Decision Theory Game Theory
Decision Theory &
Game Theory
Decision Making
choose the optimum strategy from all the
alternatives
Decision Making Situations
Perfect Information Maximize –
Minimize
Partial or Imperfect Information:
Decisions under Risk
Decisions under Uncertainty
Certainty
Risk
Uncertainty
Decisions under Risk
Based on criteria:
Expected value (of profit or loss)
Combined expected value and variance
Known aspiration level
Most likely occurrence of a future state
Expected Value Criterion
Expected Value includes the probability to
gain profit + the probability to suffer loss
Mathematical Expectation (E(x))
Expected Value and Variance
Expected Value + Variance determine the
Risk Aversion Factor (K)
Risk Aversion Factor indicates the
“importance” of an alternative
The higher value of K, the more important
alternative
Aspiration-Level Criterion
First alternative generally treats as the
“best” alternative
Decision is made in a short span of time
Role of intuition
Most Likely Future Criterion
Simplification of probabilistic problem to
deterministic
Generalization of what happen in the
(quite) similar problem
Probabilities for Under Risk
Prior Probabilities: the known probability
Posterior Probabilities: modification of
prior probabilities … Conducting
experiment …
Under Risk: Decision Tree
Nodes:
Square:
decision point
Circle:
chance event (alternative)
Decision Tree: The Example
A company considers alternatives of 10year plan (partitioned in 2-year and 8-year
plans)
Stage 1: At the beginning of the 2-year
plan
Build large plant: – 5M
High Demand (prob. = 0.75) yields 1 M/year
Low Demand (prob. = 0.25) yields 0.3 M/year
Build small plant: – 1M
High Demand (prob. = 0.75) yields 0.25 M/year
Low Demand (prob. = 0.25) yields 0.2 M/year
Decision Tree
Stage 2: at the beginning of the 8-year
plan
Expand the small plant: – 4.2 M
High Demand (prob. = 0.75) yields 0.9 M/year
Low Demand (prob. = 0.25) yields 0.2 M/year
Do not Expand the small plant:
High Demand (prob. = 0.75) yields 0.25 M/year
Low Demand (prob. = 0.25) yields 0.2 M/year
H. demand
1 M/y
0.75
2
Bu
Pl ild
an la
t
rg
e
–
5M
L. demand
Bu
P il
– lan d s
1M t m
a
0.25
ll
3
d
an
em /y
d
H. 25M
0.
0.
75
1
0.3 M/y
L.
de
ma
0.
nd
25
an
p
Ex
4
Do
–
no
tE
d
5
M
2
.
4
xp
H. demand
0.9 M/y
0.75
L. demand
0.2 M/y
0.25
H. demand
0.75
an
d
0.25 M/y
6
L. demand
0.2 M/y
0.25
0.2 M/y
Stage 1: 2 years
Stage 2 : 8 years
(2)= (10 0.75 1) + (10 0.25 0.3) = 8.25
(1) (2)= 8.25 – 5 = 3.25 (build the large plant now)
(5)= (8 0.75 0.9) + (8 0.25 0.2) = 5.8
(6)= (8 0.75 0.25) + (8 0.25 0.2) =1.9
(4) (5),(6) = 5.8 + 1.9 – 4.2 = 3.5
(3) (4) = 3.5 + (2 0.75 0.25) + (10 0.25 0.20)
= 4.375
(1) (3) = 4.375 – 1 = 3.375
Decision under Uncertainty
The Laplace Criterion optimistic
The Minimax (Maximin) Criterion less
optimistic
The Savage Criterion “less
conservative”
The Hurwicz Criterion ranging from
optimistic to pessimistic
Decision under Uncertainty
Rows : possible action (ai)
Columns: future states or condition (j)
Laplace Criterion
Based on the principle of insufficient reason
Unknown probabilities of the occurrence of
j
j = 1, 2, … n
Laplace Criterion
1 n
v(ai, j )
max
n
ai
j 1
1
Probability of
n
j = 1, 2, … n
j
Laplace Criterion: Example
Minimize total cost of actions
Customer Category
Supplies
Level
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Probability P{ =j} = ¼
j = 1, 2, 3, 4
• E{a1} = ¼ (5 + 10 + 18 + 25) = 14.5
• E{a2} = ¼ (8 + 7 + 8 + 23)
= 11.5
• E{a3} = ¼ (21 + 18 + 12 + 21) = 18.0
• E{a4} = ¼ (30 + 22+ 19 + 15) = 21.5
Laplace Criterion: Example
Minimize total cost of actions
Customer Category
Supplies
Level
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Minimax (Maximin) Criterion
Making the best out of
the worst
Minimax:
n
v(ai , j )
max
min
ai
j
j 1
Maximin:
n
v(ai , j )
max
min
j
ai
j 1
Minimax Criterion: Example
Minimax Strategy
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Minimax Criterion: Example
Minimax Strategy
1
2
3
4
max{v(ai, j)}
a1
5
10
18
25
25
v(ai, j) a2
8
7
8
23
23
j
Minimax value
a3 21
18
12
21
21
a4 30
22
19
15
30
Maximin Criterion: Example
Maximin Strategy
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Maximin Criterion: Example
Maximin Strategy
1
2
3
4
min{v(ai, j)}
a1
5
10
18
25
5
v(ai, j) a2
8
7
8
23
7
a3 21
18
12
21
12
j
Maximin value
a4 30
22
19
15
15
Savage Minimax Regret Criterion
Construct new loss or profit matrix
v(ai, j) is replaced by r(ai, j) which is
defined by
r(ai, j)
max{v(ak, j)} – v(ai, j) if v is profit
ak
v(ai, j) – min {v(ak, j)} if v is loss
ak
Only the Minimax criterion can be applied
to r(ai, j)
Savage Minimax Regret Criterion
v is loss
v(ai, j)
1
2
3
4
a1
5
10
18
25
a2
8
7
8
23
a3
21
18
12
21
a4
30
22
19
15
Savage Minimax Regret Criterion
1
2
3
4
a1
5
10
13
25
a2
8
7
8
23
a3
21
18
12
21
a4
30
22
19
15
5
7
8
15
v is loss
v(ai, j)
min {v(ak, j)}
ak
Savage Minimax Regret Criterion
v is loss
r(ai, j)
1
2
3
4
a1
0
3
10
10
a2
3
0
0
8
a3
16
11
4
6
a4
25
15
11
0
Savage Minimax Regret Criterion
v is loss
a1
1
2
3
4
max r(ai, j)
0
3
10
10
10
j
Minimax value
r(ai, j) a2
3
0
0
8
8
a3
16
11
4
6
16
a4
25
15
11
0
25
Hurwicz Criterion
Balancing between extreme pessimism
and extreme optimism
Hurwicz Criterion
v(ai, j) : profit or gain
max { max v(ai, j) + (1 – )min v(ai, j)
ai
j
j
Hurwicz Criterion
v(ai, j) : profit or gain
Most
optimistic: max max{v(ai, j)}
Most
pesimistic: max min{v(ai, j)}
ai j
weigth =
ai j
weigth = 1 –
where 0 1
= ½ (reasonable)
Hurwicz Criterion
v(ai, j) : cost
Most
optimistic: min min{v(ai, j)}
Most
pesimistic: min max{v(ai, j)}
ai j
weigth =
ai j
weigth = 1 –
where 0 1
= ½ (reasonable)
Hurwicz Criterion
v(ai, j) : cost
min { min v(ai, j) + (1 – )max v(ai, j)
ai
j
j
Minimize total cost of actions
min v(ai, j) max v(ai, j)
1 2 3 4
j
j
a1
a2
a3
a4
5
10
18
25
5
25
8
7
8
23
7
23
21
18
12
21
12
21
30
22
19
15
15
30
=½
=½
min v(ai, j)
j
a1
a2
max v(ai, j) min v(ai,
j) j
j
+
1–
max v(ai, j)
j
5
25
0.5 5 + 0.5 25 =15
7
23
0.5 7 + 0.5 23 =15
a3
12
21
a4
15
30
0.5 12 + 0.5 21
=16.5
0.5 15 + 0.5 30
=22.5
min
ai
Reading Assignment #5
Game Theory
2
person zero-sum games
Mixed Strategies
Reading Assignment #6
Introduction to Queueing Theory
The End
This is the end of Chapter 8A
Game Theory
Decision Making
choose the optimum strategy from all the
alternatives
Decision Making Situations
Perfect Information Maximize –
Minimize
Partial or Imperfect Information:
Decisions under Risk
Decisions under Uncertainty
Certainty
Risk
Uncertainty
Decisions under Risk
Based on criteria:
Expected value (of profit or loss)
Combined expected value and variance
Known aspiration level
Most likely occurrence of a future state
Expected Value Criterion
Expected Value includes the probability to
gain profit + the probability to suffer loss
Mathematical Expectation (E(x))
Expected Value and Variance
Expected Value + Variance determine the
Risk Aversion Factor (K)
Risk Aversion Factor indicates the
“importance” of an alternative
The higher value of K, the more important
alternative
Aspiration-Level Criterion
First alternative generally treats as the
“best” alternative
Decision is made in a short span of time
Role of intuition
Most Likely Future Criterion
Simplification of probabilistic problem to
deterministic
Generalization of what happen in the
(quite) similar problem
Probabilities for Under Risk
Prior Probabilities: the known probability
Posterior Probabilities: modification of
prior probabilities … Conducting
experiment …
Under Risk: Decision Tree
Nodes:
Square:
decision point
Circle:
chance event (alternative)
Decision Tree: The Example
A company considers alternatives of 10year plan (partitioned in 2-year and 8-year
plans)
Stage 1: At the beginning of the 2-year
plan
Build large plant: – 5M
High Demand (prob. = 0.75) yields 1 M/year
Low Demand (prob. = 0.25) yields 0.3 M/year
Build small plant: – 1M
High Demand (prob. = 0.75) yields 0.25 M/year
Low Demand (prob. = 0.25) yields 0.2 M/year
Decision Tree
Stage 2: at the beginning of the 8-year
plan
Expand the small plant: – 4.2 M
High Demand (prob. = 0.75) yields 0.9 M/year
Low Demand (prob. = 0.25) yields 0.2 M/year
Do not Expand the small plant:
High Demand (prob. = 0.75) yields 0.25 M/year
Low Demand (prob. = 0.25) yields 0.2 M/year
H. demand
1 M/y
0.75
2
Bu
Pl ild
an la
t
rg
e
–
5M
L. demand
Bu
P il
– lan d s
1M t m
a
0.25
ll
3
d
an
em /y
d
H. 25M
0.
0.
75
1
0.3 M/y
L.
de
ma
0.
nd
25
an
p
Ex
4
Do
–
no
tE
d
5
M
2
.
4
xp
H. demand
0.9 M/y
0.75
L. demand
0.2 M/y
0.25
H. demand
0.75
an
d
0.25 M/y
6
L. demand
0.2 M/y
0.25
0.2 M/y
Stage 1: 2 years
Stage 2 : 8 years
(2)= (10 0.75 1) + (10 0.25 0.3) = 8.25
(1) (2)= 8.25 – 5 = 3.25 (build the large plant now)
(5)= (8 0.75 0.9) + (8 0.25 0.2) = 5.8
(6)= (8 0.75 0.25) + (8 0.25 0.2) =1.9
(4) (5),(6) = 5.8 + 1.9 – 4.2 = 3.5
(3) (4) = 3.5 + (2 0.75 0.25) + (10 0.25 0.20)
= 4.375
(1) (3) = 4.375 – 1 = 3.375
Decision under Uncertainty
The Laplace Criterion optimistic
The Minimax (Maximin) Criterion less
optimistic
The Savage Criterion “less
conservative”
The Hurwicz Criterion ranging from
optimistic to pessimistic
Decision under Uncertainty
Rows : possible action (ai)
Columns: future states or condition (j)
Laplace Criterion
Based on the principle of insufficient reason
Unknown probabilities of the occurrence of
j
j = 1, 2, … n
Laplace Criterion
1 n
v(ai, j )
max
n
ai
j 1
1
Probability of
n
j = 1, 2, … n
j
Laplace Criterion: Example
Minimize total cost of actions
Customer Category
Supplies
Level
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Probability P{ =j} = ¼
j = 1, 2, 3, 4
• E{a1} = ¼ (5 + 10 + 18 + 25) = 14.5
• E{a2} = ¼ (8 + 7 + 8 + 23)
= 11.5
• E{a3} = ¼ (21 + 18 + 12 + 21) = 18.0
• E{a4} = ¼ (30 + 22+ 19 + 15) = 21.5
Laplace Criterion: Example
Minimize total cost of actions
Customer Category
Supplies
Level
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Minimax (Maximin) Criterion
Making the best out of
the worst
Minimax:
n
v(ai , j )
max
min
ai
j
j 1
Maximin:
n
v(ai , j )
max
min
j
ai
j 1
Minimax Criterion: Example
Minimax Strategy
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Minimax Criterion: Example
Minimax Strategy
1
2
3
4
max{v(ai, j)}
a1
5
10
18
25
25
v(ai, j) a2
8
7
8
23
23
j
Minimax value
a3 21
18
12
21
21
a4 30
22
19
15
30
Maximin Criterion: Example
Maximin Strategy
a1
a2
a3
a4
1
2
3
4
5
10
18
25
8
7
8
23
21
18
12
21
30
22
19
15
Maximin Criterion: Example
Maximin Strategy
1
2
3
4
min{v(ai, j)}
a1
5
10
18
25
5
v(ai, j) a2
8
7
8
23
7
a3 21
18
12
21
12
j
Maximin value
a4 30
22
19
15
15
Savage Minimax Regret Criterion
Construct new loss or profit matrix
v(ai, j) is replaced by r(ai, j) which is
defined by
r(ai, j)
max{v(ak, j)} – v(ai, j) if v is profit
ak
v(ai, j) – min {v(ak, j)} if v is loss
ak
Only the Minimax criterion can be applied
to r(ai, j)
Savage Minimax Regret Criterion
v is loss
v(ai, j)
1
2
3
4
a1
5
10
18
25
a2
8
7
8
23
a3
21
18
12
21
a4
30
22
19
15
Savage Minimax Regret Criterion
1
2
3
4
a1
5
10
13
25
a2
8
7
8
23
a3
21
18
12
21
a4
30
22
19
15
5
7
8
15
v is loss
v(ai, j)
min {v(ak, j)}
ak
Savage Minimax Regret Criterion
v is loss
r(ai, j)
1
2
3
4
a1
0
3
10
10
a2
3
0
0
8
a3
16
11
4
6
a4
25
15
11
0
Savage Minimax Regret Criterion
v is loss
a1
1
2
3
4
max r(ai, j)
0
3
10
10
10
j
Minimax value
r(ai, j) a2
3
0
0
8
8
a3
16
11
4
6
16
a4
25
15
11
0
25
Hurwicz Criterion
Balancing between extreme pessimism
and extreme optimism
Hurwicz Criterion
v(ai, j) : profit or gain
max { max v(ai, j) + (1 – )min v(ai, j)
ai
j
j
Hurwicz Criterion
v(ai, j) : profit or gain
Most
optimistic: max max{v(ai, j)}
Most
pesimistic: max min{v(ai, j)}
ai j
weigth =
ai j
weigth = 1 –
where 0 1
= ½ (reasonable)
Hurwicz Criterion
v(ai, j) : cost
Most
optimistic: min min{v(ai, j)}
Most
pesimistic: min max{v(ai, j)}
ai j
weigth =
ai j
weigth = 1 –
where 0 1
= ½ (reasonable)
Hurwicz Criterion
v(ai, j) : cost
min { min v(ai, j) + (1 – )max v(ai, j)
ai
j
j
Minimize total cost of actions
min v(ai, j) max v(ai, j)
1 2 3 4
j
j
a1
a2
a3
a4
5
10
18
25
5
25
8
7
8
23
7
23
21
18
12
21
12
21
30
22
19
15
15
30
=½
=½
min v(ai, j)
j
a1
a2
max v(ai, j) min v(ai,
j) j
j
+
1–
max v(ai, j)
j
5
25
0.5 5 + 0.5 25 =15
7
23
0.5 7 + 0.5 23 =15
a3
12
21
a4
15
30
0.5 12 + 0.5 21
=16.5
0.5 15 + 0.5 30
=22.5
min
ai
Reading Assignment #5
Game Theory
2
person zero-sum games
Mixed Strategies
Reading Assignment #6
Introduction to Queueing Theory
The End
This is the end of Chapter 8A