Decision Theory Game Theory

Decision Making

 choose the optimum strategy from all the

Decision Making Situations

 Perfect Information  Maximize –

Minimize

 Partial or Imperfect Information:

 _{Decisions under Risk}

 Decisions under Uncertainty

Decisions under Risk

 Based on criteria:

 Expected value (of profit or loss)

 _{Combined expected value and variance}

 Known aspiration level

Expected Value Criterion

 Expected Value includes the probability to

gain profit + the probability to suffer loss

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

Aspiration-Level Criterion

 First alternative generally treats as the

“best” alternative

Most Likely Future Criterion

 Simplification of probabilistic problem to

deterministic

 Generalization of what happen in the

Probabilities for Under Risk

 Prior Probabilities: the known probability  Posterior Probabilities: modification of

Under Risk: Decision Tree

 Nodes:

 Square: decision point

Decision Tree: The Example

 A company considers alternatives of

10-year plan (partitioned in 2-10-year and 8-10-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

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:

2 3 1 M/y 0.3 M/y 0.75 0.25 Build larg e Plan t Bu ild sm all Pla nt H. demand L. demand H. de man d –5M – 1M 0.75

L. d_em

and

0.2₅

0.2 M/y Stage 1: 2 years

Expa nd

1

4

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

Decision under Uncertainty

 The Laplace Criterion  optimistic

 The Minimax (Maximin) Criterion  less

optimistic

 The Savage Criterion  “less

conservative”

 The Hurwicz Criterion  ranging from

Decision under Uncertainty

 Rows : possible action (a_i)

Laplace Criterion

 _{Based on the principle of insufficient reason}

 Unknown probabilities of the occurrence of

 j = 1, 2, … n

j

Laplace Criterion















 n j j a

ai

v

n

i 1

)

,

(

1

max



n

1

Probability of



_j

Laplace Criterion: Example

15 19 22 30

a

₄ 21 12 18 21

a

₃ 23 8 7 8

a

₂ 25 18 10 5

a

₁



₄



₃



₂



₁ Customer Category Supplies Level

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

Laplace Criterion: Example

15 19 22 30

a

₄ 21 12 18 21

a

₃ 23 8 7 8

a

₂ 25 18 10 5

a

₁



₄



₃



₂



₁ Customer Category Supplies Level

Minimax (Maximin) Criterion

 _{Making the best out of}

Minimax Criterion: Example

Minimax Strategy 15 21 23 25 ₄

v(a_i, _j)

30 19 22 30 a₄ 21 12 18 21 a₃ 23 8 7 8 a₂ 25 18 10 5 a₁

max{v(a_i, _j)}

_j

₃ ₂

₁

Maximin Criterion: Example

Maximin Strategy 15 21 23 25 ₄

v(a_i, _j)

15 19 22 30 a₄ 12 12 18 21 a₃ 7 8 7 8 a₂ 5 18 10 5 a₁

min{v(a_i, _j)}

_j

₃ ₂

₁

Savage Minimax Regret Criterion

 Construct new loss or profit matrix

 v(a_i, _j) is replaced by r(a_i, _j) which is

defined by

max{v(a_k, _j)} – v(a_i, _j) if v is profit a_k

 r(a_i, _j)

v(a_i, j) – min {v(a_k, _j)} if v is loss a_k

 Only the Minimax criterion can be applied

Savage Minimax Regret Criterion

15 21 23 25 ₄

v(a_i, _j)

19 22 30 a₄ 12 18 21 a₃ 8 7 8 a₂ 18 10 5 a₁ ₃ ₂ ₁

Savage Minimax Regret Criterion

15 19 22 30 a₄ v(a_i, _j)

15 21 23 25

₄

min {v(a_k, _j)}

a_k 5 7 8

12 18 21 a₃ 8 7 8 a₂ 13 10 5 a₁ ₃ ₂ ₁

Savage Minimax Regret Criterion

0 6 8 10 ₄

r(a_i, _j)

11 15 25 a₄ 4 11 16 a₃ 0 0 3 a₂ 10 3 0 a₁ ₃ ₂ ₁

Savage Minimax Regret Criterion

0 6 8 10 ₄ 25 16 8 10

max r(a_i, _j)

_j

r(a_i, _j)

11 15 25 a₄ 4 11 16 a₃ 0 0 3 a₂ 10 3 0 a₁ ₃ ₂ ₁

v is loss

Hurwicz Criterion

 Balancing between extreme pessimism

Hurwicz Criterion

 v(a_i, _j) : profit or gain

 max { max v(a_i, _j) + (1 – )min v(a_i, _j)

Hurwicz Criterion

v

(a

_i

,



_j

) : profit or gain



Most optimistic:

max max{v(a_i, _j)} a_i _j

weigth = 



Most pesimistic:

max min{v(a_i, _j)} a_i _j

weigth = 1 – 

 where 0    1

Hurwicz Criterion

v

(a

_i

,



_j

) : cost



Most optimistic:

min min{v(a_i, _j)} a_i _j

weigth = 



Most pesimistic:

min max{v(a_i, _j)} a_i _j

weigth = 1 – 

 where 0    1

Hurwicz Criterion

 v(a_i, _j) : cost

 min { min v(a_i, _j) + (1 – )max v(a_i, _j)

15 12 7 5

min v(a_i, _j)

_j 15 21 23 25



₄ 30 19 22 30

a

₄ 21 12 18 21

a

₃ 23 8 7 8

a

₂ 25 18 10 5

a

₁

max v(a_i, _j)

_j



₃



₂



₁

Minimize total cost of actions

+ 1 – 

max v(a_i, _j)

_j

0.5  15 + 0.5  30 =22.5

0.5  12 + 0.5  21 =16.5

0.5  7 + 0.5  23 =15 0.5  5 + 0.5  25 =15

 min v(a_i,

_j) _j

30 21 23 25

max v(a_i, _j)

_j

15 12

7 5

min v(a_i, _j)

Reading Assignment #5

 Game Theory

2 person zero-sum games

Reading Assignment #6

The End