decision making types

Here’s a detailed set of notes on Decision Making Under Risk and Uncertainty, including:

• Elements of a Decision-Making Problem

• Decision Making Under Risk with:

  • Expected Monetary Value (EMV)

  • Expected Value with Perfect Information (EVwPI)

  • Expected Value of Perfect Information (EVPI)

  • Expected Opportunity Loss (EOL)

• Decision Making Under Uncertainty with:

  • Maximax / Minimin

  • Maximin

  • Hurwicz (Alpha)

  • Laplace

  • Minimax Regret

• Decision Tree with a simple example

All examples use unique tables, formulas are enclosed in $$ $$, and markdown tables are formatted for Obsidian.

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1. Elements of Decision-Making Problem

1. Decision Alternatives: Courses of action available to the decision-maker.

2. States of Nature: Possible future events not under the control of the decision-maker.

3. Payoffs: Outcomes (gains/losses) from choosing a particular action under a specific state of nature.

4. Probabilities: Likelihood of each state of nature (used in decision under risk).

5. Decision Criteria: Rules for selecting the best alternative.

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2. Decision Making Under Risk

States of nature have known probabilities.

i) Expected Monetary Value (EMV)

Formula:

$$EMV(Ai)=\sum j=1nP(Sj)\times Payoff(Ai,Sj)EMV(A_i)=\sum _{j=1}^{n}P(S_j)\times Payoff(A_i,S_j)$$

Where:

• AiA_i: Alternative

• SjS_j: State of Nature

• P(Sj)P(S_j): Probability of state SjS_j

Example:

Alternative	Profit if Demand is High (P=0.6)	Profit if Low (P=0.4)
A1 (Invest)	100	20
A2 (Do Nothing)	50	40

EMV Calculations:

• EMV(A1)=0.6×100+0.4×20=60+8=68EMV(A1) = 0.6 \times 100 + 0.4 \times 20 = 60 + 8 = 68

• EMV(A2)=0.6×50+0.4×40=30+16=46EMV(A2) = 0.6 \times 50 + 0.4 \times 40 = 30 + 16 = 46

Best Decision: Choose A1 (EMV = 68)

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ii) Expected Value with Perfect Information (EVwPI)

Formula:

EVwPI=∑j=1nP(Sj)×max⁡i{Payoff(Ai,Sj)}EVwPI = \sum_{j=1}^{n} P(S_j) \times \max_i { Payoff(A_i, S_j) }

Using the above table:

• For High demand: Max = 100 (A1)

• For Low demand: Max = 40 (A2)

EVwPI=0.6×100+0.4×40=60+16=76EVwPI = 0.6 \times 100 + 0.4 \times 40 = 60 + 16 = 76

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iii) Expected Value of Perfect Information (EVPI)

Formula:

EVPI=EVwPI−max⁡i{EMV(Ai)}EVPI = EVwPI - \max_i { EMV(A_i) } EVPI=76−68=8EVPI = 76 - 68 = 8

Interpretation: Max value worth paying for perfect information is 8 units.

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iv) Expected Opportunity Loss (EOL)

Steps:

1. Create an Opportunity Loss Table.

2. Multiply each loss with its probability.

3. Choose the alternative with the lowest EOL.

Payoff Table:

Alternative	High (0.5)	Medium (0.3)	Low (0.2)
A1	200	100	50
A2	150	120	100
A3	300	80	40

Opportunity Loss Table (Subtract from column max):

Alternative	High	Medium	Low
A1	100	20	10
A2	150	0	0
A3	0	40	60

EOL Calculations:

• A1: 0.5×100+0.3×20+0.2×10=50+6+2=580.5 \times 100 + 0.3 \times 20 + 0.2 \times 10 = 50 + 6 + 2 = 58

• A2: 0.5×150+0.3×0+0.2×0=750.5 \times 150 + 0.3 \times 0 + 0.2 \times 0 = 75

• A3: 0.5×0+0.3×40+0.2×60=0+12+12=240.5 \times 0 + 0.3 \times 40 + 0.2 \times 60 = 0 + 12 + 12 = 24

Best Decision: A3 (lowest EOL = 24)

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3. Decision Making Under Uncertainty

Probabilities are unknown.

i) Maximax (Optimistic)

Choose the alternative with the maximum of maximum payoffs.

Alternative	S1	S2	S3
A1	10	15	30
A2	20	25	10
A3	5	10	20

• Max of A1: 30

• Max of A2: 25

• Max of A3: 20

Best Decision: A1 (Maximax = 30)

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ii) Maximin (Pessimistic)

Choose the alternative with the maximum of minimum payoffs.

Alternative	S1	S2	S3
A1	10	15	30
A2	20	25	10
A3	5	10	20

• Min of A1: 10

• Min of A2: 10

• Min of A3: 5

Best Decision: A1 or A2 (Maximin = 10)

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iii) Hurwicz Criterion

Formula:

Hurwicz Value=α×Max+(1−α)×MinHurwicz\ Value = \alpha \times Max + (1 - \alpha) \times Min

Let α=0.7\alpha = 0.7

Alternative	Max	Min
A1	30	10
A2	25	10
A3	20	5

• A1: 0.7×30+0.3×10=21+3=240.7 \times 30 + 0.3 \times 10 = 21 + 3 = 24

• A2: 0.7×25+0.3×10=17.5+3=20.50.7 \times 25 + 0.3 \times 10 = 17.5 + 3 = 20.5

• A3: 0.7×20+0.3×5=14+1.5=15.50.7 \times 20 + 0.3 \times 5 = 14 + 1.5 = 15.5

Best Decision: A1

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iv) Laplace Criterion (Equal Probability)

Average all outcomes and choose the highest.

Alternative	S1	S2	S3	Average
A1	10	15	30	553=18.33\frac{55}{3} = 18.33
A2	20	25	10	553=18.33\frac{55}{3} = 18.33
A3	5	10	20	353=11.67\frac{35}{3} = 11.67

Best Decision: A1 or A2

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v) Minimax Regret Criterion

1. Create a regret table by subtracting each value from the column max.

2. Choose the alternative with minimum of maximum regrets.

Alternative	S1	S2	S3
A1	10	15	30
A2	20	25	10
A3	5	10	20

Column Maxima: S1 = 20, S2 = 25, S3 = 30

Regret Table:

Alternative	S1	S2	S3	Max Regret
A1	10	10	0	10
A2	0	0	20	20
A3	15	15	10	15

Best Decision: A1 (Minimax Regret = 10)

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4. Decision Tree (Simple Example)

A company can launch a new product (Cost = 20), with two outcomes:

• Success (Profit = 80) with P=0.6P = 0.6

• Failure (Loss = -20) with P=0.4P = 0.4

Expected Value:

EV=0.6×80+0.4×(−20)=48−8=40EV = 0.6 \times 80 + 0.4 \times (-20) = 48 - 8 = 40

Net Gain (after subtracting cost):

Net=40−20=20Net = 40 - 20 = 20

Decision Tree Structure:

[Decision: Launch or Not]
    ├── Launch
    │   ├── Success (0.6): +80
    │   └── Failure (0.4): -20
    └── Do Nothing: 0

Best Decision: Launch (Net EV = 20)

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