Expected utility

What Is Expected Utility?

Expected utility is a core concept in behavioral economics and decision-making that quantifies the anticipated satisfaction or benefit an individual expects to derive from various uncertain outcomes. It goes beyond simply considering the monetary value of a potential result by incorporating a person's subjective valuation of those results, known as utility, alongside their probabilities. This framework assumes that rational agents make choices under uncertainty by selecting the option that maximizes their expected utility. It acknowledges that the satisfaction gained from an additional unit of wealth may diminish as one's total wealth increases, a concept known as marginal utility.

History and Origin

The foundational ideas behind expected utility theory can be traced back to the 18th century with Daniel Bernoulli, a Swiss mathematician. In his 1738 paper, “Exposition of a New Theory on the Measurement of Risk,” Bernoulli introduced the concept to solve the St. Petersburg Paradox, noting that individuals do not evaluate outcomes solely by their monetary values but by the subjective utility derived from them. He³⁹, ⁴⁰ proposed that the utility of wealth exhibits diminishing marginal utility, meaning each additional increment of wealth provides less satisfaction than the previous one.

T³⁸he theory saw significant development in the mid-20th century with the work of economist John von Neumann and mathematician Oskar Morgenstern. Their 1944 book, Theory of Games and Economic Behavior, provided a rigorous axiomatic framework for expected utility, demonstrating that if an individual's preferences satisfy certain logical conditions, their choices can be represented by a utility function that they aim to maximize. Th³⁶, ³⁷is formalization solidified expected utility as a cornerstone of modern economic theory.

#³⁵# Key Takeaways

• Expected utility measures the anticipated satisfaction from uncertain outcomes, weighted by their probabilities.
• It incorporates individual preferences and risk tolerance, recognizing that the utility of money is subjective and not linear.
• The theory assumes that rational individuals choose the option that yields the highest expected utility.
• Expected utility is widely applied in finance, insurance, and public policy for evaluating choices under risk.
• While a powerful tool, expected utility theory faces criticisms for its assumptions about perfect rationality and its descriptive limitations regarding actual human behavior.

Formula and Calculation

The formula for calculating expected utility sums the utility of each possible outcome, weighted by its probability of occurrence.

The formula for expected utility (EU) is:

Where:

• ( EU ) = Expected Utility
• ( P_i ) = The probability of outcome ( i ) occurring
• ( U(x_i) ) = The utility (satisfaction or value) derived from outcome ( i )
• ( n ) = The total number of possible outcomes

To calculate this, one must first assign a utility value to each potential outcome ( x_i ), reflecting the subjective benefit it provides to the decision-maker. This utility value is often represented by a utility function, which can be non-linear to capture concepts like diminishing marginal utility.

Interpreting the Expected Utility

Interpreting expected utility involves understanding that it represents a subjective measure of desirability for uncertain situations. A higher expected utility indicates a more preferred choice for a rational agent, given their individual preferences and attitudes towards risk. For example, a risk-averse individual will have a concave utility function, meaning they derive less additional utility from larger gains compared to the utility lost from equivalent monetary losses. Conversely, a risk-seeking individual might have a convex utility function.

The power of expected utility lies in its ability to quantify how different individuals, with varying levels of risk tolerance, might make distinct choices even when faced with the same probabilities and monetary payoffs. It provides a framework for comparing various uncertain outcomes on a consistent basis.

Hypothetical Example

Consider an investor, Sarah, who has $10,000 and is deciding between two investment options. Her utility function is (U(x) = \sqrt{x}), where (x) is her wealth. This square root function indicates that Sarah is risk-averse.

Option 1: Safe Investment
This investment guarantees a return, increasing her wealth to $10,500.

• Probability (P_1 = 1) (100% certainty)
• Outcome (x_1 = $10,500)
• Utility (U(x_1) = \sqrt{10,500} \approx 102.47)

Expected Utility (Option 1) = (1 \cdot \sqrt{10,500} \approx 102.47)

Option 2: Risky Investment
This investment has two possible outcomes: a 50% chance of increasing her wealth to $12,000, and a 50% chance of decreasing her wealth to $9,000.

• Outcome A: (x_A = $12,000) (Probability (P_A = 0.5))
• Outcome B: (x_B = $9,000) (Probability (P_B = 0.5))

Utility for Outcome A: (U(x_A) = \sqrt{12,000} \approx 109.54)
Utility for Outcome B: (U(x_B) = \sqrt{9,000} \approx 94.87)

Expected Utility (Option 2) = ((0.5 \cdot \sqrt{12,000}) + (0.5 \cdot \sqrt{9,000}))
Expected Utility (Option 2) = ((0.5 \cdot 109.54) + (0.5 \cdot 94.87))
Expected Utility (Option 2) = (54.77 + 47.435 = 102.205)

Comparing the two options, the expected utility of the Safe Investment (approximately 102.47) is slightly higher than that of the Risky Investment (approximately 102.205). Based on expected utility, Sarah, as a risk-averse investor, would choose the Safe Investment despite the Risky Investment having a higher expected value ($10,500 for Safe vs. $10,500 for Risky; Note: In this specific example the expected value is the same. Let's adjust for clarity that sometimes risky choices have higher expected value but lower expected utility due to risk aversion.). If the risky investment's expected value was slightly higher, say a 50% chance of $13,000 and 50% chance of $8,000 (Expected Value = $10,500), the utility calculation still shows the preference for the safer option.

Let's re-run the risky investment example to illustrate diminishing marginal utility more clearly where the expected value is the same but utility differs:
Option 2: Risky Investment (Adjusted)

• Outcome A: (x_A = $12,000) (Probability (P_A = 0.5))
• Outcome B: (x_B = $9,000) (Probability (P_B = 0.5))
• Expected Monetary Value for Option 2: ((0.5 \cdot $12,000) + (0.5 \cdot $9,000) = $6,000 + $4,500 = $10,500).

In this scenario, both options have the same expected monetary value of $10,500. However, Sarah's calculation of expected utility shows her preference for the less volatile, certain outcome due to her risk aversion. This demonstrates how expected utility helps individuals make decisions aligning with their subjective preferences rather than just objective monetary values.

Practical Applications

Expected utility theory finds broad application across various financial domains and beyond, providing a framework for decision-making under uncertainty.

In investment decisions, individuals and organizations use expected utility to evaluate potential gains against possible losses. Portfolio managers, for instance, employ this theory to design investment portfolios tailored to an investor's risk tolerance and financial objectives. Th³³, ³⁴is involves assigning utility values to potential returns, optimizing asset allocation to maximize satisfaction rather than just monetary returns.

F³¹, ³²or the insurance industry, expected utility is crucial for policy pricing and design. Insurers assess the likelihood of events and their financial impact on policyholders, using utility functions to determine premiums that reflect perceived risk and the diminishing marginal utility of wealth for individuals seeking protection against large losses. Fo²⁹, ³⁰r instance, paying a small, certain premium has a higher expected utility for a risk-averse individual than facing the small probability of a large, catastrophic loss.

B²⁸eyond finance, expected utility theory is applied in public policy and healthcare economics to evaluate interventions and policy options by comparing expected benefits and costs to society. It²⁶, ²⁷ aids policymakers in making choices that aim to maximize overall societal welfare under uncertain conditions.

Limitations and Criticisms

While a cornerstone of rational choice theory, expected utility theory is not without its limitations and has faced significant criticisms, particularly from the field of behavioral economics.

One primary criticism is its assumption of perfect rationality. Critics argue that the theory's assumptions, such as consistency in preferences and the ability to assign precise probabilities and utility values, often do not align with how people make decisions in the real world. Em²³, ²⁴, ²⁵pirical studies have demonstrated systematic deviations from expected utility predictions, highlighting that human choices can be influenced by factors like framing effects, emotional states, and reference points, which are not fully accounted for by the theory.

A²¹, ²² notable challenge to expected utility is the Allais Paradox, which illustrates situations where individuals' choices contradict the independence axiom—a key assumption of the theory. Anot¹⁹, ²⁰her significant critique comes from economist Matthew Rabin, who demonstrated that for any concave utility function, even very little risk aversion over modest stakes implies an absurdly high degree of risk aversion over large stakes, which is empirically unrealistic. This¹⁸ suggests that expected utility theory may be an implausible explanation for appreciable risk aversion, especially for small-to-moderate risks.

The¹⁷se limitations have led to the development of alternative descriptive models, such as prospect theory, which aim to better capture the complexities and biases observed in human decision-making under uncertainty.

¹⁵, ¹⁶Expected Utility vs. Expected Value

Expected value and expected utility are two distinct concepts used in analyzing choices under uncertainty, though they are often confused. Expected value represents the probability-weighted average of the monetary outcomes of a decision. It is a purely objective, quantitative measure of the average payoff one would expect if a decision were repeated many times. For example, if a lottery ticket costs $20 and has a 0.5% chance of winning $2,000, its expected value is $10 (0.005 * $2,000). A ra¹⁴tional agent aiming to maximize expected value would not buy this ticket, as the expected value is less than the cost.

In contrast, expected utility considers the subjective satisfaction or benefit (utility) derived from those monetary outcomes, weighted by their probabilities. Unli¹³ke expected value, expected utility accounts for an individual's attitude toward risk, such as risk aversion or risk-seeking behavior. The ¹²utility of money often exhibits diminishing marginal utility, meaning that an additional dollar provides less satisfaction to a wealthier person than to a poorer one. This¹¹ distinction explains why people might choose an option with a lower expected monetary value (like purchasing insurance) if it provides greater peace of mind or protection against significant losses, thereby maximizing their expected utility.

⁹, ¹⁰FAQs

How does expected utility account for risk?

Expected utility incorporates risk by applying a utility function to monetary outcomes. This function captures an individual's subjective value for wealth, which is typically concave for risk-averse individuals. This concavity means that larger gains provide proportionately less additional utility than smaller gains, and large losses cause a greater reduction in utility, reflecting a preference for certainty over uncertainty.

###⁸ Is expected utility always rational?
Expected utility theory is based on the assumption of rational choice theory, where individuals consistently make choices to maximize their expected utility. However, real-world decision-making often deviates from this ideal due to psychological biases and inconsistent preferences, leading to criticisms of its descriptive accuracy in behavioral economics.

###⁶, ⁷ What is the difference between utility and expected utility?
Utility refers to the subjective satisfaction or benefit derived from a specific outcome or amount of wealth. Expected utility, on the other hand, is a forward-looking concept that calculates the weighted average of the utility of all possible outcomes in an uncertain situation, with the weights being their respective probabilities.

###⁴, ⁵ Why do people buy insurance if its expected value is negative?
People buy insurance because it maximizes their expected utility, even though the expected monetary value of an insurance policy is typically negative for the buyer (the premium usually exceeds the expected payout). This³ is due to risk aversion and the diminishing marginal utility of wealth. A small, certain loss (the premium) is preferred to a small probability of a very large, financially devastating loss, which would cause a significant drop in utility.¹, ²