In decision-making under risk, decision-makers have knowledge about the probabilities of different outcomes or states of nature. They can use various decision criteria to evaluate and select the best course of action based on expected values and the likelihood of different outcomes. Two commonly used criteria in decision-making under risk are:

1. Expected Monetary Value (EMV) Criterion:

The Expected Monetary Value (EMV) criterion evaluates decision alternatives based on their expected monetary payoff. It calculates the expected value of each alternative by multiplying the payoff of each outcome by its probability of occurrence and summing the results. The alternative with the highest expected monetary value is selected.

Formula:

EMV=∑i=1n​(Pi​×Xi​)

Where:

• EMV = Expected Monetary Value

• Pi​ = Probability of outcome i

  
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•  Xi​ = Monetary payoff associated with outcome i

  
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• n = Number of possible outcomes

2. Expected Opportunity Loss (EOL) Criterion:

The Expected Opportunity Loss (EOL) criterion evaluates decision alternatives based on the expected opportunity loss associated with each alternative. It calculates the expected value of the opportunity loss for each alternative by multiplying the difference between the best possible outcome and the payoff of each outcome by its probability of occurrence and summing the results. The alternative with the lowest expected opportunity loss is selected.

Formula:

EOL=∑i=1n​(Pi​×(Xbest​−Xi​))

Where:

• EOL = Expected Opportunity Loss

•  Pi​ = Probability of outcome i

• Xbest​ = Payoff of the best possible outcome

•  Xi​ = Payoff associated with outcome i

• n = Number of possible outcomes

Example:

Let’s consider a decision problem where a company is deciding whether to invest in a new product. The company estimates three possible outcomes with associated probabilities and payoffs as follows:

• Outcome 1: High demand (Probability = 0.4, Payoff = $50,000)
• Outcome 2: Moderate demand (Probability = 0.3, Payoff = $30,000)
• Outcome 3: Low demand (Probability = 0.3, Payoff = $10,000)

Using the EMV criterion: EMV = (0.4 \times 50,000) + (0.3 \times 30,000) + (0.3 \times 10,000) = $38,000

Using the EOL criterion:

$(0.4\times (50,000-{\mathrm{<br>}}_{\text{best}}))+(0.3\times (30,000-{\mathrm{<br>}}_{\text{best}}))+(0.3\times (10,000-{�}_{\text{best}}))$

EOL=(0.4×(50,000−Xbest​))+(0.3×(30,000−Xbest​))+(0.3×(10,000−Xbest​))

Based on the calculated values using both criteria, the decision-maker can select the alternative that best meets their objectives and risk preferences.