ECON3014 (Fall 2011)
24-25.11. 2011 (Tutorial 9)
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 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, -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 an infinitely repeated one?
 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






Would firms have higher incentive to cheat when interest rate is higher/lower? Why?
Factors affecting collusion in pricing games: Number of firms, history of market,
punishment mechanisms
(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 the game will end after a given play is .





Would firms have higher incentive to cheat when  is higher/lower? Why?
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(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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Sequential Move Games and Subgame Perfect Equilibrium
 A sequential move game could be represented by the extensive form (game tree) and the
subgame perfect equilibrium could be solved by backward induction
Example:
Player 1
L
R
Player 2
U
6
20
Player 2
D
U
0
0
D
5
5
10
15

Feasible strategies: Player 1: L, R
Player 2: UU, UD, DU, DD
(UU means Player 2 chooses U if Player 1 chooses L, and chooses U when Player 1
chooses R)

Generally, number of feasible strategies: Sn
(where S is the number of strategies in each information set, and n is the number of
information set)


NE: (L, UU) , (R, DD)and (R, UD)
SPE: Player 1: {R}, Player 2: {UD} and SPE outcome is Player one chooses R and Player
2 chooses D with payoffs 10 and 15 respectively
 (R, UD) is a NE as well as a SPE, while (L, UU) and (R, DD) are NE only as they
involves incredible threat. (What is the threat? Why is it incredible?) (How to solve for
NE and SPE?)

A set of strategies constitutes a subgame perfect equilibrium if
(1) It is a Nash equilibrium (How to check?)
(2) At each stage of the game (decision node) neither player can improve her payoff by
changing her own strategy
 SPE is a NE that involves only credible threats
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Example: Rotten Kid Game [Similar examples: Entry Game (P. 379), Innovation (P.380)]




Child
Go
Not go
Parents
1
1
Punish
Not punish
–1
–1







2
0
Feasible Strategies: P if G, NP if G, P if NG, NP if NG
Two NEs: (Go, Punish) and (Not go, Not Punish)
SPE: (Not go, Not Punish)
The NE (Go, Punish) is not a satisfactory one as it
involves on incredible threat.
 The threat by parents to the kid is that “if you choose go,
you will get a payoff of 1, but if you choose not go, you
will be punished and your payoff will be –1, so it is
better for you to go”.
 (Go, Punish) is a NE because given the parents’ threat,
the best response of the child is go.
Let’s consider the extensive form and we can see the threat is not credible.
Suppose the child chooses not go
Parents choose not punish instead of punish (0  –1) if they are rational.
Parents would never carry out the threat when the child chooses not go.
(Go, Punish) is not a SPE though it’s a NE.
The set of equilibria to those rely on credible threat  SPE
SPE: (Not Go, Not Punish) with payoffs (2, 0)
 The threat should be incredible if we are dealing with a one-shock game.
 If the game repeats, then the parents may act irrationally in order to build up their
reputation.
 In the rotten kid example, parents may act on the threat and punish the child if the child
refuses to go. (Although the payoff will be –1 for both parties).
Example: Entry Game (Baye, P379)
Potential Entrant
Out
In
Incumbent
0
10
Hard
Soft
–1
–1
2
0



Feasible Strategies: H if Out, S if Out, H if In, S if In
Two NEs: (Out, Hard) and (In, Soft)
SPE: (In, Soft)

(Out, Hard): Given the incumbent threat of initiating a price war if the potential entrant
enters, potential entrant’s best response is to stay out. Given that the potential entrant
stays out, the incumbent may still threaten to play hard if the potential entrant enters.
 (Out, Hard) is only a NE, but not SPE as it involves an incredible threat. If the
incumbent is rational, he will choose to accommodate instead of initiating a price war
(0> -1).

Again, the incumbent may choose to act irrationally to build up its reputation if the game
is not a one-shot game.
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Example: Sequential Bargaining (Baye, P.381)
 The manager and the labor union engaged in a negotiation over how to split a surplus of
$100. The manager moves first by making an offer ($1, 50 and $99) to the union. The
union decides whether to “Accept” or “Reject” the offer. If the offer is rejected, neither
party receives anything.
Manager
$1
$50
Union
A
99
1

$99
Union
R
0
0
A
50
50
Union
R
0
0
A
1
99
R
0
0
Feasible Strategies:
 Firm: {offer 1, offer 50, offer 99}
 Union: {AAA, ARR, ARA, RRR, RAA, RAR, AAR, RRA} (Sn = 23 = 8 strategies)

7 NEs: (offer 1, AAA), (offer 1, ARR), (offer 1, ARA), (offer 1, AAR), (offer 50, RAR),
(offer 50, RAA), (offer 99, RRA)
 The labor union threaten the manager to reject the offer made by the manager, however
all threats are incredible. (Why?)
 SPE: (offer 1, AAA). This is the only NE which does not involves incredible threat
 SPE outcome: The manager make an offer of $1 and the union Accept the offer, payoffs
to the two parties are ($99, $1)

Is there any first mover advantage in this game?
Labor Union
Manager
AAA
ARR
ARA
AAR
RRA
RAA
RAR
RRR
$1
(99, 1)
(99, 1)
(99, 1)
(99, 1)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
$50
(50, 50)
(0, 0)
(0, 0)
(50, 50)
(0, 0)
(50, 50)
(50, 50)
(0, 0)
$99
(1, 99)
(0, 0)
(1, 99)
(0, 0)
(1, 99)
(1, 99)
(0, 0)
(0, 0)
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