suppose you have a pay- off table in the below table, determine which alternative/strategies would be chosen under each of these strategies: take a=0.6 when it is necessary.

Question

A,MAXIMAX,
B Maximin,
C Laplace
D minimax regret and
e Hurwitz
f expected monetary value
g expected opportunity lose
h expected value under perfect information
possible future profit
alternatives s1 s2 s3
a1 100 40 5
a2 21 40 60
a3 70 20 20
prior probability 0.2 0.5 0.3

Answer 1

To analyze the pay-off table and determine the best alternatives under the mentioned strategies, we will follow these steps for each decision-making criterion:

Payoff Table

| Alternatives | s1 | s2 | s3 | |--------------|----|----|----| | a1 | 100| 40 | 5 | | a2 | 21 | 40 | 60 | | a3 | 70 | 20 | 20 |

Prior Probability

• s1: 0.2
• s2: 0.5
• s3: 0.3

A. MAXIMAX

The maximax criterion focuses on choosing the alternative with the maximum of the maximum payoffs.

• Maximum payoffs:
  • a1: max(100, 40, 5) = 100
  • a2: max(21, 40, 60) = 60
  • a3: max(70, 20, 20) = 70

Choosing the alternative with the highest maximum payoff:

• Choice: a1 (100)

B. MAXIMIN

The maximin criterion focuses on choosing the alternative with the maximum of the minimum payoffs.

• Minimum payoffs:
  • a1: min(100, 40, 5) = 5
  • a2: min(21, 40, 60) = 21
  • a3: min(70, 20, 20) = 20

Choosing the alternative with the highest minimum payoff:

• Choice: a2 (21)

C. LAPLAS

The Laplace criterion assumes equal probabilities for each state of nature.

• Average payoffs:
  • a1: (100 + 40 + 5) / 3 = 48.33
  • a2: (21 + 40 + 60) / 3 = 40.33
  • a3: (70 + 20 + 20) / 3 = 36.67

Choosing the alternative with the highest average payoff:

• Choice: a1 (48.33)

D. MINIMAX REGRET

Calculate the regret table first.

| Alternatives | s1 | s2 | s3 | Max Regret | |--------------|----|----|----|------------| | a1 | 0 | 20 | 55 | 55 | | a2 | 79 | 0 | 0 | 79 | | a3 | 30 | 20 | 40 | 40 |

Max regrets:

• a1: 55
• a2: 79
• a3: 40

Choosing the alternative with the lowest maximum regret:

• Choice: a3 (40)

E. HURWITZ (using a = 0.6)

• Hurwitz value is calculated as $V_H=\alpha \times \text{max}+(1-\alpha )\times \text{min}$.

Calculating:

• a1: $0.6\times 100+0.4\times 5=60+2=62$
• a2: $0.6\times 60+0.4\times 21=36+8.4=44.4$
• a3: $0.6\times 70+0.4\times 20=42+8=50$

Choosing the alternative with the maximum Hurwitz value:

• Choice: a1 (62)

F. EXPECTED MONETARY VALUE

The expected monetary value (EMV) is calculated by multiplying the payoffs by their probabilities.

$$EMV(a1)=100\times 0.2+40\times 0.5+5\times 0.3=20+20+1.5=41.5$$ $$EMV(a2)=21\times 0.2+40\times 0.5+60\times 0.3=4.2+20+18=42.2$$ $$EMV(a3)=70\times 0.2+20\times 0.5+20\times 0.3=14+10+6=30$$

Choosing the alternative with the highest EMV:

• Choice: a2 (42.2)

G. EXPECTED OPPORTUNITY LOSS

The expected opportunity loss (EOL) represents the average regret for selecting a particular alternative.

Calculate the opportunity losses for each alternative. The EOL values are calculated similarly to the EMV and the best outcomes are:

• EOL(a1) = 0.2 * 0 + 0.5 * 20 + 0.3 * 55 = 0 + 10 + 16.5 = 26.5
• EOL(a2) = 0.2 * 79 + 0.5 * 0 + 0.3 * 0 = 15.8 + 0 + 0 = 15.8
• EOL(a3) = 0.2 * 30 + 0.5 * 20 + 0.3 * 40 = 6 + 10 + 12 = 28

Choosing the alternative with the lowest EOL:

• Choice: a2 (15.8)

H. EXPECTED VALUE UNDER PERFECT INFORMATION

The expected value of perfect information is the difference between the expected value with perfect information and the expected monetary value.

Maximum outcomes for each state (s):

• s1: 100
• s2: 40
• s3: 60

Expected value under perfect information: $$EVPI=(0.2\times 100)+(0.5\times 40)+(0.3\times 60)$$ $$EVPI=20+20+18=58$$

In summary, the choices for each strategy are as follows:

Summary of Choices

A. MAXIMAX: a1
B. MAXIMIN: a2
C. LAPLAS: a1
D. MINIMAX REGRET: a3
E. HURWITZ: a1
F. EXPECTED MONETARY VALUE: a2
G. EXPECTED OPPORTUNITY LOSS: a2
H. EXPECTED VALUE UNDER PERFECT INFORMATION: 58