Using Expected Value for Better Decision Making

Feb 07, 2023

There are unknown unknowns. - Donald Rumsfeld

Decision making under uncertainty

Decision making under uncertainty refers to the process of making choices or decisions in situations where the outcome is uncertain or unknown.

Additionally, it involves situations where there is:

Lack of complete information
Multiple possible outcomes
High degree of unpredictability

By using expected value and utility theory, individuals and organizations can make informed decisions that take into account both the likelihood of different outcomes and the relative satisfaction or happiness associated with each outcome.

What is expected value

Expected value is a way to figure out which option is the best choice when you have multiple options and you're not sure what will happen. Expected value is a statistical concept that is used to calculate the average outcome of a decision based on the probability of each possible outcome.

To calculate the expected value of a decision, you would multiply the value of each outcome by its corresponding probability and then sum those products. The decision with the highest expected value is typically considered the best choice. The goal is to maximize our chances of achieving the desired outcome.

EV Formula

The formula for expected value is:

EV = (probability of event 1 * value of event 1) + (probability of event 2 * value of event 2) + ... + (probability of event n * value of event n)

Where "EV" stands for expected value, "probability of event" is the chance of that event happening, and "value of event" is the outcome or payoff of that event.

This formula can be used to calculate the expected value in a wide range of situations such as gambling, investing in the stock market, probability distributions etc.

For example, let's say you're considering investing in a stock. The stock has a probability of 0.8 of going up by 10% and a probability of 0.2 of going down by 5%. The expected value of this investment would be (0.8 * 10%) + (0.2 * -5%) = 8%. This means that on average, you can expect an 8% return on your investment.

Another EV Example

Deciding between a major in Computer Science and a major in Business Administration. Assume that the expected salary for a Computer Science major is $80,000 with a probability of 0.8 and $60,000 with a probability of 0.2. For a Business Administration major, the expected salary is $70,000 with a probability of 0.7 and $50,000 with a probability of 0.3. The expected value for Computer Science major is (0.8 * $80,000) + (0.2 * $60,000) = $76,000 and for the Business Administration major is (0.7 * $70,000) + (0.3 * $50,000) = $62,000. Therefore, the Computer Science major has a higher expected value.

What is Utility Value

Utility value, which is part of a branch of economics called utility theory, refers to the measure of satisfaction or happiness that an individual gets from a good or service. It is the numerical representation of an individual's preference for a good or service. It explains how people make choices and how they value goods and services.

Utility theory is a concept used in decision-making that takes into account not only the expected value of a decision but also the individual's preferences, or "utility," for each outcome. To use utility theory in a decision-making situation, you would assign a utility value to each possible outcome and then choose the decision with the highest expected utility.

There are several ways to calculate utility values, such as:

Cardinal utility: Cardinal utility assigns a numerical value to each outcome, typically on a scale of 0 to 1 or 0 to 100. The decision with the highest utility value is chosen.
Ordinal utility: Ordinal utility assigns a rank or order to each outcome, rather than a numerical value. The decision with the highest rank is chosen.
Risk-adjusted utility: Risk-adjusted utility takes into account the probability of each outcome and the individual's or organization's risk preferences. The decision with the highest expected utility is chosen. A

For risk adjusted utility use the following in your EU formula:

Risk averse person - use the square of the negative outcome

Risk seeking person - use the square root of the negative outcome

Risk neutral - Use the expected value with no change.

One common way to calculate risk-adjusted utility is by using the expected utility formula. The expected utility formula is as follow:

EU = Σ (p_i * u(x_i))

Where:

EU is the expected utility
p_i is the probability of outcome i
u(x_i) is the utility of outcome i

In this formula, the expected utility is calculated by summing the product of each outcome's probability and utility. By using this formula, one can compare the expected utility of different decisions and choose the one with the highest expected utility.

Combining Expected and Utility Value

For example, let's say that an individual is considering investing in two different stocks: Stock A and Stock B. The expected return for Stock A is 8% with a standard deviation of 2%, and the expected return for Stock B is 12% with a standard deviation of 4%. Using expected value, the individual would calculate the expected return for each stock by multiplying the return by the probability of that return occurring. In this case, Stock B has a higher expected return, so it would be the preferred choice based on expected value.

However, expected value alone does not take into account the individual's preferences, or "utility," for each outcome. Utility theory is a concept used in decision-making that takes into account not only the expected value of a decision but also the individual's preferences for each outcome. Utility is a measure of the relative satisfaction or happiness that an individual associates with each outcome. To use utility theory in a decision-making situation, one would assign a utility value to each possible outcome and then choose the decision with the highest expected utility.

For example, let's say that the individual is risk-averse and prefers more certain outcomes. Using utility theory, they would assign a higher utility value to a more certain outcome with a lower return, such as Stock A, and a lower utility value to a more uncertain outcome with a higher return, such as Stock B. In this case, even though Stock B has a higher expected return, the individual would choose Stock A based on the higher utility value they assigned to it.

Expected value (risk neutral):

Stock A: 8% x 0.8 + (-2%) x 0.2 = 6.4%
Stock B: 12% x 0.6 + (-4%) x 0.4 = 7.2%

Here based on the expected value you would choose stock B since the EV is 7.2% versus 6.5% for stock A.

Expected Utility for a (risk averse)

Stock A: 8 - 2² = 4
Stock B: 12 - 4² = 4

There is no preference

Expected Utility (risk seeking)

Stock A: 8 - √2 = 6.41
√2 = 1.59
Stock B: 12 - √4 = 10.41
√4 = 1.26

There is a much clearer preference for stock B if you are risk seeking.

In this example choosing stock B is the clear choice regardless of risk preference.

It is important to note that the expected value and utility values are calculated based on the probability of each outcome, and those probabilities are not always known, therefore, it is important to use decision models that consider uncertain probabilities like Bayesian decision analysis, decision trees, and influence diagrams, etc.

Remember these tools are a guide to aid our decision making, but be wary that it can never be truly accurate since the probabilities used in the calculations are not always known.

Conclusion

Expected value and utility theory can be powerful tools for making informed decisions under conditions of uncertainty. By taking into account both the likelihood of different outcomes and the relative satisfaction or happiness associated with each outcome, individuals and organizations can make choices that are well-suited to their specific needs and preferences.