Theoretical Economics Letters, 2013, 3, 317-321
Published Online December 2013 (
Open Access TEL
Non-Cooperative Collusion in Static and
Dynamic Oligopolies
Stephen B. Wolf1,2
1Tufts University, Medford, USA
2Pembroke College, Oxford, UK
Received April 17, 2013; revised May 17, 2013; accepted May 24, 2013
Copyright © 2013 Stephen B. Wolf. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor-
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This paper provides an analysis of collusion in oligopolies from a game-theoretic perspective. It first provides a basic
survey of oligopoly models an d then uses game theory to analyze non-coo perative or tacit collusion in th ese models, in
a way that should be accessible to undergraduate economics students. In this way, the author characterizes the condi-
tions under which collusive behavior might occur. Importantly, this paper draws its conclusions by using relatively ba-
sic methods with which those foreign to the subject should be able to understand.
Keywords: Game Theory; Oligopoly; Collusion
1. Introduction
Industrial organization is composed of a latticework of
interactions between agents in situations of payoff inter-
dependency, in which the optimal strategy for each agent
depends on the actions of the others. In game theory,
cooperation is said to be joint actions undertaken by agents
in order to produce mutually beneficial outcomes; the go al
of industrial economics is to examine the d etailed institu-
tional framework of industries as well as the acti on s of the
agents and firms that constitute them, and it will be highly
informative to discuss the conditions by which coopera-
tive behavior occurs between self-regarding rational actors.
The purpose of this paper is to examine the potential
for non-cooperative collusive behavior in oligopoly mar-
kets, also known as “tacit collusion”, seeking specifically
to identify if and when cooperative equilibria can occur
between otherwise competitive forces. It is important to
note the limited scope of this paper, which does not test
its conclusions by using available data; as such this paper
is best read as a survey and examination of the contem-
porary literature. Section 2 will begin by explainin g non-
cooperative games, emphasizing that complete inde-
pendence of each agent creates the potential for highly
competitive behavior. Section 3 will then discuss in terac-
tions in static oligopolies, focusing on the Bertrand
model in particular and questioning the assumption of a
one-shot game. Section 4 will examine dynamic oligo-
polies, emphasizing foresight as a major factor in the main-
tenance of a collusive equilibrium. Section 5 will summa-
rize the previous sections and discuss their conclusions.
2. Non-Cooperative Agents
For any sort of game-theoretic interaction, equilibrium
must not only arise as a result of the actions chosen by
agents but also must be a sustainable outcome if the
game were to be played repeatedly. In this paper, we
restrict our scope to non-cooperative games, or those in
which agents are not already bound by explicit agree-
ment or contract to act cooperatively and cannot make
binding commitments to each other; thus, strategies can-
not be join tly co ord in ated o r arrang ed by multip le ag en ts.
In such a system, it is the perfectly discrete nature of
agents, their payoffs, their actions, and available infor-
mation that may prevent cooperative outcomes from be-
ing reached. In other words, even though agents may
have the same constraints and may be seeking fulfillment
of the same objective, there exists potential for competi-
tive behavior and a zero-sum outcome1, simply because
1A zero-sum game is one in which a single agent wins and the others
lose without question.
each agent is a separate and distinct actor who assumes
his competitors will act according only to their own
self-interest. Thus, in this paper I will refer to coopera-
tive and collusive outcomes interchangeably, as I ignore
the possibility of contractual obligations.
Nonetheless, the potential for competitive behavior
does not preclude non-coordinated attainment of a coop-
erative equilibrium that is mutually more beneficial. Ac-
cording to Ray Rees [1], collusive behavior between co-
operative agents requires a process of communication or
exchange of information by which coordination can oc-
cur as well as some explicit mechanism of punishing
those who fail to uphold the agreement, and in the ab-
sence of such a mechanism reinforcing a cooperative
outcome, any communication or words coordination
amongst non-cooperative agents will be considered no-
thing more than “cheap talk”. Thus, collusive equilibria
in non-cooperative games must be sustained purely by
the self-interests of those agents involved, who are as-
sumed to act only in order to maximize their own utility
3. Cooperation in Static Oligopolies
Non-cooperative interactions in markets of imperfect
competition are represented at the most fundamental
level by the classic models of static oligopoly. Cleanly
put by Church & Ware [2], these models are “essentially
timeless” and act as a “simple forum to introduce the
concepts of payoff interdependency and strategic interac-
tion” ([2], p. 232); they demonstrate almost too well how
Nash equilibria—a strategy set in which each agent has
followed his or her best response strategy, and no one
can improve his or her payoff by choosing a different
individual action—can be below the optimal total payoff
that could have been achieved through cooperation.
However, as we shall see, these mod els are sev erely lim-
ited in explanatory power.
The two basic models are the Cournot and Bertrand
oligopolies; in both, the game consists of simultaneous
play during a single period under perfect information, in
which two agents (there can be multiple, but we will as-
sume only two agents engage in the static market) com-
pete by choosing a production strategy of homogeneous
goods to maximize their own payoffs given perceptions
about how rivals will act. In the Bertrand model, the
firms choose the prices of the goods they supply, while in
the Cournot model they set the levels of quantity at
which they will produce and take whatever price allowed
by the demand curve. For the sake of conciseness, we
will only examine the Bertrand model, because it makes
a cleaner statement about the limits of cooperative be-
havior in a single period.
The simplest variation of the Bertrand models sees
each firm setting the price at which it will sell the goods
it produces, and producing the quantity of goods de-
manded by consumers at that price. The strategy set
available to each firm consists of setting some price
for i = 1, 2, that has properties of
where pC is the price level at a competitive equilibrium
and pM is the monopoly price level. For firms without
fixed costs, pC is equal to marginal cost and generates
zero profit. Firms will not set
below pC because it
would imply a loss on ever y unit sold, nor above pM as it
is price at which profit is maximized. Throughout this
paper, we will assume that barriers to entry2 are suffi-
ciently high such that even if both firms were to set price
equal to pM, profits would not be enough to induce other
agents to enter the industry; thus, pM truly is the maxi-
mum level at which a firm would set its price. In practice,
such high barriers to entry are quite rare in most markets;
however, we will maintain this assumption throughout
this paper because it suits a game-theoretic approach.
Gilbert [4] provides an in-depth discussion of dynamic
markets that incorporates barriers to entry.
In this variation of the Bertrand model, there is no
product differentiation, spatial or temporal factors af-
fecting consumer preferences, asymmetric information
among consumers, nor capacity constraints on firms’
production; as a result, whichever firm sells the its prod-
uct at a lower price will supply the entire market, leaving
the other firm with zero sales (and a profit less than or
equal to zero, depending on whether production has al-
ready begun and whether marginal cost is non-refund-
able). The game becomes a slightly more complex ver-
sion of the Prisoner’s Dilemma and is a zero-sum game
for all price combinations where 12
p. Firm 1
knows that, for any 2
chosen by firm 2, it can under-
cut its competitor by setting 12
p and steal firm 2’s
entire profits, something that firm 2 is quite aware of.
Because at all price combinations above pC there is an
incentive for either firm to increase its profits and un-
dercut its competitor, the Nash equilibrium of the inter-
action occurs at
in which no agent has a profitably deviating strategy,
resulting in a total profit equal to zero. Thus, according
to the Bertrand model and the Nash equilibrium reached
through game-theoretic analysis, price competition be-
tween two agents in static game results in a complete
destruction of profits and market equilibrium equal to
2The definitions of “barriers to entry” are numerous and varied, making
a common understanding of the concept difficult, though not impossible
For our purposes, we will define them as any market condition that may
explain a difference in profitability between incumbent firms and pro-
spective entrants. For more information on barriers to entry, see Cabral
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S. B. WOLF 319
that of a perfectly competitive market, despite imperfect
competition. On the surface, this equilibrium is extre-
mely different from that of the Cournot model, where
firms exercise market power as a negative function of the
elasticity of demand and the number of firms in the mar-
ket, resulting in an equilibrium price and quantity above
and below, respectively, that of the competitive equilib-
rium3 ([2], p. 243; [1], p. 23). However, the outcomes of
the Cournot and Bertrand models are thematically inter-
changeable; regardless of whether price or quantity is
chosen as a variable, static markets under imperfect
competition will generate an equilibrium total profit well
below that reached in a monopoly.
Without prior knowledge of whether the game will be
repeated, neither firm engages in collusive behavior be-
cause both seek to maximize profits in the current period.
However, these static models are highly questionable
both in their implications and assumptions. The Bertrand
model, it could be said, implies that agents seek to beat
their competitors at all costs, even if it means the com-
plete destruction of the profitability o f the entire industry;
it therefore assumes that agent utility is a function of
profit alone. If the Bertrand model were to be examined
using utility functions that in part depended on the
strength and profitability of the industry as a whole,
would that allow for the possibility of a cooperative
equilibrium between non-cooperative ag ents? This matter
requires further investigation. Yet the most blatant criti-
cism of the static oligop oly mod els is that the assumption
of a one-shot game interaction is extremely unrealistic,
no matter the industry; therefore, to assess cooperative
behavior in a more useful way, we much analyze these
markets in a dynamic context, with the present knowl-
edge that repeated games will occur in the future.
4. Tacit Collusion in Dynamic Oligopoly
In considering dynamic oligopoly, we allow for the mul-
tiple repetitions of the game-theoretic interaction in ques-
tion; agents are aware of this rep etition, so payoffs in the
current period take into account profits from future peri-
ods. To begin, we forgo the Bertrand assumption that the
firm with the lowest price will supply the entire market,
and instead move towards a Cournot equilibrium in
which price is below monopoly level bu t above marginal
cost. Yet rather than focusing on either price or o utput as
the competitive variable, let us consider them as two
sides of the same coin; joint restriction of output and
growth of prices will increase total profit while an indi-
vidual firm lowering prices and increasing output above
the monopoly level will steal market share from its com-
petitors—though not necessarily reducing their sales to
With all this in mind, we can now examine tacit collu-
sion in and of itself as the occurrence of cooperative ac-
tions in repeated non-cooperative games. Consider an
oligopoly market in which each firm has set its price
equal to the monopo ly price level and restricted outpu t to
its share of the monopoly quantity. Industry profit is
maximized and the market is an optimum collusive equi-
librium; however, any one firm could increase its indi-
vidual profit at the cost of total profit in the next period
by setting price below the collusive level and increasing
output according to the demand curve, maximizing indi-
vidual profits given the price and output choices of its
competitors. This threat of competitive behavior is al-
ways present in collusive equilibrium. However, each
firm is aware that any deviation from the collusive out-
come will, depending on the ability of other firms to de-
tect and swiftly respond, set off a price war that will
generate heavy losses in future periods. Thus, for each
individual firm, the decision to defect from the collusive
equilibrium in a given period depends entirely on the
tradeoff between current and future profit.
There are several factors identified by the literature
that are critical to evaluating this tradeoff. First, a poten-
tial defector must be able to evaluate the profits in the
long term associated with maintaining a cooperative
equilibrium. Assuming market demand and cost condi-
tions will be roughly the same across periods, firms will
earn a collusive profit
M in every period in which all
firms maintain the monopoly price and total output.
However, it is a fundamental theme in economics that the
value of a given income in the current period is much
greater than the value of the same amount in successive
periods; the present value of expected future profits must
reflect this by discounting future cash flows. This dis-
count rate—represented by r—will often reflect the cost
of capital per period for the firm. Firms with high levels
of capital depreciation and thus larger capital costs will
more heavily discount the present value of future inco me;
these firms will be less likely to maintain a collusive
equilibrium. Furthermore, the size of the discount rate r
is significant in determining not only the present value of
maintaining a collusiv e equilibrium in future periods, but
also how costly the future losses of profit caused by pun-
ishment will be relative to the immediate gains obtain-
able by defecting.
3The reason for this is that, in the Cournot model, individual firms set
quantities of output given perceptions that its rival will sell a fixed
quantity. The demand curve faced by each firm is much less elastic
than those in the Bertrand model, where a firm can win full share o
the market by undercutting its competitors even by a miniscule
amount making for an extremely elastic demand. (Church & Ware [2],
. 270).
Next, the immediate payoff associated with defecting
from the collusive equilibrium is the increased profit
R obtained by the firm that increases output and
decreases price while other agents maintain monopoly
level price and output. Th e size of this augmented payoff
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depends “partly on cost, demand, and capacity parame-
ters and partly on the length of time for which a higher
profit… can be earned before retaliation by the other
firms takes place” ([1], p. 31). Rees further comments
that the short term gain to breaking a collusive equilib-
rium would be virtually nonexistent if production were
constrained by a maximum rate of capacity output given
technology and if the output of each firm under the col-
lusive agreement were close to capacity; conversely, if
there were an excess capacity above the collusive output
level and the monopoly price is sufficiently greater than
marginal cost, then it would be both feasible and profit-
able for a firm to defect from the collusive agreement.
Furthermore, the ability of other firms to respond puni-
tively towards the d efector and to do so with alacrity also
affects the immediate payoff of defecting. According to
Church & Ware [2], a low likelihood that non-defecting
firms will detect price cuts and increased outputs as well
as a great length of time (i.e. a high number of periods)
required for these firms to implement punitive strategies
will weaken the sustainability of a collusive equilibrium.
The final critical factor in the tradeoff facing a poten-
tial defector is the evaluation of the punitive conse-
quences of breaking collusion. One way to model these
punitive actions is explained by Martin [5] citing James
Friedman’s trig ger strategy. The trigger strategy consists
of two central features: 1) each firm produces output to
its share of the monopoly quantity—with price and thus
total profit maximized at the monopoly level—and will
continue to do so in the following period so long as all
other firms cooperated in the previous period by also
maintaining monopoly price and quantity; and 2) if in
any period the price offer ed by any firm is lower than the
monopoly level, all firms revert to their Nash equilibrium
price and output combinations—either permanently or
until such time as the short-term profit increase from
defecting is offset by the profit lost o ver multiple periods
by forgoing cooperative outcome. However, one limita-
tion to this reversion to the Nash equilibrium as a viable
form of punishment is that it may not be very severe,
since in this case the Nash equilibrium is a Cournot
quantity that is below competitive and above its share of
the monopoly level, implying only “a moderate loss of
profit relative to the collusive agreement” ([1], p. 32). A
more sever altern ative is know n th e minimax punishment.
For each firm in the market, we can define a production
and price strategy that is the firm’s best response given
actions of all other agents in the market. By identifying
the actions that generate the smallest best response profit,
or the minimax profit, for the firm in question and then
pursuing these actions, the non-defectors can punish the
defecting firm more heavily than simply reverting to the
Nash equilibrium.
Regardless of the method of punishment implemented,
the trigger strategy appears in pr actice in the form of any
conditions that lend credibility to the threat of punitive
actions, as well as increase their ability to generate heavy
future losses for the defector4. Frequent interactions be-
tween firms will ensure that they are aware of each
other’s price and output choices in a given period and
thus can detect any attempts to undercut cooperation
right after they happen. A high degree of product homo-
geneity will prevent firms from maintaining price differ-
ences that might have been enabled by product differen-
tiation; thus, a defector receives little to no protection
from punishment, as it cannot sustain a price above the
punitive level forced by its competitors. Price matching
guarantees provide credibility to threats of punishment,
ensuring that an y competitive behavior will be met with a
swift and cutthroat price war. Thus, these competitive
qualities provide a built-in trigger strategy and actually
increase the likelihood of collusive behavior.
The factors and conditions influencing the sustainabil-
ity of a collusive equilibrium are combined in what is
known as the Folk Theorem ([1,7]). Using
S to denote
the minimax punishment profit, the ith fi r m,
1, 2,,ik
, will not defect if
 
iii i
 
 .
In words, if for each firm the immediate profit gain
from defecting from the collusive equilibrium is less than
the present discounted value of future profit losses re-
sulting from minimax punishment during an infinite
number of periods, then the collusive equilibrium will
likely be sustained.
5. Conclusion and Summary
Using the separate components discussed, we can make
several statements as to the conditions under which tacit
collusion between non-cooperative agents is likely to
occur, as well as concerning the strength and sustainabil-
ity of these collusive equilibria. In a one-shot game,
agents are unlikely to collude because it will be always
one’s interest to break collusion; this conclusion can also
be applied to any market with sufficiently narrow time
horizons such that firms look no further than the single
upcoming period. In dynamic oligopolies, collusion is
likely to be sustained when the present value of future
losses exceeds the immediate payoff associated with de-
fection from a cooperative equilibriu m. While there are a
number of factors that help define this tradeoff such as
capital costs, firm interactions, capacity parameters, and
severity of minimax punishments, it is only when we
consider foresight and the role of future expectations
4An alternative analysis to the trigger strategy is reputation maintenance
in which firms cooperate in order to cultivate a reputation for coopera-
tion. For more information, see Bowles & Gintis [6].
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when these issues appear. The essence of cooperative
behavior lies in the ability to interpret future strategies
and possible events in the current period.
[1] R. Rees, “Tacit Collusion,” Oxford Review of Economic
Policy, Vol. 9, No. 2, 1993, pp. 27-40.
[2] J. Church and R. Ware, “Industrial Organization: A Stra-
tegic Approach,” Irwin McGraw Hill, London, 2000.
[3] L. Cabral, “Barriers to Entry”, In: S. N. Durlauf and L. E.
Blume, Eds., The New Palgrave Dictionary of Economics,
in New Palgrave Dictionary of Economics, 2nd Edition,
Palgrave Macmillan, 2008.
[4] R. Gilbert, “The Role of Potential Competition in Indus-
trial Organization,” Journal of Economic Perspectives, Vol .
3, No. 3, 1989, pp. 107-127.
[5] S. Martin, “Industrial Organization in Context,” Oxford
UP, Oxford, 2010.
[6] S. Bowles and H. Gintis, “Cooperation,” In: S. N. Dur-
lauf and L. E. Blume, Eds., New Palgrave Dictionary of
Economics, 2nd Edition, Palgrave Macmillan, 2008.
[7] G. Bitsakakis, “Static and Dynamic Oligopoly,” Class
Lecture, Pembroke College, Oxford University, 1-6 May