ECON241 (Fall 2010)
10. 11. 2010 (Tutorial 8)
Chapter 10 Game Theory: Inside Oligopoly
(1) Simultaneous-move, one shot game
Example: A pricing game (similar examples: Advertising game/ Quality decision)
Firm B
Low price
High price
Low price
0, 0
50, -10
Firm A
High price
-10, 50
10, 10





Both Firm A and B have a dominant strategy of charging a low price (What is a dominant
strategy?)
NE: (LP, LP), and each firm receives a payoff of 0
It resembles a prisoner dilemma (What is prisoner dilemma? Why?).
If the two firms collude and agree to charge a high price, both would receive a higher
payoff
However, collusion is not stable as both parties will have incentive to cheat. (Why? How
could we resolve the problem of cheating?)
Example: Coordination decisions (Game with multiple NE)
Firm B
120-Volt Outlets
90-Volt Outlets
120-Volt Outlets
100, 100
0, 0
Firm A
90-Volt Outlets
0, 0
100, 100



Two NEs: (120V, 120V) and (90V, 90V)
Which NE would be the final outcome? (How?)
A game of coordination rather than conflicting interest (Does any firm have incentive to
cheat?)
Example: Nash Bargaining
Union
0
50
100
0
0, 0
0, 50
0, 100
Management
50
50, 0
50, 50
-1, -1
100
100, 0
-1, -1
-1
 Three NEs: (0, 0), (50, 50) and (100, 100)
 Which NE would be the final outcome? (How?)
Example: Monitoring Employees (Game with no pure strategy NE)
Workers
Work
Shirk
Monitor
-1, 1
1, -1
Manager
Don’t Monitor
1, -1
-1, 1



No pure strategy NE
Mixed (randomized strategy) NE does exist
How would players behave in randomizing their strategies?
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(2) Infinitely repeated games
 In a one shot simultaneous move pricing game, collusion is not sustainable. What if the
pricing game is play infinitely?
 Supporting collusion with Trigger strategies
 Trigger strategies: A strategy that is contingent on the past play of a game and in which
some particular past action “triggers” a different action by a player
Example: The pricing game revisited
Firm B
Low price
High price
Low price
0, 0
50, -40
Firm A
High price
-40, 50
10, 10

for firm
if cooperate



for firm
if cheat

Would firms have higher incentive to cheat when interest rate is higher/lower? Why?
General principle:







[

One time gain of breaking the collusion  PV of cost of cheating
]
Factors affecting collusion in pricing games: Number of firms, history of market,
punishment mechanisms
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(3) Finitely repeated games
(A) Games with an uncertain final period
Example: The pricing game revisited (unknown final period)
Firm B
Low price
High price
Low price
0, 0
50, -40
Firm A
High price
-40, 50
10, 10
Suppose the probability of game will end after a given play is .

for firm
if cooperate




Would firms have higher incentive to cheat when  is higher/lower? Why?
(B) Games with a known final period: end-of-period problem
Example: The pricing game revisited (known final period)
Firm B
Low price
High price
Low price
0, 0
50, -40
Firm A
High price
-40, 50
10, 10
 Suppose the game is repeated two times, can collusion still be sustained?
 Solving the game by staring from the last period of the game
 Since the game is played twice only, in the second period (last period), each firm chooses
the strategy (Low price) as in the one shot game as there is no future period (not possible
to punish/ to be punished)
 In the first period, both parties will choose “low price” as they know their rival will
choose “low price” in the second period
 NE: Both firms choose “low price” in all periods
 Collusion does not work for games with known final period
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Example n advertising Game Kellogg’s and General Mills
 Two firms (Kellogg’s & General Mills) managers want to maximize profits.
 Strategies consist of advertising campaigns: (None, Moderate, High). The payoff
matrix is as follow:
Kellogg’s
None
Moderate
High
None
12, 12
20, 1
15, -1
General Mills
Moderate
1, 20
6, 6
9, 0
High
-1, 15
0, 9
2, 2
(a) What is the Nash Equilibrium for a one shot simultaneous move game? Can collusion
work?
NE: (High, High), each company receives a payoff of 2.
(b) Can collusion work if the game is repeated 2 times?
No. We could solve the game by backward induction
In period 2, the game is a one-shot game, so equilibrium entails High Advertising in the
last period. This means period 1 is “really” the last period, since everyone knows what
will happen in period 2.
Equilibrium entails High Advertising by each firm in both periods.
The same holds true if we repeat the game any known, finite number of times.
(c) Can collusion work if firms play the game each year, forever?
Consider the following “trigger strategy” by each firm “Don’t advertise, provided the
rival has not advertised in the past. If the rival ever advertises, “punish” it by engaging
in a high level of advertising forever after.”
In effect, each firm agrees to “cooperate” so long as the rival hasn’t “cheated” in the past.
“Cheating” triggers punishment in all future periods.
(d) Follows part (c), suppose General Mills adopts this trigger strategy. What is Kellogg’s
profit if cooperate/ cheat if interest rate is 5%? What is the NE of the game?
profit if cooperate
rofit if cheat
As the profit for cheating is lower than the profit of collusion, it doesn’t pay for
Kellogg’s to deviate from the collusion.
NE: Both firms will not advertise in every period in the infinitely repeated game
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