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Chapter 14: Price-fixing and Repeated Games
2
Collusion and cartels
• What is a cartel?– attempt to enforce market discipline and reduce competition
between a group of suppliers
– cartel members agree to coordinate their actions• prices
• market shares
• exclusive territories
– prevent excessive competition between the cartel members
Chapter 14: Price-fixing and Repeated Games
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Collusion and cartels 2
• Cartels have always been with us; generally hidden– electrical conspiracy of the 1950s
– garbage disposal in New York
– Archer, Daniels, Midland
– the vitamin conspiracy
• But some are explicit and difficult to prevent– OPEC
– De Beers
– shipping conferences
Chapter 14: Price-fixing and Repeated Games
4
Recent events
• The 1990s saw record-breaking fines being imposed on firms found guilty of being in cartels– illegal conspiracies to fix prices and/or market shares
– Archer-Daniels-Midland $100 million in 1996
– UCAR $110 million in 1998
– Hoffman-LaRoche $500 million in 1999
Chapter 14: Price-fixing and Repeated Games
5
Recent cartel violationsF. Hoffman-LaRoche Ltd. Vitamins 1999 $500 International
BASF AG (1999) Vitamins 1999 $225 International
SGL Carbon AG Graphite Electrodes 1999 $135 International
UCAR International Inc. Graphite Electrodes 1998 $110 International
Archer Daniels Midland co. Lysine and Citric Acid 1997 $100 International
Haarman & Reimer Corp. Citric Acid 1997 $50 International
HeereMac v.o.f. Marine Construction 1998 $49 International
Hoechst AG Sorbates 1998 $36 International
Showa Denko Carbon Inc. Graphite Electrodes 1998 $32.5 International
Fujisawa Pharmaceuticals Co. Sodium Gluconate 1998 $20 International
Dockwise N.V. Marine Transportation 1998 $15 International
Dyno Nobel Explosives 1996 $15 Domestic
F. Hoffman-LaRoche Ltd. Citric Acid 1997 $14 International
Eastman Chemical Co. Sorbates 1998 $11 International
Jungblunzlauer International Citric Acid 1997 $11 International
Lonza AG Vitamins 1998 $10.5 International
Akzo Nobel Chemicals BV & Glucona BV Sodium Gluconate 1997 $10 International
Chapter 14: Price-fixing and Repeated Games
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Recent cartel violations 2• Fines steadily grew during the 1990s
0100
200300
400500600
700800
9001000
Cri
min
al F
ines
($ m
illi
on)
1994 1995 1995 1996 1997 1998 1999
Fiscal Year
Chapter 14: Price-fixing and Repeated Games
7
Cartels
• Two implications– cartels happen
– generally illegal and yet firms deliberately break the law
• Why?– pursuit of profits
• But how can cartels be sustained?– cannot be enforced by legal means
– so must resist the temptation to cheat on the cartel
Chapter 14: Price-fixing and Repeated Games
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The incentive to collude
• Is there a real incentive to belong to a cartel?
• Is cheating so endemic that cartels fail?
• If so, why worry about cartels?
• Simple reason– without cartel laws legally enforceable contracts could be written
De Beers is tacitly supported by the South African government• gives force to the threats that support this cartel
– not to supply any company that deviates from the cartel
• Investigate– incentive to form cartels
– incentive to cheat
– ability to detect cartels
Chapter 14: Price-fixing and Repeated Games
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An example• Take a simple example
– two identical Cournot firms making identical products
– for each firm MC = $30
– market demand is P = 150 - Q
– Q = q1 + q2 Price
Quantity
150
150
Demand
30 MC
Chapter 14: Price-fixing and Repeated Games
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The incentive to collude
Profit for firm 1 is: 1 = q1(P - c)
= q1(150 - q1 - q2 - 30)
= q1(120 - q1 - q2)
To maximize, differentiate with respect to q1:
1/q1 = 120 - 2q1 - q2 = 0
Solve this for q1Solve this for q1
q*1 = 60 - q2/2
The best response function for firm 2 is then:
q*2 = 60 - q1/2
This is the best responsefunction for firm 1
This is the best responsefunction for firm 1
Chapter 14: Price-fixing and Repeated Games
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The incentive to collude 2
• Nash equilibrium quantities are q*1 = q*2 = 40
• Equilibrium price is P* = $70
• Profit to each firm is (70 – 30)x40 = $1,600.
• Suppose that the firms cooperate to act as a monopoly– joint output of 60 shared equally at 30 units each
– price of $90
– profit to each firm is $1,800
• But– there is an incentive to cheat
• firm 1’s output of 30 is not a best response to firm 2’s output of 30
Chapter 14: Price-fixing and Repeated Games
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The incentive to cheat
• Suppose that firm 2 is expected to produce 30 units
• Then firm 1 will produce qd1 = 60 – q2/2 = 45 units
– total output is 75 units
– price is $75
– profit to firm 1 is $2,025 and to firm 2 is $1,350
• Of course firm 2 can make the same calculations!
• We can summarize this in the pay-off matrix:
Chapter 14: Price-fixing and Repeated Games
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The Incentive to cheat 2
Firm 1
Fir
m 2
Cooperate (M)
Cooperate (M)
Deviate (D)
Deviate (D)
(1800, 1800) (2250, 1250)
(1250, 2250) (1600, 1600)
This is the Nashequilibrium
This is the Nashequilibrium
(1600, 1600)
Both firms have theincentive to cheat on
their agreement
Chapter 14: Price-fixing and Repeated Games
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The incentive to cheat 3
• This is a prisoners’ dilemma game– mutual interest in cooperating
– but cooperation is unsustainable
• However, cartels to form
• So there must be more to the story– consider a dynamic context
• firms compete over time
• potential to punish “bad” behavior and reward “good”
– this is a repeated game framework
Chapter 14: Price-fixing and Repeated Games
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Finitely repeated games
• Suppose that interactions between the firms are repeated a finite number of times know to each firm in advance– opens potential for a reward/punishment strategy
• “If you cooperate this period I will cooperate next period”
• “If you deviate then I shall deviate.”
– once again use the Nash equilibrium concept
• Why might the game be finite?– non-renewable resource
– proprietary knowledge protected by a finite patent
– finitely-lived management team
Chapter 14: Price-fixing and Repeated Games
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Finitely repeated games 2• Original game but repeated twice
Firm 1
Fir
m 2
Cooperate (M)
Cooperate (M)
Deviate (D)
Deviate (D)
(1800, 1800) (2250, 1250)
(1250, 2250) (1600, 1600)
• Consider the strategy for firm 1– first play: cooperate
– second play: cooperate if firm 2 cooperated in the first play, otherwise choose deviate
Chapter 14: Price-fixing and Repeated Games
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Finitely repeated games 3
• This strategy is unsustainable– the promise is not credible
• at end of period 1 firm 2 has a promise of cooperation from firm 1 in period 2
• but period 2 is the last period
• dominant strategy for firm 1 in period 2 is to deviate
Firm 1
Fir
m 2
Cooperate (M)
Cooperate (M)
Deviate (D)
Deviate (D)
(1800, 1800) (2250, 1250)
(1250, 2250) (1600, 1600)
Chapter 14: Price-fixing and Repeated Games
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Finitely repeated games 4• Cooperation in the second period is based on a worthless
promise– but suppose that there are more than two periods
• with finite repetition over T periods the same problem arises– in period T any promise to cooperate is worthless– so deviate in period T– but then period T – 1 is effectively the “last” period– so deviate in T – 1– and so on
• Selten’s Theorem– “If a game with a unique equilibrium is played finitely many times
its solution is that equilibrium played each and every time. Finitely repeated play of a unique Nash equilibrium is the equilibrium of the repeated game.”
Chapter 14: Price-fixing and Repeated Games
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Finitely repeated games 5
• Selten’s theorem applies if there is a unique equilibrium to the game
• What if this condition is not satisfied?– suppose that there is more than one Nash equilibrium
– both firms agree that one equilibrium is “good” and the other “bad”
– both firms benefit from cooperation
• Then the possibility of cooperation is opened up– reward early cooperation by moving to the “good” equilibrium in
later periods
– punish early deviation by moving to the “bad” equilibrium
Chapter 14: Price-fixing and Repeated Games
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An example
• Consider the following pricing game
Firm 2
Fir
m 1
$105 $130
$105
$130
$160
$160
(7.31, 7.31) (8.25, 7.25)
(7.25, 8.25)
(9.38, 5.53)
(5.53, 9.38)
(8.5, 8.5) (10, 7.15)
(7.15, 10) (9.1, 9.1)
Chapter 14: Price-fixing and Repeated Games
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An example 2
• There are two Nash equilibria
Firm 2
Fir
m 1
$105 $130
$105
$130
$160
$160
(7.31, 7.31) (8.25, 7.25)
(7.25, 8.25)
(9.38, 5.53)
(5.53, 9.38)
(8.5, 8.5) (10, 7.15)
(7.15, 10) (9.1, 9.1)
(7.31, 7.31)
(8.5, 8.5)
Both agree that this is “bad”
Both agree that this is “good”
Both agree that this is best
Chapter 14: Price-fixing and Repeated Games
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An example 3
• Suppose that the game is played twice– and that second period profits are not discounted
• Consider the strategy for both firms– First period: Set a price of $160
– Second period: If history from the first period is ($160, $160) then set price of $130, otherwise set price of $105
• Depends on history of play– equilibrium path is ($160, $160) then ($130, $130)
– why?• second period is easy
– ($130, $130) is a Nash equilibrium for the second period subgame
• what about the first period?
Chapter 14: Price-fixing and Repeated Games
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An example 4• Best response to a price of
$160 is $130
• This gives profit of 10
• So deviation is period 1 gives the profit stream 10 + 7.31 = $17.31
• Cooperation in period 1 gives the profit stream 9.1 + 8.5 = $17.6
• Undercutting does not pay
Firm 2
Fir
m 1
$105 $130
$105
$130
$160
$160
(7.31, 7.31) (8.25, 7.25)
(7.25, 8.25)
(9.38, 5.53)
(5.53, 9.38)
(8.5, 8.5) (10, 7.15)
(7.15, 10) (9.1, 9.1)
Chapter 14: Price-fixing and Repeated Games
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Extensions
• Two extensions– more than two periods: say T periods
• no discounting
• play ($160, $160) for T – 1 periods and ($130, $130) in period T
• cooperation can be sustained in all but the last period
– discounting: suppose that discount factor is R• two periods
– cooperation gives 9.1 + 8.5R
– deviation gives 10 + 7.31R
– cooperation is profitable if and only if R > 0.756
• three periods– cooperation gives 9.1 + 9.1R + 8.5R2
– deviation gives 10 + 7.31R + 7.31R2
– cooperation is sustainable if and only if R > 0.398
Chapter 14: Price-fixing and Repeated Games
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Lessons from finite games
• If Nash is unique – cooperation cannot be sustained in a finite game
• If Nash is not unique– cooperation can be sustained by a credible strategy
• part of the time
• depending upon discount factor
• and number of repetitions
– by a credible threat to punish deviation
Chapter 14: Price-fixing and Repeated Games
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Repeated games with an infinite horizon
• With finite games the cartel breaks down in the “last” period– assumes that we know when the game ends
– what if we do not? • some probability in each period that the game will continue
• indefinite end period
• then the cartel might be able to continue indefinitely– in each period there is a likelihood that there will be a next period
– so good behavior can be rewarded credibly
– and bad behavior can be punished credibly
Chapter 14: Price-fixing and Repeated Games
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Valuing indefinite profit streams
• Suppose that in each period net profit is t
• Discount factor is R
• Probability of continuation into the next period is • Then the present value of profit is:
– PV(t) = RR2+…+ Rttt + …
– valued at “probability adjusted discount factor” R– product of discount factor and probability of continuation
Chapter 14: Price-fixing and Repeated Games
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Trigger strategies
• Consider an indefinitely continued game– potentially infinite time horizon
• Strategy to ensure compliance based on a trigger strategy– cooperate in the current period so long as all have cooperated in
every previous period
– deviate if there has ever been a deviation
• Take our first example– period 1: produce cooperative output of 30
– period t: produce 30 so long as history of every previous period has been (30, 30); otherwise produce 40 in this and every subsequent period
• Punishment triggered by deviation
Chapter 14: Price-fixing and Repeated Games
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Cartel stability
• Profit from sticking to the agreement is:– PVM = 1800 + 1800R + 1800R2 + … = 1800/(1 - R)
• Profit from deviating from the agreement is– PVD = 2025 + 1600R + 1600R2 + … = 2025 + 1600R/(1 - R)
• Sticking to the agreement is better if PVM > PVD
– this requires 1800/(1 - R) > 2025 + 1600R/(1 - R)
– or R(2.025 – 1.8)/(2.025 – 1.6) = 0.529• if = 1 this requires that the discount rate is less than 89%
• if = 0.6 this requires that the discount rate is less than 14.4%
Chapter 14: Price-fixing and Repeated Games
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Cartel stability 2• This is an example of a more general result
• Suppose that in each period – profits to a firm from a collusive agreement are C
– profits from deviating from the agreement are D
– profits in the Nash equilibrium are N
– we expect that D >C >N
• Cheating on the cartel does not pay so long as:
R > D - C
D - N
• The cartel is stable– if short-term gains from cheating are low relative to long-run losses
– if cartel members value future profits (low discount rate)
This is the short-run gainfrom cheating on the cartel
This is the short-run gainfrom cheating on the cartel
This is the long-run lossfrom cheating on the cartel
This is the long-run lossfrom cheating on the cartel
There is always a value of R < 1 for which
this equation issatisfied
Chapter 14: Price-fixing and Repeated Games
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Cartel stability 3
• Consider the price game with two Nash equilibria– two possible strategies to sustain cooperation
• trigger with deviation to the “good” Nash
• trigger with deviation to the “bad” Nash
• the latter entails a harsher punishment so is sustainable with a lower probability adjusted discount factor
Firm 2
Fir
m 1
$105 $130
$105
$130
$160
$160
(7.31, 7.31) (8.25, 7.25)
(7.25, 8.25)
(9.38, 5.53)
(5.53, 9.38)
(8.5, 8.5) (10, 7.15)
(7.15, 10) (9.1, 9.1)
Chapter 14: Price-fixing and Repeated Games
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Trigger strategies
• With infinitely repeated games– cooperation is sustainable through self-interest
• But there are some caveats– examples assume speedy reaction to deviation
• what if there is a delay in punishment?– trigger strategies will still work but the discount factor will have to be
higher
– harsh and unforgiving• particularly relevant if demand is uncertain
– decline is sales might be a result of a “bad draw” rather than cheating on agreed quotas
– so need agreed bounds on variation within which there is no retaliation
– or agree that punishment lasts for a finite period of time
Chapter 14: Price-fixing and Repeated Games
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The Folk theorem
• Have assumed that cooperation is on the monopoly outcome– this need not be the case
– there are many potential agreements that can be made and sustained – the Folk Theorem
Suppose that an infinitely repeated game has a set of pay-offs that exceed the one-shot Nash equilibrium pay-offs for each
and every firm. Then any set of feasible pay-offs that are preferred by all firms to the Nash equilibrium pay-offs can be
supported as subgame perfect equilibria for the repeated game for some discount factor sufficiently close to unity.
Chapter 14: Price-fixing and Repeated Games
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The Folk theorem 2
• Take example 1. The feasible pay-offs describe the following possibilities
$2100
$2100$1500 $1600
$2000
$2000
If the firms colludeperfectly they share
$3,600
If the firms colludeperfectly they share
$3,600
$1600
If the firms competethey each earn
$1600
If the firms competethey each earn
$1600
The Folk Theorem statesthat any point in thistriangle is a potentialequilibrium for the
repeated game
The Folk Theorem statesthat any point in thistriangle is a potentialequilibrium for the
repeated game
Collusion onmonopoly giveseach firm $1800
Collusion onmonopoly giveseach firm $1800
$1800
$1800
$1800 to each firm may not be sustainable but something less
will be
Chapter 14: Price-fixing and Repeated Games
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Balancing temptation
• A collusive agreement must balance the temptation to cheat
• In some cases the monopoly outcome may not be sustainable– too strong a temptation to cheat
• But the folk theorem indicates that collusion is still feasible– there will be a collusive agreement:
• that is better than competition
• that is not subject to the temptation to cheat
Chapter 14: Price-fixing and Repeated Games
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Summary
• Infinite or indefinite repetition introduces real possibility of cooperation– stable cartels sustained by credible threats
• so long as discount rate is not too high
• and probability of continuation is not too low
• There are good reasons for the Justice department and the Competition Directorate to be concerned about cartels– but they have limited resources
– so where should they look?
– how might they detect cartels?