics predictions, while formally highlight-
11
The Model
2.
For simplicity,
we
focus primarily on a highly stylized model with two firms and two cost
two
types. Formally,
customers with valuation
denoted
firms
1
=
-
and
{L,
H}
2,
produce perfect substitutes, and
Each firm has possible
r.
any pricing
realized costs prior to
its
and
firms, 1
decisions.
costs Oi
by 6
2 are given
=
and 9
6j
=
2
a unit mass of
and Oh and privately observes
Thus, the state space in any period
x {L, H}, and we index these states as
1
sell to
6k in state
(j,
(j, k).
k)
The
G
-
,
where the costs of
probability of the cost
draw j G {L, H} in any period is denoted Pt(9 = 6j) = rjj, where rjj > and r} L + r]H
this is assumed independent over time and across firms. To simplify the exposition
z
few of the
this
results,
assumption
we maintain the assumption
is
is
equal to
profit to
ne
tt
=
—
— 6 H ),
(Oh
—
a
1
sit
G
when
For each
while the low type mixes, receiving
=
each firm
A = {L,H,N};
is
O^VhIIl- This payoff can be contrasted with the first-best level of
fb
tt
\(r
- E[min(^
cannot make monetary side-payments.
(i)
will note
Thus, ex ante expected profit to each firm in this equilibrium
1
,^
2
)]).
In our basic repeated-game model, firms can
each period:
we
of a
Ol)vh, the expected profit from just undercutting the price charged
types.
each firm,
1/2; however,
12
a symmetric mixed strategy equilibrium.
firm, the high type charges price equal to cost (p
by the high-cost
>
1;
game (without any communication, com-
to the one-shot pricing
or transfer possibilities)
profit equal to (6h
tjl
=
used.
The Nash equilibrium
mitment
that
is
(iii)
i
observes
each firm
meet and communicate
their types but
Formally the firms play the following game in
its
type
l
;
(ii)
each firm
i
makes an announcement
l
i
then selects a price p and makes a market-share
out for a specified number of periods in the future. Athcy, Bagwell and Sanchirico (1998) also describe
differs by characterizing optimal asymmetric collusive
a similar asymmetric scheme. The present paper
schemes using dynamic programming.
11
Our work can be contrasted with the macroeconomics literature (e.g., Green (1987) or Atkeson and
Lucas (1992)) on repeated games with hidden information, which studies the game between a central
planner and a continuum of agents. Only a few papers consider small numbers of agents (i.e. Wang (1994),
and, independently, Cole and Kocherlakota (1998)), and typically the focus of these papers is on existence
methods rather than theoretical characterizations.
Given that we restrict attention to pure strategy equilibria in the repeated game, the lack of pure
strategy equilibria in the stage game creates some tension for the formulation of the model. However, our
emphasis on Parcto optimal equilibria implies that in the characterizations we highlight, the best equilibria
or computational
12
are pure.
p
proposal q
1
=
2
p = (p\p ) and q
(iv) for
;
2
(g\<?
then market shares are allocated according to
m *(P) q)
=
9*
^
+9 D
1
J
Q
1)
and
=
"i'(p, Q)
This description of the stage
and then
then q l
1
m
min(g
(p,q)
l
max(|,
,
r,
=
8
1
1;
and
—
J
g
))
=
p
if
0; if
=
l
r,
p < p3
p> U r, then
l
'
otherwise.
We
game warrants a few comments.
interpret this
make announcements concerning
as describing an environment in which firms meet,
respective cost types
p >
if
),
and make market-share proposals.
select prices
We
game
their
allow
may announce that it has low costs (L),
(H) and a firm may decide not to make an
each firm three possible announcements: a firm
may announce
a firm
that
We
announcement (N).
it
has high costs
include the latter option, since, while
and communicate, they are under no obligation to do
we permit
share proposals,
market shares when they
firms'
announced cost
market share
allocate
make proposals
firms to
set the
same
we
allow firms to meet
In our formalization of market-
so.
that jointly determine their respective
market-share proposals follow the
price. Since the
may
positions, our formalization allows that equally-priced firms
in a state-dependent fashion.
We
can imagine that consumers divide
themselves evenly between two equally-priced firms, unless one firm chooses not to
sell
additional units at the posted price, at which point the consumers go to the other firm.
We
do not allow, however, both firms to produce positive quantities
rules out the possibility that the firms divide the
charge different prices in each segment.
Our next
notation
is
goal
is
is
-
{a*
:
i
-
typical policy for firm
component
,
-> ,4} x {p*
is
i
denoted
follows:
i
-
:
6
=
1
A
l
l
->
K} x
=
{a
l
{tp*
l
(8
2
),
we
a(0)
=
a)
=
<p(0,
7
a-
is
:
l
),
the announcement function, the second
is
(9 ,9
x
s (9 ,a^)
the market-share proposal function and
letting
Although the
we begin with an abstract repre1
2
{L, H} and a = (a a ), we define the space of
=
%
from which a firm might choose as
i
first
example, geographically) and
(for
simplified in later sections, for completeness
S =
A
This
13
to formally define firm strategies for the stage game.
sentation of these strategies. Letting
policies
market
at different prices.
is
-
l
l
x
A
j
(9 ,a ),(p
->
l
3?}.
(9\a
:i
)},
where the
the pricing function, the third
firm j's realized announcement. Further,
define the following vectors:
(a
1
1
(
1
(9
p (9\a
(
2
),a (9
2
2
2
),
p(0,a)
));
2
1
p (9 ,a
(
));
= {p\0\a% p\9\o}))
s(0)
= (s\9\a 2 ),s 2 (9 2 ,a
1
))
A benefit of this stage game is that it offers a simple framework (e.g., all transactions occur at the
same price and so a rationing rule need not be specified) within which to allow for the possibility that firms
communicate and allocate market share in a state-contingent fashion. Of course, other assumptions are
possible concerning market-share division and the ability of firms to restrict output and set prices. Our
analysis below highlights how the results change if we modify this (somewhat ad hoc) description of market
13
behavior.
Notice that a policy vector s(0) determines announcements as well as the price and marketshare proposal responses to these announcements. Accordingly, a policy vector determines
may
a path through the stage game, and we
on a realization of cost types as
=
7p( S )
Now
7r'(s,
Eg[^(s,e)}.
The
consider notation for the repeated game.
discounted stream of profit, given the
firms' cost types are
A
0),with expected stage-game payoffs then given as
game described above, where each
stage
therefore write stage-game payoffs conditional
firms
firm has the objective of maximizing its expected
common
discount factor
announce costs types and then
further that,
As mentioned above, the
S.
across firms and over time, so that the environment
i.i.d.
novel feature of our model, formally described in the stage
Assume
meet each period to play the
game
above,
is
is
stationary.
that firms can
and make market-share proposals accordingly.
set prices
upon entering a period
of play, a firm observes only the history
of:
own cost draws, and (ii) realized announcements, prices and market-share proposals.
Thus, we assume that a firm does not observe rival types or rival policy functions sJ
We would like our dynamic game to have a recursive structure, so that dynamic(i) its
.
programming techniques may be
We
applied.
thus follow Fudenberg, Levine and Maskin
(1994) and restrict attention to those sequential equilibria in which firms condition only
on the history of realized announcements, prices and market-share proposals and not on
own
their
private history of types
and such sequential
strategies
familiar arguments,
any history
is itself
we may
a
PPE
and policy schedules. Such
strategies are called public
Using
equilibria are called perfect public equilibria (PPE).
verify for our
in the full
game
game and
that the continuation of any
PPE
after
yields payoffs in the continuation equal to
what would have been obtained had the strategy
profile
been used from the
start.
Formally, the public history of realized prices, announcements and market-share proposals
up to date
t is
histories at period
be a strategy
t
n
=
1,
1
{cr t
...,
oo.
{h t ))
.
t.
specified as follows: h t
A
strategy for firm
profile in period
t,
and
Then, given a history h
t
,
H
{a ,p ,qj. Let
(
t
t
in period t
i
let cr
=
is
denoted
be the
o\
:
set of potential
H
t
— S
l
>
Let
.
<r t
represent a sequence of such strategy profiles,
the expected per-period payoff in period
t
for firm
i is
Given the mapping from strategies to prices and market shares outlined above,
each strategy induces a probability distribution over game play in each period, resulting in
an expected payoff
for firm
%
i,
denoted as v (a)
=
t
E[Y^1\ 6
~1
i
7r
(a t (h t ))], where hi
is
the
null history.
To simplify our
the literature:
analysis,
we assume
we make a
final
assumption that
is
commonly invoked
in
that after every period, firms can observe the realization of
some public randomization device and
select continuation equilibria
on
this basis.
This
assumption has the consequence of convexifying the set of equilibrium continuation values
available at a particular point in time.
While we believe that
this
assumption
is fairly
innocuous, convexity of the set of continuation values plays an important role in some
Thus, below
parts of our analysis.
we analyze conditions under which
(in Section 4.3)
we do not introduce
convexity obtains without resorting to randomization. For simplicity,
randomization process.
explicit notation for the
we can now
Following Abreu, Pearce and Stacchetti (1986, 1990),
T(V) which
set.
define an operator
PPE values, V*, as the largest invariant, or "self-generating,"
S and v = {v ,v 2 ), the operator is defined as follows:
yields the set of
S= S x
2
1
Letting
x
(u\u 2 )
T(V)
S and v A 2 x
3 s G
:
for
=
i
=
1, 2,
and, for each
>
u*
3?
:
=
u*
i
and
+6Ev
f^s*,^')
i
i
[s
s
-» co(V) such that:
+ 6Ev*(B(0)y,
7f*(s)
l
4
=
(a l ,p
l
G S
l
,
(p
)
l
,
j
i
(8 ,d>(9 )),s>(6 &*(&))]
i
,
This operator effectively decomposes equilibrium play into two components, strategies
for
We
the current period and continuation values drawn from the convex hull of the set V.
record immediately a useful property of T:
Lemma
1.
T maps
compact
compact
sets to
and the
sets,
PPE
set of
values,
V*
is
,
compact.
Proof.
weak
All of the constraints entail
utility functions
compact; and
inequalities; the feasible set is
and the functions defining the constraints
are real-valued, continuous
and
bounded.
T
Given compactness, the operator
can also be used to compute the set of equilibrium
which
values, using the following algorithm. Begin with a set Vo
discounted value of
we could take
V
all
to
is
larger than the present
possible per-period realizations of utility for each firm. For example,
be
[0,
r/(l
-
8)}
x
[0,
r/(l
-
8)].
Then, construct
Vn+1 = T(Vn n>0.
),
Fudenberg, Levine and Maskin (1994) extend Abreu, Pearce and Stacchetti (1990) to show
that lim n _ 00
Vn =
V* the
,
set of equilibrium values v(<r) of the
game.
We
apply this
computational algorithm later in the paper.
Our
We
goal
is
to characterize the operator
T
and thereby the
set of equilibrium values.
are particularly interested in characterizing the Pareto frontier of this set, in order to
understand optimal collusive arrangements. From there,
it
will
be possible to back out the
strategies that generate the equilibrium values.
To present our
findings,
we
distinguish between two kinds of equilibria, based on the
quality of information transmission between firms. In an informative
PPE,
firms
employ
equilibrium strategies in which they always share their cost information with one another:
for all
i
Section
G
{1, 2}
and j G {L, H},
5, this restricts
the set of
additional requirement on
s.
unable) to communicate and
if
9
%
=
9j,
then
PPE, and we
Similarly, in
a
l
1
{9
)
= j.
As we
define the operator
discuss in
T
1
more
detail in
(V) to represent this
an uninformative PPE, firms are unwilling
make market-sharing
plans,
and we may capture
(or
this situation
by focusing upon
equilibria in
and j E {L,H}, a
requirement on
s.
=
l
(6j)
which firms never share cost information:
N.
We
for all i € {1,2}
U
use the operator T (V) to capture this additional
PPE
Informative and uninformative
the juxtaposition of these two classes of
PPE
are of independent interest, and
serves to highlight the benefits
and costs
of successful information sharing for colluding firms. Furthermore, the characterization of
such equilibria contributes to our understanding of the
equilibria of the unrestricted
some
3.
histories
and not
PPE
class
may
PPE
full
set,
V*, since optimal
involve informative communication following
others.
The Mechanism Design Approach
we consider the
In this section,
optimal informative
we
In addition,
PPE
class of informative
can be recast
in
PPE
and show that the search
for the
terms of a static mechanism design program.
establish the solution to this
program
in
two special
cases.
The
cases
correspond to familiar and easily interpretable assumptions in static mechanism design
(budget-balanced transfers, no transfers), and they provide useful benchmarks for a more
general analysis.
3.1.
We
Mechanism Notation and Incentive Constraints
rely
on an application of the revelation principle to simplify our characterization of
informative
PPE. Since the
the operator
T
1
,
set of informative
every utility vector
u
PPE
values,
V1
,
is
in the set of informative
the largest invariant set of
PPE
values has associated
current-period strategies and continuation- value functions, s and v, that yield expected
utility of u.
When
following these strategies, firms report their cost types truthfully
receive the corresponding prices
and market-share
allocations.
Our approach
is
and
to introduce
notation for such state-contingent prices, market-share allocations and continuation values,
and then formalize the corresponding incentive constraints that these must
satisfy in order
to be implementable as equilibrium play.
We
begin with a general description of the incentive constraints. In an equilibrium of
the repeated game, there are two kinds of deviations that a firm might entertain.
a firm with cost type 9
equilibrium specifies
l
may
when
its
consider a deviation in which
cost type
is
instead 6
l
it
First,
adopts the policy that the
j,
therefore sets this price
then firm
i's
price
may
if it
differ
makes
positive sales
from pjk, but
it
)
l
(i.e., if
qj k
snl
tascin
>
If
cannot be lower.
11
0).
firm
i
re pjk prevails. Firm i
makes no sales in state {j, k),
U*(L, L; z)
We
to
>
U*(H, L\
next formally represent the incentive that a firm has to charge a price not assigned
any cost type, that
to undertake
is,
an off-schedule deviation. In an informative PPE,
The
there are two types of off-schedule incentive constraints.
of a firm to deviate
from the assigned price
concerns the incentive
1
might
place. If
undercut firm
2's price
and capture the
satisfied, neither of these deviations
8{v)k
where v
tion,
l
=H
l
(V)
=
-
> max
l
)
inf{V
(q%(pjk
v G V}.
:
(We
as a
it
It is
parameter
As v
l
is
-
q)^ - p]k ))
6j),
will write v
the following constraint
l
(IC-Offljfc )
.
rather than v l (V) to conserve nota-
relative to the set of values
reached only
in the analysis.
If
might wish to
1
profitable:
and take the off-schedule constraints
in a particular context.)
treat
v
is
16
entire market.
both firms
above firm
like to price slightly
to avoid producing in that state; alternatively, at higher prices, firm
slightly
is
first
communication takes
after
are assigned a price less than firm l's cost, firm
2,
(IC-On^)
z)
is
we can
equilibrium path,
off of the
IC-Off2L
under consideration
essentially
defined analogously.
worth pausing to notice how the form of these constraints follows from our assump-
tions about
communication and timing. The constraints must hold state by
when communication among
firms
is
informative each firm knows
cost type (and thus the price) of its opponent.
when
state, since
setting
its
price the
These constraints can be contrasted with
the case without informative communication (analyzed in more detail below), whereby an
off-schedule price deviation
is
contemplated without knowledge of a
other assumptions about timing are also possible.
The second type
of off-schedule deviation
is
or (H,L). If firm
cost, in order to
high price.
draws a low
cost, firm 1
Such an undercut might be attractive
and firm
final decision to
U\L, L- z) >
in state (L, L) or (L,
Y,
Vk-
1
1
for exactly
one of firm
undercut. Deviations of this kind are dissuaded
BL )
follows.
H) than
might then undercut firm
can wait to learn the realization of firm
max (4(pffl -
course,
in
might be tempted to report a high
induce firm 2 to price high, so that firm
or perhaps for both,
making a
1
Of
an interim deviation, described as
Suppose that the collusive scheme assigns a lower price
(H,H)
rival's cost.
17
+ 6v\qHk (p Hk l
9L )
2's
2's cost types,
2's
type before
if:
+ 8v Hk )
l
,
(IC-Off-Ml)
ke{L,H}
16
Given that unit costs are constant in output, a firm best deviates by claiming all market share or
all market share. In either event, a small change in price serves the purpose. We therefore
need not concern ourselves with the possibility that a firm deviates by maintaining the price and adjusting
up or down its proposed market share.
17
For example, if the firms are bidding in an auction where only one firm can win, they might use a
randomization device to select the winner, where q* k is the probability i wins in state (j, k); then, the onrelinquishing
schedule constraints would be identical, but the off-schedule constraints would require that unless
player
i
must be
willing to "sit out" entirely
this more-restrictive alternative,
if
the randomization turns against her.
but the qualitative results are similar.
12
We
q'
jk
=
1,
do not analyze
M
where the
analogously.
is
It
mnemonic
The
for "misrepresentation."
constraint for firm 2
can be easily verified that, so long as IC-Offl^
is
satisfied for k
is
defined
E {L, H},
the high type never has the incentive to engage in this type of misrepresentation. Further,
it is
same
3.2.
We
be the case
clear that (as will
in
many
each state, this type of deviation
in
The Repeated Game
as a
of our characterizations below)
=
z
f
7
Mechanism
set of continuation values
(p,q,v)&Z(V): Ebrallt
when
V
=
firms use informative
.
l,2,
1
\ IC-Ontu, lC-Oniu, IC-OffiJ and IC-Off-Mi hold.
fc
With
Lemma
we present the
this notation in place,
Given a set
2.
VC
3?
'
=T
The lemma
I
to
J
following lemma.
let
=
i
(p,q,v)e.F'(V0
= 1, 2,
u*
=
1
C/*(z).
J
(V).
follows
by a comparison of constraints and an application of standard rev-
elation principle arguments.
gramming
,
{uW): 3z
l
| such that for
f I (V) =
Thenf\V)
2
prices are the
(weakly) dominated.
is
introduce notation for the feasible set of policy vectors
communication, given an arbitrary
if
Intuitively,
decompose the problem
once we have invoked the tools of dynamic pro-
into a choice of current-period strategies
and future
continuation equilibria, the revelation principle applies to the "staticized" representation of
the game. For the class of informative
the repeated
we can
game and the mechanism
characterize the operator
constraints of a fairly standard
restriction (u] fc ,Wj fe )
An
£
T
1
PPE, Lemma
2 formalizes the relationship
design problem we have just defined.
as generating the set of
all utilities
It
between
states that
that satisfy the
mechanism design problem, with the addition
of the unusual
V.
important consequence of this result
is
that for any informative
PPE
utility vector
u, there exists a policy vector (p, q, v) th
tsV.
These examples are motivated by the
of available
monetary
static
mechanism design
transfers. In the first example,
V is a line of slope
"budget-balanced" transfers of utility that incur no efficiency
13
literature
loss.
where
— 1;
V is the set
this represents
In the second example,
we
consider sets of the form
(K,K)
values except
V =
1
{(u ,f
2
)
The
are Pareto inefficient.
benchmarks allow us to develop some basic
consider sets
V
^
v1
:
2
K};
clearly the analogy to the static
1
fi (v)
° n[ '
In discussing schemes,
=
l
pricing
pj k
if
=
2
+
q) k
,
qj k
r for
=
I
fa ,*
~\ such
mechanism design
pair (v ,v
To
2
)
G
begin,
Lemma
3.
V
1
all
These
later
)
3z
:
u
1,2,
define:
G^JF)
(p,q,v)
=
that fori
and q\ H
(j, k) G
= qHL =
2
.
1.
We
Similarly, the
ignore
then refer to the set of constraints
Tb n (V), and we
=
we
literature,
2.
i
=U
i
\
(z).
J
efficiency
for every state
if
say that a scheme uses efficient
scheme
is
characterized by Pareto
every (j,k), there does not exist a continuation value
2
that Pareto dominates (Vj k ,v' k ).
we record the
Any
2
We
we say that a scheme uses productive
efficient continuation values if for
l
1.
on which we build when we
intuition,
excluding off-schedule incentive constraints as
-
continuation
cases are illustrated in Figure
the off-schedule incentive constraints in this section.
G
sets, all
with more general shapes, such as the convex set illustrated in Figure
To draw most
(j,k)
such
for
following standard lemma:
z satisfying IC-Oni D and IC-Oniy also satisfies
H U
q^
q\. If
IC-Oni D
binds, then
U\H, H; z) = U
l
(L,
H;
z)
=U
l
(L, L; z)
Market-share monotonicity follows since our model
-
<?L {6
satisfies
the low-cost type has a higher marginal return to market share.
relationship between the expected utility of each type follows
says that the low-cost type earns an "efficiency rent"
when IC-Omo binds
l
(z)
=U
l
a single-crossing property:
The
representation of the
by algebraic manipulation;
of ^(^// —
each firm, then the ex ante expected
for
(H, H- z)
This reveals that
(3.1)
it
L ) over the high-cost type.
a discrete- type variation on the classic "revenue equivalence theorem":
It also illustrates
U
H - 6L ).
among
+ n L <?L (e H -
6L )
i is
given by
9L ).
(3.2)
utility for firm
= W(H, H; z) + 8&H + n L ^L {6 H -
the set of allocation rules where IC-Onz^ binds, firm
between providing incentives with low prices or low continuation values
i
is
indifferent
for its low-cost type.
This follows because (in contrast to market share) neither the price nor the continuation
value interacts directly with the firm's type in the firm's objective function.
cartel has a preference over the use of low-cost prices
and continuation values
Thus, the
for
which
a firm's on-schedule incentive constraint binds, only insofar as these instruments generate
efficiency losses or gains for the other firm.
firms.
In contrast,
continuation value
when
may
Lowering price decreases the
utility of
both
cross-firm transfers of utility are available, lowering one firm's
allow an increase in that of the other firm, and continuation values
are then a superior instrument for maintaining on-schedule incentives.
14
To better
some
highlight
we turn now
of these themes,
suppose that the set of feasible continuation values
Lemma
any
and
K
>
K
For
4.
0,
G
the Pareto frontier of
%
a line of slope
is
V(K) = {(v\v 2
suppose that
3?,
n {V{K))
to two special cases. First,
)
+ v 2 = 2K}. Then, for
u + u 2 = 62K + 2tt fb },
v1
:
{(u\u 2 )
is
-l.
1
:
can be implemented with a policy vector z that
this frontier
properties: productive efficiency, pricing efficiency
we
18
and Pareto
satisfies the following
efficient
continuation values
H
(Vj k + v k = 2K for all j,k); v\iL — v\ H = (r — 9 )/8; the downward IC-On constraints
bind; the upward IC-On constraints are slack; and v\ H < vjj < v HL for j G {L, H}.
2
l
This result merits some interpretation. As expected, first-best
ward IC-On constraints bind,
and market share
is
since
it is
the low-cost type
desirable for both firms.
who has
is
attained.
The down-
the higher market share,
Thus, the relevant consideration
is
to dis-
suade the high-cost type from mimicking the low-cost type; as lower cost types have a higher
marginal benefit to high market share,
if
the high-cost type
is
just indifferent
between the
high and low announcement, the low-cost type strictly prefers the low-cost announcement.
The optimal mechanism
announcing high
for
ing,
line,
requires transfers through continuation values that reward a firm
costs. In particular,
and continuation- value
the
IC-Om^
constraints bind
discounted equals the
Next,
from a
efficiency,
the policy vector satisfies productive, pric-
and the continuation values
and only
maximum amount
we consider a second
set in
if
when
if
v HL —
l
v\ H =
on a budget-balanced
lie
(r
— 9 H )/8, which when
that a high-cost firm could gain by misreporting.
special case, wherein the firms receive continuation values
which each firm can receive at most K. To state the
we
result,
refer to the
following condition:
(r
Lemma
5.
Suppose that V(K)
=
- 9H )/(6H {(v l ,v 2 )
and
this frontier
:
u
1
+ u2 = r -
E[d]
(Pool)
rjH
v l ,v 2
:
Then, for any K, the Pareto frontier off^ n (V(K))
{{u\u 2 )
>
6L )
K}.
(i)
Suppose that (Pool) holds.
+ 62K,
u*
>
is
0},
can be implemented with a policy vector (p, q, v) that
efficiency, Pareto efficient continuation values
lowing properties: pricing
i,j, k)
and productive
Then the Pareto
{{u\ u
18
frontier
2
)
inefficiency with c?H
:
ofTb n (V{K))
u i+ u 2
=
rlH
is
(r- H )
=
cfi
for
i
=
1,2.
(ii)
satisfies the fol(Vj k
=
K
for all
Suppose that (Pool)
fails.
given by
+
rj
L {\
+ Vh )(6 h -
6L )
+ 82K, u >
{
0}.
For the public goods problem with private information, d'Aspremont and Gerard- Varet (1979) show
when the mechanism must satisfy
McAfee and McMillan (1992) specialize this
that the first-best can be attained using budget-balanced transfers,
incentive compatibility but not participation constraints.
result for the case of first-price auctions,
this problem.
The
continuation values to
sum
showing that participation constraints
in fact
can be
satisfied in
a further specialization to the two-type model, rephrased to allow for
to a constant other than zero.
following result
is
15
implemented using productive
his can be
(v'jf.
Oh)
= K for all i,j, k), pricing efficiency in
+ @h m the remaining states.
Lemma
an environment
5 refers to
in
efficiency,
state (H,
Pareto
H) (phh
efficient
=
continuation values
^
and a price of
f)
(r
which the only instruments available (reduced
continuation values, low prices) with which to achieve incentive compatible productive
When
ficiencies are wasteful.
(Pool) holds, so that the profit to the high-cost type
Lemma
advantage of the low-cost type,
relative to the efficiency
—
is
large
5 establishes that the
Pareto frontier entails productive inefficiency: the loss in profit from either Pareto
cient continuation values or inefficient pricing then
ef-
ineffi-
overwhelms any potential productive
efficiency gain.
To
see the logic of the condition (Pool), consider raising productive efficiency
creasing q HL (and therefore decreasing q HL
understanding the
effects of this
).
The
by
utility
—
r)i(r
in-
subtle aspect of the intuition entails
change when prices and continuation values must adjust
The change
to maintain the on-schedule incentive constraints.
expected
by
decreases firm l's ex ante
and firm
Oh), since firm l's high type bears the cost directly
l's
low type must now charge a lower price or receive a lower continuation value to avoid vio-
The change
lating IC-Onl£>.
increases firm 2's ex ante expected utility by 7]lT)h(Qh
the higher "efficiency rent" (0 H
—
@l),
0£) available to firm 2's low-cost type in state (H,L).
(Pool) guarantees that the cost to firm
outweighs the efficiency benefit to firm 2
This result introduces a theme that
that there
—
1,
incurred across both states (H,L) and (L,L),
in state
(H,L).
will recur
throughout our analysis.
It illustrates
a "tax" on productive efficiency: improving productive efficiency tightens the
is
on-schedule incentive constraint, leading to further distortions.
If,
in contrast, (r
— 0h)/{Oh —
Ol)
<
Vh, the firms prefer to incur the incentive costs
that the achievement of productive efficiency requires. These costs
low continuation values or low prices
for the low-cost types.
may be
In the environment under
consideration, the firms are indifferent between these two alternatives,
it is
manifested as
and
in particular,
always possible to achieve the optimal collusive payoffs using the highest available
continuation values and low prices for the low-cost types. Further, notice that the pricing
scheme outlined
r
when
its
own
market share
Whether
in the
cost
is
lemma can be implemented decentrally: each firm charges a price of
high, and selects a price p when its own cost is low. This allocates
efficiently
and achieves the
price of
{(v ,v
2
)
:
v
1
^
2
V =
H).
{(K,K)}
where
to a situation
K}. In other words, wasteful continuation values are not
instruments for providing incentives. In a repeated
set of equilibrium values
case of
in all states except (H,
firms choose to produce efficiently or not, expected profit to a cartel
improved by moving from a situation where
l
p
is
restricted to take the
Symmetric PPE, optimal
game
form
V=
{(v
^
2
)
:
v
1
=
v
2
},
not
V =
useful as
context, this implies that
1
is
if
the
as in the
equilibria are stationary: "price wars" are not employed.
16
.
We
elaborate on this result further in Section
6.
More
general oligopoly models also have
similar forces present: the on-schedule incentive constraints create cross-type externalities
19
that favor pooling of cost types.
This result can be related to the existing literature, where types are typically taken
In their analysis of "weak cartels,"
continuous, in several ways.
(1992)
show that when
types, F(8),
V=
{(u ,^
n
rj
)
v1
:
=
2
i>
},
as motivated
(r
{(0,0)}),
all
and the distribution over
firms pricing at the reser-
by Symmetric PPE. The continuum- type approach
easily
be the probability that cost type n
(Pool):
=
Athey, Bagwell and Sanchirico (1998) extend this result to the case where
2
and our two-type approach can be most
Let
(V
log-concave, the best collusive scheme entails
is
vation price.
1
transfers are prohibited
McAfee and McMillan
- 9 N )rjm
-rj N Yl™=i Vn(0 n +i
nondecreasing in m.
The
first
-
is
realized.
6n )
expression
connected with reference to an A-type model.
is
>
Then, the following conditions replace
for all
m; and
-
0n)/Vm
is
the analog of (Pool); the second condition
is
J2n=i Vn(0 n+i
the analog of log-concavity of the distribution F(8).
4.
Characterization of Informative
we
In this section,
PPE
characterize the set of informative
PPE
the insights developed in the benchmark cases of Section
lytically
some key
findings,
and we then
values.
3.3.
Our
analysis builds
on
Throughout, we develop ana-
illustrate additional subtleties
with computational
examples.
Before beginning the formal analysis, we outline some of the central tradeoffs. Suppose
that the firms attempt to implement first-best profits. Consider
plished,
and the
factors that
how
this
might be accom-
might prevent the firms from reaching this
goal. In the first
period of the game, a first-best scheme must implement productive efficiency and pricing
efficiency; thus,
from the perspective of current-period
profits, high-cost firms are
to mis-report their costs in order to achieve greater market share.
porting, the agreement therefore
favors from firm
1
must provide that firm 2
and consider the scheme
a first-best collusive scheme, productive efficiency
H)
is
truthful re-
receives future market- share
following a realization of the state (L, H). Suppose then that (L,
realized in the first period,
(L,
To ensure
is
in the
if
the firms experience the
same
H)
is
second period of the game. In
again required; consequently,
once more realized, then firm 2 must again receive zero market share.
other hand,
tempted
if
state
On
the
costs in the second period, then the collu-
19
When the monopoly price varies with the cost type, firms seek to tailor the price to the cost realization.
For the case of Symmetric PPE, Athey, Bagwell and Sanchirico (1998) show that if demand is sufficiently
downward-sloping, the firms find it optimal to partially sort types. In a two-type model, if (Pool) holds,
optimal schemes entail productive efficiency only
of the
demand
function
is
large
enough
if
the cost types are sufficiently different and the slope
(or if the firms are too impatient to
collusive equilibrium)
17
support the high prices of the
sive
is
may
arrangement
favor firm 2 while simultaneously delivering first-best profits. This
and (H, H)
(L, L)
firms
still
market shares are appropriately chosen, both
states are realized. If these
have the incentive to report truthfully.
What might
The
prevent such a scheme from succeeding?
patient so that firm
its
assigned market share
is
may
require low prices, or
some market share
must be
is
low when
Pulling these themes together,
in a given period, the firms seek
may
which
in the state in
an off-schedule deviation
it
its
call
it is
What
low.
accomplished? Then, asymmetric treatment introduces new
the scheme
firms
sufficiently
dissuaded from undertaking an off-schedule deviation following
is
1
a realization of (L,L), when
is
when the
achieved by giving firm 2 more than 1/2 of the market in the second period
this
if
inefficiencies.
cannot be
In particular,
upon the disadvantaged firm to
most
temptation to undertake
efficient, as its
assigned market share
we may summarize the
relinquish
is
high.
central tradeoffs as follows.
If
productive efficiency "today," then asymmetric treatment
required "tomorrow." Productive and pricing efficiency tomorrow, however, can then be
maintained only
the asymmetric treatment
if
among
share assignments
this
is
possible only
if
may
implemented through asymmetric market-
equally-efficient firms pricing at the reservation value.
tomorrow the disadvantaged firm
assigned low market share.
patient firms
is
sufficiently patient to
is
In turn,
endure
its
In view of these tradeoffs, a cartel comprised of moderately
assign market shares today without achieving
we
productive efficiency,
burden.
in order to lessen the future transfer
In the next two subsections,
full
derive conditions on the discount factor under which
firms are able to implement a given level of efficiency (such as first-best) in every period of
the game. Subsequently,
we explore
between current and future
4.1.
A
efficiency faced
Linear Informative
In this subsection,
we
in greater
PPE
Set
depth the optimal resolution of the tradeoffs
by firms of moderate patience.
With
First-Best Profits
than one above which the
identify a discount factor strictly less
cartel can achieve first-best profits in every period. Recall that Section 3.3 analyzes Pareto
optimal schemes
directly the
for
an exogenously given
set of continuation values.
endogenous nature of the continuation-value
existence of a set of Informative
profits to the cartel as
PPE
a whole, and
values,
(ii).
where
(i).
We now
confront
set.
Our
each
utility pair yields first-best
goal
when implementing any
is
to establish the
point in the
set,
only
other elements of the set are used as continuation values on the equilibrium path (by
contrast, off of the equilibrium path,
punishment outside the
set is required.)
"self-generating" set comprised only of values that yield first-best profits
segment with slope
We
(y,x).
Clearly, a
must be a
line
— 1.
seek to construct a self-generating line segment with endpoints denoted (x, y) and
The
first
step
is
to ignore off-schedule constraints
18
and
identify conditions under
which the endpoint
segment. 20
(x, y)
Formally,
weu
we
[0> 1])
(x,y)
= (U l (z),U 2 (z))
line
explore conditions under which there exist market shares qHH
as continuation values v] k G [x,y] with v
111 e
as
can be implemented using only continuation values on the
and IC-Om^
binds for each
k
when
i,
=
2
x
+
—
y
ujfc
,
such that
,
the firms use pricing and
productive efficiency
This step can be challenging
r — Oh,
cost firm,
U
1,
l
=
(z)
is
for certain
x, while
E\n}(J,j;z)]
D
for firm 1 to reveal its
by
v^ k (l
each (j,k). Further, in order to provide incentives
5) for
type when
:
is
it
high cost, the future must look relatively better
Lemma 4, a necessary
— 6 H )/8, and so vHL must
following the logic of
IC-Om^ to bind is that v HL — v\H = (r
least (r — 6 H )/8. This requirement places additional downward
expected
profit,
which must be low enough to
—
v\
0l
is
too small,
y. Intuitively,
profits derived
achieved
when
can be
it
U
satisfy
bound on today's
pricing efficiency impose a lower
term, Oh
profit for a high-
per-period profits derived from each of the continuation
its
—
each
at
monopoly
it
following a realization of (H, L)
for
If
may be difficult to achieve the desired level of profit for firm
maintaining vj k > x. To see why, observe that firm l's average profit
too large,
today must then be worse than
values:
parameter values.
difficult to
1
=
(z)
profit.
x.
implement
U
pressure on today's
{z)
=
if
the efficiency-rent
y while maintaining
firm 2's average profit today must then be greater that
from each of
continuation values.
its
the efficiency rent 6 H
—
9L
can be significantly increased by raising g| L while
,
per-period
its
Recalling (3.2), this
is
more
then current-period profit
large, as
is
exceed x
However, productive and
Similarly,
2
condition
still
easily
for firm 2
respecting the on-schedule incentive
constraints.
In the Appendix,
we show that there
the constraints listed above are
than
exists a critical discount factor less
all satisfied, if (recalling
our assumption
r]
L
>
1/2)
vl>i^T^VL-i).
We may
verify that (4.1)
sufficiently close to 1/2.
20
We
22
is
satisfied if r
Under
argue in Section 4.3 that
y.
if
z
(4.1),
— Oh < Oh —
we show
Ql\
in the
1
where
21
:
(4.1)
more
generally,
it
holds
if tjl is
Appendix that implementation
is
and z
Although we do not formally analyze
this case,
it
can be shown that
for t/l
<
i
,
a different
but analogous condition must hold.
22
parametric restriction fails, then the firms may not be able to implement exactly first-best
However, it is possible to construct self-generating sets composed of three connected line segments,
If this
profits.
interior line segment has slope — 1, and all points on that segment are implemented using
productive efficiency. In contrast, some productive inefficiency is used on the exterior segments. As firms
become more patient, the width of the interior fine segment grows, and so first-best can be approximated
where the
arbitrarily closely as 6 approaches
1.
19
e
feasible
6
if
is
e
<
greater than a critical discount factor, 6
cFon
step
(o H
-o L
denned as
-°H
+ 2(r-e H )(i-
follows:
W
L
VL )
to identify a second critical discount factor, above which the
is
schedule incentive constraints hold.
off-schedule deviation leads to
standard,
,
r
_
"
The second
Fon
an
It is sufficient for
our construction to specify that an
infinite reversion to the static
Nash
In the Appendix,
we
As
equilibrium.
firms are sufficiently patient, then an off-schedule deviation
if
off-
is
unattractive.
is
describe the computation of the critical discount factor, 8 Foif above
,
which the off-schedule constraints are cleared:
ro/f
=~
23
(r-0 H )(l-T,L )+T} L (e H -OL)
- H )(i - VL + VL (9 H - L + vl(r- e H
(r
)
)
)
Suppose that r -6 H < 9 H -0l, and define 8 fb = max{6 Fon ,6 Foff ) G (0, 1).
such that x + y = 2ir FB /(l - 6),
Then, for all 6 e (6 FB 1], there exist values y > x >
and the line segment [(x, y), (y, x)] is in the set of informative PPE values, V 1 Each utility
Proposition
1.
,
.
u on
the segment can be implemented using a policy vector (p, q, v) such that pricing
and productive efficiency hold, and Vjk € [(x, y), (y,x)] for each (j, k).
pair
For the computational example we explore below, where r
r)
L
=
.6,
we
find that 8
Proposition
1
Fon
=
.66,
and
6 Foff
=
=
2.5, 6jj
=
2,
0l
=
I
and
.82.
can be thought of as a generalization of Fudenberg, Levine and Maskin's
(1994) folk theorem.
Instead of resorting to infinite patience, we compute a discount
factor strictly less than
one
(.82 in the previous
result further provides
an
explicit characterization of the behavior associated
first-best
example) where
first-best is achieved.
with this
arrangement: since real inefficiencies might be associated with the provision of
incentives,
and
since
some firms may be
at the
margin of deviating from the agreement,
it is
important that the cartel design the collusive agreement to circumvent these
The
cartel achieves this
efficiency
23
Our
is
by shifting market share among firms
not sacrificed.
This discount factor
may be
may
where productive
24
it is computed using a particular method of implementparameter regions under consideration. For particular
alternative self-generating sets that can be implemented with lower levels of
conservative, since
ing the self-generating set, one that applies for
parameter regions, there
in states
issues.
all
patience.
24
themes from the review-strategy
games with infinitely-patient firms. Under this
approach, after many periods, there is a "review," whereby the distribution of observables is compared to
the expected distribution if each firm had followed the specified strategy. Any firm whose behavior over
that period of time does not conform is punished as harshly as possible. In the limit, patient firms view the
small probability of punishment as sufficiently bad that they conform to the required strategies. Further,
It is
useful to contrast our characterization of collusive behavior with
literature (Radner, 1981) as developed in "hidden action"
the review periods can be taken to be arbitrarily distant, so that in the limit, punishments occur with
zero probability. In our model, by contrast, firms are not infinitely patient, and so real inefficiencies may
be required to provide incentives. Our analysis
is
concerned with the optimal manner
such incentives.
20
in
which to provide
Computational Example
4.1.1.
Throughout Section
We now
detail.
offer
4,
we use a computational example
to develop our insights in further
some general remarks about our computational approach, and we next
consider an explicit parameterization that relates to Proposition
Our approach
is
V° and then compute
to specify a set
iterating until the distance
between the
operationalize this algorithm, a natural
sets
1.
V = T 1 (V 1-1
1
At the
Wang
start of the algorithm,
(1994),
we
method
is
we use a
divide the set
The
denote the vector of points in this grid.
values for firm
2,
of the algorithm,
to divide each set
[0,
r/(l
V into a grid, and then
V t+1 This approach is
.
which speeds up the computations. 25
— 6)]
where we
into a fixed grid,
and worst continuation values
best
To
1
are the only values ever permitted for firm
we compute the
1, ...,
level.
let
x
grid represents the set of feasible continuation
further ease the computational burden,
than compute the lower bound of the set
bound with
trick
=
2.
On
each iteration
for firm 1
which can
for each Xk in this grid.
be sustained
To
and these
t
becomes lower than a given tolerance
check which members of this grid survive to become members of
slow, however. Following
for
)
static
Nash equilibrium
V
1
we impose two
on each
The
payoffs.
iteration,
First, rather
restrictions.
we approximate
restriction to static
this lower
Nash punishments
does not directly affect the qualitative characterization of the efficiency frontier, since (as
the computations show) the firms only leave the efficiency frontier off of the equilibrium
path.
The
available punishment, however, does affect the extent to
asymmetric equilibrium values. Thus,
point on the frontier.
it
indirectly affects the continuation values for every
Second, we restrict the firms to use pricing efficiency. 26 Given the
we have imposed, our computations should be
restrictions
which firms can sustain
interpreted as lower bounds on
the Pareto frontier of equilibria.
Figures
3,
We number
4 and 5 illustrate our computation for a particular set of parameter values.
the states on the frontier.
The
proceeds from every point on the frontier.
table below the figure shows
how
the
game
Observe that the computations yield slightly
asymmetric continuation values across the two
firms; this arises as a result of the
tational algorithm, which treats one firm's profit as discrete
Further, on the region where the continuation value frontier
compu-
and the other's as continuous.
is
approximately
linear, there
many ways to implement a given value; but, due to the discretization
the firms may have a strict preference among alternative collusive schemes
are often
of the
frontier,
giving
25
Wang's (1994) approach builds in turn on Phelan and Townsend (1991). Recently, Judd (1995) has
developed several approaches to computation that could be substantially more efficient. As our aim with
the computations is to illustrate the theoretical results, we have not pursued this approach, though it is
an interesting avenue for future work.
26
For impatient firms, this restriction
is
perhaps more onerous; we show in later subsections that for
inefficiency may be desirable on the Pareto frontier
some parameter values, even if (Pool) holds, pricing
when the off-schedule incentive constraints bind.
21
approximately the same
among nearby
states.
utility.
Thus, occasionally behavior appears to "jump" drastically
This does not qualitatively
affect the
computation of the equilibrium
set.
Using the table of transitions,
it is
possible to trace the path of play given any sequence
of cost realizations, taking any state
on the Pareto
The
frontier as the starting point.
market-share table illustrates the extent of productive efficiency in each state.
Figure 3 illustrates the implementation of an equilibrium yielding first-best profits for
the cartel at every point in time.
and productive
efficiency
The region of states between
11
and 24
used (approximately) in each state.
is
is
self-generating
— v LH =
l
l
-
B„)/6.
particularly useful to consider the implementation of the equilibrium value cor-
It is
responding to state
11, since
the off-schedule constraints for the extreme values of the
linear self-generating set are the
above,
most
we
most demanding. Consistent with the theoretical analysis
see that in state 11, firm 2 produces (essentially) only in state (H, L), where
efficient.
not produce, but does receive a
"rewards" accrue to firm 2
following the realization
when
(H,L)
announces high
it
entails
when
value. In other words, larger
cost, while the future
remaining in state
when implementing
payoffs to firm 2
is
is
Linear Informative
PPE
(L,L), but the prospect
is
PPE
any lower; thus, the only way to lower
set,
is
high.
The remainder
is
of Section 4 analyzes collusive
on characterizing the Pareto
before proceeding in that direction,
we
briefly
only partial productive efficiency
and
we analyze them
for
is
two reasons.
must be approximated numerically,
may be
24, 12
Even though such
used.
24,
and 13 and
in
frontier of the
PPE
values,
where
sets typically are not Pareto
First, in contrast to the
Pareto frontier, which
linear self-generating sets are tractable to analyze
verified that self-generating sets are not unique: the regions of states
and
schemes
extend the approach
of the last section to consider linear self-generating sets of Informative
It
equi-
Set with Partial Productive Efficiency
such circumstances. Although our main focus
Informative
if its
firms are less patient, their ability to use market-share favors
in future "tie" states is limited.
27
rel-
to lower firm 2's market share in state (H,L), where the off-schedule
As mentioned above, when
often
must be
state 7, firm 2 cannot adhere to the agreement
constraints are slack since firm 2's market share
optimal,
"punishment"
Clearly, firm 2
11.
the cost realization
librium profit following a realization of (L, L)
A
other cost realizations, firm 2 does
to state 15 in the following period induces firm 2 to adhere to the agreement.
In contrast,
4.2.
all
more favorable continuation
atively patient to endure sitting out
moving
it is
Further, following the cost realization (H,L), the equilibrium specifies that
the firms return to state 11. In contrast, following
of
,
Incentives for truthful
revelation of the cost type are provided using continuation values, with v HL
(r
27
24, respectively, are also self generating.
22
between 8 and
24, 10
in closed form. Second, a self-generating set of equilibrium values provides a lower
on the
bound
can be sustained by colluding firms in an optimal PPE. This
level of profits that
lower bound, phrased in terms of exogenous parameters,
useful
is
when we attempt
to
characterize the Pareto frontier in the next section.
Depending on parameter
values, a variety of different constraints
plementation of a linear self-generating
In this section,
set.
we
may
bind
in the im-
take a conservative ap-
proach, constructing a linear set [(x,y),(y, %)] where the off-schedule constraints bind in
two states for firm
by specifying a
when implementing
1
set of constraints that bind,
Formally, suppose that,
isfied.
and we then
when implementing
IC-Offl[ L IC-OfflJjj IC-Onln and IC-On2 D
,
,
To construct the
(x,y).
.
We
=
vlh
=
vll
(x,y),
=
restrictions, q\ h
€
q\ H
and
(0, 1). It
q HL — q HH =
l
segment, we proceed
verify that the others are sat-
(x,y), the following constraints bind:
Nash equilibrium
use the static
threat point in the off-schedule constraints. Further,
Vhh = v H l,
line
we
l
0.
It
specify that Pjk
—
as the
r for all
(j, k),
can be verified that under these
l
remains to verify that v HH
D
y.
Straightforward cal-
culations (presented in the Appendix) establish that this condition can be satisfied
when
firms are sufficiently patient:
Proposition 2. There exists a 5 hn < 1 such that, for all 6 > 6 hn there exist values y >
x > such that the line segment [(x, y), (y, x)] is in the set of informative PPE values, V 1
Each utility pair u on the segment can be implemented using a policy vector (p, q, v) such
,
.
l
2
andq LH
+qHL
that pricing efficiency holds, vjk G [(x,y), (y,x)] for each (j,k),
where
=
(r
- 6 H )/(0 H -
For the parameter values we use in our computations
Vl
=
hn
-6),
value of
b~
~
-7,
and
In fact,
2.
scheme never yields
it
is
straightforward to verify
first-best profits,
condition that IC-Offl£ H
restriction
=
(r
2.5,
is
even when 6
+ q\ L ~ 1.5, less
that q\ H + qjiL <
= 1;
this
is
—
6H
at that discount factor, q\ H
is
= l+ j^rgfa
,
B L ).
2,
6L
than the
1
+
6,
=
1
and
first-best
and so
this
because we have imposed the
binding when implementing (x,y).
When
6 gets larger, this
too strong; a more profitable linear self-generating region exists where only
IC-Offl£ L binds when implementing the endpoints. Although the Pareto superior sets can
be computed as
4.3.
We
set.
well,
The Shape
proceed
now
we do not pursue
it
here.
of the Pareto Frontier
to consider the shape of the Pareto frontier of the informative
As discussed
at the start of Section 4,
when
PPE
utility
firms attempt to implement highly asym-
metric equilibrium values, the off-schedule incentive constraints bind and some inefficiency
may be
required.
We
thus anticipate that total cartel profits
asymmetric, indicating that the frontier
we
is
fall
as values
become more
typically nonlinear. In the present subsection,
establish conditions under which a subset of the Pareto frontier of the set of informative
23
PPE
we formally
characterize the
which the off-schedule constraints determine the boundaries of
this linear subset
values
manner
in
a line with slope equal to —1.
is
boundaries of the frontier
(as well as the
To
we
begin,
In addition,
itself).
our assumption that firms can randomize between continuation
recall
equilibria,
which ensures that firms have available a convex
any point
in time.
Figure 2 illustrates the general shape of a symmetric, convex set of
continuation values.
or vn, vs, ve, vw,
corners, the
is
The
=
where vn
boundary of the
values. In this context,
When
if
for
some
and West,
set has four "corners," labelled as North, South, East,
(v}f,v%)
set is
of particular interest to us, since
inefficiency
set of continuation values at
we say
(J,
and
likewise for the other corners.
Between two
monotone. The part of the boundary between v^ and Ve
represents the Pareto frontier of the set of continuation
it
that a scheme
k), (vj k ,Vj k ) is
is
characterized by continuation value Pareto
not on the Pareto frontier of V.
describing the Pareto frontier of the set of feasible continuation values given a
we use the notation
set V,
max{u?
/(«}*)
-C
{
some
/.
We
,
v%) E co{V)}
if
v%
\
for
{v)k
:
fc
[
=
large constant C.
define the function
{v) k
- vE
Of course, convexity
)
of the set
v) k
e
vE
[vj,,
if
v) k
if
v) k
]
l
< vN
> vE
V implies concavity of the frontier
/ outside the domain of the Pareto
frontier in order to simplify
the statement of some of our results about the slope of the frontier.
Given our assumption that firms are symmetric, f(y) + v is maximized at v 1 = vl,
where /(f]) = v]. We may thus say that a scheme is characterized by future inefficiency
+
if Vj k
fail
to
<
f{vj k )
maximize
2v\ for
some
(j,k), so that
under some state the continuation values
As mentioned above,
total cartel future profits.
future inefficiencies are
associated with highly asymmetric values, and represent an efficiency cost that
when
firms attempt to provide incentives with such values. Thus,
it is
— 1,
may make some
loss.
For the informative
PPE
set
V
1
,
provides equal utility to both firms.
V1
that
Consider a policy vector that implements
v[,
and
for simplicity that pricing efficiency is used.
by
e/r]L
and (H, H) by
e/tjh,
it is
By
in states (L,
lowering firm
L) and (H,H). Suppose
market share
l's
also feasible.
While firm
in state
possible to transfer market share from firm 1 to
firm 2 without upsetting any of the on-schedule incentive constraints.
is
so that the firms
be the point on the Pareto frontier of
let v{
assume that the off-schedule constraints do not bind
(L, L)
incurred
important to identify
conditions under which a subset of the Pareto frontier has slope of
use of future market-share favors without efficiency
is
l's profit is lower, total cartel profit is
This new scheme
unchanged, and so the
Pareto frontier has an interval with slope equal to —1.
How
can we ensure that the off-schedule constraints do not bind in states (L, L) and
(H,H)7 Without
loss of generality,
when implementing
24
v{,
we may
specify that
q^
=
1/2
and
=
Vjj
r
v s for j G {L,H}.
With
this specification in place,
demanding circumstance from the perspective
(L, L)
when pll
=
we
r,
see that
suffices to
it
of off-schedule constraints arises in state
check the following condition:
{r-eL )/{2S)<v s I -v
1
Of
course, v]
1
is
endogenously determined.
and observing that the most
We
1
(OffSS)
.
illustrate
a range of discount factors where
(OffSS) holds in our computational examples. Further, Proposition 2 establishes a lower
bound
v]
1
for v]
1
:
if
we
define
must be greater than
that
if
(r
- 6H )/(dH ~
x and y
(x
BL )
+ y)/2
>
.275,
as in Section 4.2, since [(x, y), (y, x)]
is
Using straightforward computations,
.
28
then (OffSS) holds
for all 6
>
6
lin
.
contained in
it
V1
,
can be shown
In other words,
(OffSS) holds whenever the linear self-generating set constructed in Section 4.2 exists.
When
(OffSS) holds,
we have the
following initial characterization of the shape of the
Pareto frontier:
Proposition 3. Assume (OffSS).
1
(i) The Pareto frontier ofV
has an open interval with slope equal to — 1.
(ii) Let (x, y) and (y, x) denote the endpoints of this open interval, and let (p, q, v) implement (x,y). Then at least one of the following two conditions hold: (a)forsomej G {L,H},
for some
IC-Offljj binds; (b) vjj Q x for some j G {L,H}, and either pjj
6j or q^ =
je{L,H}.
Part
confirms the existence of a subset of the Pareto frontier with slope
(i)
— 1.
Part
(ii)
then identifies the factors that limit this subset. Formally, either an off-schedule constraint
binds, or else the firms run out of market-share favors
values in the event of
ties.
and the
ability to shift continuation
In either case, the firms cannot implement any further transfer
away from firm one and towards firm two without a loss of efficiency. For firms of
moderate patience (6 < 8 FB ), the off-schedule constraints typically bind first. It is possible
of utility
to further characterize the implementation of (x, y) for different parameter regions, but
will
not pursue
Next,
it
we
here.
we observe
entire Pareto frontier
that the off-schedule constraints also determine the endpoints of the
and that they
when implementing
force the firms to bear inefficiency
those endpoints.
Proposition
then there
each firm
is
i,
4.
T
Suppose that (p,q, v) implements vN
(i)
.
If
IC-Oniu
either pricing inefficiency, productive inefficiency or both,
at least one of the following two conditions hold:
1
IC-Offljj binds; (b) v^
D
vjj for
some
j
G {L, H}, and
(a) for
either pjj
0j
is
slack for each
i,
lf(£L >qH for
some j € {L, H},
or q^ =
for some
(ii)
L
je{L,H}.
The lower bound on (r — 6h)/{8h — #l)
and so the bound is relaxed when r\h > 1/2.
28
is
calculated
25
when rj L
=
1/2. This
bound
is
decreasing in
tjl,
To understand part
(i),
observe that
if
the firms are using productive and pricing
then IC-Offl£ H and IC-Off-Ml are
ficiency,
incentive constraints are slack (as they are in
can given up some market share
in state (L,
So long as the upward on-schedule
slack.
all
H)
ef-
of our computational examples), 29 firm
1
without violating any incentive constraints.
In other words, the off-schedule constraints create a force in favor of sacrificing productive
when
efficiency
firms attempt to implement asymmetric continuation values.
similar to Proposition 3
If
(ii).
neither of the off-schedule constraints bind,
Part
it
should be
possible to shift market shares or continuation values in the event of ties from firm
firm
an
In contrast to Proposition 3
2.
(ii),
however, shifting continuation values
efficiency loss; indeed, the frontier will typically
is
(ii)
may
to
1
entail
have slope greater than —1 near the
r
corner, v N
.
Finally,
we consider whether the
and constraints are nonlinear
(1
—
qj k )
pjk), J-
J
{V)
same
prices are the
in
is
in
set of equilibrium values
convex. Since payoffs
market shares and prices (they depend on
and
not generally convex,
two
is itself
V
1
may
gj fc
,
gj fc
-p^ and
not be convex either.
When
distinct equilibria, however, the nonlinearity does not pose a
problem, and the convex combination of two equilibrium values can be implemented using
a convex combination of the two associated policy vectors.
In the next subsection,
analyze conditions under which prices are always equal to the reservation value r
implementing values on the Pareto frontier of the equilibrium
4.4.
Pricing and Continuation Value Efficiency
we consider whether, when implementing Pareto
values, pricing
and continuation-value Pareto
efficiency are used.
circumstances under which we expect these properties to hold.
properties imply that the
To
begin,
downward on-schedule
we consider whether
when implementing equilibrium
far
when
set.
In this section,
and
we
(ii)
the Pareto frontier
is
pricing
sufficiently wide,
frontier,
constraints are typically binding.
(i)
discount factors
.
efficiency hold
the off-schedule constraints are slack,
and the equilibrium value
!
v N and v E
when implementing
that exceed 8 FB
PPE
We identify two distinct
We then show that these
,
is
sufficiently
that the firms implement the
r
equilibrium value using continuation values strictly between v N
and v E
indicates, these properties hold
informative
and continuation-value Pareto
values such that:
from the corners of the Pareto
efficient
.
As Proposition
1
values in the neighborhood of v* for
However, conditions
(i)
and
(ii)
apply in a wider set
29
To further understand the role of the assumption that IC-Omjy is slack, first observe that if both
downward and upward on-schedule constraints bind for firm i, then q^ = (pH which requires productive
inefficiency. Part (ii) does not cover the case where IC-Om'j/ binds but IC-Onio is slack for one or both
firms. Although we do not formally analyze this case, we establish below conditions under which the
upward on-schedule constraints are slack at the start of the game, when implementing v[ adjusting the
scheme so that the upward on-schedule constraints bind typically requires new inefficiencies, consistent
,
;
with the proposition.
26
of circumstances.
more moderate discount
In particular, for a range of
factors, the off-
schedule constraints are slack at the begi
)
implement a Pareto
and
the off-schedule constraints hold with slack,
V
efficient utility pair in
vjj
<
vjk
1
< v^
for all
1
Suppose that
.
(j,
that firms use Pareto
efficient
Suppose
value
is
continuation values. In establishing this result,
that there exists a cost state j for firm
first
always
(i.e.,
in
2's
more
Suppose that there
subtle.
firm 2's continuation value
example, we might have
and increase
Provided that the
frontier, this
(j,
(ii)
L
f(vj H ) and
and
rj
H
,
v"j
L
initial
continuation values
frontier.
continuation
In this case,
frontier.
experiences cost
1
incentive constraints.
The second
exists a cost state j for firm 1
<
f(Vj L ). In this event,
such that
for firm 2.
we may reduce
2's
l's
lie
ex ante
For
Vj H
strictly
utility or incentive constraints.
between the corners of the Pareto
continuation values, in states
(j,
H)
as well as
But we may now apply the argument developed
adding a constant to firm
and thereby increase firm
we
respectively), while holding the continuation values
without disturbing firm
below the
first possibility,
Part
H=
rj
its
possibilities.
2's
below the frontier in exactly one cost state
maneuver positions firm
L), strictly
cost state j,
is
v"j
vj L (at rates
for firm 2 constant,
such that firm
continuation value whenever firm
state j. This raises firm 2's utility without disturbing
is
1
both states (j,L) and (j,H)) below the
can simply add a constant to firm
possibility
we must examine two
2's
continuation value whenever firm
1
for
our
experiences
2's utility.
establishes the optimality of efficient pricing.
The
result follows because
it is
always possible to raise prices and lower continuation values to keep the firms indifferent;
but we have just established that continuation values below the frontier can be
strictly
improved upon.
Our next
objective
is
to relax the constraint that the continuation values
tween the corners of the Pareto
frontier.
We find that
lie
strictly be-
additional restrictions on parameters
are required to maintain our efficiency results in this case.
In a computational example,
Figure 5 illustrates our approximation of the Pareto frontier for a set of parameters such
that, at the start of the
game, when the firms attempt to implement equal
utility,
the
off-schedule constraints do not bind (in particular, the off-schedule constraints are slack for
27
the (p, q, v) implementing states 7-17). However, the Pareto frontier
not wide enough to
is
productive efficiency using Pareto-efficient transfers (since the off-schedule
implement
full
constraints
do bind
for the "corner" continuation values, that
at the edges of the Pareto
is,
frontier).
Proposition
6.
efficient utility
Suppose that (Pool) holds. Suppose that
pair in
V
1
,
implements a Pareto
(p, q, v)
the off-schedule constraints hold with slack,
and further
either
both v H h and v LL are on the interior of the line segment on the Pareto frontier ofco(V)
< g|- < 1 for j G {L,H}. Then: (i) continuation values
with slope equal to — 1, or (b)
(a)
are Pareto efficient,
and
(ii)
prices are efficient.
This result generalizes Proposition
stand the restrictions, recall
Lemma
increase the total collusive profit
efficiency of prices
implies that
under several additional
5,
5,
when the continuation
To under-
which establishes that when (Pool) holds, firms
by decreasing productive
and continuation
restrictions.
values.
30
As an extension
value frontier
is
and increasing the
efficiency
of
Lemma
5,
this result
narrow, and firms maximize the
of cartel profits, they choose productive inefficiency over inefficient prices
sum
and continuation
values.
Continuing with the case
natural question
is
in
which the off-schedule incentive constraints are
whether the Pareto
frontier
is itself
self-generating.
slack, a
Are there circum-
stances in which the implementation of a Pareto efficient equilibrium utility pair requires
for
some
state a Pareto inefficient continuation value
cumstance were to
asymmetric
arise,
utility pair,
to transfer utility
then
it
(i.e.,
a "price war")?
such a
cir-
would be associated with an attempt to implement an
with the continuation- value inefficiency then employed as a means
between agents. Furthermore, as Proposition 6 Part
efficient utility pair,
If
(i)
indicates, for
an
Pareto inefficient continuation values would b
,
however,
it is
conceivable
that an optimal cartel might implement a utility transfer with an inefficient continuation
value, particularly in the (L, L) state, so as to
relaxing that firm's
downward on-schedule
possibility does not arise in
So
far,
draw
from a firm while simultaneously
incentive constraint.
We
constraints. Before proceeding,
effects of off-schedule constraints
we make
on pricing and continuation-
30
is
note, though, that this
any of our computational examples.
we have considered only the on-schedule
two observations about the
utility
Although we do not formalize this, it can be shown that when (Pool)
maximized using productive efficiency, even at the expense of
fails,
the
sum
of firm profits
i
plementing asymmetric
vectors, the firms
may
use both pricing and continuation value inefficiency.
further here.
28
We
will not
utility
pursue this case
.
value efficiency. First, the off-schedule constraints place
The
off-schedule constraints
may
downward pressure on the
prices.
require lower prices for low-cost firms, in order to reduce
the incentive that such firms have to undercut the equilibrium price. However, as lowering
prices reduces total cartel profits, lowering prices does not necessarily help firms achieve
Pareto
When prices
equilibrium values.
efficient
are inefficient, in principle, the
we have not discovered any examples where
schedule constraints might then bind, although
we employed
this occurs. Second, in establishing continuation- value Pareto efficiency,
ments
which firms
in
upward on-
shift profits across states of the world.
When
bind, the profit of a firm in a particular state of the world
argu-
off-schedule constraints
may be
Thus,
constrained.
our characterizations are more limited. However, in the next subsection, we establish a set
of characterizations of productive efficiency that hold even in the presence of off-schedule
constraints.
Returning to the case where the off-schedule constraints are
consequences of pricing and continuation- value Pareto
that the
downward on-schedule
slack,
we now examine the
lie
on a
segment of the
linear
establish
constraints then bind, unless the firms are indifferent
a range of equally-desirable implementation schemes (as might occur
values
we
efficiency. In particular,
if
among
the continuation
frontier)
u be a Pareto efficient utility pair in V 1 Whenever there exists a
policy vector implementing u such that prices are efficient and continuation-value Pareto
efficiency holds, then there exists a (p, q, v) that implements u such that either: (a) at
Proposition
least
(fL
>
7.
Let
.
one off-schedule constraint
(fa for
each
i,
then v\ H
D
binding, or (b) (1)
is
l
D vHL
uj-
for each j
IC-Onio binds
for each
and
i,
(2) if
G {L, H}.
This result explores the consequences of concavity of the
frontier, f(v).
Concavity
implies that asymmetric continuation values are associated with future inefficiency. Thus,
unless behavior
is
constrained by off-schedule considerations,
it
is
only optimal to use
asymmetric continuation values to the extent that they provide a mechanism to enhance
productive efficiency benefits.
If
the frontier
is
strictly
concave in the relevant region,
the firms use asymmetric continuation values up to the point that the
To
constraints bind.
l
see this, suppose that v HL
the incentives are stronger than necessary:
we can
>
v\ L and IC-Onl^
frontier, this
If
we simultaneously move
adjustment
slack.
is
strictly concave.
In contrast,
if
U
2
slack.
Then,
bring vHL and v\ L toward one another
undisturbed
{j]i
and
rjn,
firm l's continuation values along the Pareto
not disturb firm
Such an adjustment leaves
is
will
will
is
1
at exactly the rates that leave firm 2's on-schedule incentives
respectively).
downward IC-On
l's
on-schedule incentives because IC-Oni^
unchanged, and increases
the frontier
is
U
1
whenever the
frontier
linear in the relevant region, the firms
be indifferent to the adjustment. In any case, only the off-schedule constraints would
prevent such an adjustment.
29
Finally, observe that the ranking of continuation values in (b)(2)
intuition that firm
1 is
rewarded in state (H,L) and punished
The
deter a (downward) on-schedule deviation.
each
is
i
downward on-schedule
same time
> tfH
for
downward on-schedule
a given firm. Since (b)(1) specifies
for
upward
31
Computational Examples
Consider Figure 4 in relation to Proposition
for the policy vectors that
implement states 11 to
Proposition 5 apply for those states.
to another state
The
by Proposition
on the Pareto
and thus our characterizations from
20,
frontier.
7:
for firm
1,
Given that
this property holds, the
The continuation
frontier.
the realization (L,H)
continuation value, while the realization (H, L)
is
when implementing
is
values are ranked
followed by a low
followed by a high continuation value.
Similar results hold in the example illustrated in Figure
Proposition 6 are satisfied
frontier are binding
table of transitions illustrates that continuation-
diagram only labels and represents the Pareto
as predicted
illustrated in the diagram, neither the
in every state, after every realization of cost types, the
value Pareto efficiency holds:
move
As
5.
on the width of the Pareto
off-schedule constraints nor the constraints
firms
in order to
constraints bind, the assumption implies that the
on-schedule incentive constraints are slack.
4.4.1.
(L,H),
additional assumption that q\
included because under that assumption, the upward and
incentive constraints cannot bind at the
that the
in state
conforms with the
5.
In this case, the conditions of
states 9-17.
Productive Efficiency
4.5.
Having considered the use of pricing and continuation-value
efficiency,
whether firms use productive efficiency when implementing Pareto
Our
tions.
first
It
result applies to all points
on the Pareto
frontier,
we next explore
efficient utilities.
with no additional
restric-
hinges upon the tradeoff between productive efficiency today and future
ciency tomorrow.
To
<
see the logic, suppose q\ H
engineer a Pareto superior equilibrium.
The
1,
ineffi-
and consider whether firms could
firms could increase productive efficiency to-
day by increasing q\ H To maintain on-schedule incentive compatibility for firm 1, v\ H
could then be reduced so that the combined adjustments in q\ H and v\ H do not increase
.
in
U
l
(L,
H;z). Since a firm values an increase
assured
this
is
this
adjustment
if
the adjustments are
is
made
in
market share
made, and now consider firm
2.
q*
i,
its
costs are high,
.
2
This firm experiences a reduction in qLH
As market share is more valuable when
costs are low, the positive effect of the increase in q\ H
If (p =
L
H for some
formally delineate them.
when
so as to leave f/^L, L; z) unchanged. Suppose
and, moving along the frontier, an increase in v\ H
31
less
on
U
l
(L,L;z)
is
greater than the
the rankings of continuation values are not quite as sharp, and we do not
30
complementary negative
value reduction that leaves
U
2
(H, H;
z), if
2
the decrease in qLH
on
effect of
U
1
(L, L; z)
future inefficiency
raises
U
therefore sufficient to strictly increase
is
is
not too great
A
(H, H;z).
2.
Indeed, even
the frontier
if
the frontier
(i.e., if
Pareto superior equilibrium
is
not too
is
strictly concave, if the
then this maneuver
flat),
thus generated, provided that the
is
arrangement continues to satisfy the off-schedule incentive constraints. 32
is
continuation-
firm l's continuation- value reduction were to translate, dollar for dollar, into
a continuation- value gain for firm
2
unchanged
U 2 (H,H;z). The
assured, since from the perspective of the off-schedule constraints,
new
this, too,
does not matter
it
whether market share or continuation values are used to provide profits
But
in a given state (in
contrast, higher prices raise the incentive to deviate).
Formally,
we have the
Proposition
8.
following proposition:
Choose any Pareto
efficient utility pair in
Then productive
policy vector that implements this pair.
q LH =
l
(i.e.,
1
1)
ifpLH > @h and
+ A:f(vlH = 1 +
)
if
there exists e
>
-
-
[f(vl H )
f(vl H
Productive efficiency holds in state (H,L)
e
>
(i.e.,
V1
and
let (p, q,
v) be the
efficiency holds in state (L,
H)
such that
e)]/e
l
q HL
=
<
(9 H
-
0) if Phl
9L )/(pLH
-
> 6h and
9 L ).
if
(4.2)
there exists
such that
1
+
A+/«L
)
=1+
[Hv'hl
+ e) -
/(«k)]/e
> -(fe - fe)/(pm -
(4-3)
fe).
Consistent with the description above,
we
is
used when im-
plementing Pareto
the continuation values in the (L,
H) and (H, L)
states are
efficient utilities, if
drawn from regions
too greatly from
see that productive efficiency
of the frontier at
which the frontier slope does not depart
— 1.
Recall that by Proposition
3,
when
(OffSS) holds, the interior of the Pareto frontier
has slope equal to —1. Thus, Proposition 8 can be used to show that,
some productive
efficiency
see why, observe that
is
when
(OffSS) holds,
used by optimally colluding firms at the start of the game. To
(OffSS) holds,
it is
feasible to
implement a symmetric scheme
with no productive efficiency using pricing efficiency and Vjk
is
when
=
v^ for all
(J,
k); clearly, this
the best scheme with no productive efficiency. However, Proposition 8 implies that the
latter
scheme
is
Pareto dominated by a scheme with at least some productive
implemented using vlh and v hl
with slope
It is
necess
(at least
efficiency,
weakly) outside the interval of the Pareto frontier
— 1.
important to emphasize, though, that Proposition 8 provides
sufficient,
but not
constraints are slack (as
when
ferable without efficiency
and
the conditions of Proposition 7 hold), and utility
loss, as in
sufficient conditions for
Proposition
6.
productive efficiency,
33
is
trans-
However, in order to state necessary
we must
confront the possibility that a
range of policy vectors implements the same equilibrium values. In such cases, we focus on
equilibria that satisfy
two
we
of productive efficiency. Second,
as firm 2 in state (H, L).
utility is transferrable,
utility in states
symmetry
To
Qlh
q\ H
see that such a
scheme
exists,
sum
in state (L,
1
H)
the
observe that as long as
of firm profits,
and
transfers
(L,L) and (H,H); the symmetry of the model then implies the desired
Suppose that
9.
Qhli v lh
=
1
implements an equilibrium value
(p, q, v)
ofV 1 and either assumption
Then q\ H G
.
the schemes should treat firm
the optimal scheme maximizes the
for each
i,
that
plh
v hli an d that no other such policy vector implements
(0, 1)
and qHL G
+ Atf(vl H D
)
u on
the Pareto
(a) or (b) of Proposition 6 is satisfied. Further,
and IC-Oniy are slack
that the off-schedule constraints
—
is,
of policy vectors.
Proposition
frontier
select the policy vector with the highest level
focus on schemes that are symmetric outside of states
where the cost types are equal; that
same
we
criteria. First,
(0, 1) if
and only
if for all e
jtt <
Y^=\
+ Vl{Vh -
Plh — Vh
1
Vl)
>
u
suppose
= Phl >
&h,
using a larger
0,
+ Kfivln)-
(4-4)
Further, q\ H = 1/2 and q\iL = 1/2 if and only if the second inequality holds at v HL
l
if and only if the second inequality fails.
while qLH = 1 and qHL =
v lh — vs
=
>
One
implication of this result
is
that
v^
if
—
<
vjj
(r
— 9 H )/8
(the
minimum width
required to implement productive efficiency using pricing and continuation-value Pareto
efficiency)
ing v
!
s
,
,
there
is
productive inefficiency even in the
so long as the off-schedule constraints
1
first
period of play,
when implement-
do not bind. Further, we see that
if
future
=
is extreme, so that A^/^f
0, the second inequality in (4.4) holds when
=
r if and only if (Pool) holds. Thus, we can interpret Proposition 9 as a generalization
Plh
of Lemma 5. In general, firms implement some productive efficiency; however, they stop
inefficiency
short of
full
)
productive efficiency
if
the slope of the frontier gets too steep or too
in particular, if the frontier is too narrow.
33
The approach
of Proposition 9 can be extended to the case where utility
firms; however, the conditions are
34
flat,
and
34
more cumbersome and more
is
not transferrable across
stringent.
if the upward on-schedule constraints are
l
= 1, and if A+/(vL // ) > -1, then qHH = 0.
then for e sufficiently small, A7/(vL H ) < -1, then q HH
In other words, the firms take the market shares to the extreme before incurring future inefficiency in state
Additional characterizations can be provided. For example,
slack,
(H,H).
It is also possible to show that, when off-schedule constraints are slack, / is differentiable, and
l
continuation values are interior, Pareto efficient points require }'{v
LH ) f'{v\j L ) = f'{v LL ) f'{v HH ).
Intuitively, the continuation values are chosen to balance the inefficiencies incurred in each state of the
all
world.
32
now summarize our
Let us
characterizations of Pareto efficient collusive schemes for
we
firms of moderate patience. First,
find that firms are willing to bear
when
ciency to gain productive efficiency in the present. However,
do not bind, and either the Pareto frontier
tain pricing
ciency
and continuation-value Pareto
In contrast, the firms
efficiency.
is
when they attempt
to
may
some future
ineffi-
off-schedule constraints
wide enough or (Pool) holds, the firms maineven at the expense of productive
efficiency,
and continuation- value Pareto
sacrifice pricing
implement asymmetric equilibrium values,
if
effi-
the off-schedule
incentive constraints directly or indirectly prevent the use of future market-share favors in
the event of
ties.
fairly efficient
may
Thus,
schemes
for firms of
moderate patience, we expect to
(at worst, there is
some productive
start the
inefficiency),
game using
but the schemes
incorporate additional inefficiencies following a series of one or more cost realizations
whereby one firm has lower cost than another.
4.5.1.
Computational Examples
Consider again Figure
productive efficiency
Notice that for a wide range of states (9 to 22), (approximately)
4.
is
implemented, as predicted by Proposition
tinuation values (in states
the frontier
21 use the
is
(L,H) and (H,L)) associated with these
always within the bounds specified in
same extreme continuation values (vlh
on the diagram.
Due
shift
efficiency loss
the frontier
is
(4.3).
—
24 and vhl
states, the slope of
Further, states 9 to
3);
these are labeled
to greater future inefficiency. Thus, in order to implement
a range of equilibrium values, firms find
and
=
and
(4.2)
extreme con-
to concavity of the frontier, shifting these continuation values to
become asymmetric would lead
fixed,
at the
8:
it
optimal to hold the extreme continuation values
the market shares and continuation values in
from shifting market shares
linear (as predicted
ties.
Naturally, there
is
no
in the event of ties; likewise, since the inter
by Proposition
3),
there
is
no
efficiency loss in shifting
continuation values along the linear portion of the frontier.
Further,
we observe that the
extreme values of the Pareto
firms use productive inefficiency
frontier. In particular, states 3
when implementing the
and 24 use productive
ineffi-
ciency; since these states (or less efficient ones) are reached with positive probability
from
every starting point, even the most profitable points on the Pareto frontier yield less than
the first-best profits. This
=
vlh
is
26, the corner of the
is
consistent with Proposition
Pareto frontier. Thus,
8:
when implementing
(4.2) fails, so that
state 24,
productive efficiency
not necessarily optimal.
Now
consider Figure
productive efficiency
1.57,
and incentives
to state
1
is
5.
Across
all
but the most extreme states, the overall level of
approximately the same, with qLH
+ qHL
approximately equal to
for truth-telling are provided in a similar fashion,
following a realization of (H, L)
and to state 20 following a
These states correspond to the corners of the
33
frontier,
and so the
with the firms going
realization of (L, H).
fact that firms achieve
only partial productive efficiency
consistent with Proposition
is
Along the
8.
interior of
the frontier, the slope of the frontier stays fairly close to —1, and this indicates that the
firms are willing to incur
efficiency.
35
some future
inefficiency in order to
However, consistent with Proposition
inefficiency to
implement
implement the extreme
productive efficiency.
higher levels of
These features imply that the
states,
6,
implement partial productive
the firms do not use continuation-value
interior of the frontier
somewhat
is
optimally colluding firms capture some productive efficiency in the
but at the cost of greater inefficiency in the future.
However, to
nearly linear.
greater productive inefficiency
first
is
required. Thus,
period of the game,
36
Outside of states 7-17, the off-schedule constraints bind, and at the extreme points of
the frontier
we have computed,
it is
impossible to decrease the utility of the disadvantaged
As our computational algorithm does
firm any further while maintaining pricing efficiency.
not allow for pricing inefficiency, we have not determined
lowering prices.
5.
Informative
We now
if
the firms could do better by
37
Uninformative Communication
v.
consider the role of communication.
mative communication with the opposite
We
begin by contrasting the case of infor-
where firms are unable or unwilling
possibility,
to communicate. Recall that Section 2 specifies a particular extensive-form
communication occurs
and then firms make pricing
first,
However, market-share proposals only
posals.
are equal.
The main
affect
decisions
outcomes
game
in
which
and market-share pro-
in the event that prices
insights of Section 4 are robust to alternative specifications of the
extensive-form game, so long as the firms have an instrument to divide the market on a
state-contingent basis, and so long as pricing decisions are taken after communication takes
place.
However, when we consider how the game progresses in the absence of communicatiming assumptions play a
tion, the
To
much more important
begin, let us take the specific extensive-form
game
role.
of Section
2.
Interpreted
the absence of communication implies that the firms use the announcement
35
When
implementing states
thus, the firms
36
1
and
x
2,
q LH
«
1
a
literally,
l
l
(8
but the overall level of productive efficiency
go to state 18 or 19 following state (L, H) rather than state
)
is
=
N
lower;
20.
Relative to the linear self-generating set constructed in Section 4.2, however, the level of productive
efficiency
is
higher (1.57 in the computation versus 1.5 for the strictly linear set). Intuitively, within the
linear set, the level of productive efficiency
contrast, in Figure 5, states
in those states.
With those
1, 2,
and 20 are
(less efficient)
is
the
same when implementing each point
in the set.
In
available only because the firms sacrifice productive efficiency
equilibrium values in the
share favors available, and thus greater productive efficiency
37
is
set, firms
have greater future market
attained at the start of the game.
However, since the advantaged firm produces at least half the time in each state, lowering the price
any state would apparently lower profits for both firms, and thus leave the Pareto frontier. But, a more
rigorous argument is required to establish such a result, since we must consider more radical adjustments
in conjunction with a price decrease, such as reallocating market shares across states of the world.
in
34
now
(which we will
suppress in our notation) in
It
linked to communication;
we could formally accommodate such an
alternative assumption
=
by requiring that following the no-communication announcement N, ^(Oj)
make the
consider
might be argued that market-share proposals are inherently
market-share proposals.
Although the main qualitative ideas described
for all j, k.
Now
states of the world.
all
latter assumption, for consistency
we
2
<p
{Bk)
=
1/2
apply when we
in this section
instead follow explicitly the model of Section
whereby equally-priced firms are able to unilaterally withhold quantity from the market
2,
even
The
absence of explicit communication.
in the
difference
is
that a firm's price and
market-share proposals must then be decentralized, as these variables depend only on the
firm's
own
cost type.
In this context,
how
tive? This requirement has
both a downside and an upside. The downside
market-sharing arrangements that can be implemented
ever, is
perhaps
less severe
is
still
can be allocated
a state-contingent fashion, provided
in
38
consistent with decentralized decision-making.
2
simplest no-communication scheme might require p (0h)
ip
l
{6j)
—
ip
2
(9 k )
=
1/2 for
all j, k:
=
1
P (Bh)
>
this yields productive efficiency,
and even pricing
in the event of ties
that the set of
This restriction, how-
restricted.
is
is
than might be expected: in our model, even without informative
communication, market share
that the allocation
announcements to be uninforma-
are the firms affected by requiring
For example, the
2
=
1
P (9l), and
equal market shares
P {Ql)
Bertrand setting, arbitrarily small
efficiency (in our
by setting p 2 (B H ) > P 1 {Bh) > P 2 (8l) >
achieve productive and pricing efficiency, but now firm 1
price differences can allocate consumers). Similarly,
1
P {Ql)i the firms
wins all ties.
may
continue to
Notice that the two schemes just described could be implemented even
the market-share proposals altogether,
if
if
we eliminated
we instead maintained the assumption that con-
sumers divide themselves evenly among two firms charging the same
However, when
price.
decentralized quantity restrictions are included, the firms might specify a scheme whereby
p>(e H )
=
l
p
{B H )
> p 2 (9 L =
)
l
p {6 L ), and </(%)
=
2/3,
to share the market unequally in the event of ties.
pi{e H )
firm
1
market
=
2
p
(B L )
= p\B H >
to serve the
)
p (BL ), and tp^Bj)
market in
its
=
Compare our model
if
2/3,
=
p
2
(B k )
1/3. This allows the firms
also use a
=
communication.
scheme whereby
1/3. This
1
scheme allows
receives 2/3 of the
many market-sharing arrangements
are
39
to a Cournot model, where without communication, colluding firms typically suffer
the opponent turns out to be low cost. In contrast,
able to select the
(B k )
They could
large inefficiencies: a high-cost firm produces, just in case the
wasteful
2
entirety in state (L,H), while firm
in other states. Despite these possibilities,
infeasible without informative
38
1
tp
monopoly output
for the cost
opponent
when
is
high-cost as well, but this
is
the firms can communicate, they are
type of the firm that actually produces, and other firms
from producing. Thus, our base model presents a conservative case for communication.
39
This discussion highlights the role of our assumption that the game has simultaneous moves: the firms
might wish to move sequentially, if that were possible, to restore the ability to assign market shares in a
fully state-contingent manner.
refrain
35
An
environment without informative communication also admits an upside
for the col-
luding firms, once the off-schedule constraints are considered. In the absence of informative
communication, each firm must be dissuaded from deviating from the collusive agreement
after observing its
own
knowing the type
type, but before
of the other firm. In other words,
the off-schedule incentive constraints bind at the interim stage. For example, suppose that
an equilibrium
=
specifies q\ L
=
1/2, q\ H
=
x
q HL
1,
x
and
q HH
=
1/2. This market-share
allocation can be achieved without informative communication. The off-schedule incentive
constraint might bind for a low-cost firm
it
would then be tempting to cut price
By
contrast,
if
a low-cost firm
the entire market with probability
tion can
firm 1 were to
tive
then the off-schedule
then already anticipates winning
In this way, the absence of informative
communica-
promote cooperation, by preserving uncertainty about opponent play and thereby
softening the off-schedule incentive constraint.
relieves
that firm 2 has low cost:
2's cost type,
easily met, since firm 1
tjh-
know
and pick up the remaining 1/2 of the market.
slightly
were unaware of firm
1
more
incentive constraint would be
1, if
some
PPE,
of the pressure to give
since
it
becomes
40
Notably, uninformative communication
up productive
efficiency that
is
easier to lower q\ L while maintaining q\ H
present in an informa-
=
1
without violating
the off-schedule constraints.
We turn now to a formal representation of the incentive constraints
With appropriate
communication.
definitions,
we can
inally, let
for firm 1
in place,
v
under uninformative
represent the
x
represent the continuation value
k
and p2 (9k)- With these definitions
the on-schedule incentive constraints are again represented by IC-Oni^ and ICthat
is
induced by the price selections p
Oniy. To define the off-schedule constraints,
(6j)
somewhat
7
H}, IC-Offl^
decentralized pricing strategies. For j G {L,
£
it is
1
Vk[qjk(Pjk-9j)
is
easier to refer directly to the
defined as:
+ fivliA
k€{L,H}
> max
2
-
((p (0 L )
2
Vh(p (0h) ~
9j),
while the corresponding constraint for firm
now
For a given continuation- value set V, we
3?+
x
[0, l]
4
x
V —
4
>
{0, 1}.
We
let
IC-Off2^,
2,
C(p,q, v)
=
Oj),
is
o)
+ 8v\
defined analogously.
define a function C(p, q, v), where
1 if
decentralized strategies (a (8
40
In other words,
when
l
)
=
TV
{
',
p
i
l
(9
)
,
,i
ip (6 ))
:
the specified prices, market shares or
continuation values require informative communication, while C(p, q, v)
l
C
=
if
there exists
that can induce the specified prices,
firms do not communicate, they can pool off-schedule incentive constraints across
different information states.
This is broadly analogous to Bernheim and Whinston's (1990) finding that
multi-market contact between perfectly-informed firms serves to pool incentive constraints and facilitate
cooperation.
36
market shares and continuation values. Notice that continuation values can only be statecontingent to the extent that the state of the world
When
divisions.
C(p,
q, v)
=
1,
revealed by pricing and market-share
is
the off-schedule constraints defined above, IC-Offz 7 and
u
IC-Off-Mi, are appropriate, while the IC-Offi constraints are appropriate
The
when
feasible set for
C(p,
if
firms use uninformative communication can
q, v)
=
0.
be written as
follows:
u (v)
[
_
"
I
'
z
=
(P. q>
v £Z(V)
)
C(p,
:
=
For
0;
all i
IC-Omu and IC-OflW
IC-Onix,,
I
q, v)
=
j G {L, H},
1, 2,
hold.
Notice that, since the off-schedule constraints are different from the case of informative
PPE,
neither
Tv (V)
nor
^{V)
is
can then be characterized as the largest invariant
f u (v) =
1
An
initial
observation
reservation price
J-
U (V)
is
'
and the
is
'
The
a subset of the other.
set of uninformative
set of the following operator:
= (p.q.v)e^(v)
for i = 1, 2, «* = U\z).
{v}y): 3a
l
| such that
PPE
1
J
PPE, even
that in an uninformative
if
firms collude at the
off-schedule constraints are slack, the set of feasible policy vectors
may not be convex, so that we rely
can randomize among continuation equilibria. 41
not convex. Thus, the set of equilibrium values
more heavily on our assumption that firms
We now
hurt firms
if
discuss several circumstances under which restricting communication might
l
the off-schedule constraints do not bind. First consider the choice of qLH In
.
regions where the continuation value frontier
tion value frontier
is
is
too steep or too
or
flat,
if
the continua-
too narrow, Propositions 8 and 9 establish that the firms implement
productive inefficiency. In such cases, intermediate values of q\ H are typically optimal, so
that the restriction to uninformative communication
examples
and 4
in Figures 3
illustrate
may be
The computational
costly.
a range of equilibrium values where, for the (p, q, v)
that implement these values, C(p, q, v)
=
1.
In Figure 4, the (p,q, v) that implement the
middle states clear the off-schedule constraints with slack, and thus restricting attention to
U
(p,q, v) € !F (V) unambiguously lowers cartel profits.
More
formally, observe the tradeoff
can be characterized
q\ H
=
in
i
Of
course,
it
+
efficiency
a manner analogous to Proposition
1/2 can be improved upon by a scheme where q\ H
IC-Onlu (holding the
41
between productive
rest of the
[f{v\ H )
-
scheme
=
1
A
scheme
and v\ H
is
inefficiency
(p, q,
v) where
chosen to satisfy
fixed), if
f(vl H )]/(vl H
-
v\H )
<
H-
(B.
e L )/(PLH
-
could be argued that randomizing between continuation equilibria
firms cannot communicate; however, from a formal perspective, there
on an ongoing
8.
and future
basis in order to randomize.
37
is
no need
eL ).
is
less plausible
for firms to
when
communicate
However,
if
the frontier
is
productive efficiency even
too narrow, or
if
eventually becomes too steep, firms sacrifice
if it
were
(4.2) holds so that,
available, a small increase in q\ u
it
(holding the market shares in other states fixed) would increase profits. Thus, restricting
communication can lead to greater productive
Similarly,
inefficiency.
it
would often be
l
optimal to choose intermediate values of qHH and q\ L typically not equal to one another,
,
if
those were available.
Thus, we have argued that there exist circumstances under which restricting commu-
A
nication to be uninformative lowers the profits available to the cartel.
second question
concerns whether these losses accrue for every discount factor, or whether the losses disap-
pear when firms are sufficiently patient or impatient. Clearly, restricting communication
does not hurt in the static Nash equilibrium, or
if
the firms are too impatient to implement
nonstationary market shares. Restricting communication also cannot hurt
patient enough to implement first-best in an uninformative
establishes that, even
when communication
exists a critical discount factor strictly less
ticular, the linear self-generating
is
prohibited,
than
1
PPE. The
we
still
above which
segment constructed
if
the firms are
following proposition
have the result that there
first-best
in Proposition
1
is
attained. In par-
can be implemented
without communication.
Proposition 10. Assume that r — 9 H < 9 H — 9 L and define 6 FB as in Proposition 1.
For all 6 G [6 FB 1], there exists an uninformative PPE that yields first-best profits to the
FB /(l - <5),tt fb
u
cartel: (n
/(1 - 8)) E V
,
,
.
Thus, restricting communication hurts collusive ventures, but only when firms are moderu we note that most of our results about
ately patient. For further characterizations of V
,
V
1
can be modified to characterize V u but we
,
will not
pursue that here.
Building on these results, we turn finally to a discussion of the qualitative features
of the set of unrestricted
PPE, where the
communication can depend on
period, the firms
first
choice of whether or not to use informative
One way
history.
to understand the
choose whether or not to communicate.
and face the IC-OfK
7
If
game
is
that in each
they communicate, they
they do not, they choose from
a restricted set of market-share and price policies, but they face the relaxed IC-Offi u
reveal their types
constraints. Formally, they can select
f(v)
=
{
|
(ul '" 2):
constraints;
if
from two distinct
3z
=
(p>q.
such that for
i
sets of feasible policy vectors:
v )e{^/ GOu.F'0/)}
=
1, 2,
u*
=U
l
1
(z).
J
Consider a qualitative example where firms choose to refrain from communication. Suppose that the firms implement productive
efficiency, so
that firm
deviate in state (L, H), and they wish to implement a scheme that
2.
If
is
1
has no incentive to
biased in favor of firm
the firms attempt to reduce the market share or continuation value of firm
38
1
in state
By
(L, L) too far, IC-Offl£ L will bind.
refraining from communication, the firms can pool
the (L, L) off-schedule constraint with the non-binding (L,
allowing for a lower profit for firm
1
in state
(L,L)
(e.g.,
H)
off-schedule constraint, thus
market share equal to
0).
More generally, our analysis suggests that firms use communication in order choose a
r
policy vector from J- (V), so as to implement market-sharing arrangements that are not
when
available using decentralized schemes; however,
and when
ing the off-schedule constraints,
there
significant gain
is
from relax-
the "ideal" market shares are close enough to
a scheme that can be implemented without communication, firms refrain from communicating, choosing a policy vector
state
at r
from
TU (V).
For example, in Figure
5,
the utility pair of
could be approximated using a decentralized scheme, where firm
1
—
e,
and firm
when
2 prices at r
its
cost
Then, firm 2 receives no market share when
with firm
1
when
its
from communication,
frontier,
cost
it
is
Once the
low.
may be
is
its
high and at r
cost
utility pair to
e
is
1
when
high, while
off-schedule constraint
possible to shift v\ L
producing an equilibrium
is
—
always prices
its
cost
splits the
it
is
low.
market
relaxed by refraining
l
and vHH to the Southeast along the
the Southeast of state
equilibrium value could be used to provide future market-share favors
1.
In turn, such an
when implementing
other states, increasing productive efficiency across the frontier.
6.
Symmetric
In this section,
PPE
we explore
market-share favors.
We
firms can communicate;
the role of our assumption that firms have access to future
examine the way
(ii)
in
which collusive behavior changes when
no bribes are permitted; and
(iii)
(i).
firms are restricted to use
Symmetric PPE. Although there may be only limited circumstances where firms can communicate but not track behavior over time, an examination of
fies
we
this possibility further clari-
the consequences of the institutional environment for collusive conduct.
42
In any case,
when it is available, informative communication is always weakly
a Symmetric PPE, and is strictly dominated for an intermediate range of
establish that even
dominated
in
discount factors.
Working with a continuous-type model
allowed, Athey, Bagwell
Lemma
which communication between firms
same
is
not
stationary, and, under a wide range of parameter values,
price
and win with equal
probability.
5 to confirm the analogous result for the discrete-type
In this section,
we
all
use
model with communication.
Consider the content of the restriction to Symmetric PPE. Formally, the set
42
is
and Sanchirico (1998) have previously established that the optimal
symmetric collusive scheme
cost types charge the
in
Vs
of
an example where communication might be possible, consider the common practice in art
A group of agents could meet before
an auction without revealing their clients. If clients can use different agents in different auctions, their
identities can remain concealed.
To
see
auctions of using agents to represent buyers, often anonymously.
39
Symmetric
PPE
values
is
the additional restriction u
Lemma
2,
we
1
=
u
2
on the
V
Consider a set
(u ,u
G
)
V
is
Lemma
apply
that
is
=
3z
:
(p,q,v) G {J#{V)
V
=
{(ti
1
,^
(V)
is
1, 2,
2
:
)
V
logic of
u
=
u
U F{V)}
=U
2
V
l
'(z).
any
(v},v?)
S
D sup{« G V}}.
T
(V)
satisfying
Lemma
since
,
u1
=
u2
for all
Thus, we can
5 implies that
available, the firms simply use the best continuation values in
implemented using iaax.{(v},u
2
)
Lemma
Lemma
5 indicates that the largest point in
G V} as a continuation
a stationary equilibrium of the repeated game
since the schemes of
1
Clearly,
5 to characterize the best point in
the set regardless of the state. Formally,
T
=
i
closed and bounded.
contained in
even with a larger set
S
Applying the
set of values permitted.
such that for
2
which imposes
seek to analyze the operator
(u\u 2 )
l
T S (V),
the largest invariant set of the operator
(i.e.
Vjk
=
value, corresponding to
maxV s for
all j, k).
Further,
5 can be implemented without informative communication,
the firms choose not to communicate.
Proposition 11. There exists a 8 s < 1 such that, for all 6 G (6 s 1] 43 (i) If (Pool) holds,
1
then the optimal Symmetric PPE is stationary, and it yields value to each firm ofu — (r —
E[9])/(l — 6). This is implemented as follows: each firm prices at r in every period, communication is uninformative and the firms divide the market evenly, (ii) If (Pool) fails, then the
optimal Symmetric PPE yields values u = (n H (r - 6 H ) + n L (l + r]H )(8 H - 6L )) /(2(l-<5)).
This can be implemented using a stationary equilibrium with uninformative communication
and the pricing scheme outlined in Lemma 5.
:
,
{
Since informative communication
it
provides no benefit;
exist discount factors
it
is
not necessary to implement the best collusive schemes,
simply tightens the off-schedule incentive constraints. Thus, there
under which maximal collusive profits can be implemented only
if
firms refrain from informative communication.
The optimal
equilibria for less patient firms are analyzed in greater
depth in Athey,
Bagwell and Sanchirico (1998), so here we simply highlight a few brief points.
(Pool) holds, the low-cost type
6
<
6
s
,
it
is
is
most tempted to deviate from the
rigid-price rule.
necessary to "separate" the cost types and specify a lower price for a low-
cost firm, in order to mitigate the incentive of such a firm to undercut. 44
(Pool) implies that the collusive profit
43
In particular,
expression
is
When
When
when
is
lower. In the computational
(Pool) holds, 6 s satisfies
^
more complicated (but straightforward
to
r ~ £e
2
=
\(r
-
6L )
compute given a
+
In this case,
example of Figure
5,
<5V. When (Pool) fails, the
parameter values), since
set of
might bind first for different parameter values.
which is developed further in Athey, Bagwell and Sanchirico (1998), is reminiscent of
arguments made by Rotemberg and Saloner (1986), although they emphasize that all firms must reduce
different off-schedule constraints
44
price
This
logic,
when a high and
public
demand shock
is
encountered.
40
6
=
6
s
=
.71; yet, in
the self-generating set of informative
PPE
values depicted in Figure
5,
the firms are able to achieve partial productive efficiency, and thus strictly greater profits,
than
in
the best Symmetric
Section 4.2 exists for 6
From the perspective
>
PPE.
and
.7001,
of policy,
Similarly, the linear self-generating set constructed in
we
this
scheme
also improves
upon the
rigid-price rule.
therefore conclude that regulations that might appear to
hinder collusion, such as withholding the identities of the bidders in a procurement auction,
might have the perverse consequence of leaving prices unchanged but lowering productive
efficiency.
7.
Bribes
we extend the model
In this section,
The
model as the Bribes model.
game:
firm
(v)
i
sends b
>
l
communication as defined
and we
to allow for bribes,
following stage
model
is
extended
added to the extensive form stage
is
to firm j; firm j receives jb
in our
refer to the
1
Observe that (ex ante)
.
not necessary to implement bribes; the firms
can condition bribes on the market shares realized ex post. To simplify the exposition, we
communication; however, our main results carry over to
restrict attention to informative
endogenous communication. The exogenous parameter 7 £
of the bribe;
7
=
1
corresponds to the use of
incentive constraints,
we
both firms send bribes,
first
describes the inefficiency
money withou
observe that optimal collusion never requires a state in which
since,
a single bribe
efficiently if
[0, 1]
is
with 7
1,
the desired net transfer can be achieved most
made from one
firm to the other.
With
this observation
imposed, and under informative communication, the off-schedule constraint
be written
is
2, it
b) k
+ 8{v)k -
v
1
)
>
max(<z?fc (pjfc
-
OJ,
defined analogously; likewise, IC-Off-Mz s
natural way.
firm
can
as:
7 b% IC-Off2^.
for firm 1
Notice that
when
IC-Offl^. holds,
if
q}^ - pjk
is
)).
(IC-Offlf
fc
)
constructed from IC-Off-Mz in the
firm
1
is
assigned to send a bribe to
never has the incentive to withhold the bribe after production takes place. After
41
production takes place, firm
v})
>
0.
But
1
wishes to adhere to the equilibrium play
by IC-Offl^.. In words,
this inequality holds
it is
if
7^ — tfk + 5( vjk~
more tempting to deviate
from the agreement before production takes place, as the deviating firm can capture the
market and avoid paying a bribe
if
that
is
required, than after production takes place,
when
the firm can only avoid paying the bribe.
IB
Let
(V) be defined as .F (V), once the utility functions and constraints from the
F
base model are replaced with the utility functions and constraints with bribes.
vector
is
now
(z,b),
where z
=
(p,q, v). Finally, with ex ante utility given as
The
U
Bl
policy
(z,
b),
let
f B (v) =
u2)
(ttl
-
3(p q v b) eTlB{v)
:
'
J
\ such that for
i
'
'
=
1, 2,
u*
= U Bi (z, b).
\
J
We denote the set of PPE values in the Bribes model as V B which following our previous
arguments is the largest invariant set of T B Let vf be the point on the Pareto frontier of
V B that gives equal utility to both agents.
,
.
Suppose that (r - 6 L )/(28) < v\ B - v} Then for all 7 < 1, if any
productive efficiency is implemented (qi H > 1/2), the equilibrium is non-stationary. If 7 =
B (ii) Assume r — 9 <
1, there exists a non-stationary (p, q, v) that implements v
H
9h — 9lB
B
For all 7 < (=)1, there exists 6 < 1 such that, for all 6 € [8 1], bribes are never used
(resp. not necessary) along the equilibrium path in the most profitable equilibria for the
Proposition 12.
(i)
.
.
,
firms.
Proposition 12 examines the extent to which bribes replace market-share favors in imple-
menting Pareto
efficient
equilibrium values. Recall that
so long as the off-schedule constraints
frontier of the set of informative
Thus,
ther,
it is
when
initially
more
PPE
efficient to
sufficiently patient firms
are dominated.
we have already
established that,
do not bind at the symmetric point on the Pareto
values, the frontier has
an open interval of slope
— 1.
use market-share favors than inefficient bribes. Fur-
can attain
first-best
without bribes,
inefficient bribes
45
Proposition 13. Fix 6 and 7, and suppose that (p, q, v, b) implements a Pareto efficient
B
value u G V
Suppose that p H L, Plh > 6 H If 1 - 7 < (9 H - 6 l )/(plh - 6l) and
—
—
1
7 < (9h
9l)/(phl — 9l), then the scheme uses productive efficiency.
.
.
Proposition 13 characterizes the use of bribes to provide incentives for productive
ficiency,
when
firms are not patient
the set of informative
PPE
values,
enough to implement
first-best.
ef-
Bribes can expand
whenever the equilibrium without bribes
calls for
the
45
In a continuum-type model, the Pareto frontier has slope equal to — 1 at the Pareto efficient equilibrium
providing equal utility to both firms, if firms are sufficiently patient such that the off-schedule constraints
permit small future market-share favors. Thus, for any 7
not fully replace future market-share favors.
42
<
1,
even
in
a continuum-type model, bribes do
use of continuation values that entail greater future inefficiency than the inefficiency of
bribes. Figure 6 illustrates
how
bribes
augment the
set of continuation values available to
the firms. Recall that Proposition 8 establishes that firms use productive efficiency unless
the continuation values are not available or require too
much
future inefficiency, even
if
the
off-schedule constraints bind.
Thus, so long as bribes are suitably
main
analysis highlights an important theme: the
firms
is
firms use productive efficiency.
efficient,
factor limiting productive efficiency of
the availability of an instrument for transferring
constraints. Intuitively, in the absence of bribes,
then a market-share transfer
share allocation
is
may be
if
today's productive efficiency, and, ideally,
4,
utility,
rather than off-schedule
firms achieve productive efficiency today,
required tomorrow; therefore, tomorrow's market-
burdened with twin objectives:
have shown in Section
This
it
also
it
must embody the required transfer
must be productively
When
these objectives can sometimes clash.
however, they can be used to achieve the
efficient itself.
for
As we
bribes are available,
objective, leaving tomorrow's market-share
first
allocation free to achieve the second. Bribes can thus enable a substantial
improvement
in
productive efficiency for firms attempting to effect transfers, provided that the associated
transactions cost (1
—
7)
is
low enough.
of contrast, now consider the use of bribes in the case of Symmetric PPE.
1
2
B
BS
Let T
(V) be defined from T (V), imposing the additional constraint that u = u and
BS be the largest invariant set of T BS From the perspective of designing collusive
let V
As a point
,
.
we can think of bribes as enlarging
lemmas from Section 4.4, we have:
agreements,
Using the
Proposition 14.
(i)
the set of continuation values beyond
The Pareto Optimal equilibrium value
using a stationary collusive scheme;
sufficiently patient, the
(ii)
If
1
— 7 <
optimal collusive scheme
efficiency, pricing efficiency
and the use of
is
it is
it
critical
@l)
and
it
entails productive
and firms are
critical
discount factor for support-
critical
when
1
When
the firms do not communi-
discount factor for sustaining collusion with
—7 <
(Oh
— &L)l( r — #l)
is
strictly lower
discount factor for sustaining rigid prices and productive inefficiency
These
results have
two main implications
In terms of applications, there
may be
(i.e.,
than the
when 7
for applied analysis of collusion.
observe that firm-specific reputations and non-stationarity
46
—
important to note that this optimal collusive scheme can
can be easily shown that the
productive and pricing efficiency
can be implemented
Ol)/(t
stationary,
be implemented without informative communication.
cate,
—
V BS
.
bribes.
Although we do not include the computation of the
ing efficiency in this model,
(Oh
in
V s 46
=
First,
0.
we
market-share favors) are
only limited circumstances where firms cannot track indi-
vidual behavior over time, but they can use individual-specific bribes as conditioned on current-period
announcements; however, an example
is
an auction market where the bidders use agents
art auctions), agents represent multiple bidders
(as
is
common
and bidders can switch agents without observation.
43
in
robust features of collusive ventures, so long as bribes are inefficient and individual firm
behavior can be tracked over time.
path price wars) are not useful
in
Non-stationary equilibria (including on-equilibrium-
two extreme
with side-payments
cases: legalized cartels
cartels facing informational or other (unmodelled) constraints that force
and
Symmetric PPE.
them
into
47
Second, our results have a somewhat perverse policy implication. To the extent that
bribes are used, in either symmetric or asymmetric
efficiency of the cartel. For
collusion at high prices,
On
increase the productive
discount factors and parameter values, firms can sustain
and the only
can implement productive
reduces welfare.
many
PPE, they
efficiency.
issue for the cartel
Thus, for
many
is
the extent to which they
discount factors, prohibiting bribes
the other hand, there do exist moderate discount factors such that
prohibiting bribes lowers the collusive profits enough that collusion takes place only at
substantially lower prices. For moderately patient firms, therefore, a prohibition on bribes
may
8.
raise
consumer
welfare.
Policy and Empirical Implications
Our
analysis has a variety of implications for policy.
One theme
ing firms tolerate productive inefficiency before lowering prices.
welfare, that
may
is
bad news:
policies that limit firms' ability to
that successfully collud-
is
48
From the perspective
communicate and use bribes
serve mainly to limit productive efficiency, without having a large effect on prices.
These
policies increase
consumer welfare only
if
firms are sufficiently impatient that remov-
ing these instruments destroys their ability to collude at high prices.
bribes
all,
of
may
not have
much
effect at all, unless the firms
49
Further, prohibiting
have moderate patience.
our findings therefore provide some formal support for those
(e.g.,
Bork (1965,
Over1966),
who are attentive to the possible efficiency gains that restraints of trade may
and who therefore question the unqualified application of the per se rule against
Sproul (1993))
afford,
price fixing.
50
Similar results hold
when we
consider designing markets (such as auction markets for
47
Of course, in reality, tracking a complicated, non-stationary agreement, and potentially communicating
about cost information, also has costs, so our comparison between inefficient bribes and future marketshare favors may be biased against bribes; nonetheless, our results indicate that market-share favors can
be quite effective as a collusive mechanism for sufficiently patient firms.
48
This theme arises in the base model, as a comparison of Propositions 6 and 9 confirm. It also appears
when we
consider policies that prohibit communication (see the discussion preceding Proposition 10),
impose anonymity
49
Of
(see Propositions 11
and 14) or prohibit bribes
course, in our inelastic-demand model, social welfare
is
(see Propositions 13
independent of
and
14).
price. Qualitatively similar
would also arise in a model with downward-sloping demand, however.
At the same time, we note that our findings depend somewhat on the ability of firms to use small
differences in prices to allocate market share. As discussed in footnote 38, in a Cournot model, restricting
communication might improve consumer welfare.
findings
50
44
government contracts or internet auction markets) to withhold firm
identities.
51
If
firms are
patient enough to support the fixed-price equilibrium, policies that impose anonymity only
reduce productive
efficiency.
On
the other hand, there are discount factors under which
PPE entails productive efficiency and lower
best asymmetric PPE entails some productive
the best Symmetric
prices
when
firms have low
cost, while the
inefficiency
but monopoly
prices.
Another implication of our model concerns merger
policy.
Although our formal model
considers only two firms that draw from identical cost distributions, the modeling frame-
work can be
As
is
easily
extended to allow
for multiple firms
customary, with more firms, the
critical
with asymmetric cost distributions.
discount factor for supporting cooperation
goes up. But asymmetries also frustrate collusion
among impatient
pects higher cost and/or lower market share in the future
a collusive agreement.
It
thus
difficult for
more tempted
who
ex-
to deviate from
asymmetric firms to implement
Accordingly, a merger that creates asymmetries across firms
future market-share favors.
may impede
may be more
is
firms: a firm
the provision of market-share favors and thereby the attainment of productive
efficiency.
In terms of empirical implications, our results suggest the following taxonomy:
very
impatient firms use low prices that vary significantly with costs, and implement productive
efficiency; slightly
firms
who
are
more patience leads
more patient
still,
but
to higher prices, but
who
still
prices that vary with cost;
face informational or coordination constraints
that prevent tracking individuals over time, use high prices that do not vary with cost, and
each firm's market share
is
independent over time; and
finally,
firms
who
are patient
and
sophisticated use high prices that do not vary with cost, but they implement productive
efficiency
analysis
and a
is
firm's
market share
is
negatively correlated over time.
An implication of our
thus that market-share volatility and productive efficiency are non-monotonic
in the "extent" of collusion, being greatest
when
firms interact noncooperatively or
employ
sophisticated collusive schemes.
Our
results also provide
some
insight into the role of institutions, such as industry asso-
ciations, industry trade publications
and the celebrated "smoke-filled room."
a potential benefit to communication:
it
We
identify
allows firms to allocate market share in a state-
contingent fashion. Clearly, this only has value in collusive schemes that implement some
productive efficiency. 52
51
On
the other hand,
we have seen
that firms
may
prefer to avoid
Indeed, an interesting question for policy concerns the design of formal auction markets: should the
auctioneer reveal the identities of the bidders?
For example, an internet market can be designed using
can be designed so that participants are required to publicly reveal their identities. In the
former case, the market-maker provides the service of allowing participants to meet anonymously and still
conduct transactions; in the latter case, the market-maker provides the service of verifying identities (for
aliases, or
it
example, using credit cards).
5
In alternative models, communication might offer other benefits; for example, firms might share
information to better forecast future demand.
45
communication following some
after
histories, in order to
they learn that they will not produce.
We
about the use of bribes; they are mainly useful
keep impatient firms from deviating
have also provided some negative results
for firms of
expect difficulty in tracking identities over time.
They
moderate patience, or firms who
typically
do not
fully replace
market-
share favors as a mechanism for providing incentives, and thus an optimally-designed cartel
allocates current production in a
Our
manner that
is
sensitive to firm-specific histories.
results provide a lens to guide future empirical
lusive behavior.
may be
However, market-share favors
not necessarily generate the same kind of "paper
where side-payments were permitted
But there are more recent examples
lusion
among
difficult to
come from
series of side-payments.
historical
examples of
legal-
Stocking and Watkins (1946)).
(see, e.g.,
as well. For example,
col-
document, since they do
might a
trail" as
Indeed, most of the detailed case studies of cartels
ized cartels
and case-based research about
McMillan (1985) describes
col-
firms in the J
ith our formal analysis,
and sometimes even
ture market-share favors,
McMillan reports that firms use
bribes, as a
means
fu-
of providing incentive for
honest communication so that greater productive efficiency can be achieved.
With the
recent increase in anti-trust enforcement of price-fixing cases in the U.S. (and
the incentives
it
may be
it
provides to conspirators to divulge details of the workings of the cartel),
53
possible to gain further insight into the institutional design of individual contem-
porary cartels and to document the kinds of behavior studied in this paper. In
many
of the
recent cases, market-share allocation was a primary concern of the cartel; examples include
citric acid, feed additives,
marine
services, graphite electrodes,
tels differed in their sophistication
of explicit
communication
rules.
for these cartels, particularly
and used side-payments and
Interestingly, other cartels used a
favors: in the case of citric acid, cartel
year,
car-
and design. According to prosecutors, the main topic
peared to be market-share allocation. Some of the
individual customers
and vitamins. 54 These
cartels,
among
high-level executives, ap-
such as marine services, allocated
profit-sharing, following a written set of
mechanism
closer in spirit to our market-share
members examined market
and any company who sold more than a
pre-specified
the following year by purchasing from other cartel
shares at the end of each
amount had
to
members whose market
make
shares
it
fell
up
in
below
the target.
More
generally, the existing literature
on collusion typically does not distinguish between
market-share allocation schemes that implement productive efficiency and those that use
pre-determined (and thus
53
The
allocation rules.
U.S. Justice Department began a "leniency"
price-fixing case to confess
54
inefficient)
is
immune from
program
For example, in the case of bid-
in 1993,
whereby the
first
conspirator in a
criminal charges.
See Business Week (July 27, 1998), "Justice's Cartel Crackdown," pp. 50-51 for more discussion.
46
common, but many
rigging at auctions, bid rotation schemes are
involve randomized market-share allocation
Schankerman (1975) analyzed
and they showed that
cartels
all
famous examples
of the
"Phases of the Moon").
(i.e.,
Comanor and
prosecuted cases of bid rigging over a 50-year period,
with a smaller number of bidders were more
likely to use
bid rotation schemes than "identical bidding." Although they did not provide
about the nature of the bid rotation, further study of the legal testimony
discern those rotation schemes that
made
much
may
detail
allow us
use of market-share favors or bribes to implement
productive efficiency. In other applications,
it
may be
possible to verify the prediction of
negative correlation in the market shares of firms using sophisticated rotation schemes as
opposed to randomized rotation schemes.
9.
Conclusions
While the
results in this
applied to a
number
paper are motivated by the problem of collusion, they can be
of other contexts.
At a general
level,
our model considers interactions
between agents - such as family members, workers in a firm or politicians - who must
repeatedly take actions in an environment with two main characteristics:
payments.
each firm's
from period to period, and the actual change
cost or benefit of taking the action changes
private information;
first,
and second, there are
limits
Essentially, the repeated play of
on the
is
firms' ability to use explicit side-
any of the standard multi-agent mechanism
design problems (public goods, auctions, bargaining)
fits
into the framework, with the
additional assumption of restricted transfers.
Private information
may be privately
it is
is
easy to motivate in
many
contexts. For example, family
informed about how tired they are on a particular day, and thus
to perform household work. Likewise, division heads within a firm
information about the efficiency of access to a resource, and politicians
information about the costs of legislation.
transfers also arise frequently: families
share a
common
members
how costly
may have
may have
private
private
Limits or transaction costs associated with
may
common budget, division heads may
may be illegal. Social norms may also
share a
resource and payments for votes
prohibit monetary transfers in certain circumstances. In this broader context, our paper
can be interpreted as providing a formal analysis of the hierarchy of relationships available,
as well as the extent of efficiency achieved
by each type
establish that the promise of future favors
may
use inefficient bribes in the present.
55
of relationship. For example,
we
eliminate the need and desire of agents to
55
Holmstrom and Kreps (1996), who study the use
monetary payments for favors. Our analysis differs from theirs in that we
bring together the tools of dynamic programming and mechanism design to characterize optimal equilibria
for firms for a given discount factor, and we explicitly model the tradeoff between different kinds of sideIn this regard, our results are similar to those of
of "promises" as opposed to
payments.
47
^From a methodological
paper
is
the
first
perspective, our analysis highlights several points.
which we are aware to provide tools
of
use of market-share favors by impatient firms.
comparison between
them both
for characterizing the
The problem
among
We
those instruments.
identify these tradeoffs
static
and dynamic analyses of
and explore
it is
possible
Second, we develop the
to provide rich characterizations of optimal collusive schemes.
between
optimal
and impatient
theoretically as well as using computational examples, showing that
precise connections
our
of cartel design motivates our
a variety of instruments available to the colluding firms,
firms face real tradeoffs
First,
making
collusion,
clear the
similarities and differences, and laying the groundwork for treating other repeated-game
problems within the mechanism design framework. Third, our work motivates some new
questions for static
them.
56
mechanism
design,
and takes some
towards addressing
initial steps
Finally, our comparison of symmetric and asymmetric equilibria indicates that
symmetry can be a
significant restriction
eliminating side-payments in static
on the
set of equilibria,
mechanism design problems
much
as restricting or
yields drastically different
answers than allowing for budget-balanced side payments. This observation motivates more
two kinds of
careful exploration of the classes of environments suited to the
Appendix
10.
Lemma
Proof of
For any (u 1 ,u 2 ) 6 T'{V), there exist s G
2:
that satisfy incentive compatibility conditions and yield u
tions uniquely define prices,
can define the associated
(p, q, v).
constraints of
l
=
1
tt
It is
game
is
l
(Q
)
^
6
l
,
T
(V) are
T / (F)
satisfied.
S and v
(s)
+
in
6v
1
:
A2
(s(6)).
x
K4
-> co(V)
As these func-
each state of the world, we
straightforward to verify that any (p,q, v) that can be
Z(V), up to payoff-irrelevant choices
in the feasible set
example, prices greater than r that receive no market share).
the incentive constraint in
a
l
market shares and continuation values
attained in the extensive- form
(for
equilibria.
It
then remains to verify that
implies that the on- and off-schedule incentive compatibility
Any
desired on-schedule deviation can be achieved
by using
while any off-schedule deviation can be achieved by choosing the appropriate 6? and
2
2
p\ Thus, (p,q,v) E^iV), and thus (u\u ) G f^V). Now consider a (u\u ) € f J (V). Again,
any (p,q,v) €Z(V) can be achieved with the appropriate s € S and v
A 2 x 5J4 —» co(V).
l
l
%
The on-schedule incentive constraints imply that a (6 ) —
and the off-schedule incentive conl
l
x
straints imply that the most profitable deviations, a ,p ,(p give lower profits than the proposed
:
,
,
equilibrium.
Proof of
Lemma
4:
Imposing pricing
efficiency,
productive efficiency and Pareto efficient
56
For example, we examine how restrictions on transfers (for instance, to a convex set) affect optimal
mechanisms. In the literature on collusion, only a limited class of restrictions on transfers have received
Cramton and Palfrey (1990) and Kihlstrom and Vives (1992) study the role of participation
McAfee and McMillan (1992) distinguish between "weak" and "strong" cartels, where
weak cartels cannot implement any side-payments and strong cartels can write enforceable contracts and
make side-payments. The model of Symmetric PPE can be interpreted as a generalized model of weak
attention.
constraints; while
which firms can impose symmetric "penalties" in response to the placement of a low bid in a
procurement auction. The strong-cartel model is closer in spirit to the model of Asymmetric PPE: transfers
can be made from one firm to another, but only imperfectly, using asymmetric continuation equilibria.
cartels, in
48
continuation values
+ vjk = 2K
v lk
(i.e.,
-
and IC-On2£i respectively bind
=
=
for all (j, k)),
and only
if
straightforward to show that
it is
IC-Onl^
if
- H }{q H {qHH - 1) - VlQll} + 6 {Vh(vH h ~ v\ H ) + Vl(v hl - v LL )}, and
A r - e H]{mqHH + nL{.l-ql L )}-5{ilH(v]{H -v}{L + L {vl H -vl L )}
l
l
[r
ri
)
Adding the constraints
— vjjH =
yields the necessary condition v HL
(r
— H )/6, and we may choose
the remaining market shares and continuation values to respect the additional conditions in the
lemma while satisfying each of
Proof of Lemma 5: We
in
from
(3.1) for
„2
U
l
(L, L; z).
the above constraints.
posit that the
We
2
max
„2
„2
(PjH
QjH
„2
PLH, PHH, PHL^r;
+A
downward IC-On
constraints bind, and substitute
solve a relaxed program:
- Oh) +
Sv H
+ VL
2
q L (9 H
- 6L )
v^^jjOK
(1
(PHk
QHk)
-0H +Sv h
)
+t] L
U
(1
k€{L,H}
k£{L,H}
W
E
E
VkQHk
ke{L,H}
+ ^-\ql-qn}
mqlk
k£{L,H}
which is non-negative on the Pareto frontier.
The multipliers on the monotonicity constraints are denoted ip l and these are also non-negative.
By inspection, it is clearly optimal to set pnk — r, Pjh = r and v H = K; then, differentiating
Let A be the multiplier on firm
l's utility constraint,
,
with respect to the market-share variables, we
d
(i
-
rf
L (eH
A)
-
eL )
get:
- iiP-n L + v>V;
*?LL
r\hir
2
dqLH
d
9 Thl
(i)
Notice
unless
ip
1
first
>
if
2
>
ip
Notice that these
Oh)
- ^VlVh{0h -
VlVh(0h - L ) -
At? l (t-
L)
-
-
A)[?7tf (r
- 6 H )} + ^VH ~
ip
2
VH
^Vh - ip 2 r]L
- H + ip^rjL + ^"Hh
)
0,
1)
productive inefficiency (q^ = q^). Now suppose
and consider asymmetric solutions. Still, there will be no
>
0.
monotonicity constraints are binding, and there
that
(1
we maximize the sum of the firms' utilities (A = 1), (Pool) implies that
2
we increase qLH
and decrease q HL until the boundary constraints bind.
variables move in the direction of inefficiency. At a symmetric solution, both
that
or
~
d
2
dqHH
we weight the
firms evenly (A
productive efficiency unless
ip
1
+
=
ip
2
is
Suppose that
2
> and
2
qHii = 1 and
tp
2
1
i(j
— 0. Then, the objective
= 0. But then, firm 2's
increasing in q UH and decreasing in q\ L so we take
q\L
monotonicity constraint implies tjh + "nhq\n < VHqHL> a contradiction.
is
,
Thus, we have established that the largest joint
utility available to the firms is achieved by
and
that
allowing
allocations of utility will not improve
for
asymmetric
cfa,
the sum of utilities. This scheme satisfies all of the constraints in ^o„(V). So, an upper bound
for the sum of utilities is given by r — E[0\. Now observe that for any a G [0, 1], we can allocate
a(r — E[0}) to firm 1 and (1 — a)(r — E[9\) to firm 2 by simply changing the market shares of the
"pooling,"
where q^
=
49
firms while maintaining q\
—
(fa.
IC-On
Since this satisfies the
constraints, the Pareto frontier
is
given as in the statement of the lemma.
Under the alternative assumption that (Pool) fails, inspection of the program shows that
when q\ H = and q\ h = 1 The monotonicity constraints do not bind. At
program is independent oipLH and phl- This scheme can be implemented
relaxed
these values, the
by using pricing efficiency in state (H,H) (j>hh = f), Vj k = K for all i,j, k, and productive
efficiency. The truth-telling constraints can be satisfied as follows: find p < r to be used by all
low-cost types. With q LL = q HH = 1/2, truth-telling by a high-cost firm requires:
(ii)
profits are highest
.
l
-Vh(t - Oh)
yielding the price stated in the lemma.
We
=
It is
{vh +!jVl)(p
now
- H
)
direct to derive the utility frontier.
To
compute 6 Fon consider the following system (imposing pricing efficiency, productive efficiency,
l
= x,
and v]k = 2-k fb /{1 -8)- v)k for all (j,k)): IC-Onl D IC-On2 D U 1 = x, tP = y, v LH
vjf H = x, q\j H = 0, and q\ L — 0. It can be verified that under our assumption that tjl >
r^a Since
In particular, v\ L - v\ H =
1/2, v\ H < v\ L
^f- G (0,1), this implies
Proof of Proposition
1:
solve a set of equations for the critical discount factor.
,
,
^1
.
,
.
< v ll < v hl-> wnere the latter inequality follows since the downward on-schedule constraints
l
= v\ H + r ~® ti It remains to verify that given our solutions to these equations,
imply that v HL
T
v hl = x + ~6 H < V- We can compute:
v lh
.
y-* =
rrj(9 H
-e L )-(r-e H )(2T) L -l)
—6
always positive under our restrictions tjl > 1/2 and r — Oh <
T~
as stated in the text, solves the equation y — x = s H It can be verified that y —
Fon
This expression
is
.
5
>
6
Oh —
x>
r~
0l- 6
6
Fon
H for
,
all
.
To compute
8
Fo
^
,
we observe first that IC-Offl£#
is slack,
as
implementation described above, IC-Offl^ implies IC-Offl^ L
for qjj
and w 1
computed above, and
.
is
IC-Off-Ml. Further, using the
We
then substitute
in the values
verify (using tedious algebraic manipulation) that IC-Offl£ L
when 6 > 8 Fo ^ where 8 Fo ^ is given in the text. Finally, since for
the endpoints of the interval, we have described a policy vector that meets all of our constraints
and uses as continuation values other values on the same interval, we can then construct the
and IC-Offl^^ are
satisfied
,
segment using convex combinations of policy vectors to implement convex
combinations of equilibrium values. This is possible because, when pricing efficiency is imposed,
the constraints and utilities are linear in market shares and continuation values. Thus, we have
remainder of the
line
constructed a self-generating set of equilibrium values with first-best profits to the cartel.
Proof of Proposition
and
q HL = q HH =
l
l
0.
2:
To implement
Further, set q\ H
—
q\L
(x, y)
=
,
let
prg~hr
pj k
'
—
r,
vlh
=
^ll
=
{x, y),
Finally, use the static
as the off-schedule threat point. Notice first that IC-0n2£>
vhh =
vhl,
Nash equilibrium
and IC-On2[/ both hold with equality
given these values. Given the specified market shares and off-equilibrium-path threat points, the
l
that
v LH
satisfies
IC-Offl£ L and IC-Offl£ H
is
uniquely determined. Next,
it
can be shown that
IC-Onl£> holds with these parameter values if and only if vjiL — v\jH — q^xir — 0h)/8. Thus,
it remains to verify that this distance is feasible using v
y, where y is determined as firm
HL
scheme. Given pricing efficiency and the specified level of productive efficiency
and since all continuation values he on a line with slope -1 through (x,y) it is
= x + y, as a function of the exogenous
compute the sum of firm profits at (x,y),
2's profit in this
for the cartel,
possible to
,
K
50
l
parameters. Then, since v LH
K — v\ H
and
.
It
—
x, v HL
can be verified that this
solving, the constraint binds for 8
VJ
~ qz+riH) +
\/(a(l
+)~
l
l
and only if vLH
){r — 0#)/(2<5)
+ {q\ H + q LL
constraint becomes relaxed as 8 increases. Substituting
y holds
—
VLJVL
8 hn
+
if
=
D)) 2
+4D(1 +
d)([?
2{& + rJl+r i + UnL {l +
]
Proof of Proposition 3:
implemented with g!-- = 1/2 and
(i)
uj-
The symmetric
=
lower vjj by
e/{28) until
we
2rlL ))
T(V) can be
we observe that we can takepjj > 6j
— pjj > 0, we can raise pjj by e and
point of the Pareto frontier of
v 2j. Before beginning,
without loss of generality. To see why, observe that
l
+ rjljl + rjj) + ^(1 + 2^))
arrive at pjj
and
Vjj,
if 8j
where
9j
=
pjj,
without affecting any
utilities
or incentive constraints (since an optimal off-schedule deviation would ensure zero market-share
in state jj).
To
see that the resulting
assumption that IC-Offrj
is
preserve this inequality, the
feasible continuation values
may employ a
t)*-j
is feasible,
observe that given market share of 1/2, our
slack implies that ^(9j
—
new continuation
v^ must
is
value
convex and symmetric,
similar adjustment, unless
u!-
=
v
l
.
But
Pjj)
it is
<
&{ v)j
—H
1
satisfy
feasible.
)',
since the adjustments
v^ > v
Finally,
in this case (OffSS)
is
l
.
if
Since the set of
8j
—
pjj
—
0,
we
violated.
an alternative utility pair, with no
2
1
efficiency loss, in which U is decreased and U is increased. We define two perturbations. In
l
Perturbation 1, we lower qHH
by e/{{phh — 6l)vh) and lower q\L by e/({pll — 9l)vl)- For
l
each firm i, IC-Onif/ is unaltered by this perturbation. In Perturbation 2, we lower qHH
by
l
—
—
lower
by
For
each
firm
i,
this
perturbation
leaves
and
e/({pll
e/((Phh
0h)vl)&h)vh)
q LL
l
2
unaltered IC-Om/j. Both perturbations lower U and increase C?
There are several cases to consider. Suppose first that, for a given A € {£/, £>}, IC-Oni^ is
slack for each i. If A = U, we use Perturbation 2 to engineer the desired utility transfer without
violating on-schedule incentive constraints. Likewise, if A = D, we use Perturbation 1. Next,
we modify the argument for the case where the on-schedule constraints are slack in different
directions. First, take the case where IC-Onlo and IC-On2[/ are slack. If pll
Phh, we use
Perturbation 1, which relaxes IC-On2£> by {phh-Qh)/{phh-Ql)-{vll-Qh)/{pll-Ql), which
is positive by our assumption that pll
Phh- ^Pll > Phh, we use Perturbation 2. This relaxes
IC-Onl[/ by (p HH - 9l)/(phh - Qh) - (pll - 8l)/(Pll - Oh), which is positive. Similarly, in the
second case, where IC-Onlj/ and IC-On2£> are slack, we proceed as follows: if pll
Phh, we
use Perturbation 2, which relaxes IC-On2[/ by (p LL - 6 L )/(p LL - 6 H ) - (p HH - &l)/(phh - H ),
which is positive; and if pll > Phh, we use Perturbation 1, which relaxes IC-Onlo by (pll ~
Qh)/(pll - 9l) - (phh - 8h)/(phh - 0l), which is positive.
(ii) Suppose that (a) and the first part of (b) fail: IC-Offl£
L and IC-Offl|^^ are slack, and
vll > x an ^ v hh > x Then, lower v HH by de/r]H and lower v\ L by de/rjL, and raise the
corresponding values for firm 2 by the same amount (this is possible because uj > x and the
set of available continuation values is convex). This does not affect any on-schedule incentive
constraints, relaxes firm 2's off-schedule constraints, and decreases U 1 and increases U 2 with no
efficiency loss, contradicting the hypothesis that (x, y) is the end point of the region with slope
equal to —1. Now consider the case where (a) and the second part of (b) fail: IC-Offl LL and
l
IC-Offl^ are slack, q\L >
and qHH
> 0, and pjj > 0j for each j. We will show below in
Proposition 8 that under the assumptions of this proposition, q\ > q\, which implies that for
each i, one of IC-OniD and IC-Oniu is slack. We may now apply the algorithm used in the proof
Starting from this point, our approach
is
to implement
.
-
51
of Part
to
(i)
efficiency loss,
implement a
without
utility pair that yields lower (higher) utility for firm 1 (2),
contradicting the hypothesis that (x, y)
the endpoint of a region with slope equal
is
to -1.
Proof of Proposition 4: (i) If each IC-Oniy is slack, prices are efficient, and q\ H = 1, then
IC-Offl£# and IC-Off-Ml are slack, and we can decrease q^ H and give the market share to firm 2
without violating any on-schedule constraints. But this makes firm 1 worse off and firm 2 better
off, violating the hypothesis that the scheme implements a corner of vj^.
and the first part of (b) fail: IC-Offl£ L and IC-Offl## are slack, and
Then, for e\ small enough, there exists an e 2 > such that it is possible
2
l
and
lower v\ L by £1/771,, and raise v Hli by £2/vh and raise v\ L by £2/%
to lower v HH by ei/tih
(this is possible because the set of available continuation values is convex and by the definition of
r
vN
as the north corner of the Pareto frontier). This does not upset any on-schedule constraints,
and makes firm 1 worse off and firm 2 better off. There is potentially an efficiency loss, however.
Next, we consider the case where (a) and the second part of (b) fail. We may then argue as in
the proof of Proposition 3 and arrive at a contradiction.
Proof of Proposition 5: The proof proceeds in a series of lemmas. Part (i) follows by
Lemma 7, and part (ii) follows by Lemma 8
Suppose that
(ii)
(a)
l
>vjj
v\ L >v]J and v HH
.
Lemma 6. Consider a scheme (p, q, v). (Tlj) If we subtract -q^dv from vj H and add r\ndv to
U 1 IC-Onlo and IC-Only are unaffected. (T2j) If we add r\ndv to i>£- and subtract rudv
2
from v H
U 2 IC-On2r) and IC-On2\j are unaffected.
vh,
,
-,
Lemma
,
If (p, q, v) satisfies
7.
^o n {V) and
vjj
>
> vE
l
l
v k
-
2
'
scheme is Pareto dominated by another scheme that satisfies
continuation values on the Pareto frontier of V and uses the same prices.
for
any
(j, k),
this
v 2H
<
for all (j,k), then if v k
Fb n {V),
f{vj k )
has
all
2
<
f(vj H ) and v L < f(vjL ). Then, we can hold fixed
firm l's continuation values and increase v H and v? L by the same amount without affecting
Proof. Suppose that
for
some
j,
2
IC-On2, thus increasing
Lemma
6,
U2
.
Then, suppose that v 2H
(Tlj), so that neither continuation value
can be increased, again increasing
Lemma
ifvjj
>
Suppose that
8.
l
v k
j
>
Vg, then pjk
U
(p, q, v)
=
Proof. Suppose pjk <
2
=
is
without affecting
and v 2L < f(vjL ). Then, apply
on the frontier. Then, both v 2H and v 2L
f(Vj H )
-
U
implements a Pareto
1
.
The other
efficient
case
point in
is
TQn (V). For
all (j, k),
r.
r.
Then we can
increase pjk by de and decrease
without affecting payoffs or on-schedule IC constraints. Then, we can improve
the continuation values to the frontier as in
Proof of Proposition 6: Lemma
{(u^u 2 ) u
K}, the total
the shape
analogous.
l
:
Lemma
vL and
utility
v?j
k
by ^de
by returning
7.
when the
down when
5 establishes that
continuation value set has
cartel profits go
firms use Pareto inefficient
This logic can be applied directly here, observing that we are
maximizing total profits because utility can be transferred across the firms (as in Proposition 3)
under the conditions stated in the proposition.
Proof of Proposition 7: The proof is established by applying parts (ii) and (iii) of the
continuation values or prices.
following lemma.
52
Lemma
exists
u be a Pareto
Let
9.
T^V), and
point in
efficient
let
(p,q, v) implement u.
(i)
There
an implementation of u,(p,q, v), such that: either (a) at least one off-schedule constraint
2
and, either (a), (d) IC-0n2r)
v 2iL and v\ H G v HH
IC-Onlo binds, or else (c) v\ L
binds, (b)
;
x
l
v HH (ii) If (p,q, v) satisfies pricing efficiency, or, more
v\ H and v HL
binds, or (e) v\ L
l
generally,
(H,H;z)
IP(L, H;z) for each i, and further continuation-value Pareto efficiency
.
H
an implementation of u, (p, q, v), such that q = q and either (a) at least
one off-schedule constraint binds, or (b) IC-Onio binds for each firm, and unless q\ — qfa, ICOniu is slack. (Hi) If the assumptions of part (ii) hold, and if in addition, q^ > (?H for each
firm i, there exists an implementation of u, (p, q, v), such that q — q and either (a) at least one
holds, then there exists
IC-Onio binds
off-schedule constraint binds, or (b)
for each firm
l
and v LH
i
1
v^
v BL for
je{L,H}.
Proof. Define
= ft - fH Rb =
L*
and
k.
RA
likewise for
can be written
I
.
Let
>L
PC-
*(«£
~
RA =
«ff)i
J2
l
l
.
6l.
Suppose that
(i)
for (p, q, v),
2
according
Then bringing together v\ k and v Hk
Given the convexity of the
can be written
IC-Onl/?
to trick (T2j) of
set of feasible continuation values,
R
l
values of v\ k and v Hk whereby
incentives for firm 1 since IC-Onl^
- QHkPHk)
ke{L,H}
R = R\ + R B Then IC-Oni^
1
VkiQLkPLk
1
and
is
U
slack.
1
is
L 6h >
1
slack
R
increase weakly; these
,
while IC-Onify
and v\ k > v\ik
Lemma 6 does
we can
1
for
some
not affect firm
do not
violate on-schedule
Thus, we can perform such adjustments until either
(b) or (c) holds, or else (a) does, so that the off-schedule constraints prevent the adjustments.
similar logic holds for the case
(ii)
Lemma
where IC-On2o
> (pH
> qnH
3 implies q^
IC-Oniu bind, while
if
q^
,
.
2.
find corresponding feasible
is
A
slack.
=
fy, then simple calculations verify that IC-Onir) and
only one on-schedule constraint can bind for each firm. Now
If (pL
L 9h — R\ — L (6h — r) < 0, so that IC-Om/j implies
{
R B < 0. Since U (L,H;z) \Y(H,H;z) = U9H - R\, the condition lY(L,H;z) D W(H,H;z)
also imphes R B < 0. Consider firm 1. If (p, q, v) is compatible with R B < 0, then we cannot have
l
v\ L > vHL and v\ H > v\j H Using continuation- value efficiency, the latter pair of inequalities
suppose q\
> qH
L
If
.
1
x
prices are efficient,
.
are equivalent to those in part
that either (i)(a) or (i)(b) holds.
(fa
>
for
(fa
each
i,
(i)(c).
The
then each IC-Oniu
But, by part
case of firm 2
is
there must then exist a (p,q, v) such
analogous. Finally, if IC-Onio binds and
(i),
is
slack.
Apply part (ii) to derive a scheme whereby either the downward on-schedule constraints
bind and the upward on-schedule constraints are slack, or else the off-schedule constraints bind.
Call this scheme (p,q,v). Suppose that the off-schedule constraints are slack. Suppose further
that Vj L < Vj H (and thus, v? L > v? H ). Following the logic of part (ii), we must have v 2_j L < v 2_- H
(iii)
(and thus v]_j L > v]_j H ), or else IC-On2£> would be violated. Then, decreasing vjH by Vl(vjh~ vjl)
and increasing vh by tih(vj H — vh) keeps U 1 and firm l's on-schedule constraints unaffected,
while moving the continuation values for firm 2 along the frontier weakly increases
is
concave, and
rate t)h
until
it
IC-On2o (though it may violate IC-On2[/). Next, decrease
v}_j H at rate rji, and move firm 2's continuation values along the
relaxes
and increase
U2
IC-On2£> binds again (by the logic of part
(ii),
since
v\_
/
L at
frontier,
bind with v 2, L < v 2_ jH ). This
unchanged, while it weakly increases U 2
IC-On2£>
will
and U 1
since / is concave. For this new scheme, either an off-schedule constraint binds, or else we have
a feasible, weakly Pareto dominant scheme where the downward on-schedule constraints bind,
leaves firm l's on-schedule incentive constraints
53
—
—
and vh
—
i)j
H and
>
u* L
l
v _H
Similar arguments establish that
.
if
v\ k
> vHk we
,
can find a
weakly Pareto dominant scheme whereby the downward on-schedule constraints bind, and further
2
2
> i) Hk
for each k, and thus v\ k D v Hk establishing our result.
v Lk
,
suppose q\ H < 1. Add - rre~£ to q\ H and subtract e
from v\ H Hvlh is on the Pareto frontier and A~ f(v\ H ) > — 1, move v\ H along the frontier of V.
Otherwise, raise v\ H by e (this is possible by convexity of the set of feasible continuation values,
Proof of Proposition
8:
First,
.
y
This does not affect U
Consider now the
and since satisfaction of (4.2) implies vjjH > v N
).
l
l
1
effect on the interim expected utility of both firms: U {H,H;z), U {H,L;z), and C/' (L, L;z) are
2
2
l
unchanged; and U {L,H;z) decreases. U (L,H;z) and U (L,L;z) are unchanged. U 2 (H,H;z)
goes up if -(plh — 9h)/(plh — &l)~ max(— 1, &.~ f(v\ H )) > 0, which when rearranged gives (4.2).
U 2 (H, L; z) goes down if v\ H increases by no more than e, which it does by construction. Thus,
Finally, none of firm l's
all of the on-schedule incentive-compatibility constraints are relaxed.
off-schedule constraints are affected by this shift, and firm 2's off-schedule constraints are relaxed.
To see the result for q HL we perform an analogous trick, subtracting - ^-de from qHL and
.
,
adding de to v HL
,
and noting that
satisfaction of (4.3) implies that v\j L <v\. (recalling that in
specified a large negative slope for / when vj k >u]j).
we
Proof of Proposition
the definition of /,
9: Under the assumptions of the proposition, utility is fully transand we can simply maximize the sum of firm utilities. Doing so leads
to a symmetric scheme across states (L, H) and (H, L), with one firm being favored over another
l
in states {H,H) and (L,L), if at all. Now imagine lowering q LH and raising q\j L by e, and then
l
2
adjusting v LH and v HL upward by ( until both firms' downward IC constraints bind again. The
opponents' continuation values are moved along the frontier. Solving for £, we obtain:
ferable across the firms,
1
£(plh ~ Oh)
l- VL (l-Atf(vl H ))
8
can be shown using algebraic manipulation that the first inequality in the statement of the
is necessary and sufficient for this change to lower total firm profit.
The second
It
proposition
inequality
is
necessary and sufficient for the firms' joint profit to decrease
if
we
reverse this
change.
Proof of Proposition
proof of Proposition
1
The
10:
linear self-generating set of equilibria constructed in the
implements the endpoints of the segment,
1, q^ k —
(9l) = r — 2e,
that have market shares q\ H
=
for all other (j, k).
(x, y)
and
(y,x), using
schemes
Consider a scheme whereby firm
chooses p^iOn) — r and p 1
and p2 (0h) — r — e and p2 (0l) = r — 3e. Using this
scheme, the market shares are assigned appropriately. Further, each firm's announced price differs
1
by
is
state, so that continuation values
can be contingent purely on prices. Thus, communication
not required to implement the scheme. Since firms can draw from a convex set of continuation
values, all continuation values in
between
(x, y)
and
(y,
x) are available to the firms,
and the
linear
set is self-generating.
Symmetric PPE
and Lemma 5 indicates
that the highest point in T (V) is implemented as described in (i) and (ii). Since the scheme
described in (i) is still feasible when we take V s as the set of feasible continuation values, it must
be optimal. The rigid-price scheme clearly requires no communication; the scheme in (ii) can be
implemented without communication, using p {6u) = r and p {6i) = PProof of Proposition 12: (i) Under the stated conditions, the frontier has an open interval
with slope — 1, and thus it is more efficient to use continuation values rather than inefficient bribes.
Proof of Proposition
values
is
11:
contained in the set
This follows from
V=
{(u 1 ,u 2 )
Lemma
(u ,^ 2 )
1
:
i
5,
i
54
since the set of
D sup{u G
V s }},
Proposition
(ii)
1
establishes a critical discount factor
without bribes; thus,
if
bribes are at
all inefficient,
beyond which
first-best
they are not used even
Proof of Proposition 13: This follows as in Proposition 8.
Proof of Proposition 14: (i) This follows as in Lemma 5 and
(ii)
When
all
implemented using
as in Proposition
—
6h)/{2
constraints.
+
The optimality
of bribes for
The
7
ties
—
(1
the availability
continuation
in the specified region follows exactly
Consider a scheme that uses
and productive efficiency, and b\ H —
replacing continuation values with bribes.
equal market-sharing and no bribes in the event of
(r
8;
inefficient
of the off-schedule constraints are cleared, productive efficiency can be
bribes.
8,
they are available.
Proposition
of bribes does not change the tradeoff between productive inefficiency and
values,
if
can be attained
i)vh)- It can be verified that this scheme satisfies all of the on-schedule IC
critical discount factor can be computed simply by checking the off-schedule
constraints.
11.
References
Abreu, Dilip, David Pearce and Ennio Stacchetti, "Optimal Cartel Equilibria with ImperMonitoring", Journal of Economic Theory, 39.1, June 1986, pp. 251-69.
Abreu, Dilip, David Pearce and Ennio Stacchetti, "Toward a Theory of Discounted Refect
peated
Games with
Imperfect Monitoring," Econometrica, 58.5, September 1990,
pp. 1041-63.
Aoyagi, Masaki, "Bid Rotation in a Model of Repeated Auctions," Mimeo, 1998, University
of Pittsburgh.
Athey, Susan, Kyle Bagwell, and Chris Sanchirico, 1998, "Collusion and Price Rigidity,"
MIT Working
Atkeson,
Paper no. 98-23.
Andrew and Robert
tion,"
E. Lucas,
"On
Review of Economic Studies,
Efficient Distribution
with Private Informa-
59.3, July 1992, pp. 427-53.
Bernheim, B. Douglas and Michael Whinston, "Multimarket Contact and Collusive Behav-
RAND
ior,"
Journal of Economics, 21, 1990, 1-26.
Bork, Robert H., "The Rule of Reason and The Per Se Concept: Price Fixing and Market
Division," Yale
Law
Journal, 74.5, April 1965, 775-847.
Bork, Robert H., "The Rule of Reason and
Division," Yale
Law
The Per Se Concept:
Price Fixing
and Market
Journal, 75.3, January 1966, 373-475.
Business Week, "Justice Cartel Crackdown," July 27, 1998, pp. 50-51.
Cole, Harold L., and Narayana Kocherlakota, "Dynamic Games with Hidden Actions and
Hidden States," Federal Reserve Bank of Minneapolis Research Department Staff
Report 254, 1998.
Comanor, W.S. and M.A. Schankerman, "Identical Bids and Cartel Behavior," Bell Journal
of Economics,
Compte,
Olivier,
7,
1976, 281-6.
"Communication
Econometrica, 66.3,
May
in
Repeated Games with Imperfect Private Monitoring,"
1998, 597-626.
Cramton, Peter C. and Thomas R.
Palfrey, "Cartel
Enforcement with Uncertainty about
Costs." International Economic Review, 31 (1) February 1990, pp. 17-47.
Damania, Richard and Bill Z. Yang, "Price Rigidity and Asymmetric Price Adjustment in
a Repeated Oligopoly," Journal of Institutional and Theoretical Economics, 154,
1998, pp. 659-679.
55
Doyle, Maura, and Christopher Snyder,
"Information Sharing and Competition in the
Mimeo,
George Washington University, 1998.
Motor
d'Aspremont, Claude and L.A. Gerard- Varet, "Incentives and Incomplete Information,"
Vehicle Industry,"
Journal of Public Economics, 11, 1979, 25-45.
Fudenberg, Drew, David J. Levine and Eric Maskin, "The Folk Theorem with Imperfect
Public Information," Econometrica, 62.5 September 1994, pp. 997-1039.
Genesove, David and Wallace Mullin, "Narrative Evidence on the Dynamics of Collusion:
The Sugar
Institute Case,"
Mimeo, 1998, Michigan State
University.
Genesove
Minnesota Studies in Macroeconomics series, vol. 1 Minneapolis: University of
Minnesota Press 1987, pp. 3-25.
Green, Edward J. and Robert H. Porter, "Noncooperative Collusion under Imperfect Price
Information," Econometrica; 52.1 January 1984, pp. 87-100.
Holmstrom, Bengt and David Kreps, "Promises," mimeo, MIT, 1996.
Judd, Kenneth, and James Conklin, "Computing Supergame Equilibria," Mimeo, Hoover
Institution, Stanford University, 1995.
Kandori, Michihiro and Hitoshi Matshushima, "Private Observation, Communication and
Collusion," Econometrica, 66.3,
May
1998, 627-52.
Kihlstrom, Richard and Xavier Vives, "Collusion by Asymmetrically Informed Firms,"
Journal of Economics and Management Strategy, 1 (2) 1992, pp. 371-96.
Kuhn, Kai-Uwe and Xavier Vives, "Information Exchanges Among Firms and Their Impact
on Competition," mimeo, Barcelona: Institut d'Analisi Economica, June 1994.
Levinsohn, James, "Competition Policy and International Trade," in Fair Trade and Harmonization J. Bhagwati and R. Hudec, eds., MIT Press, pp. 329-356, 1995.
McAfee, Preston and John McMillan, "Bidding Rings," American Economic Review, 82.3,
June 1992, pp. 579-599.
McMillan, John
"Dango: Japan's Price-Fixing Conspiracies," Economics and Politics,
,
,
November
1991, (3): 201-218.
McCutcheon, Barbara, "Do Meetings
of Political
Economy 105
in Smoke-Filled
(2)
Rooms
Facilitate Collusion?" Journal
April 1997, pp. 330-50.
Phelan, Christopher, and Robert M. Townsend, "Computing Multi-period, Information-
Constrained Optima." Review of Economic Studies 58 (5), 1991, pp. 853-81.
Radner, Roy, "Monitoring Cooperative Agreements in a Repeated Principal- Agent Relationship," Econometrica, 49, 1981, pp. 1127-1148.
Roberts, Kevin, "Cartel Behavior and Adverse Selection," Journal of Industrial Economics,
33, 1985, pp. 401-413.
Rotemberg, Julio and Garth Saloner, "A Supergame-Theoretic Model of Price Wars During
Booms," American Economic Review, 76, 1986, pp. 390-407.
56
Shapiro, Carl, "Exchanges of Cost Information in Oligopoly," Review of Economic Studies,
53, 1986, pp. 433-446.
Sproul, Michael, "Antitrust
and
Prices,"
Journal of Political Economy, 101.4, 1993, pp.
741-754.
Stocking, George
W. and Myron W.
tional Business Diplomacy,
Watkins, Cartels in Action. Case Studies in Interna-
New
York:
The Twentieth Century Fund,
Vives, Xavier, "Duopoly Information Equilibrium: Cournot
1946.
and Bertrand," Journal of Eco-
nomic Theory, 34, 1984, 71-94.
Wang, Cheng, "Dynamic Insurance with Private Information and Balanced Budgets," Review of Economic Studies 62.4, October 1995, pp. 577-95.
;
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