### Document

```Extensive Form Games
With Perfect Information
(Illustrations)
Extensive Form Games with
Perfect Information: Illustrations
• Ultimatum Game
– Two people using the following procedure to split \$k.
Person A offers person B some positive \$x less than
or equal to k. If B accepts, A and B get k-x and x
respectively. If B does not accept, neither gets any
money. Each person only cares about their own
payoff.
– Players: A and B
– Strategies: Player A: an offer between 0 and k.
Player B: Accept or Not accept.
– Payoffs: (k-x,x) or (0,0)
Extensive Form Games with
Perfect Information: Illustrations
• Subgames: Every possible offer by person A
is a subgame. If the offer, x, is a continuous
variable (ie, offers of fractions of cents are
possible), then this game has an infinite
number of subgames.
– In the subgame following an offer x>0, the optimal
response of player B is to accept. (not accepting
yields a payoff of zero).
– In the subgame following an offer x = 0, player B
is indifferent between accepting and not
accepting.
Extensive Form Games with
Perfect Information: Illustrations
 So what should player A do? If player B
accepts all possible offers (including x =
0), then clearly offering x = 0 and attaining
payoffs of (k,0) is our subgame perfect
Nash equilibrium. If player B accepts all
offers except x=0, then there is NO
subgame perfect Nash equilibrium !
Offering \$1 is dominated by offering
\$0.50. Offering \$0.01 is dominated by
offering \$0.005. And so on.
Extensive Form Games with
Perfect Information: Illustrations
 So does this result violate our theorem
that all finite games have a Nash
equilibrium, possibly in mixed strategies?
No, because even though there are a finite
number of players, there are an infinite
number of strategies.
Extensive Form Games with
Perfect Information: Illustrations
• Suppose the smallest unit of currency was a
cent and player B always rejects an offer of
x = 0, then x=0.01 would be the offer in a
subgame perfect Nash equilibrium.
• We could also complicate things further by
allowing player B to make counteroffers.
This will be developed in chapter 16 on
repeated games.
• Experiments in the ultimatum game: pg 183.
Extensive Form Games with
Perfect Information: Illustrations
• Exercise 185.1 in Osborne. Two people use the following procedure to
allocate two desirable identical indivisible objects.
• One person proposes an allocation (both objects go to person 1, both
go to person 2, one goes to each), which the other person then either
accepts or rejects. In the event of rejection, neither person receives
either object. Each person cares only about the number of objects she
obtains.
• Construct an extensive game that models this situation and find its
subgame perfect equilibria.
• Does the game have any NE that are not SPE?
• Is there any outcome that is generated by a NE but NOT by any SPE ?
Extensive Form Games with
Perfect Information: Illustrations
Oligopoly: market structure characterized by a few firms that recognize their strategic
interdependence. Middle ground between monopoly and perfect competition.
Duopoly: special case of oligopoly with 2 firms competing strategically with each other in
the market(s) for the good(s) they both produce.
Dynamic Quantities: Firm A is the leader and firm B is the follower. In period 1, firm A
chooses its quantity and in period 2, firm B observes firm A's choice and then chooses
its own quantity. Called "Quantity Leadership" or "Stackelberg in Quantities."
• Who has more power in this setting?
Dynamic Prices: Firm A is the leader and firm B is the follower. In period 1, firm A chooses
its price and in period 2, firm B observes firm A's choice and then chooses its own price.
Called "Price Leadership" or "Stackelberg in Prices."
• What will be the feature of an equilibrium set of prices in this setting?
Simultaneous Quantities: Firm A and firm B set their quantities at the same time given
their beliefs about what the other firm will do. Called Cournot competition.
• What will be the feature of an equilibrium in this setting?
Simultaneous Prices: Firm A and firm B set their prices at the same time given their
beliefs about what the other firm will do. Called Bertrand competition.
• What will be the feature of an equilibrium in this setting?
Collusion: a situation where firms jointly determine their prices or quantities to maximize
joint profits. When firms get together and attempt to set prices and outputs so as to
maximize total industry profits, they are known as a cartel.
• OPEC.
Extensive Form Games with
Perfect Information: Illustrations
Stackelberg’s Model of Duopoly.
• Firms move sequentially choosing quantities of production. Firm A chooses first
and B chooses second, having observed firm A’s choice. Both firms seek to
maximize their individual profits given their costs, Ci(qi). Total quantity is Q = qA
+ qB. Demand is P(Q).
– At t=1, A chooses qA. At t=2, B chooses qB, At t=3, payoffs realized.
•
Who has more power in a game like this? (eg, which firm will end up with a
higher level of profits if their costs are identical?) What are the strategies of the
two firms (formally) ?
•
Example 6.2.2. Page 188 in Osborne
•
Comparing Cournot, Bertrand, and Stackelberg.
– Next Time!
I
0,2
2,0
1,1
II
y
n
2,0
0,0
yyy
II
II
y
1,1
n
y
n
0,0
0,2
0,0
yyn
yny
ynn
nyy
nyn
nny
nnn
(2,0) 2,0
2,0
2,0
2,0
0,0
0,0
0,0
0,0
(1,1) 1,1
1,1
0,0
0,0
1,1
1,1
0,0
0,0
(0,2) 0,2
0,0
0,2
0,0
0,2
0,0
0,2
0,0
I
0,2
2,0
1,1
II
y
n
2,0
0,0
yyy
II
II
y
1,1
n
y
n
0,0
0,2
0,0
yyn
yny
ynn
nyy
nyn
nny
nnn
(2,0) 2,0
2,0
2,0
2,0
0,0
0,0
0,0
0,0
(1,1) 1,1
1,1
0,0
0,0
1,1
1,1
0,0
0,0
(0,2) 0,2
0,0
0,2
0,0
0,2
0,0
0,2
0,0
I
0,2
2,0
1,1
II
y
n
2,0
II
II
0,0
y
1,1
n
y
n
0,0
0,2
0,0
SPNE: { (2,0),yyy); ((1,1),nyy) }
yyy
yyn
yny
ynn
nyy
nyn
nny
nnn
(2,0) 2,0
2,0
2,0
2,0
0,0
0,0
0,0
0,0
(1,1) 1,1
1,1
0,0
0,0
1,1
1,1
0,0
0,0
(0,2) 0,2
0,0
0,2
0,0
0,2
0,0
0,2
0,0
NE: { All Strategies corresponding to the 9 Highlighted Payoffs Above are NE }
I
0,2
2,0
1,1
II
y
n
2,0
II
II
0,0
y
1,1
n
y
n
0,0
0,2
0,0
SPNE: { (2,0),yyy); ((1,1),nyy) }
yyy
yyn
yny
ynn
nyy
nyn
nny
nnn
(2,0) 2,0
2,0
2,0
2,0
0,0
0,0
0,0
0,0
(1,1) 1,1
1,1
0,0
0,0
1,1
1,1
0,0
0,0
(0,2) 0,2
0,0
0,2
0,0
0,2
0,0
0,2
0,0
NE: { All Strategies corresponding to the 9 Highlighted Payoffs Above are NE }
```