Extensive Forms with Perfect Information

1
Extensive Forms with Perfect Information

• Solution Concepts

2
Individual Decision Making ????????????????
180, Two and a half hours
??
????
200, Two hours
??
??
195, Two hours and 15 mins
??
??
??
??
110, Three hours
??
??
????
130, Two and a half hours
??
125, Two hours and 30 mins
??
3
Key Backward Induction (???)
180, Two and a half hours
??
????
200, Two hours
??
??
195, Two hours and 15 mins
??
??
??
??
110, Three hours
??
??
????
130, Two and a half hours
??
125, Two hours and 30 mins
??
4
Key Backward Induction (???)
180, Two and a half hours
??
????
200, Two hours
??
??
195, Two hours and 15 mins
??
??
??
??
110, Three hours
??
??
????
130, Two and a half hours
??
125, Two hours and 30 mins
??
5
Multi-person decision Making Game with perfect
information
Key No strategic uncertainty
6
Multi-person decision Making Game with perfect
information
Key No strategic uncertainty
2, 0
u
Player 2
U
Player 1
d
-1, -1
D
0, 3
7
Multi-person decision Making Game with perfect
information
Key No strategic uncertainty
2, 0
u
Player 2
U
Player 1
d
-1, -1
D
0, 3
Your Thought?
8
Key Backward Induction
2, 0
u
Player 2
U
Player 1
d
-1, -1
D
0, 3
9
Key Backward Induction
2, 0
U
Player 1
D
0, 3
10
Key Backward Induction
Complete Solution
2, 0
u
Player 2
U
Player 1
d
-1, -1
D
0, 3
11
Key Backward Induction
Complete Solution
Play or Path A path from the root to a
terminal Equilibrium Path The predicted path
2, 0
u
Player 2
U
Player 1
d
-1, -1
We called the red line equilibrium path.
D
0, 3
12
Exercise Solve the following game
1
2
1
1
2
2
R
R
R
R
R
R
(8,5)
L
L
L
L
L
L
(1,0)
(0,1)
(3,0)
(2,4)
(6,3)
(4,7)
13
Answer
1
2
1
1
2
2
R
R
R
R
R
R
(8,5)
L
L
L
L
L
L
(1,0)
(0,1)
(3,0)
(2,4)
(6,3)
(4,7)
Full Description Player 1 L, L, L Player 2 L,
L, L
14
A somewhat realistic case
Merger and Acquisition
Some Interesting Terms Takeover directly
purchase shares to gain control (???) Leveraged
buyout Takeover through outside
borrowing Merger Two firms voluntarily agree to
consolidate To prevent merger poison pills in
corporate charters Green Mails to
bribe outside company Golden parachute
Buyout package White Knights An
outside, friendly, company outbids the hustle
company to win the take over bid.
15
Piper Pickled Peppers Case (Case 4,
Ch.13) There are five members in the board of
directors 1. Staggered Five Year Term of
Directors Can only replace one director in a
year. 2. Only directors can propose to change
the board of directors. Voting is in a
clockwise fashion. If a propose fails, then
the one who proposes and its supporters are
deprived from its board membership and stock
holdings. Their stock holdings are distributed
evenly to remaining members. Can you find a
method to replace the entire board in a year,
if you have gained one seat in the board?
16

• A Method
• 1. If the proposal passes unanimously, then the
  new
• member choose an entirely new board. Each
• board member replaced is given a small
• compensation.
• 2. If the proposal passed 4 to 1, then the person
  voting
• against is removed from the board, and no
• compensation is made.
• 3. If the proposal passed 3 to 2, then the new
  member
• transfer its share (51) to the two yes
  voters evenly.
• The two no voters are removed without
  compensation.

17

• Why? Backward Induction
• Consider the last voters problem.
• Three situations 1 to 3, 2 to 2, 3 to 1, or 4 to
  0.
• 1. 4 to 0 or 3 to 1 If vote no, then shares will
  be deprived .
• So vote yes.
• 2. 2 to 2
• If vote no, then gets two no voterss shares,
  which
• equals to (5112.25)/321.1.
• If vote yes, then gets the new members share
• 51/225.5.
• So vote yes.
• 3. 1 to 3 vote no.

18

• The 4th voters problem
• 1. 1 yes votes
• If vote no, then 1 to 3, and the last voter
• will vote no. Get 51/412.25 shares.
• If vote yes, then 2 to 2, and the last voter
  will vote yes,
• gets 51/3 shares.
• 2. Two yes votes
• If vote no, then 2 to 2, and the last voter will
  vote yes.
• Will be kicked out.
• If vote yes, then 3 to 1, and the last voter
  will vote yes.
• Will not be kicked out.
• 3. Three yes votes Obviously vote yes.

19

• The third voters problem
• 1. One yes votes
• If vote yes, then 2 to 1, the fourth voter and
  the last
• voter will vote yes. The proposal passes.
• If vote no, then 1 to 2, the fourth voter will
  vote yes
• Will be kicked out from the board.
• 2. Two yes votes
• If vote yes, then 3 to 0. The proposal passes.
• If vote no, then 2 to 1. The 4th and 5th voters
  will vote
• yes, the proposal passes, and will be kicked out.

20

• The second voters problem
• If vote yes, then there are two yes votes, the
  following
• voters will all vote yes.
• If vote no, then there are only one yes votes,
  the 3rd voter
• will vote yes Then will be kicked out from the
  board.
• Outcome (Equilibrium Path)
• The proposal passes unanimously!!!

21
Exercise Draw the tree of the situation above
and solve it using backward induction.
22
Exercise Draw the tree of the situation above
and solve it using backward induction.
Solution
1
X
O
2
2
X
X
3
3
O
O
3
3
X
X
X
X
O
O
O
O
4
4
4
4
4
4
4
4
X
X
X
X
X
X
X
X
O
O
O
O
O
O
O
O
5 0
4 1
3 2
4 1
3 2
2 3
3 2
2 3
3 2
1 4
2 3
3 2
1 4
3 2
4 1
2 3
23
1
X
O
2
2
X
X
3
3
O
O
3
3
X
X
X
X
O
O
O
O
4
4
4
4
4
4
4
4
X
X
X
X
X
X
X
X
O
O
O
O
O
O
O
O
50 0 0 50
0 33 33 33
50 0 50 0
25 0 25 25
0 0 50 50
33 33 0 33
25 25 25 25
33 33 33 0
0 50 0 50
33 0 33 33
0 25 25 0
0 25 25 25

50 50 0 0
25 25 0 25
25 25 25 0
24
1
X
O
2
2
X
X
3
3
O
O
3
3
X
X
X
X
O
O
O
O
4
4
4
4
4
4
4
4
X
X
X
X
X
X
X
X
O
O
O
O
O
O
O
O
50 0 0 50
0 33 33 33
50 0 50 0
25 0 25 25
0 0 50 50
33 33 0 33
25 25 25 25
33 33 33 0
0 50 0 50
33 0 33 33
0 25 25 0
0 25 25 25

50 50 0 0
25 25 0 25
25 25 25 0
25
1
X
O
2
2
X
X
3
3
O
O
3
3
X
X
X
X
O
O
O
O
4
4
4
4
4
4
4
4
X
X
X
X
X
X
X
X
O
O
O
O
O
O
O
O
50 0 0 50
0 33 33 33
50 0 50 0
25 0 25 25
0 0 50 50
33 33 0 33
25 25 25 25
33 33 33 0
0 50 0 50
33 0 33 33
0 25 25 0
0 25 25 25

50 50 0 0
25 25 0 25
25 25 25 0
26
Example from a legal case ???????????????,??????
?? ????????,????????????? Key ???????????,??
(induce) ??????
27
Three antagonists, Larry, Mo, and Curly, are
engaged in a three- way duel. There are two
round, each player is given one shot first
Larry, then Mo, and then Curly. After the first
round, any survivors are given a second shot,
again beginning with Larry, then Mo, and then
Curly. For each duelist, the best outcome is to
be the sole survivor. Next best is to be one of
two survivors. In the third place is the outcome
in which no one gets killed. Dead last is that
you get killed. Larry is a poor shot, with
only a 30 percent chance of hitting a person at
whom he aims. Mo is a much better shot, achieving
80 percent accuracy. Curly is a perfect shot
he never misses. Predictions?
28
Larrys Move (hit rate 0.3)
Mos Move (hit rate 0.8)
Curlys Move (hit rate 1)
Nature Move
0.7
0.3
M dies, C shoots L with 100 hit rate, C dies
Shoots M
Shoots C
29
Larrys Move (hit rate 0.3)
Mos Move (hit rate 0.8)
Curlys Move (hit rate 1)
Nature Move
Shoots L
0.7
Shoots C
0.3
M dies, C shoots L with 100 hit rate, C dies
Shoots M
Shoots C
30
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Nature Move
Shoots L
0.2
Shoots C
0.7
0.3
M dies C shoots L with 100 hit rate C dies
Shoots M
Shoots C
31
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
Shoots C
0.7
0.3
Shoots M
Shoots C
32
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
0.2
Curly Shoots
Shoots C
0.7
0.3
Shoots M
Shoots C
33
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
0.2
Curly Shoots
Shoots C
0.7
0.3
0.8
Shoots M
0.3
C, M dies
0.7
Shoots C
L shoots, C has dies, shoots M
34
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
0.2
Curly Shoots
Shoots C
0.7
0.3
0.8
Shoots M
0.3
C, M dies
0.7
Shoots C
M shoots, C has dies, shoots L
35
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
0.2
Curly Shoots
Shoots C
0.7
0.3
0.8
Shoots M
0.3
C, M dies
0.8
0.7
Shoots C
M survives
M shoots, C has dies, shoots L
0.2
M, L, survive
36
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
0.2
Curly Shoots
Shoots C
0.7
0.3
0.8
Shoots M
0.3
C, M dies
0.8
0.7
Shoots C
M survives
0.7
This part of game tree is identical to the tree
above.
0.2
M, L, survive
37
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
0.2
Curly Shoots
Shoots C
0.7
0.3
0.8
Shoots M
0.3
C, M dies
0.8
0.7
Shoots C
M survives
0.7
0.2
M survives
M, L, survive
0.3
0.8
L survives
0.3
M survives
C dies
0.8
0.2
0.7
M, L survives
0.2
38
Curlys Game Curly is the last mover. He can
move if he survived from Mos and/or Larrys
shoot.
Shots Mo
Curly
Shots Larry
39
Curlys Game Curly is the last mover. He can
move if he survived from Mos and/or Larrys
shoot.
Only L survives
Shoots C, hit with probability 0.3
Larry
Shoots Mo
Curly
Prob 0.7, miss
Shoots Larry
40
Curlys Game Curly is the last mover. He can
move if he survived from Mos and/or Larrys
shoot.
Only L survives
Shoots C, hit with probability 0.3
Larry
Shoots Mo
Then C shoots, L dies
Curly
Prob 0.7, miss
Shoots Larry
41
Curlys Game Curly is the last mover. He can
move if he survived from Mos and/or Larrys
shoot.
Only L survives
Shots Curly hits with probability 0.3
Larry
Shoots Mo
Then C shoots L dies
Curly
Prob 0.7, misses
Shoots Larry
Mo
Shoots Curly hits with probability 0.8
M survives
42
Curlys Game Curly is the last mover. He can
move if he survived from Mos and/or Larrys
shoot.
Only L survives
Shots Curly hits with probability 0.3
Larry
Shoots Mo
Then C shoots L dies
Curly
Prob 0.7, misses
C shoots M dies
Shoots Larry
Misses
Mo
Shoots Curly hits with probability 0.8
M survives
43
Larrys Move (hit rate 0.3)
L dies C shoots M with 100 hit rate M dies
Mos Move (hit rate 0.8)
0.8
Curlys Move (hit rate 1)
Curly Shoots
Nature Move
Shoots L
0.2
0.2
Curly Shoots
Shoots C
0.7
0.3
0.8
Shoots M
0.3
C, M dies
0.8
0.7
Shoots C
M survives
0.7
0.2
M survives
M, L, survive
0.3
0.8
L survives
0.3
M survives
C dies
0.8
0.2
0.7
M, L survives
0.2
44
Curlys Game Curly is the last mover. He can
move if he survived from Mos and/or Larrys
shoot.
Only L survives
Shots Curly hits with probability 0.3
Larry
Shoots Mo
Then C shoots L dies
Curly
Prob 0.7, misses
C shoots M dies
Shoots Larry
Misses
Mo
Shoots Curly hits with probability 0.8
M survives