ECON7322
Tutorial Week 7
7th September 2023
Decision Theory and Games
(Part B)
Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
10
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s2: Incorrect
Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
10
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s2: Incorrect
Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
10
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s2: Incorrect

Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
10
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s1: Correct
(0.4)(900)+(0.6)(0)
=$360
Start
Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
10
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s1: Correct
(0.4)(900)+(0.6)(0)
=$360
Choose $400
Start
Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s1: Correct
(0.4)(900)+(0.6)(0)
=$360
Start
10
(0.6)(400)+(0.4)(0)
=$240
Choose $400
Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
10
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s1: Correct
(0.4)(900)+(0.6)(0)
=$360
Choose $400
Start
(0.6)(400)+(0.4)(0)
=$240
Choose $240
Decision Tree
1
a1: Answer
MATHS
a2: Stop
2
3
4
5
6
7
a1: Answer
ECON
a2: Stop
$100
8
9
$0
$0
$0
a1: Answer
STAT
a2: Stop
10
11
12
13
$400
0.4
s1: Correct
0.6
s2: Incorrect
$900
$0
0.6
s1: Correct
0.4
s2: Incorrect
0.8
s1: Correct
0.2
s1: Correct
(0.4)(900)+(0.6)(0)
=$360
Choose $400
(0.6)(400)+(0.4)(0)
=$240
Choose $240
(0.8)(240)+(0.2)(0)
=$192
Start
I should start by answering the maths question. If I get the
maths question right, I should answer the economics question.
If I get the economics question right, I should stop. The
expected value of this strategy is $192.
Excel QM – Decision Analysis (Decision Trees)
Uncheck this box
Excel QM – Decision Analysis (Decision Trees)
The numbers beneath the decision nodes (the squares)
indicate the optimal path.
I should start by answering the maths question. If I get the
maths question right, I should answer the economics question.
If I get the economics question right, I should stop. The
expected value of this strategy is $192.
Numbers represent gains to the player on the left, and losses to the player on the right
Player 1
(Gain)
Player 2
(Loss)
Prefer
larger
number
Prefer smaller number
Using Dominance
Using Dominance
R1, R3 and R4 are all dominated by R2.
Player 1
(Gain)
Using Dominance
Player 2
(Loss) Prefer smaller number
Using Dominance
C2 and C4 are now dominated by C1 and C3.
Thus, the player at the left should choose R2, and the player at the top should
choose C1 or C3. The value of the game is 6.
Player 2
(Loss)
Player 1
Player 2
Saddle Point Theorem
Excel QM – Games (Zero Sum)
Excel QM – Games (Zero Sum)
Note that this screen indicates that the player at the top should choose C3.
However, C1 is an alternative optimal strategy.
Row player should choose R2
Column player should choose C3
Value of the game
Alternative optimal Strategy
Not zero-sum game
Payoff Matrix
• This is not a zero- sum game, so the elements in this payoff matrix do not
represent losses to the player at the top.
• E.g., when both firms choose High, Firm 2 does not lose $600, but rather
gains 1000 – 600 = $400
• However, it can still be solved using our usual methods.

Firm 1’s profit
1000 – 500
1000 – 300
1000 – 400
1000 – 600
Firm 2’s profit
Using Dominance
Are there any strategies dominated by other strategies? No
Saddle Point
This condition is not satisfied. There no saddle point.
The players will adopt mixed strategies.
400 500≠
Mixed strategies
If Firm 2 choose Low, the expected value of Firm 1's gains would be: (p)(500) + (1– p)(300)
If Firm 2 choose High, the expected value of Firm 1's gains would be: (p)(400) + (1– p)(600) 500 + 1 − 300 = 400 + 1 − 600500 + 300 − 300 = 400 + 600 − 600400 = 300 = 300400 = 0.75

The value of the game is (0.75)(500) + (0.25)(300) = $450.
Firm 1 can expect to end up with $450.
Firm 2 can expect to end up with 1000 – 450 = $550.
p
1 - p
Let p be the proportion of time Firm 1 chooses
Low.
⟹ Firm 1 should choose Low 75% of the time
Mixed strategies
If Firm 1 choose Low, the expected value of Firm 2's gains would be: (p)(1000 – 500) + (1– p)(1000 – 400)
If Firm 1 choose High, the expected value of Firm 2's gains would be: (p)(1000 – 300) + (1– p)(1000 – 600) 1000 − 500 + 1 − 1000 − 400 = 1000 − 300 + 1 − 1000 − 600500 + 600 − 600 = 700 + 400 − 400400 = 200 = 200400 = 0.5

The value of the game is (0.5)(1000 – 500) + (0.5)(1000 – 400) = $550.
Firm 2 can expect to end up with $550.
Firm 1 can expect to end up with 1000 – 550 = $450.
p 1 - p
Let p be the proportion of time Firm 2 chooses
Low.
⟹ Firm 2 should choose Low 50% of the time
Excel QM – Games (Zero Sum)
• Firm 1 should choose Low 75% of the time and High 25% of the time.
• Firm 2 should choose Low 50% of the time and High 50% of the time.
• The value of the game is $450.
Value of the game
Firm 1
Firm 2
Thank you!
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