2.2 Risk attitudes and expected utility theory

Risk attitudes and expected utility theory provide the formal tools for analyzing how people make decisions when outcomes are uncertain. Rather than assuming everyone evaluates a risky bet the same way, these concepts explain why two rational people can look at the same gamble and make opposite choices. That difference comes down to their utility functions and how those functions curve.

Risk Attitudes and Decision-Making

Factors Influencing Risk Attitudes

A person's risk attitude describes their willingness to accept uncertainty when making decisions. Several factors shape this attitude:

• Personal preferences determine baseline risk tolerance. Some people are naturally more comfortable with uncertainty than others.
• Past experiences shift attitudes over time. Someone who lost money in a bad investment may become more cautious, while someone who profited from a risky bet may seek out similar opportunities.
• Cultural background plays a role as well. Research suggests that collectivist cultures tend toward more risk-averse group decisions, while individualistic cultures may tolerate more risk.
• Situational context matters enormously. The same person might take a risk on a $10 bet but play it safe with their retirement savings.

Utility and Risk Attitudes

Utility is the subjective value or satisfaction a person assigns to an outcome. It's not the same as the dollar amount; $100 means something very different to a billionaire than to a college student.

The shape of a person's utility function reveals their risk attitude:

• Concave utility function → risk aversion. Each additional dollar of wealth adds less utility than the last (diminishing marginal utility). Losing $100 hurts more than gaining $100 helps, so the person avoids fair gambles.
• Linear utility function → risk neutrality. Each dollar adds the same utility regardless of current wealth. This person cares only about expected monetary value.
• Convex utility function → risk seeking. Each additional dollar adds more utility than the last (increasing marginal utility). The thrill of a big win outweighs the pain of a loss, so this person gravitates toward gambles.

Risk-averse individuals tend to prefer safer options with more predictable outcomes, while risk-seeking individuals accept lower expected returns for a shot at higher rewards.

Expected Utility Theory

Normative Model for Decision-Making Under Risk

Expected utility theory (EUT) is a normative model, meaning it prescribes how a rational decision-maker should choose under uncertainty. It doesn't claim people always behave this way; it defines a standard of rational consistency.

The core idea: calculate the expected utility of each option by weighting the utility of every possible outcome by its probability, then pick the option with the highest expected utility.

Formally, for an option with outcomes $x_1, x_2, \ldots, x_n$ occurring with probabilities $p_1, p_2, \ldots, p_n$:

$EU = \sum_{i=1}^{n} p_i \cdot u(x_i)$

Notice that this formula uses $u(x_i)$, the utility of each outcome, not the raw dollar value. That distinction is what makes EUT sensitive to risk attitudes.

Both the utility assignments and the probability estimates are subjective and can vary across individuals, which is why two people using EUT can rationally reach different decisions.

Applications and Assumptions

The option with the highest expected utility is the rational choice under EUT. This framework applies broadly:

• Investment decisions: comparing portfolios with different risk-return profiles.
• Insurance choices: deciding whether the premium is worth the reduction in uncertainty. A risk-averse person may buy insurance even when the expected monetary value of going uninsured is higher.
• Gambling: evaluating whether a bet is worth taking given your utility function. A risk-averse person rejects a fair coin flip for $1,000 because the utility lost from losing exceeds the utility gained from winning.

EUT rests on several key assumptions:

• Individuals have complete and transitive preferences (they can rank all outcomes and do so consistently).
• They can assign meaningful utilities to outcomes.
• They can accurately estimate probabilities.

These assumptions are idealized. Real-world decision-making often violates them, which is why behavioral alternatives like prospect theory exist. But EUT remains the foundational benchmark.

Risk Aversion vs. Risk Seeking

Characteristics and Utility Functions

Measuring Risk Aversion

The Arrow-Pratt measure of absolute risk aversion quantifies how risk-averse a person is at a given wealth level:

$r(w) = -\frac{u''(w)}{u'(w)}$

Here, $u''(w)$ is the second derivative of the utility function (capturing curvature) and $u'(w)$ is the first derivative (marginal utility). A higher value of $r(w)$ means greater risk aversion. For a risk-neutral person, $u''(w) = 0$, so $r(w) = 0$.

Risk attitudes aren't always consistent across contexts:

• A person might be risk-averse with financial decisions but risk-seeking in recreational choices.
• The same person may be risk-averse for small stakes but risk-seeking for large stakes. Prospect theory, developed by Kahneman and Tversky, formalizes this pattern by showing that people tend to be risk-averse over gains but risk-seeking over losses.

Expected Utility Calculation

Steps to Calculate Expected Utility

1. List all possible outcomes for the decision, along with the probability of each outcome occurring. Probabilities must sum to 1.
2. Assign a utility value to each outcome using the decision-maker's utility function $u(x)$.
3. Multiply each utility by its probability to get the weighted utility contribution: $p_i \cdot u(x_i)$.
4. Sum all weighted utilities to get the expected utility: $EU = \sum p_i \cdot u(x_i)$.
5. Repeat for each available option, then compare. The option with the highest $EU$ is the rational choice.

Determining the Optimal Choice

Consider a decision between two investments:

• Investment A: payoff of $100 with probability 0.6, payoff of $0 with probability 0.4
• Investment B: payoff of $50 with probability 0.8, payoff of $0 with probability 0.2

For a risk-neutral decision-maker (where $u(x) = x$):

• $EU(A) = 0.6 \times 100 + 0.4 \times 0 = 60$
• $EU(B) = 0.8 \times 50 + 0.2 \times 0 = 40$

Investment A is optimal because $EU(A) > EU(B)$.

Now consider a risk-averse decision-maker with $u(x) = \sqrt{x}$:

• $EU(A) = 0.6 \times \sqrt{100} + 0.4 \times \sqrt{0} = 0.6 \times 10 = 6.0$
• $EU(B) = 0.8 \times \sqrt{50} + 0.2 \times \sqrt{0} = 0.8 \times 7.07 \approx 5.66$

Investment A still wins here, but the gap has narrowed. With different payoff structures, a risk-averse person could prefer the "safer" option even when it has a lower expected monetary value. That's the whole point of using utility rather than raw dollars.

Sensitivity analysis is also useful: check how the optimal choice changes if you adjust the probabilities or utility values slightly. If a small shift in one probability flips the decision, that tells you the choice is fragile and depends heavily on getting that estimate right.