5 Risk, Value, and Choice

Choices under risk and uncertainty

Choices under risk or uncertainty depend on the possible outcomes, how likely those outcomes are, and how the decision-maker values them. Risk usually means probabilities are known or can be estimated. By contrast, uncertainty often means the probabilities themselves are unclear or unavailable, so calculations that assume precise probabilities may be misleading.

Expected utility provides a formal way to compare options when their outcomes and probabilities can be specified. It evaluates each outcome by its utility, then weights that utility by the outcome’s probability:

Here, $A$ is an option, $x_i$ is an outcome, $p_i$ is its probability, and $u(x_i)$ is its utility. The theory recommends choosing the option with the highest expected utility.

Utility and expected value

Utility represents how desirable an outcome is to the decision-maker; it is not necessarily the outcome’s monetary amount. The same dollar loss may matter more to someone with little savings than to someone wealthy.

For example, compare a guaranteed $\$40$ with a gamble offering a $\frac{1}{2}$ chance of $\$100$ and a $\frac{1}{2}$ chance of $\$0$. The gamble’s expected monetary value is:

That expected monetary value does not automatically make the gamble preferable under expected utility. If the person’s utility for money increases at a diminishing rate, the certain $\$40$ may have greater expected utility.

Expected utility is a powerful standard for coherent choices when probabilities and outcomes can be specified. It is not, by itself, a complete description of how people actually choose: people may have difficulty estimating probabilities or judging how much an outcome matters.

Prospect theory and reference points

Prospect theory describes some systematic patterns in risky choices that expected utility, used as a descriptive model, does not readily capture. People commonly assess outcomes as gains or losses relative to a reference point, such as their current situation or what they expected. Prospect theory also allows probabilities to be psychologically weighted rather than treated as their stated numerical values.

A typical prospect-theory value curve has several features:

• It is often concave for gains, encouraging risk aversion. For example, someone may prefer a sure gain to a gamble with a higher average payoff.

• It is often convex for losses, which can encourage risk seeking. For example, someone may gamble to avoid accepting a sure loss.

• Near the reference point, it is steeper for losses than for comparable gains, reflecting loss aversion.

These are common patterns, not rules that predict every person’s choice in every situation. How an option is framed and where the reference point is set can change whether it feels like a gain or a loss.

Loss aversion and framing

Loss aversion is the tendency for a loss to feel more significant than an equal-sized gain. A person may reject a coin toss that offers a chance to win $\$100$ or lose $\$100$, even though its expected monetary value is $\$0$. The refusal can reflect the disproportionate impact of the possible loss, rather than a mathematical error about the odds.

Framing can also shift choices. Someone may be cautious when a choice is described as securing a gain, yet more willing to gamble when the same choice is described as avoiding a loss. Prospect theory helps explain this pattern by combining reference-dependent value with different responses to gains and losses.

Comparing the approaches

The two approaches address different questions about choice:

• What is evaluated? Expected utility evaluates the utility of outcomes, typically in terms of final states. Prospect theory evaluates gains and losses relative to a reference point.

• How are probabilities handled? Expected utility weights outcomes by their stated or believed probabilities. Prospect theory allows probabilities to be transformed into decision weights.

• What is the main use? Expected utility is a normative framework for comparing risky options. Prospect theory is a descriptive account of common departures from that framework.

• What is the central insight? Expected utility recommends the option with the greatest expected utility. Prospect theory highlights how framing, reference points, and loss aversion can shape choices.

When probabilities are unclear, neither a simple expected-value calculation nor a model built on exact probabilities should be treated as more precise than the available evidence. A practical comparison is to identify the possible outcomes, estimate probabilities only as confidently as the evidence permits, state the reference point, and check whether the description emphasizes gains or losses.