decision making under risk

Decision Making Under Risk

In decision-making under risk, the decision maker knows the possible outcomes and the probability of occurrence of each state of nature. This allows for quantitative evaluation of each alternative.

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i) Expected Monetary Value (EMV) Criterion

• Definition: The EMV of an alternative is the sum of possible payoffs for each state of nature, each weighted by its probability.

• Formula:

$$EMV=\sum (Payoff\times Probability)$$

• Decision Rule: Choose the alternative with the highest EMV.

Alternative	State A (p = 0.6)	State B (p = 0.4)	EMV
Option X	₹5000	₹2000	5000×0.6+2000×0.4=₹3800
Option Y	₹4000	₹3000	4000×0.6+3000×0.4=₹3600
ans. option X

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ii) Expected Value with Perfect Information (EVwPI)

• Definition: EVwPI is the expected payoff when a decision maker has perfect knowledge of which state of nature will occur.

• Calculation:

  1. For each state of nature, choose the best payoff across all alternatives.

  2. Multiply each best payoff by the probability of its state.

  3. Sum the results.

Formula:

$$EVwPI=\sum (BestPayoff\times Probability)$$

Example:
If:

• State A (0.6): Max payoff = ₹5000

• State B (0.4): Max payoff = ₹3000

EVwPI=5000×0.6+3000×0.4=₹4200EVwPI = 5000×0.6 + 3000×0.4 = ₹4200EVwPI=5000×0.6+3000×0.4=₹4200

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iii) Expected Value of Perfect Information (EVPI)

• Definition: EVPI measures the maximum amount a decision maker would be willing to pay for perfect information.

• Formula:

$$EVPI=EVwPI-EMVbest$$

Example:

• EVwPI = ₹4200

• Best EMV = ₹3800

$$EVPI=4200-3800=₹400$$

→ You would not pay more than ₹400 for perfect information.

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iv) Expected Opportunity Loss (EOL)

• Definition: Also called Expected Regret. It is the expected amount of loss due to not choosing the best action for each state.

• Steps to Calculate:

  1. Construct an Opportunity Loss Table:
     For each state, subtract each alternative’s payoff from the best payoff in that state.

  2. Multiply each opportunity loss by the probability of the corresponding state.

  3. Sum for each alternative.

• Decision Rule: Choose the alternative with the minimum EOL.

Example:

State	Best Payoff	Option X	Option Y	Regret X	Regret Y
A (0.6)	5000	5000	4000	0	1000
B (0.4)	3000	2000	3000	1000	0

$$EOLX=(0\times 0.6+1000\times 0.4)=₹400$$ $$EOLY=(1000\times 0.6+0\times 0.4)=₹600$$

→ Choose Option X (lower EOL).