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Cheating on collusive agreements
International
Journal
of Industrial
CHEATING
Organization
ON
8 (1990) 519-543.
COLLUSIVE
North-Holland
AGREEMENTS*
Margaret E. SLADE The University of British Columbia, Vancouver, B.C., Canada V6T 1 WS G.R.E.Q.E. 13002 Marseille, Final version
France
received January
1990
This paper addresses the question of why firms might incur great expense and individuals risk prison sentences to enter a collusive agreement and then not abide by it. In the model, firms must make pricing or production decisions at frequent intervals. Explicit coordination is therefore not possible in each decision period. Players have private information about their own costs which they can use in making their choices. This private information can cause defection and precipitate the punishment phase. Nevertheless, participants fmd it in their own best interest to enter the collusive arrangement.
1. Introduction
Collusive agreements have a very long history, perhaps thousands of years.’ Unfortunately, from the participants point of view, cheating on such agreements is just as old. Nevertheless, in the economics literature there is a paucity of formal models that explain why firms will enter an agreement and then renege. This paper addresses the question of why colluders cheat. In particular, we wish to know why lirms incur great expense and individuals risk prison sentences to enter a collusive arrangement and then do not abide by it. A question of secondary importance is when firms will cheat. For example, are economic upturns or downturns more likely to lead to cartel breakdown? And finally, it asks what range of collusive price-quantity outcomes we expect to see. Many forms of collusive arrangement exist. For example, tacit collusion involves no formal agreement or meeting. With explicit collusion, in contrast, formal agreements are possible. These can be either legal or illegal. The *This research was supported by a grant from the Canadian Social Sciences and Humanities Research Council. I would like to thank the following people for helpful comments and suggestions: Marcel Boyer, Brian Copeland, Peter Cramton, Partha Dasgupta, Mukesh Eswaran, Paul Geroski, Norman Ireland, Tracy Lewis, Jean-Francois Mertens, Francoise Schoumaker, John Weymark, two anonymous referees, and participants in the Canadian Economic Theory Study Group, Montreal. ‘For example, Corder0 and Tarring (1960) state that ‘Pliny, writing perhaps in A.D. 70, mentions a decree which limited the output of British lead. It is suggested that a quota system was imposed at the request of owners of lead mines in Spain when the export of British lead began to result in severe international competition.’ 0167-7187/90/$03.50
G 1990-Elsevier
Science Publishers
B.V. (North-Holland)
520
M.E. Slade. Cheating on collusice agreements
model developed here is designed to explain cheating in the context of the first two but not the third type of arrangement. That is, we deal with pure tacit collusion and illegal price or quantity-setting conspiracies within the context of noncooperative game theory. Examples of the three types of collusion abound. For example, tacitly colluding firms are illustrated by the U.S. producers of lead-based antiknocks (gasoline additives).2 No cheating occurred in this industry until demand fell drastically due to government intervention in the form of environmental regulation. At this point, however, firms began to offer discounts off the (tacitly) agreed price, causing strife in the industry [U.S. Federal Trade Commission ( 198 1)I. Perhaps the most famous example of an illegal price-fixing agreement is the electrical-equipment conspiracy of the 1950’~.~ This cartel, which involved twenty-nine U.S. companies selling heavy-electrical equipment, fixed prices on standardized items as well as pricing formulas for custom products. In spite of the risks associated with participating in such an agreement, due to the criminal nature of these activities under U.S. antitrust law, agreeing parties chiseled repeatedly and even touched off price wars (Smith, 1961). Legal but not legally enforceable cartels are extremely prevalent in naturalresource industries.* Present day examples are principally organizations of governments, often both producer and consumer. Many earlier agreements, however, were among producer firms. The 1930s were the heyday of private commodity agreements. These cartels, however, were generally unsuccessful and the dominant theme was the inability of private organizations to limit exports (McNicol, 1978). Their successors today experience similar problems. Although not the subject of this paper, legal arrangements which cannot be supported by binding commitment to enforcing an agreed-upon strategy are also plagued by breakdowns. The model developed here, a parametric example, is one where symmetric duopolists produce a homogeneous product under conditions of linear demand and cost. Simplifying linearity assumptions, which are made for tractability, allow precise predictions to be made. Firms in the model must make pricing or production decisions at frequent intervals. Even when collusion is explicit, it must also be covert, and so highly complex arrangements are undesirable. Participants must thus agree upon simple repeated-game strategies which are self enforcing. In each period, firms have private information about their own costs. It is this private information that can cause cartel breakdown. sThis is only one of many possible examples. Empirical studies of industries where a few firms produce a homogeneous product almost always find evidence of tacit collusion [see for example Applebaum (1982). Roberts (1984). Spiller and Favaro (1984), and Slade (1986) and (1987a). ‘For numerous examples of illegal price-lixing agreements, see Hay and Kelly (1974) and Asch and Seneca (1976). 4For accounts of other legal cartels, see Stocking and Watkins (1946).
M.E.
Slade,
Cheating
on collusive
agreements
521
Supergame models which exhibit seemingly disequilibrium behavior such as price wars generally rely on some form of uncertainty.5 For example, in the Green and Porter (1984) model, the common demand intercept is a random variable, implying that rivals’ actions are not observable. And in the Slade (1989) model, the common economic environment shifts unpredictably, implying that own and rival payo$s are unknown. Here, in contrast, each firm’s marginal cost is priwte information, implying that the payoffs of its rioals are not known. The Green and Porter (1984) model is therefore characterized by imperfect monitoring or moral hazard, the Slade (1989) model is one of incomplete information, and the model developed here is characterized by asymmetric-incomplete information or adverse selection. The organization of the paper is as follows. In the next section, an example with no uncertainty is developed in the context of the simple carrot-and-stick repeated-game strategies introduced by Abreu (1986). Carrot-and-stick strategies are of course not the only class that can sustain tacit collusion. Nevertheless, they have two important advantages. First, with the certainty model, these strategies can be optimal (the punishments can be most severe) and are thus capable of sustaining the highest level of collusion.6 And second, carrot-and-stick strategies are extremely simple to communicate and to calculate. When collusion is purely tacit, it is difficult to imagine transmitting very complex information. If collusion is to be sustained, however, it is essential that firms understand the rules of the game and know what would happen if they were to deviate from these rules. Even when collusion is explicit, simple strategies have enormous appeal. In section 3, which is the heart of the paper, uncertainty is introduced. The model with private-cost information is developed for the case of quantity competition. The modifications required for price competition with differentiated products can be found in Slade (1987b). With the introduction of uncertainty, it is necessary to choose a solution concept. If this were a legally enforceable cartel, it would be natural to seek optimal incentive-compatible contracts. This is the approach taken by Roberts (1985), Kilstrom and Vives (1987) and Cramton and Palfrey (1990) in the context of one-shot cartel games. In the context of the current paper, however, this route has two disadvantages. First, for the revelation principle to be valid, it is necessary that the cartel sAn exception is the Rotemberg and Saloner model (1986). where all variables including the random shocks are observable. In their model, however, the punishment is never triggered. Instead, they assess the difftculty of colluding over the business cycle. 6This class of strategy ceases to be optimal for very low values of the discount factor. Under these circumstances, one must resort to asymmetric punishments, which are (much) more complex (see Abreu 1986). In addition, when uncertainty is introduced, they are no longer optimal. As long as strategies satisfy the incentive constraints discussed below, however, they are credible in the sense of Selten (1975).
522
M.E.
Slade, Cheating
on collusive
agreements
be able to commit to the cartel rule. That is, the cartel must be able to guarantee that it will not use the private-cost information once it is truthfully revealed. In the current context, the ability to commit does not seem to be a reasonable assumption. Second, an optimal state-contingent contract can be very complex, involving a different action for every possible cost realization. Here, however, we wish to emphasize the need to transmit only simple strategies. If collusion is purely tacit, no verbal communication is possible, and if it is illegal, no written communication is advisable. For these reasons, we limit attention to simple strategies which are perfect-Bayesian-Nash equilibria but which would not be optimal if binding contracts were feasible.
2. The repeated game with no uncertainty In this section, symmetric duopolists engage in an infinitely repeated game of quantity competition. Collusion in the repeated or supergame is supported by the symmetric carrot-and-stick punishment strategies introduced by Abreu (1986). These strategies, which are developed formally below, have a simple intuitive structure. Suppose that xi, i= 1,2, is the choice variable for player i in a repeated game. A carrot-and-stick strategy is a pair (.?,z?) where .? is the carrot or collusive value of xi and X is the stick or punishment value of xi. Supergame strategies consist of playing .? in the first period and in every subsequent period as long as both players played 2 in the previous period. If, however, someone deviated in the previous period, both players play X for one period and then resume collusion. Abreu (1986) shows that such two-phase punishments are optimalsymmetric punishments in the sense that they can sustain the highest level of symmetric collusion (see, however, the caveat in the previous footnote). In general, this highest level of collusion is supported by sticks that are more severe than Nash reversion.
The one-shot game
In each period, symmetric duopolists produce a homogeneous product and choose quantity as a strategic variable. The industry inverse-demand function is linear and can be written as p=A-B(q’+q2),
where p is price and qi is the output of the ith firm. Without generality, units can be chosen such that B= 1.
loss of
M.E.
Slade, Cheating
on collusice
523
agreements
Each firm produces at constant marginal cost, c. Profit for the ith firm is then,
where a=,4 -c is the demand intercept net of marginal cost. In the one-shot game, players choose qi to maximize their single-period profit or payoff, rti. It is straightforward to show that the Cournot-Nash solution to the one-shot game is qN = a/3,
TC~= a2/9,
(3)
and the joint-profit-maximizing qM = a/4,
or monopoly solution is
nM = a2/8.
(4)
Let n(q) be the profit to each player when they play symmetrically, 7r(q):=ni(q,q)=(u-2q)q.
(5)
n(q)
is plotted in fig. 1. It is zero at q=O, increases to a maximum at q =q”, and then falls until it reaches zero at q = 2q” =qc, the competitive output. n(q) is negative for all production levels greater than 4’. Finally consider qd(q), the best response to q or optimal single-period defection given that an opponent is producing q. qd(q) is defined by
q’(q) = argyx
x’(q’,
The profit corresponding
4) =
(a-4)/2 o
if q_la. otherwise.
to qd, red(q), is defined by
;(a-q)‘2’2
r?(q) = 7c’[q”(q), q] =
Zt;$tise
(7)
i
nd(q), which is the highest single-period profit that one firm can earn given that the other is producing q, is also graphed in fig. 1. It has a maximum of .~ 27~“’at q = 0 and falls off monotonically as q increases.
The repeated
Suppose
game
now that the one-shot
game is repeated
a countably
infinite
524
M.E. Slade, Cheating
Fig. 1. One-shot
on collusire
profits
agreemenrs
to loyal and defecting
lirms.
number of times. If players are colluding tacitly, they never communicate directly.’ For explicit colluders, it is assumed that, due to the illegal nature of the agreement, complex communication is difficult. Participants therefore must choose self-enforcing strategies that can be used currently and in the indefinite future. The payoff to player i in the repeated game is the discounted-profit stream
7Li=f s’rr’(q:,q:),
(8)
O
r=o
where 6 is the discount factor. In general, a strategy for this game is an infinite sequence of functions {sf} t=O,. . . , x, where sf maps the set of histories up to period t- 1 into R+. Here, however, we limit attention to a very simple class of strategies that have a Markov structure. A carrot-andstick supergame strategy is an output pair (G,@ and a rule
4b=4,
i= 1,2
(9) ‘When strategies.
Iirms never meet, they must resort
to devices
like press releases
to communicate
their
A4.E. Slade, Chearing
525
on co~/usive agreemenrs
For (&4) to be subgame perfect, players must have an incentive to remain loyal both in the collusive and in the punishment phases.’ Credibility therefore implies that < and 4 satisfy the following incentive constraints:
and
Constraint IC, says that the gain from defection during the collusive phase, nd(G)--z(@), must be less than or equal to the loss due to the punishment that is triggered by defection, n(g)-n(q). Because the loss is incurred in the following period, it is discounted by the factor 6. Constraint IC2, which applies to the punishment phase, can be interpreted in a similar fashion. The set C,[C,] of carrot-and-stick pairs which satisfy the incentive constraint IC,[IC2] can be written in terms of the demand and cost parameters as
C,(a,S):=((rj,cj)~R+ xR+Iu2+(-6a-4aS)(i’+4u6q+(9+86)ij2 -86&O}
(10)
and C,(U,S):={(&~)ER+
xR+l u2 - 4~66 + ( - 6u + 4aS)4 + 8&j2
Elementary techniques of analytical geometry show that, for all a >O and 0~6 < 1, the boundary of C, is a hyperbola in <-_4 space with center of symmetry [(3a+2&)/(9+86),q”]. This hyperbola is shown in fig. 2 for the case a= 1 and 6=0.875. If (&Q) lies inside the hyperbola, there is no incentive to cheat during a collusive phase. sWith such defections.
simple
time dependence,
it is suflicient
to check
the desirability
of one-shot
M.E. Slade, Cheating on collusice agreements
526
s
\
/
.’
/’
--I
./“I C,
9
@
qN
qM .-‘, /
*\
/’
‘\4C,
/.
\
/’
-, 9 Fig. 2.
/
\
qM qN Incentive
constraints a=1
qc
G
and perfect equilibria.
6 =0.875
Similarly, for all a>0 and Oc 6 < 1, the boundary of C2 is an ellipse with center of symmetry [q”,(3a-2a6)/(9-86)]. This ellipse is also graphed in fig. 2. If (&q) lies inside the ellipse, there is no incentive to cheat during a punishment phase. The hatched area in the figure is therefore the set of subgame-perfect equilibria, PE(a, a), where
PE(a, 6): = C,(a, 6) n C,(a, 6).
(11)
M.E. Slade, Cheating on collusiceagreements
527
The two conic sections which form the boundaries of C, and C2 intersect at (qN, qN) and at some (G’,4’) with 4’ $ qM and 4’ 2 qN. Equality holds only when both conic sections degenerate into the single point (qN,qN). In the certainty case, the cartel will rationally choose @= max {q”, @‘}. Fig. 3 shows the set of subgame-perfect equilibria for a = 1 and 6 =0.25. As 6 approaches zero, PE(a,6) shrinks towards {(qN,qN)} for all a. It is obvious from the figures that corresponding to most sustainable carrots 4’ there is a large number of credible punishments or sticks 4. The most severe punishment that can sustain q is the max {q[(G,4) E PE(a, S)}. As long as the stick will never be used, however, the choice of punishment seems somewhat arbitrary.
3. The repeated-game with uncertain costs Uncertainty is now introduced into the model. It is assumed that risk-neutral firms have private information about their costs. As the cartel is not legally enforceable, the duopolists cannot enter into a legal contract which specifies an output pair for each realized cost pair. Moreover, as coordination is tacit or unwritten, the agreement is assumed to be conditional only on the commonly known mean of the distribution of costs, Z. The sequence of events is as follows. In period 0, players enter the agreement described by eq. (9). At the beginning of each period t, t = 1,. . . , cc, firms marginal costs cf, i= 1,2, are independently drawn from a common distribution. Firms then choose output simultaneously, having observed their own but not their rival’s cost. More formally, suppose that cf is i.i.d. (the same for both players) and serially uncorrelated and that the covariance between c: and c: is zero. In addition, the distribution of c, which is symmetric about its mean and has a finite support, is common knowledge. Let f and F be density and cumulative density functions for a’:=A-c’, where the time subscripts has been dropped to simplify notation. The support of ui, Asup,is then A sup:={ui~f(ui)>O}=[u-,uC],
u-20
and
u+SA.
(12)
* The mean of ui, Z, is (a+ +a-)/2. Equilibrium
without cheating
At first we assume that the cartel is sustainable for all values of a (in a sense that is made precise below). Each player therefore assumes that his rival will not defect when evaluating the constraints IC, and IC,. The assumption of sustainability for all values of a is relaxed later. A strategy for the game with uncertain costs is an infinite sequence of functions {sf} t = 0,. . . , 00, where of maps the history of play up to period
528
M.E. Slade, Cheating on collusice agreemenrs
a
/
\
s”ii’ qN Fig. 3.
Incentive
constraints a=1
-
Q
qc
and perfect equilibria.
6 = 0.250
t- I and the private information u: into Rf. Attention is initially confined to carrot-and-stick strategies which are not state contingent. That is, in this subsection players do not condition their choices on their private information and cheating never occurs. We begin with the simplest case where 4 and 4 are determined exogenously. At the beginning of every decision period, each firm observes its own ci. The gain from defection is evaluated at ui. The loss, however, because it occurs in the next period, must be evaluated in expected-value terms. With a slight abuse of notation, the dependence of rc, rrd,’and rri on a is made explicit. Because 7~is linear in a, E[n(q, a)] = n(q, 5). Incentive constraints ICi and sets Ci, i= 1,2 therefore become
529
M.E. Slade, Cheating on collusive agreements
=
nd(q,a’)- Tc(q,a’)- s[iT(& a) -
UC;)
rt(tj, fi)] 5 0,
C;(a,61u’):={(~,q)ER+ x R+I (ui)2-6q'ui+(9+86)q'2-86q2+4ciS(q-ij)~0),
and
(13) C;(&6/u’):=((&q)~R+
xR+l
Strategies (&q) which satisfy the incentive constraints IC:, i= 1,2, for all ui E Asup are perfect-Bayesian-Nash equilibria for the game.’ Formally, the set of carrot-and-stick equilibria is
PW&,,
6): = ((47@)I
(ij,cj)~C~(ti,if(u)n C2(ti,G(u)Vu~A,up).
(14)
These strategies, however, are no longer optimal. In section 2, the credible (Q,q) pairs were determined for fixed u (figs. 2 and 3). We now adopt the reverse procedure. That is, we fix (@,q) and determine the values of a for which the incentive constraints are satisfied. Fig. 4 shows IC\ and IC2 plotted as functions of a’. These constraints are quadratic in a’. Let u; and uh be the roots of IC; =O, with a;suL. ai and ui are defined similarly for lC2. When the roots are imaginary, we adopt the convention that ~~[a~] =a:[&] = 3<[3q]. It is straightforward to show that a:, ai, and u: are positive. Only a; can be negative.
‘Here as before, since strategies desirability of one-shot defections.
1.10
B
have
such
a short
memory,
it is sufficient
to check
the
530
M.E.
Fig. 4.
Slade, Cheating
Incentive
on collusice
constraints
agreements
as functions
of cost.
Define max (a;, at},
(15)
aLi:= min {a:, a:},
(16)
aL: =
and (17)‘
A,,,(q,g) is the set of all a’s such that the incentive constraints are satisfied; that is, it is the set of u’s such that (&~)EC~(~~,S[U)n C2(&6/u). We assume that (G,@ was chosen such that A,,, is nonempty. There are two possibilities: the support of a may or may not be contained in the sustainable set. When A,,,(q’,g)~A,,,, (6, ij) is a perfect-Bayesian-Nash equilibrium of the game. Under these circumstances, the cartel is sustainable over the entire range of costs, firms will want to enter into a collusive agreement, and cheating will never occur (4 will never be invoked). This is the case shown in fig. 4.
A4.E. Slade, Cheating
on collusice agreemenrs
531
Normally, there will be a large number of equilibrium pairs (&q). An optimal pair (GO,@‘) can be chosen to maximize expected payoffs, E(n’), subject to the constraints IC;, i= 1,2. Finding an explicit solution to this maximization, however, is not an easy task. Instead, we turn to a different problem. We wish to examine the effect of the size of the support of a on the mostdesirable sustainable collusive outcome. To do this, we consider the two extreme cases. The first is the certainty case, which corresponds to Asup= {Z}. This is the situation depicted in figs. 2 and 3. With the example of fig. 2, the most-desirable sustainable collusive output is q.“. And with example of fig. 3, it is 4’. Next, we consider the largest Asup for which the cartel is sustainable using carrot-and-stick strategies. With this in mind, some new notation is introduced. For given & let q*((i) be the stick that maximizes the symmetric region around ti for which the cartel is sustainable. Formally, define (18) Then, for any @, cj*(ij) = arg max cc*(&4).
(19)
4
Figs. 5 and 6 reproduce figs. 2 and 3. The new graphs in addition show 4*((i) for all sustainable carrots 4’(the line with alternating dots and dashes). Similarly, for given 4, let g*(Q) be the carrot that maximizes the symmetric region around ti for which the cartel is sustainable. That is, for any i, ij*(ij) = arg max cr*(&q).
(20)
B
i*(g) is also shown in figs. 5 and 6 (the solid line). G*(4) ranges over all possible punishments and q*(i) ranges over all collusive production levels that are sustainable under conditions of certainty. The two functions must therefore intersect, and it is possible that this intersection is unique. Figs. 5 and 6, however, show that not only is the intersection not unique, the overlap is surprisingly large. This leads to a new definition. Let
maxa*(&@)/g=<*(q) 3.4
andq=q*(q)
.
(21)
532
M.E.
Slade, Cheating
on collusioe
agreements
s
‘\ \
IS . .
I
.?
I I
I I I 0
/
./
.’
1’
I
.I’ ,.’
I
Carrots
-\
/
\
qN
4
and sticks which maximize a=1
a qc ’
u “.% q
\
I
’
;,
Fig. 5.
I
I
7
security.
6 = 0.875
[a--r**, a+~**] is the largest support for which the cartel is sustainable using carrot-and-stick strategies. We denote the carrot and stick that ;z-e;gnd to a** (4”‘, qMS), where MS stands for maximum_sMuspport.GM’ 1s strictly are marked with *‘s in figs. 5 and 6. In the figures, q greater than both q”’ and 4’. This turns out to be a general result which is stated as Fact 1. Fact 1.
GM’>
qM
and ifus >$for
all a>0
The verification of Fact 1 is by simulation.
and all 0<6<
1.
,M.E. Slade, Chearing on collusive
agreements
533
s
Fig. 6.
Carrots
and sticks which maximize a=1
security.
6 = 0.250
Fact 1 implies that as the support of a increases, after some point the set of sustainable collusive output levels becomes less desirable. An obvious consequence is that the cartel is less profitable. It is therefore reasonable to ask if the cartel can do better. Equilibrium with cheating In this subsection we examine what happens if the carte1 allows cheating. That is, we wish to know if firms can improve their long-run profitability by occasionally cheating (playing myopically) rather than always following their
534
,M.E. Slade, Cheating
on collusive agreemenrs
carrot-and-stick strategies. Under these circumstances, firms will enter a collusive agreement expecting to cheat. That is to say, when they observe an a outside A,,,, they will defect. To examine this possibility, suppose that for given 4’ and 4 the support of a is not contained in the set of a’s for which the cartel is sustainable. Each player will know that there is some probability that the other will defect. The evaluation of constraints IC’, and lCz therefore becomes more complex. It is still true, however, that a player will not defect during a collusive [punishment] phase if the expected value of the first [second] incentive constraint is negative. Before evaluating the constraints, it is necessary to introduce some notation. Up to now, the output observed in a collusive [punishment] phase was a fixed quantity, G[@], where a collusive phase is a period t such that qf_ I = 4, k = 1,2, or q:- 1= q, k = 1,2. Now, however, because there is a positive probability of observing a’s outside of A,,,, the output observed in each phase becomes a random variable. Denote the realized values of these quantities by GA and 4” respectively, and let the expected value of GA [4^] be 4’ [$I. 4’ and 4’ are defined implicitly by”
(22) and
+
“j 4w.m
qd(cj’,u)f(u)du.
(23)
Eq. (22) can be interpreted as follows. With probability F(uy)--F(uf), a rival will not defect and will play 4. If an a is observed in the tails of the distribution, however, a rival will defect. When defection occurs, a player will choose to produce the most lucrative q, given that his opponent is expected to produce @‘.The interpretation of (23) should be obvious.
“In
(22) and (23) it is assumed
that ff=O
when z>/L
M.E.
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535
agreements
These equations are highly nonlinear. Nevertheless, it is possible to show that Proposition 1. Q’>=E(qN).
There exists a solution to (22) and (23) with cj’sE(q”)
and
Proof. The proof involves a straight-forward application of Brouwer’s fixed-point theorem. Eqs. (22) and (23) define a continuous function H(q’,@‘). Let the domain of this function, A, be
A=
0,; II
1
x[O,ci].
H maps A, a compact and convex subset of R2,into itself and therefore has a fixed point. If i’> E(qN) and qe
The existence of pairs ($,qe) satisfying (22) and (23) is cold comfort if the solution is not unique. With nonuniqueness, pairs of expected values are not interchangeable. That is, if ($‘, @) and (rj”‘, 4”“) are fixed points, (G”, @“) will in general not be one. i1 Fortunately, solutions may be unique. For example, Proposition 2.
When f(a)
is uniform, there is a unique solution to (22) and
(23). Proof.
See appendix A.
The intuition behind proposition 2 is as follows. With well behaved density functions, the partial derivatives of H are also well behaved. For fixed q’[G’], therefore, the right and left-hand sides of eq. (22) [23] cross only once. It seems reasonable to assume that the agreed-upon. carrot is at least as desirable as the single-period Cournot-Nash outcome. Therefore, because qd(q)>q for all qq. In other words, the effective carrot, $‘, is less desirable than the agreed-upon carrot, 4. Similarly, the effective stick, $, is less dire than the agreed-upon stick, 4. We can now evaluate the constraints.
=7rd(fj’,ai) -n’(g,
cj’, a’)-Gl[rc(@,
ci)-n(qe,Z)]
+ CEB, 50
“This is an example of the (often not mentioned) problem that plagues noncooperative theory. When games have multiple equilibria, strategy pairs are not usually interchangeable.
game
536
M.E.
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agreements
(25)
=~d(~‘,~i)-n’(~,q’,ai)-6,[n(q”,~)-~(~,~)]
+ CEB, SO,
where expectations are taken over iA, 4^, and a, CEB, is the certaintyequivalence bias for constraint j, and 6j is the effective discount factor for constraint j. The certainty-equivalence biases are given byI
-Cov(u,tjA)} and
(26) CEB,=Var(~“)/4+6,{2[Var($A)-Var(q”)]+Cov(a,q”) -
cov(a, GA))
and the effective discount factors are given by
sj=G[F(&-F(u’,)].
(27)
Eq. (27) has the following interpretation. The next-period loss due to defection must be weighed by the discount factor, 6. In addition, firm one’s rival will defect in the current period with probability [l -F(ui,) + T(u’,)]. If firm two defects, punishment will ensue regardless of firm one’s choice and there is no loss to firm one if it also defects. The next-period loss must therefore also be weighted by the probability that the rival remains loyal in the current period. The effective discount factor is therefore smaller than 8. The signs of the certainty-equivalence biases are ambiguous. For a large range of reasonable parameter values, however, CEBj is positive. This will be the case, for example, whenever Var($“)zVar(d”) and COV(~“,U)Z COV(q”,u).‘3 If the support of a is not contained in the set of sustainable u’s, there are thus four consequences. First, the effective carrot is less desirable than the agreed-upon carrot. Second, the effective stick is less severe than the agreedupon stick. Third, the effective discount factor is smaller than 6. And finally, certainty-equivalence bias implies that the expected value of the gain due to 12The certainty-equivalence biases are obtained by expanding the formulas for n, nd, and ~’ and taking their expected values. 13Through the use of numerical simulations, it is possible to show that when f(a) is uniform, CEBj is always positive.
.V.E. Slade, Cheating
on collusice
agreements
537
defection is usually greater than the gain evaluated at the expected values of the random variables. The combination of these four effects reduces the profitability of collusion. Nevertheless, if the present value of collusion with cheating is greater than z(q’)/( l-6), the highest payoff that each could earn if they adhered to their carrot-and-stick strategies with no cheating, firms will want to choose a 6~4’. This leads to the definition of a new strategy, the cheating carrot-andstick strategy. For any 4 and q, let a collusive phase be a period t when q:_ 1=ij or q:_ I =& k = 1,2. All other periods are then punishment phases. The cheating strategy is
qa=4,
i=l,2
4 qf =
qd(& a’) 4 qd(@9 a’)
(28)
I
if collusive
if punishment
phase and
phase and
This strategy, which is state dependent, is a perfect-Bayesian-Nash equilibrium for the game. It remains to be seen if, for some parameter values, the expected value of each firm’s profit stream when both use cheating strategies is higher than n(G’),!( l-6). An example shows that this is possible. Let a= 1.0 and 6 =0.7 and suppose 1.15361. With that a is distributed uniformly on Asup= 1.0+0.1536=[0.8464, these parameter values, Go = gnrs = 0.28, 4’ = $MS =0.39, c(** = 0.1536, and n(GO)=0.1232. With no cheating, the highest payoff that firms can earn is 7r(~0)/(l-6)=o.4107. Now suppose that firms decide to allow cheating and select $=0.25 = E(q”) and $=0.42 =q*($), where c stands for cheating. The corresponding A,,,= [0.8577,1.1523]. Clearly, it is possible to observe values of a for which cheating pays. Player i will defect in a collusive phase if 1.1523 caf 5 1.1536. Similarly, player i will defect in a punishment phase if 0.84645 af<0.8577. Appendix B shows that under these circumstances expected profit from playing the cheating strategy exceeds 0.4107. The example demonstrates that if colluders are constrained to using carrot-and-stick strategies then, due to private information about production costs, firms can improve their long-run profitability by occasionally cheating (playing myopically). That is to say, they can be made better off than if they were constrained to choose a strategy ((i,@ for which A,,,(& 4) 2 Asup. In some sense, the carrot-and-stick strategy is a strawman. It does not
538
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on collusive
agreements
allow the necessary flexibility when there is incomplete information about costs and is therefore not optimal. It is my contention, however, that collusive agreements that are more complicated than carrot-and-stick strategies may be too complex to be a feasible means of tacit collusion.
4. Summary and conclusions In the model developed here, symmetric duopolists must make frequent pricing or production decisions. Even when collusion is overt, therefore, it is not possible for them to coordinate in each decision period. Firms have private information about their own costs which they use in making their decisions. It is this private information that can cause cheating. The collusive agreement will be stable under normal conditions. However, either unexpectedly high or low costs can-cause a breakdown. With the linear demand and cost model developed here, defection is more likely to occur in good times. That is, firms are more tempted to cheat and increase output when costs are 10w.l~ Can we really say that firms cheat, given that cheating is part of an equilibrium strategy? Perhaps it would be better to say that, to the outside world, firms appear to cheat. For example, it is common for cartels to announce output quotas. Announcing that violations are to be expected, however, would be much more unusual behavior. Nevertheless, it may be understood among cartel members that cheating will occur under abnormal economic conditions. Cartel discipline, of course, requires that cheaters be punished. For if cheaters were not punished, members would have an incentive to misrepresent their costs. That is, they would increase output when costs were normal. For tractability, the model developed here makes use of several restrictive assumptions. Many extensions and generalizations are therefore possible. Obvious modifications include using profit functions other than quadratic and supergame strategies other than carrot and stick. It seems clear, however, that if cheating occurs in this model, it will also occur when the less severe punishment of Nash reversion is used. In spite of the simplicity of the model and the variety of possible generalizations and extensions, it is hoped that it offers insights concerning the age-old problem of cheating on collusive agreements.
14With the present model, an unusually low cost is more apt to cause breakdown in a collusive phase. If the opposite were true, however, there would be no need to punish firms for playing too cooperatively, given that both players benefit from a unilateral output reduction. The result of breakdown when cost is low is not the same as breakdown when demand is high, as in Rotemberg and Saloner (1986).
539
M.E. Slade, Cheating on coilusice agreements
Appendix A Proof
of Proposition
2
First I will show that for q’ fixed at an arbitrary cje’zE(qN), unique rj” such that I?‘(<“, a”) =Lj”.
H’(cj’,q’j=‘i
(I-
there exists a
qd(ie,a)f(a)da-~qd(cje,a)f(a)dn+Lj[F(a~~)-F(u~)] 4
- ;
wdIl)2-(4)213
where
the support of a, A,,,=[G--!x,G+r]. When a:=~;,H’(@~,q~‘)= and 4” = E(qN) =ti/3. Assume therefore that a~>~~. In this case, there exists an interval [qtqt] such that when rje lies in this region, defection does not occur for some realized values of a. To find ai, at., q:, and q:, apply the quadratic formula to eq. (13) in the text to obtain q’(<‘, ci)
aL=34’-fi
and
a:=3~Yj~+,/O,
(A.l)
where (8;i’
s>
if this expression is > 0 otherwise.
(A.2) shows that qt=min{(i/2-q”,O)
and
qh=q’+
In addition. +4=2JG, and
(a;S)2-(at)2=
12lj’Jo,
(A.2)
.Cf.E. Slade, Cheating on collusioe agreements
540
H’(q’,q’‘)=qd(q-e,,-)
(A.3) shows H’(tj’,
that
4”) 5 qd($,
-
when
$F(<‘_~).
4’s 4, H’(Ge, de’) 1
(A.3) qd(ie, 5)
and
when
4’26,
5).
Now consider t?H’/&j’=:H:,
where 26(fi-4~7 Jo 0
ifO2O -otherwise.
For g’IqL, O=jl=O, H’=qd(qe,fi), and H:=-l/2. For q,?=
Using extremely similar reasoning, it is possible to show that for 4’ fixed at an arbitrary 4” 6 E(q,‘), H’(cj”, 4’) crosses the 4%degree line 4’= 4’ exactly once on D,. The details are left up to the reader. Call the point where the two cross q=‘= @(4”‘).
We must now determine the slope of G’(tj’j. From (A.l) Sak/Z@e’=
26(-4@+5) Jo 0
if oro *otherwise.
This implies that, ?aQ&j” 2 0
for all
4” 2 E(q”).
M.E. Slade, Cheating
on collusice agreements
541
In addition,
S&/g = - C?at/$h 0 for all
4” 2 E(q”).
The middle term in (22) (the sustainable region) therefore increases as 4” increases and the two end terms (the defection regions) decrease. This implies that G”(@)~O. By very similar reasoning, it can be shown that G”($+‘) 60. The two functions G’ and G*, therefore, cross at most once on A. By proposition 2, however, they cross at least once on A. Q.E.D. Appendix B The example
In our example, a= 1.0 and 6=0.7. Suppose that a is distributed uniformly on Asup= 1.0&0.1536=[0.8464, 1.15361. With these parameter values, Go= GMS= 0.28, 4’ = QMS= 0.39, u **=0.1536, and ~(@‘)=0.1232. We wish to show that, by allowing cheating, the expected value of profit in each period is greater than 0.1232. With the cheating strategy, players select $=0.25 =qM and q’=O.42 = q*($), where c stands for cheating. The corresponding A,,,= [0.8577, 1.15231. Let pI[p2] be the probability that firm i defects in a collusive [punishment] phase, pi =prob {a’# A,,,lcollusive phase) =prob{l.1523
J!Lza!_ 2
=0.0113/l ASUP/=0.0368,
phase, there are four possibilities: both players j remains loyal, firm j defects while i remains d’(q) to the profit to the nondefecting firm who and plays qd(q).
(B.1)
If p’ is the probability of being in a collusive phase and p” is the probability of being in a punishment phase then the expected value of km i’s profit in any period, conditional on using the cheating strategy, is
542
M.E.
Slade, Cheating
on collusice
agreements
+Pa(l -P2) [n”(F) + ~dw)l +p:4?d(~‘))}.
U-3.2)
In (B.2), for each phase each of the four possibilities is weighted by its respective probability. p’ and pp can be found as solutions to the equations Pc=Pc(1-p,)2+pp(l-p2)2,
pp=
1 -PC.
(B.3)
For example, the first part of (B.3) says that the probability of being in a collusive phase equals the probability that you were in a collusive phase last period times the probability that both players remained loyal plus the probability of being in a punishment phase last period times the probability that both punished. With our parameter values, p’=O.9910
and
pP=O.O090
7T($)=O.1250
x(qC)= 0.0672
Ge=0.2510
qe = 0.4052
n”((1’)= 0.0940
nd(4’) = 0.1403
n”(Lj’)= 0.1205
nd(@) = 0.0884
n(qd(4"')) =0.0940
n(qd(if)) =0.1205.
Values for Lieand $ were obtained by solving eqs. (22) and (23) numerically. All other values were obtained by straight-forward application of the definitions of the functions. When numerical values are substituted into (B.l), the result is E(n’lcheating) =0.1245 >O.l232=E(n’lcarrot
and stick).
References Abreu, D., 1986, External equilibria of oligopolistic supergames, 191-225. Applebaum, E., 1982, The estimation of the degree of monopoly 19. 287-299.
Journal
of Economic
power, Journal
Theory
39,
of Econometrics
M.E.
Slade, Cheating
on collusice
agreements
543
Asch, P. and J.J. Seneca, 1976, Is collusion profitable?, The Review of Economics and Statistics 58, l-12. Cordero, H.G. and L.H. Tarring, 1960, Babylon to Birmingham (Quin Press Ltd., London). Cramton, P.C. and T.R. Palfrey, 1990, Cartel enforcement with uncertainty about costs, International Economic Review 31, 17-48. Green, E.J. and R.H. Porter, 1984, Noncooperative collusion under imperfect price information, Econometrica 52. 87-100. Hay, G.A. and D. Kelly, 1974, An empirical survey of price-tixing conspiracies, Journal of Law and Economics 17, 13-38. Kilstrom, R.E. and X. Vives, 1987, Collusion by asymmetrically informed firms, University of Pennsylvania working paper. McNicol, D.L.. 1978, Commodity agreements and price stabilization (D.C. Heath, Lexington, MA). Roberts, K., 1985, Cartel behavior and adverse selection, Journal of Industrial Economics 33, 33-45. Roberts, M.J., 1984, Testing oligopolistic behavior: An application of the variable-profit function, International Journal of Industrial Organization 2, 367-383. Rotemberg, J.J. and G. Saloner, 1986, A supergame-theoretic model of price wars during booms, American Economic Review 76, 390-407. Selten. R., 1975, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 2, 25-55. Slade, M.E., 1986, Conjectures, firm characteristics, and market structure: An empirical assessment, International Journal of Industrial Organization 4, 347-370. Slade, M.E., 1987a. Intertirm rivalry in a repeated game: An empirical test of tacit collusion. Journal of Industrial Economics 35, 483-498. Slade, M.E., 1987b, Cheating on collusive agreements, Department of Economics Discussion paper no. 87-27, University of British Columbia. Slade, M.E., 1989, Price wars in price-setting supergames, Economica 56, 295-310. Smith, R.A.. 1961, The incredible electrical conspiracy, Fortune, May, p. 224. Spiller, P.T. and E. Favaro, 1984, The effects of entry regulation on oligopolistic interaction: The Uruguayan banking sector, Rand Journal of Economics 15, 244-254. Stocking, G.W. and M.W. Watkins, 1946, Cartels in action (Twentieth Century Fund, New York). U.S. Federal Trade Commission, 1981, Decision in re Ethyl Corp. et al., Docket no. 9128.
Journal
of Industrial
CHEATING
Organization
ON
8 (1990) 519-543.
COLLUSIVE
North-Holland
AGREEMENTS*
Margaret E. SLADE The University of British Columbia, Vancouver, B.C., Canada V6T 1 WS G.R.E.Q.E. 13002 Marseille, Final version
France
received January
1990
This paper addresses the question of why firms might incur great expense and individuals risk prison sentences to enter a collusive agreement and then not abide by it. In the model, firms must make pricing or production decisions at frequent intervals. Explicit coordination is therefore not possible in each decision period. Players have private information about their own costs which they can use in making their choices. This private information can cause defection and precipitate the punishment phase. Nevertheless, participants fmd it in their own best interest to enter the collusive arrangement.
1. Introduction
Collusive agreements have a very long history, perhaps thousands of years.’ Unfortunately, from the participants point of view, cheating on such agreements is just as old. Nevertheless, in the economics literature there is a paucity of formal models that explain why firms will enter an agreement and then renege. This paper addresses the question of why colluders cheat. In particular, we wish to know why lirms incur great expense and individuals risk prison sentences to enter a collusive arrangement and then do not abide by it. A question of secondary importance is when firms will cheat. For example, are economic upturns or downturns more likely to lead to cartel breakdown? And finally, it asks what range of collusive price-quantity outcomes we expect to see. Many forms of collusive arrangement exist. For example, tacit collusion involves no formal agreement or meeting. With explicit collusion, in contrast, formal agreements are possible. These can be either legal or illegal. The *This research was supported by a grant from the Canadian Social Sciences and Humanities Research Council. I would like to thank the following people for helpful comments and suggestions: Marcel Boyer, Brian Copeland, Peter Cramton, Partha Dasgupta, Mukesh Eswaran, Paul Geroski, Norman Ireland, Tracy Lewis, Jean-Francois Mertens, Francoise Schoumaker, John Weymark, two anonymous referees, and participants in the Canadian Economic Theory Study Group, Montreal. ‘For example, Corder0 and Tarring (1960) state that ‘Pliny, writing perhaps in A.D. 70, mentions a decree which limited the output of British lead. It is suggested that a quota system was imposed at the request of owners of lead mines in Spain when the export of British lead began to result in severe international competition.’ 0167-7187/90/$03.50
G 1990-Elsevier
Science Publishers
B.V. (North-Holland)
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M.E. Slade. Cheating on collusice agreements
model developed here is designed to explain cheating in the context of the first two but not the third type of arrangement. That is, we deal with pure tacit collusion and illegal price or quantity-setting conspiracies within the context of noncooperative game theory. Examples of the three types of collusion abound. For example, tacitly colluding firms are illustrated by the U.S. producers of lead-based antiknocks (gasoline additives).2 No cheating occurred in this industry until demand fell drastically due to government intervention in the form of environmental regulation. At this point, however, firms began to offer discounts off the (tacitly) agreed price, causing strife in the industry [U.S. Federal Trade Commission ( 198 1)I. Perhaps the most famous example of an illegal price-fixing agreement is the electrical-equipment conspiracy of the 1950’~.~ This cartel, which involved twenty-nine U.S. companies selling heavy-electrical equipment, fixed prices on standardized items as well as pricing formulas for custom products. In spite of the risks associated with participating in such an agreement, due to the criminal nature of these activities under U.S. antitrust law, agreeing parties chiseled repeatedly and even touched off price wars (Smith, 1961). Legal but not legally enforceable cartels are extremely prevalent in naturalresource industries.* Present day examples are principally organizations of governments, often both producer and consumer. Many earlier agreements, however, were among producer firms. The 1930s were the heyday of private commodity agreements. These cartels, however, were generally unsuccessful and the dominant theme was the inability of private organizations to limit exports (McNicol, 1978). Their successors today experience similar problems. Although not the subject of this paper, legal arrangements which cannot be supported by binding commitment to enforcing an agreed-upon strategy are also plagued by breakdowns. The model developed here, a parametric example, is one where symmetric duopolists produce a homogeneous product under conditions of linear demand and cost. Simplifying linearity assumptions, which are made for tractability, allow precise predictions to be made. Firms in the model must make pricing or production decisions at frequent intervals. Even when collusion is explicit, it must also be covert, and so highly complex arrangements are undesirable. Participants must thus agree upon simple repeated-game strategies which are self enforcing. In each period, firms have private information about their own costs. It is this private information that can cause cartel breakdown. sThis is only one of many possible examples. Empirical studies of industries where a few firms produce a homogeneous product almost always find evidence of tacit collusion [see for example Applebaum (1982). Roberts (1984). Spiller and Favaro (1984), and Slade (1986) and (1987a). ‘For numerous examples of illegal price-lixing agreements, see Hay and Kelly (1974) and Asch and Seneca (1976). 4For accounts of other legal cartels, see Stocking and Watkins (1946).
M.E.
Slade,
Cheating
on collusive
agreements
521
Supergame models which exhibit seemingly disequilibrium behavior such as price wars generally rely on some form of uncertainty.5 For example, in the Green and Porter (1984) model, the common demand intercept is a random variable, implying that rivals’ actions are not observable. And in the Slade (1989) model, the common economic environment shifts unpredictably, implying that own and rival payo$s are unknown. Here, in contrast, each firm’s marginal cost is priwte information, implying that the payoffs of its rioals are not known. The Green and Porter (1984) model is therefore characterized by imperfect monitoring or moral hazard, the Slade (1989) model is one of incomplete information, and the model developed here is characterized by asymmetric-incomplete information or adverse selection. The organization of the paper is as follows. In the next section, an example with no uncertainty is developed in the context of the simple carrot-and-stick repeated-game strategies introduced by Abreu (1986). Carrot-and-stick strategies are of course not the only class that can sustain tacit collusion. Nevertheless, they have two important advantages. First, with the certainty model, these strategies can be optimal (the punishments can be most severe) and are thus capable of sustaining the highest level of collusion.6 And second, carrot-and-stick strategies are extremely simple to communicate and to calculate. When collusion is purely tacit, it is difficult to imagine transmitting very complex information. If collusion is to be sustained, however, it is essential that firms understand the rules of the game and know what would happen if they were to deviate from these rules. Even when collusion is explicit, simple strategies have enormous appeal. In section 3, which is the heart of the paper, uncertainty is introduced. The model with private-cost information is developed for the case of quantity competition. The modifications required for price competition with differentiated products can be found in Slade (1987b). With the introduction of uncertainty, it is necessary to choose a solution concept. If this were a legally enforceable cartel, it would be natural to seek optimal incentive-compatible contracts. This is the approach taken by Roberts (1985), Kilstrom and Vives (1987) and Cramton and Palfrey (1990) in the context of one-shot cartel games. In the context of the current paper, however, this route has two disadvantages. First, for the revelation principle to be valid, it is necessary that the cartel sAn exception is the Rotemberg and Saloner model (1986). where all variables including the random shocks are observable. In their model, however, the punishment is never triggered. Instead, they assess the difftculty of colluding over the business cycle. 6This class of strategy ceases to be optimal for very low values of the discount factor. Under these circumstances, one must resort to asymmetric punishments, which are (much) more complex (see Abreu 1986). In addition, when uncertainty is introduced, they are no longer optimal. As long as strategies satisfy the incentive constraints discussed below, however, they are credible in the sense of Selten (1975).
522
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agreements
be able to commit to the cartel rule. That is, the cartel must be able to guarantee that it will not use the private-cost information once it is truthfully revealed. In the current context, the ability to commit does not seem to be a reasonable assumption. Second, an optimal state-contingent contract can be very complex, involving a different action for every possible cost realization. Here, however, we wish to emphasize the need to transmit only simple strategies. If collusion is purely tacit, no verbal communication is possible, and if it is illegal, no written communication is advisable. For these reasons, we limit attention to simple strategies which are perfect-Bayesian-Nash equilibria but which would not be optimal if binding contracts were feasible.
2. The repeated game with no uncertainty In this section, symmetric duopolists engage in an infinitely repeated game of quantity competition. Collusion in the repeated or supergame is supported by the symmetric carrot-and-stick punishment strategies introduced by Abreu (1986). These strategies, which are developed formally below, have a simple intuitive structure. Suppose that xi, i= 1,2, is the choice variable for player i in a repeated game. A carrot-and-stick strategy is a pair (.?,z?) where .? is the carrot or collusive value of xi and X is the stick or punishment value of xi. Supergame strategies consist of playing .? in the first period and in every subsequent period as long as both players played 2 in the previous period. If, however, someone deviated in the previous period, both players play X for one period and then resume collusion. Abreu (1986) shows that such two-phase punishments are optimalsymmetric punishments in the sense that they can sustain the highest level of symmetric collusion (see, however, the caveat in the previous footnote). In general, this highest level of collusion is supported by sticks that are more severe than Nash reversion.
The one-shot game
In each period, symmetric duopolists produce a homogeneous product and choose quantity as a strategic variable. The industry inverse-demand function is linear and can be written as p=A-B(q’+q2),
where p is price and qi is the output of the ith firm. Without generality, units can be chosen such that B= 1.
loss of
M.E.
Slade, Cheating
on collusice
523
agreements
Each firm produces at constant marginal cost, c. Profit for the ith firm is then,
where a=,4 -c is the demand intercept net of marginal cost. In the one-shot game, players choose qi to maximize their single-period profit or payoff, rti. It is straightforward to show that the Cournot-Nash solution to the one-shot game is qN = a/3,
TC~= a2/9,
(3)
and the joint-profit-maximizing qM = a/4,
or monopoly solution is
nM = a2/8.
(4)
Let n(q) be the profit to each player when they play symmetrically, 7r(q):=ni(q,q)=(u-2q)q.
(5)
n(q)
is plotted in fig. 1. It is zero at q=O, increases to a maximum at q =q”, and then falls until it reaches zero at q = 2q” =qc, the competitive output. n(q) is negative for all production levels greater than 4’. Finally consider qd(q), the best response to q or optimal single-period defection given that an opponent is producing q. qd(q) is defined by
q’(q) = argyx
x’(q’,
The profit corresponding
4) =
(a-4)/2 o
if q_la. otherwise.
to qd, red(q), is defined by
;(a-q)‘2’2
r?(q) = 7c’[q”(q), q] =
Zt;$tise
(7)
i
nd(q), which is the highest single-period profit that one firm can earn given that the other is producing q, is also graphed in fig. 1. It has a maximum of .~ 27~“’at q = 0 and falls off monotonically as q increases.
The repeated
Suppose
game
now that the one-shot
game is repeated
a countably
infinite
524
M.E. Slade, Cheating
Fig. 1. One-shot
on collusire
profits
agreemenrs
to loyal and defecting
lirms.
number of times. If players are colluding tacitly, they never communicate directly.’ For explicit colluders, it is assumed that, due to the illegal nature of the agreement, complex communication is difficult. Participants therefore must choose self-enforcing strategies that can be used currently and in the indefinite future. The payoff to player i in the repeated game is the discounted-profit stream
7Li=f s’rr’(q:,q:),
(8)
O
r=o
where 6 is the discount factor. In general, a strategy for this game is an infinite sequence of functions {sf} t=O,. . . , x, where sf maps the set of histories up to period t- 1 into R+. Here, however, we limit attention to a very simple class of strategies that have a Markov structure. A carrot-andstick supergame strategy is an output pair (G,@ and a rule
4b=4,
i= 1,2
(9) ‘When strategies.
Iirms never meet, they must resort
to devices
like press releases
to communicate
their
A4.E. Slade, Chearing
525
on co~/usive agreemenrs
For (&4) to be subgame perfect, players must have an incentive to remain loyal both in the collusive and in the punishment phases.’ Credibility therefore implies that < and 4 satisfy the following incentive constraints:
and
Constraint IC, says that the gain from defection during the collusive phase, nd(G)--z(@), must be less than or equal to the loss due to the punishment that is triggered by defection, n(g)-n(q). Because the loss is incurred in the following period, it is discounted by the factor 6. Constraint IC2, which applies to the punishment phase, can be interpreted in a similar fashion. The set C,[C,] of carrot-and-stick pairs which satisfy the incentive constraint IC,[IC2] can be written in terms of the demand and cost parameters as
C,(a,S):=((rj,cj)~R+ xR+Iu2+(-6a-4aS)(i’+4u6q+(9+86)ij2 -86&O}
(10)
and C,(U,S):={(&~)ER+
xR+l u2 - 4~66 + ( - 6u + 4aS)4 + 8&j2
Elementary techniques of analytical geometry show that, for all a >O and 0~6 < 1, the boundary of C, is a hyperbola in <-_4 space with center of symmetry [(3a+2&)/(9+86),q”]. This hyperbola is shown in fig. 2 for the case a= 1 and 6=0.875. If (&Q) lies inside the hyperbola, there is no incentive to cheat during a collusive phase. sWith such defections.
simple
time dependence,
it is suflicient
to check
the desirability
of one-shot
M.E. Slade, Cheating on collusice agreements
526
s
\
/
.’
/’
--I
./“I C,
9
@
qN
qM .-‘, /
*\
/’
‘\4C,
/.
\
/’
-, 9 Fig. 2.
/
\
qM qN Incentive
constraints a=1
qc
G
and perfect equilibria.
6 =0.875
Similarly, for all a>0 and Oc 6 < 1, the boundary of C2 is an ellipse with center of symmetry [q”,(3a-2a6)/(9-86)]. This ellipse is also graphed in fig. 2. If (&q) lies inside the ellipse, there is no incentive to cheat during a punishment phase. The hatched area in the figure is therefore the set of subgame-perfect equilibria, PE(a, a), where
PE(a, 6): = C,(a, 6) n C,(a, 6).
(11)
M.E. Slade, Cheating on collusiceagreements
527
The two conic sections which form the boundaries of C, and C2 intersect at (qN, qN) and at some (G’,4’) with 4’ $ qM and 4’ 2 qN. Equality holds only when both conic sections degenerate into the single point (qN,qN). In the certainty case, the cartel will rationally choose @= max {q”, @‘}. Fig. 3 shows the set of subgame-perfect equilibria for a = 1 and 6 =0.25. As 6 approaches zero, PE(a,6) shrinks towards {(qN,qN)} for all a. It is obvious from the figures that corresponding to most sustainable carrots 4’ there is a large number of credible punishments or sticks 4. The most severe punishment that can sustain q is the max {q[(G,4) E PE(a, S)}. As long as the stick will never be used, however, the choice of punishment seems somewhat arbitrary.
3. The repeated-game with uncertain costs Uncertainty is now introduced into the model. It is assumed that risk-neutral firms have private information about their costs. As the cartel is not legally enforceable, the duopolists cannot enter into a legal contract which specifies an output pair for each realized cost pair. Moreover, as coordination is tacit or unwritten, the agreement is assumed to be conditional only on the commonly known mean of the distribution of costs, Z. The sequence of events is as follows. In period 0, players enter the agreement described by eq. (9). At the beginning of each period t, t = 1,. . . , cc, firms marginal costs cf, i= 1,2, are independently drawn from a common distribution. Firms then choose output simultaneously, having observed their own but not their rival’s cost. More formally, suppose that cf is i.i.d. (the same for both players) and serially uncorrelated and that the covariance between c: and c: is zero. In addition, the distribution of c, which is symmetric about its mean and has a finite support, is common knowledge. Let f and F be density and cumulative density functions for a’:=A-c’, where the time subscripts has been dropped to simplify notation. The support of ui, Asup,is then A sup:={ui~f(ui)>O}=[u-,uC],
u-20
and
u+SA.
(12)
* The mean of ui, Z, is (a+ +a-)/2. Equilibrium
without cheating
At first we assume that the cartel is sustainable for all values of a (in a sense that is made precise below). Each player therefore assumes that his rival will not defect when evaluating the constraints IC, and IC,. The assumption of sustainability for all values of a is relaxed later. A strategy for the game with uncertain costs is an infinite sequence of functions {sf} t = 0,. . . , 00, where of maps the history of play up to period
528
M.E. Slade, Cheating on collusice agreemenrs
a
/
\
s”ii’ qN Fig. 3.
Incentive
constraints a=1
-
Q
qc
and perfect equilibria.
6 = 0.250
t- I and the private information u: into Rf. Attention is initially confined to carrot-and-stick strategies which are not state contingent. That is, in this subsection players do not condition their choices on their private information and cheating never occurs. We begin with the simplest case where 4 and 4 are determined exogenously. At the beginning of every decision period, each firm observes its own ci. The gain from defection is evaluated at ui. The loss, however, because it occurs in the next period, must be evaluated in expected-value terms. With a slight abuse of notation, the dependence of rc, rrd,’and rri on a is made explicit. Because 7~is linear in a, E[n(q, a)] = n(q, 5). Incentive constraints ICi and sets Ci, i= 1,2 therefore become
529
M.E. Slade, Cheating on collusive agreements
=
nd(q,a’)- Tc(q,a’)- s[iT(& a) -
UC;)
rt(tj, fi)] 5 0,
C;(a,61u’):={(~,q)ER+ x R+I (ui)2-6q'ui+(9+86)q'2-86q2+4ciS(q-ij)~0),
and
(13) C;(&6/u’):=((&q)~R+
xR+l
Strategies (&q) which satisfy the incentive constraints IC:, i= 1,2, for all ui E Asup are perfect-Bayesian-Nash equilibria for the game.’ Formally, the set of carrot-and-stick equilibria is
PW&,,
6): = ((47@)I
(ij,cj)~C~(ti,if(u)n C2(ti,G(u)Vu~A,up).
(14)
These strategies, however, are no longer optimal. In section 2, the credible (Q,q) pairs were determined for fixed u (figs. 2 and 3). We now adopt the reverse procedure. That is, we fix (@,q) and determine the values of a for which the incentive constraints are satisfied. Fig. 4 shows IC\ and IC2 plotted as functions of a’. These constraints are quadratic in a’. Let u; and uh be the roots of IC; =O, with a;suL. ai and ui are defined similarly for lC2. When the roots are imaginary, we adopt the convention that ~~[a~] =a:[&] = 3<[3q]. It is straightforward to show that a:, ai, and u: are positive. Only a; can be negative.
‘Here as before, since strategies desirability of one-shot defections.
1.10
B
have
such
a short
memory,
it is sufficient
to check
the
530
M.E.
Fig. 4.
Slade, Cheating
Incentive
on collusice
constraints
agreements
as functions
of cost.
Define max (a;, at},
(15)
aLi:= min {a:, a:},
(16)
aL: =
and (17)‘
A,,,(q,g) is the set of all a’s such that the incentive constraints are satisfied; that is, it is the set of u’s such that (&~)EC~(~~,S[U)n C2(&6/u). We assume that (G,@ was chosen such that A,,, is nonempty. There are two possibilities: the support of a may or may not be contained in the sustainable set. When A,,,(q’,g)~A,,,, (6, ij) is a perfect-Bayesian-Nash equilibrium of the game. Under these circumstances, the cartel is sustainable over the entire range of costs, firms will want to enter into a collusive agreement, and cheating will never occur (4 will never be invoked). This is the case shown in fig. 4.
A4.E. Slade, Cheating
on collusice agreemenrs
531
Normally, there will be a large number of equilibrium pairs (&q). An optimal pair (GO,@‘) can be chosen to maximize expected payoffs, E(n’), subject to the constraints IC;, i= 1,2. Finding an explicit solution to this maximization, however, is not an easy task. Instead, we turn to a different problem. We wish to examine the effect of the size of the support of a on the mostdesirable sustainable collusive outcome. To do this, we consider the two extreme cases. The first is the certainty case, which corresponds to Asup= {Z}. This is the situation depicted in figs. 2 and 3. With the example of fig. 2, the most-desirable sustainable collusive output is q.“. And with example of fig. 3, it is 4’. Next, we consider the largest Asup for which the cartel is sustainable using carrot-and-stick strategies. With this in mind, some new notation is introduced. For given & let q*((i) be the stick that maximizes the symmetric region around ti for which the cartel is sustainable. Formally, define (18) Then, for any @, cj*(ij) = arg max cc*(&4).
(19)
4
Figs. 5 and 6 reproduce figs. 2 and 3. The new graphs in addition show 4*((i) for all sustainable carrots 4’(the line with alternating dots and dashes). Similarly, for given 4, let g*(Q) be the carrot that maximizes the symmetric region around ti for which the cartel is sustainable. That is, for any i, ij*(ij) = arg max cr*(&q).
(20)
B
i*(g) is also shown in figs. 5 and 6 (the solid line). G*(4) ranges over all possible punishments and q*(i) ranges over all collusive production levels that are sustainable under conditions of certainty. The two functions must therefore intersect, and it is possible that this intersection is unique. Figs. 5 and 6, however, show that not only is the intersection not unique, the overlap is surprisingly large. This leads to a new definition. Let
maxa*(&@)/g=<*(q) 3.4
andq=q*(q)
.
(21)
532
M.E.
Slade, Cheating
on collusioe
agreements
s
‘\ \
IS . .
I
.?
I I
I I I 0
/
./
.’
1’
I
.I’ ,.’
I
Carrots
-\
/
\
qN
4
and sticks which maximize a=1
a qc ’
u “.% q
\
I
’
;,
Fig. 5.
I
I
7
security.
6 = 0.875
[a--r**, a+~**] is the largest support for which the cartel is sustainable using carrot-and-stick strategies. We denote the carrot and stick that ;z-e;gnd to a** (4”‘, qMS), where MS stands for maximum_sMuspport.GM’ 1s strictly are marked with *‘s in figs. 5 and 6. In the figures, q greater than both q”’ and 4’. This turns out to be a general result which is stated as Fact 1. Fact 1.
GM’>
qM
and ifus >$for
all a>0
The verification of Fact 1 is by simulation.
and all 0<6<
1.
,M.E. Slade, Chearing on collusive
agreements
533
s
Fig. 6.
Carrots
and sticks which maximize a=1
security.
6 = 0.250
Fact 1 implies that as the support of a increases, after some point the set of sustainable collusive output levels becomes less desirable. An obvious consequence is that the cartel is less profitable. It is therefore reasonable to ask if the cartel can do better. Equilibrium with cheating In this subsection we examine what happens if the carte1 allows cheating. That is, we wish to know if firms can improve their long-run profitability by occasionally cheating (playing myopically) rather than always following their
534
,M.E. Slade, Cheating
on collusive agreemenrs
carrot-and-stick strategies. Under these circumstances, firms will enter a collusive agreement expecting to cheat. That is to say, when they observe an a outside A,,,, they will defect. To examine this possibility, suppose that for given 4’ and 4 the support of a is not contained in the set of a’s for which the cartel is sustainable. Each player will know that there is some probability that the other will defect. The evaluation of constraints IC’, and lCz therefore becomes more complex. It is still true, however, that a player will not defect during a collusive [punishment] phase if the expected value of the first [second] incentive constraint is negative. Before evaluating the constraints, it is necessary to introduce some notation. Up to now, the output observed in a collusive [punishment] phase was a fixed quantity, G[@], where a collusive phase is a period t such that qf_ I = 4, k = 1,2, or q:- 1= q, k = 1,2. Now, however, because there is a positive probability of observing a’s outside of A,,,, the output observed in each phase becomes a random variable. Denote the realized values of these quantities by GA and 4” respectively, and let the expected value of GA [4^] be 4’ [$I. 4’ and 4’ are defined implicitly by”
(22) and
+
“j 4w.m
qd(cj’,u)f(u)du.
(23)
Eq. (22) can be interpreted as follows. With probability F(uy)--F(uf), a rival will not defect and will play 4. If an a is observed in the tails of the distribution, however, a rival will defect. When defection occurs, a player will choose to produce the most lucrative q, given that his opponent is expected to produce @‘.The interpretation of (23) should be obvious.
“In
(22) and (23) it is assumed
that ff=O
when z>/L
M.E.
Slade,
Cheating
on collusive
535
agreements
These equations are highly nonlinear. Nevertheless, it is possible to show that Proposition 1. Q’>=E(qN).
There exists a solution to (22) and (23) with cj’sE(q”)
and
Proof. The proof involves a straight-forward application of Brouwer’s fixed-point theorem. Eqs. (22) and (23) define a continuous function H(q’,@‘). Let the domain of this function, A, be
A=
0,; II
1
x[O,ci].
H maps A, a compact and convex subset of R2,into itself and therefore has a fixed point. If i’> E(qN) and qe
The existence of pairs ($,qe) satisfying (22) and (23) is cold comfort if the solution is not unique. With nonuniqueness, pairs of expected values are not interchangeable. That is, if ($‘, @) and (rj”‘, 4”“) are fixed points, (G”, @“) will in general not be one. i1 Fortunately, solutions may be unique. For example, Proposition 2.
When f(a)
is uniform, there is a unique solution to (22) and
(23). Proof.
See appendix A.
The intuition behind proposition 2 is as follows. With well behaved density functions, the partial derivatives of H are also well behaved. For fixed q’[G’], therefore, the right and left-hand sides of eq. (22) [23] cross only once. It seems reasonable to assume that the agreed-upon. carrot is at least as desirable as the single-period Cournot-Nash outcome. Therefore, because qd(q)>q for all q
=7rd(fj’,ai) -n’(g,
cj’, a’)-Gl[rc(@,
ci)-n(qe,Z)]
+ CEB, 50
“This is an example of the (often not mentioned) problem that plagues noncooperative theory. When games have multiple equilibria, strategy pairs are not usually interchangeable.
game
536
M.E.
Slade, Cheating
on collusice
agreements
(25)
=~d(~‘,~i)-n’(~,q’,ai)-6,[n(q”,~)-~(~,~)]
+ CEB, SO,
where expectations are taken over iA, 4^, and a, CEB, is the certaintyequivalence bias for constraint j, and 6j is the effective discount factor for constraint j. The certainty-equivalence biases are given byI
-Cov(u,tjA)} and
(26) CEB,=Var(~“)/4+6,{2[Var($A)-Var(q”)]+Cov(a,q”) -
cov(a, GA))
and the effective discount factors are given by
sj=G[F(&-F(u’,)].
(27)
Eq. (27) has the following interpretation. The next-period loss due to defection must be weighed by the discount factor, 6. In addition, firm one’s rival will defect in the current period with probability [l -F(ui,) + T(u’,)]. If firm two defects, punishment will ensue regardless of firm one’s choice and there is no loss to firm one if it also defects. The next-period loss must therefore also be weighted by the probability that the rival remains loyal in the current period. The effective discount factor is therefore smaller than 8. The signs of the certainty-equivalence biases are ambiguous. For a large range of reasonable parameter values, however, CEBj is positive. This will be the case, for example, whenever Var($“)zVar(d”) and COV(~“,U)Z COV(q”,u).‘3 If the support of a is not contained in the set of sustainable u’s, there are thus four consequences. First, the effective carrot is less desirable than the agreed-upon carrot. Second, the effective stick is less severe than the agreedupon stick. Third, the effective discount factor is smaller than 6. And finally, certainty-equivalence bias implies that the expected value of the gain due to 12The certainty-equivalence biases are obtained by expanding the formulas for n, nd, and ~’ and taking their expected values. 13Through the use of numerical simulations, it is possible to show that when f(a) is uniform, CEBj is always positive.
.V.E. Slade, Cheating
on collusice
agreements
537
defection is usually greater than the gain evaluated at the expected values of the random variables. The combination of these four effects reduces the profitability of collusion. Nevertheless, if the present value of collusion with cheating is greater than z(q’)/( l-6), the highest payoff that each could earn if they adhered to their carrot-and-stick strategies with no cheating, firms will want to choose a 6~4’. This leads to the definition of a new strategy, the cheating carrot-andstick strategy. For any 4 and q, let a collusive phase be a period t when q:_ 1=ij or q:_ I =& k = 1,2. All other periods are then punishment phases. The cheating strategy is
qa=4,
i=l,2
4 qf =
qd(& a’) 4 qd(@9 a’)
(28)
I
if collusive
if punishment
phase and
phase and
This strategy, which is state dependent, is a perfect-Bayesian-Nash equilibrium for the game. It remains to be seen if, for some parameter values, the expected value of each firm’s profit stream when both use cheating strategies is higher than n(G’),!( l-6). An example shows that this is possible. Let a= 1.0 and 6 =0.7 and suppose 1.15361. With that a is distributed uniformly on Asup= 1.0+0.1536=[0.8464, these parameter values, Go = gnrs = 0.28, 4’ = $MS =0.39, c(** = 0.1536, and n(GO)=0.1232. With no cheating, the highest payoff that firms can earn is 7r(~0)/(l-6)=o.4107. Now suppose that firms decide to allow cheating and select $=0.25 = E(q”) and $=0.42 =q*($), where c stands for cheating. The corresponding A,,,= [0.8577,1.1523]. Clearly, it is possible to observe values of a for which cheating pays. Player i will defect in a collusive phase if 1.1523 caf 5 1.1536. Similarly, player i will defect in a punishment phase if 0.84645 af<0.8577. Appendix B shows that under these circumstances expected profit from playing the cheating strategy exceeds 0.4107. The example demonstrates that if colluders are constrained to using carrot-and-stick strategies then, due to private information about production costs, firms can improve their long-run profitability by occasionally cheating (playing myopically). That is to say, they can be made better off than if they were constrained to choose a strategy ((i,@ for which A,,,(& 4) 2 Asup. In some sense, the carrot-and-stick strategy is a strawman. It does not
538
M.E.
Slade, Cheating
on collusive
agreements
allow the necessary flexibility when there is incomplete information about costs and is therefore not optimal. It is my contention, however, that collusive agreements that are more complicated than carrot-and-stick strategies may be too complex to be a feasible means of tacit collusion.
4. Summary and conclusions In the model developed here, symmetric duopolists must make frequent pricing or production decisions. Even when collusion is overt, therefore, it is not possible for them to coordinate in each decision period. Firms have private information about their own costs which they use in making their decisions. It is this private information that can cause cheating. The collusive agreement will be stable under normal conditions. However, either unexpectedly high or low costs can-cause a breakdown. With the linear demand and cost model developed here, defection is more likely to occur in good times. That is, firms are more tempted to cheat and increase output when costs are 10w.l~ Can we really say that firms cheat, given that cheating is part of an equilibrium strategy? Perhaps it would be better to say that, to the outside world, firms appear to cheat. For example, it is common for cartels to announce output quotas. Announcing that violations are to be expected, however, would be much more unusual behavior. Nevertheless, it may be understood among cartel members that cheating will occur under abnormal economic conditions. Cartel discipline, of course, requires that cheaters be punished. For if cheaters were not punished, members would have an incentive to misrepresent their costs. That is, they would increase output when costs were normal. For tractability, the model developed here makes use of several restrictive assumptions. Many extensions and generalizations are therefore possible. Obvious modifications include using profit functions other than quadratic and supergame strategies other than carrot and stick. It seems clear, however, that if cheating occurs in this model, it will also occur when the less severe punishment of Nash reversion is used. In spite of the simplicity of the model and the variety of possible generalizations and extensions, it is hoped that it offers insights concerning the age-old problem of cheating on collusive agreements.
14With the present model, an unusually low cost is more apt to cause breakdown in a collusive phase. If the opposite were true, however, there would be no need to punish firms for playing too cooperatively, given that both players benefit from a unilateral output reduction. The result of breakdown when cost is low is not the same as breakdown when demand is high, as in Rotemberg and Saloner (1986).
539
M.E. Slade, Cheating on coilusice agreements
Appendix A Proof
of Proposition
2
First I will show that for q’ fixed at an arbitrary cje’zE(qN), unique rj” such that I?‘(<“, a”) =Lj”.
H’(cj’,q’j=‘i
(I-
there exists a
qd(ie,a)f(a)da-~qd(cje,a)f(a)dn+Lj[F(a~~)-F(u~)] 4
- ;
wdIl)2-(4)213
where
the support of a, A,,,=[G--!x,G+r]. When a:=~;,H’(@~,q~‘)= and 4” = E(qN) =ti/3. Assume therefore that a~>~~. In this case, there exists an interval [qtqt] such that when rje lies in this region, defection does not occur for some realized values of a. To find ai, at., q:, and q:, apply the quadratic formula to eq. (13) in the text to obtain q’(<‘, ci)
aL=34’-fi
and
a:=3~Yj~+,/O,
(A.l)
where (8;i’
s>
if this expression is > 0 otherwise.
(A.2) shows that qt=min{(i/2-q”,O)
and
qh=q’+
In addition. +4=2JG, and
(a;S)2-(at)2=
12lj’Jo,
(A.2)
.Cf.E. Slade, Cheating on collusioe agreements
540
H’(q’,q’‘)=qd(q-e,,-)
(A.3) shows H’(tj’,
that
4”) 5 qd($,
-
when
$F(<‘_~).
4’s 4, H’(Ge, de’) 1
(A.3) qd(ie, 5)
and
when
4’26,
5).
Now consider t?H’/&j’=:H:,
where 26(fi-4~7 Jo 0
ifO2O -otherwise.
For g’IqL, O=jl=O, H’=qd(qe,fi), and H:=-l/2. For q,?=
Using extremely similar reasoning, it is possible to show that for 4’ fixed at an arbitrary 4” 6 E(q,‘), H’(cj”, 4’) crosses the 4%degree line 4’= 4’ exactly once on D,. The details are left up to the reader. Call the point where the two cross q=‘= @(4”‘).
We must now determine the slope of G’(tj’j. From (A.l) Sak/Z@e’=
26(-4@+5) Jo 0
if oro *otherwise.
This implies that, ?aQ&j” 2 0
for all
4” 2 E(q”).
M.E. Slade, Cheating
on collusice agreements
541
In addition,
S&/g = - C?at/$h 0 for all
4” 2 E(q”).
The middle term in (22) (the sustainable region) therefore increases as 4” increases and the two end terms (the defection regions) decrease. This implies that G”(@)~O. By very similar reasoning, it can be shown that G”($+‘) 60. The two functions G’ and G*, therefore, cross at most once on A. By proposition 2, however, they cross at least once on A. Q.E.D. Appendix B The example
In our example, a= 1.0 and 6=0.7. Suppose that a is distributed uniformly on Asup= 1.0&0.1536=[0.8464, 1.15361. With these parameter values, Go= GMS= 0.28, 4’ = QMS= 0.39, u **=0.1536, and ~(@‘)=0.1232. We wish to show that, by allowing cheating, the expected value of profit in each period is greater than 0.1232. With the cheating strategy, players select $=0.25 =qM and q’=O.42 = q*($), where c stands for cheating. The corresponding A,,,= [0.8577, 1.15231. Let pI[p2] be the probability that firm i defects in a collusive [punishment] phase, pi =prob {a’# A,,,lcollusive phase) =prob{l.1523
J!Lza!_ 2
=0.0113/l ASUP/=0.0368,
phase, there are four possibilities: both players j remains loyal, firm j defects while i remains d’(q) to the profit to the nondefecting firm who and plays qd(q).
(B.1)
If p’ is the probability of being in a collusive phase and p” is the probability of being in a punishment phase then the expected value of km i’s profit in any period, conditional on using the cheating strategy, is
542
M.E.
Slade, Cheating
on collusice
agreements
+Pa(l -P2) [n”(F) + ~dw)l +p:4?d(~‘))}.
U-3.2)
In (B.2), for each phase each of the four possibilities is weighted by its respective probability. p’ and pp can be found as solutions to the equations Pc=Pc(1-p,)2+pp(l-p2)2,
pp=
1 -PC.
(B.3)
For example, the first part of (B.3) says that the probability of being in a collusive phase equals the probability that you were in a collusive phase last period times the probability that both players remained loyal plus the probability of being in a punishment phase last period times the probability that both punished. With our parameter values, p’=O.9910
and
pP=O.O090
7T($)=O.1250
x(qC)= 0.0672
Ge=0.2510
qe = 0.4052
n”((1’)= 0.0940
nd(4’) = 0.1403
n”(Lj’)= 0.1205
nd(@) = 0.0884
n(qd(4"')) =0.0940
n(qd(if)) =0.1205.
Values for Lieand $ were obtained by solving eqs. (22) and (23) numerically. All other values were obtained by straight-forward application of the definitions of the functions. When numerical values are substituted into (B.l), the result is E(n’lcheating) =0.1245 >O.l232=E(n’lcarrot
and stick).
References Abreu, D., 1986, External equilibria of oligopolistic supergames, 191-225. Applebaum, E., 1982, The estimation of the degree of monopoly 19. 287-299.
Journal
of Economic
power, Journal
Theory
39,
of Econometrics
M.E.
Slade, Cheating
on collusice
agreements
543
Asch, P. and J.J. Seneca, 1976, Is collusion profitable?, The Review of Economics and Statistics 58, l-12. Cordero, H.G. and L.H. Tarring, 1960, Babylon to Birmingham (Quin Press Ltd., London). Cramton, P.C. and T.R. Palfrey, 1990, Cartel enforcement with uncertainty about costs, International Economic Review 31, 17-48. Green, E.J. and R.H. Porter, 1984, Noncooperative collusion under imperfect price information, Econometrica 52. 87-100. Hay, G.A. and D. Kelly, 1974, An empirical survey of price-tixing conspiracies, Journal of Law and Economics 17, 13-38. Kilstrom, R.E. and X. Vives, 1987, Collusion by asymmetrically informed firms, University of Pennsylvania working paper. McNicol, D.L.. 1978, Commodity agreements and price stabilization (D.C. Heath, Lexington, MA). Roberts, K., 1985, Cartel behavior and adverse selection, Journal of Industrial Economics 33, 33-45. Roberts, M.J., 1984, Testing oligopolistic behavior: An application of the variable-profit function, International Journal of Industrial Organization 2, 367-383. Rotemberg, J.J. and G. Saloner, 1986, A supergame-theoretic model of price wars during booms, American Economic Review 76, 390-407. Selten. R., 1975, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 2, 25-55. Slade, M.E., 1986, Conjectures, firm characteristics, and market structure: An empirical assessment, International Journal of Industrial Organization 4, 347-370. Slade, M.E., 1987a. Intertirm rivalry in a repeated game: An empirical test of tacit collusion. Journal of Industrial Economics 35, 483-498. Slade, M.E., 1987b, Cheating on collusive agreements, Department of Economics Discussion paper no. 87-27, University of British Columbia. Slade, M.E., 1989, Price wars in price-setting supergames, Economica 56, 295-310. Smith, R.A.. 1961, The incredible electrical conspiracy, Fortune, May, p. 224. Spiller, P.T. and E. Favaro, 1984, The effects of entry regulation on oligopolistic interaction: The Uruguayan banking sector, Rand Journal of Economics 15, 244-254. Stocking, G.W. and M.W. Watkins, 1946, Cartels in action (Twentieth Century Fund, New York). U.S. Federal Trade Commission, 1981, Decision in re Ethyl Corp. et al., Docket no. 9128.