ECON241 (Fall 2010)
17. 11. 2010 (Tutorial 9)
Chapter 10 Game Theory: Inside Oligopoly
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
involives 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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Problems and Examples:
Question 4 (Baye, P.387)
Player 2
Player 1
A
B
C
10, 10
-5, 60
D
60, -5
50, 50
(a) Identify the one-shot NE
(b) Suppose the players know this game will be repeated exactly 3 times. Can they achieve
payoffs that are better than the one-shot NE? Explain.
(c) Suppose this game is infinitely repeated and the interest rate is 5%. Can the players
achieve payoffs that are better than the one-shot NE? Explain.
(d) Suppose the players do not know exactly how many times this game will be repeated, but
they do know that the probability the game will end after a given play is . If  is
sufficiently low, can players earn more than they could in the one-shot NE?
(a) NE is (A, C) and each player gets a payoff of 10.
(b) No. Collusion of any would not be stable for any game with certain finite time horizon.
As eventually it will come to a point that both players are certain there is no tomorrow
and there is no way to punish/ can be punished for having broken the promise, both
players will cheat in every period up to the known last stage.
(c) If firms adopt the trigger strategies, higher payoffs can be achieved if
 Cheat   Coop 1
 .
 Coop   N
i
Given an interest rate of 5%, πCheat = 60, πCoop = 50, πN = 10,
 Cheat   Coop 60  50 1
1 1

  0.25 < 
 20
Coop
N


50  10 4
i .05
Thus, each firm can indeed earn a payoff of 50 in each period via the trigger strategies.
(d) Yes.
If the probability the game will end after a given play is sufficiently low (  0.8), then
players can earn more than they could in the one-shot NE
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Question 8 (Baye, P.388)
Player 1
A
B
Player 2
X
Y
5, 5
0, -200
-200, 0
20, 20
(a) Determine the NE outcomes that arise if the players make decisions independently,
simultaneously, and without any communication. Which of these outcomes would you
consider most likely? Explain.
(b) Suppose Player 1 is permitted to “communicate” by uttering one syllable before the
players simultaneously and independently make their decisions. What should Player 1
utter, and what outcome do you think would occur as a result?
(c) Suppose player 2 can choose its strategy before Player 1, that Player 1 observes Player 2’s
choice before making her decision, and that this move structure is known by both players.
What outcome would you expect? Explain.
(a) There are two Nash equilibria: (A, X) and (B, Y).
The (A, X) equilibrium would seem most likely since the other equilibrium entails
considerable risk if the players don’t coordinate on the same equilibrium.
(b) “B”. This would signal to player 2 that player 1 is going to use strategy B, and therefore
permit the players to coordinate on the (20, 20) equilibrium.
(c) Player 2 would choose Y and player 1 would follow by choosing B. This is the subgame
perfect equilibrium.
Question 17 (Baye, P.391)
Ata time when demand for ready-to-eat cereal was stagnant, a spokesperson for the cereal
maker Kellogg’s was quoted as saying, “…for the past several years, our individual company
growth has come out of the other fellow’s hide.” Kellogg’s has been producing cereal since
1906 and continues to implement strategies that make it a leader in the cereal industry.
Suppose that when Kellogg’s and its largest rival advertise, each company earns $0 billion in
profits. When neither company advertises, each company earns profits of $8billion. If one
company advertises and the other does not, the company that advertises earns $48 billion and
the company that does not advertise loses $1 billion. Under what conditions could these firms
use trigger strategies to support the collusive level of advertising?
Kellogg’s
Advertise
Not Advertise
Advertise
8, 8
48, -1
Rival
Not Advertise
-1, 48
0, 0
The NE is for both firms not to advertise if it is a one-shot game.
If it is not a one shot game, collusion is profitable under the usual trigger strategies if
$48  $8
1
 Cheat   Coop 1
 $5  . Thus, one requirement is for the interest rate to be
 , or
Coop
N
$8  0
i


i
less than 20 percent. Another requirement includes the ability of firms to monitor (observe)
potential deviations by rivals.
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Question 13 (Baye, P.389)
Coca-cola and PepsiCo are the leading competitors in the market for cola products. In 1960
Coca-cola introduced Sprite, which today is the worldwide leader in lemon-lime soft drink
and ranks fourth among all soft drinks worldwide. Prior to 1999, PepsiCo did not have a
product that competed against Sprite and had to decide whether to introduce such a soft drink.
 By not introducing a lemon-lime soft drink, PepsiCo would continue to earn$200 million
profit and Coca-cola could continue to earn a $300 million profit.
 By introducing a new lemon-lime soft drink, one of these two possible strategies could be
pursued:
(1) PepsiCo could trigger a price war with Coca-cola in both the lemon-lime and cola
markets. Each company earns $100 million in this case.
(2) Coca-cola could acquiesce and each firm maintain it’s current 50/50 split of the cola
market and split the lemon-lime market 30/70 (PepsiCo/Coca-cola). In this case, PepsiCo
earns $227 million and Coca-cola earns $275million
If you are the manager at PepsiCo, would you try to convince your colleagues that
introducing the new soft drink is the most profitable strategy? Why or why not?
The game could be represented by the following extensive form:
Pepsi
Not Introduce
Introduce
CocaCola
200
300
Price War
100
100
Accomodate
227
275
Notice that Coca-Cola’s best response if Pepsi introduces is to acquiesce to earn $275 million
rather than to start a price war and earn $100. Thus, while Coca-Cola might threaten to start a
price war in an attempt to keep you out of the market, this threat isn’t credible; your best
option is to introduce.
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