Expected utility theory

What Is Expected Utility Theory?

Expected utility theory (EUT) is a foundational concept within decision theory that describes how individuals make choices when faced with uncertain outcomes. It posits that people do not simply choose the option with the highest expected value in monetary terms, but rather choose the option that maximizes their "expected utility." Utility, in this context, represents the satisfaction or happiness an individual derives from a particular outcome or level of wealth. The theory suggests that rational agents evaluate the potential outcomes of various choices under uncertainty by weighing the utility of each possible outcome by its probability, and then selecting the option with the highest weighted sum.

History and Origin

The conceptual roots of expected utility theory can be traced back to the 18th century, primarily through the work of Swiss mathematician Daniel Bernoulli. In 1738, Bernoulli proposed a solution to the "St. Petersburg Paradox," a famous problem in probability theory that highlighted a discrepancy between the expected monetary value of a gamble and the amount individuals were willing to pay to play it. The paradox demonstrated that while a certain gamble had an infinite expected monetary payoff, no rational person would pay an arbitrarily large sum to participate.²³,²²

Bernoulli resolved this paradox by introducing the concept of moral value or utility, arguing that the subjective value of money diminishes as wealth increases. This idea, known as diminishing marginal utility of wealth, suggested that an additional dollar provides less satisfaction to a wealthy person than to a poor one.²¹, Therefore, individuals make choices based on the expected utility derived from wealth, not solely on its monetary value.²⁰ This groundbreaking insight formed the basis of expected utility theory, influencing economic thought for centuries.¹⁹

Key Takeaways

• Expected utility theory posits that individuals make decisions under uncertainty by maximizing their expected utility, not just expected monetary value.
• Utility represents the subjective satisfaction or value an individual derives from an outcome.
• The theory incorporates an individual's risk tolerance, explaining why some prefer safer options even if they have lower expected monetary returns.
• A key concept within EUT is the idea of diminishing marginal utility of wealth, meaning that additional wealth provides progressively less additional satisfaction.
• Expected utility theory assumes rationality and consistent preference orderings in decision-making.

Formula and Calculation

The formula for expected utility (EU) is a sum of the utility of each possible outcome, weighted by its probability of occurrence.

For a gamble or choice with (n) possible outcomes, (X_1, X_2, \ldots, X_n), each with a corresponding probability (p_1, p_2, \ldots, p_n), the expected utility is calculated as:

Where:

• (EU) = Expected Utility
• (p_i) = Probability of outcome (i)
• (U(X_i)) = Utility derived from outcome (X_i)

The utility function (U(X)) represents an individual's preferences over different levels of wealth or outcomes. For a risk-averse individual, the utility function is typically concave, reflecting diminishing marginal utility. Conversely, a risk-seeking individual would have a convex utility function, and a risk-neutral individual would have a linear utility function.¹⁸,¹⁷

Interpreting the Expected Utility Theory

Expected utility theory provides a framework for understanding how individuals make choices under conditions of risk and uncertainty, moving beyond simple monetary expectations. It suggests that a person's willingness to take on risk depends on how much additional happiness or satisfaction they gain from an increase in wealth versus the potential unhappiness from a loss. For example, a wealthy individual might be less bothered by a potential loss of $1,000 than someone with very limited funds, because the marginal utility of that $1,000 is lower for the wealthy person.¹⁶

In essence, expected utility theory helps interpret why individuals might choose an option with a lower mathematical expected value if it offers higher certainty or a more desirable utility profile, aligning with their personal preference for risk.¹⁵

Hypothetical Example

Consider an investor, Sarah, who has $10,000 and is deciding between two investment options for a short period:

• Option A: Invest in a conservative bond. This option has a 100% chance of yielding a certain $100 profit, bringing her total wealth to $10,100.
• Option B: Invest in a volatile stock. This option has a 50% chance of yielding a $1,000 profit (total wealth $11,000) and a 50% chance of yielding a $500 loss (total wealth $9,500).

First, let's look at the expected monetary value (EMV) of each option:

• EMV of Option A: (1.00 \times $100 = $100)
• EMV of Option B: ((0.50 \times $1,000) + (0.50 \times -$500) = $500 - $250 = $250)

Based purely on expected monetary value, Option B appears more attractive. However, Sarah makes her investment decisions based on expected utility. Suppose her utility function is (U(W) = \ln(W)), where (W) is her wealth.

• Utility of current wealth: (U($10,000) = \ln(10,000) \approx 9.21)

• Expected Utility of Option A:

  • If she chooses Option A, her wealth is $10,100.
  • (EU(A) = 1.00 \times U($10,100) = 1.00 \times \ln(10,100) \approx 1.00 \times 9.22 = 9.22)

• Expected Utility of Option B:

  • If she chooses Option B, there are two outcomes: $11,000 (profit) or $9,500 (loss).
  • (EU(B) = (0.50 \times U($11,000)) + (0.50 \times U($9,500)))
  • (EU(B) = (0.50 \times \ln(11,000)) + (0.50 \times \ln(9,500)))
  • (EU(B) \approx (0.50 \times 9.30) + (0.50 \times 9.16))
  • (EU(B) \approx 4.65 + 4.58 = 9.23)

In this hypothetical example, Option B has a slightly higher expected utility (9.23) than Option A (9.22) for Sarah, despite the risk involved. Therefore, according to expected utility theory, Sarah would choose Option B, reflecting a slight willingness to take on risk when the potential utility gain outweighs the disutility of potential loss, as depicted by her logarithmic utility function. If her utility function were more concave (e.g., square root), she might exhibit higher risk aversion and choose Option A.

Practical Applications

Expected utility theory finds widespread application across various financial and economic domains:

• Insurance: EUT helps explain why individuals purchase insurance. Even though the expected monetary value of an insurance policy is negative (premiums exceed expected payouts), risk-averse individuals derive higher expected utility from the certainty of avoiding a large potential loss. Insurance acts as a mechanism for risk management, effectively smoothing out potential large fluctuations in wealth.
• Portfolio Management: Investors use the principles of expected utility theory, often implicitly, when constructing investment portfolios. Modern portfolio management strategies, such as mean-variance optimization, aim to maximize return for a given level of risk, or minimize risk for a given return, aligning with an investor's risk preference.
• Public Policy and Regulation: Policymakers consider expected utility when designing social safety nets, healthcare policies, and regulatory frameworks. Understanding how individuals weigh risks and benefits underpins decisions regarding pensions, unemployment benefits, and mandatory insurance, aiming to maximize societal welfare by accounting for people's utility from various outcomes. Research also suggests that macroeconomic announcements can reveal significant deviations from expected utility, influencing how investors perceive and react to new information.¹⁴
• Corporate Finance: Businesses apply expected utility concepts in capital budgeting decisions, evaluating projects not just on their expected financial returns, but also on the risk profiles and how those risks affect the utility of shareholders or the firm itself.

Limitations and Criticisms

While expected utility theory is a cornerstone of economic theory, it faces several limitations and criticisms, particularly from the field of behavioral economics:

• Descriptive vs. Normative: EUT is often seen as a normative theory, describing how rational agents should behave, rather than a descriptive theory explaining how real people actually behave.¹³
• Allais Paradox and Ellsberg Paradox: These famous thought experiments demonstrate systematic violations of the axioms of expected utility theory, such as the independence axiom. They show that individuals often make inconsistent choices when presented with similar probabilistic scenarios, contradicting the strict rationality assumed by EUT.
• Reference Dependence: Expected utility theory assumes that utility is derived from absolute levels of wealth. However, research by Daniel Kahneman and Amos Tversky, for which Kahneman received a Nobel Prize, suggests that people evaluate outcomes relative to a reference point (e.g., their current wealth or an expected outcome) rather than absolute wealth.,¹²,¹¹ Losses loom larger than equivalent gains, a phenomenon called "loss aversion," which EUT struggles to explain.¹⁰,⁹
• Framing Effects: The way a choice is presented or "framed" can significantly influence decisions, even if the underlying probabilities and outcomes are objectively the same. EUT does not account for such psychological biases.⁸

Expected Utility Theory vs. Prospect Theory

Expected utility theory and prospect theory are both frameworks for understanding decision making under risk, but they differ significantly in their approach and assumptions.

Expected utility theory is a normative model that assumes individuals are rational utility maximizers. It suggests that people calculate the weighted average of the utilities of all possible outcomes and choose the option that yields the highest expected utility, regardless of their current financial position. It focuses on the final states of wealth.

In contrast, prospect theory, developed by Daniel Kahneman and Amos Tversky, is a descriptive model that aims to explain observed human behavior, which often deviates from EUT's predictions. Prospect theory introduces several key concepts:

• Reference Dependence: Individuals evaluate outcomes as gains or losses relative to a specific reference point, not in terms of absolute wealth.⁷,⁶
• Loss Aversion: The psychological impact of a loss is felt more intensely than the pleasure of an equivalent gain.,⁵
• Diminishing Sensitivity: The marginal impact of gains or losses decreases as their magnitude increases.
• Probability Weighting: People tend to overweight small probabilities and underweight moderate to high probabilities.

Essentially, EUT describes how a perfectly rational agent should behave, while prospect theory describes how actual people do behave, often exhibiting cognitive biases and emotional influences not captured by EUT.⁴

FAQs

Q: Does Expected Utility Theory imply that everyone is risk-averse?
A: No. Expected utility theory can account for risk aversion, risk-neutrality, and even risk-seeking behavior. The shape of an individual's utility function determines their attitude towards risk. A concave utility function indicates risk aversion, a linear function indicates risk neutrality, and a convex function indicates risk-seeking behavior.³,²

Q: How is expected utility different from expected value?
A: Expected value is a purely mathematical concept that calculates the average monetary outcome of a gamble or decision by multiplying each possible monetary outcome by its probability and summing the results. Expected utility, however, incorporates an individual's subjective value or satisfaction (utility) from those monetary outcomes. A choice with a lower expected monetary value might be preferred if it offers a higher expected utility due to an individual's preference for certainty or aversion to risk.¹

Q: Is Expected Utility Theory still relevant in finance today?
A: Yes, despite its criticisms, expected utility theory remains a fundamental concept in finance and economics. It provides a robust theoretical foundation for understanding risk management, asset pricing, and portfolio management under idealized conditions. While behavioral economics offers alternative descriptive models, EUT serves as a crucial benchmark for rational decision-making and is often used as a starting point for more complex models.