Unlocking Decision-Making: How Utility Theory and Risk Evaluation Guide Choices

1. Utility Theory and Risk Evaluation

Every decision is made by a decision‑maker whose experience, intelligence, courage and judgment inevitably affect the choice of a plan. The decision‑maker’s attitude toward risk is also crucial; a conservative decision‑maker and a risk‑seeking one will often choose very different options for the same problem. Moreover, the same amount of money can have different "value" for different economic agents. A loss of tens of thousands of yuan may be trivial to a large‑enterprise manager but catastrophic to a small‑enterprise manager. Even for the same agent, the perceived value of a monetary amount can vary across time and circumstances.

For example, a person may face a choice: receive a guaranteed 25 yuan, or gamble with a coin toss—heads yields 150 yuan, tails requires paying 50 yuan. Most people would take the safe 25 yuan, though some might gamble. If the gamble were repeated many times (10, 100, or more), the expected‑value rule would favor the gamble because its expected payoff is 150×0.5 + (‑50)×0.5 = 50 yuan.

When only a single opportunity exists, such a gamble does not match typical risk preferences. This illustrates that the same monetary amount can carry different utility values in different contexts. The utility value of a monetary amount reflects the decision‑maker’s subjective valuation, while "utility" denotes the decision‑maker’s interest, feeling, and trade‑off regarding expected gains and losses. Utility captures how personal values manifest in decision activities and represents the decision‑maker’s attitude toward risk. The following sections introduce utility theory, including the definition of utility functions, their shapes, and applications in risk‑based decisions.

2. Definition and Composition of Utility Functions

In economics, utility refers to the ability of a good or service to satisfy a person’s desires or needs. Whether a good has utility and how much depends on how well it meets those desires. Utility varies across individuals, time, and place; the same good may provide different utility to different consumers under different circumstances. Thus, utility is a concept used to describe the degree to which a product or service fulfills consumer needs, primarily for consumer‑behavior analysis. In decision theory, we similarly need to discuss how alternative outcomes satisfy the decision‑maker’s wishes and preferences, which leads to the introduction of utility and its measurement.

2.1 Definition of Utility Function

Assume a decision problem has several possible outcome values. Based on the decision‑maker’s subjective wishes and value orientation, each outcome has a different degree of importance. The variable that reflects the magnitude of this importance for the decision‑maker is called utility, denoted by u.

In decision theory, utility is both a concept—reflecting how well an outcome satisfies the decision‑maker’s wishes—and a measurable quantity that can be quantified and used as the basis for decision analysis.

2.2 Types of Utility Functions

Because utility functions depend on the decision‑maker’s risk attitude, they can take different forms.

1. Linear Utility Function

A linear utility function shows a proportional relationship between monetary amount and utility. The decision‑maker is risk‑neutral, using expected profit as the sole criterion without needing a utility function.

2. Conservative Utility Function

A conservative utility function increases with monetary amount but at a decreasing rate. The decision‑maker reacts slowly to gains but is highly sensitive to losses, preferring to avoid risk. The curve is concave (upper‑convex), indicating strong risk aversion.

3. Risk‑Seeking Utility Function

A risk‑seeking utility function also rises with monetary amount, but the increase accelerates. The decision‑maker pursues large gains and is relatively indifferent to losses. The curve is convex (upper‑concave), reflecting a strong preference for bold, high‑risk choices.

4. Aspirational Utility Function

An aspirational utility function shows an initially convex (risk‑seeking) segment for small amounts, followed by a concave (risk‑averse) segment after a turning point (c, h). Below the turning point the decision‑maker prefers risky actions; above it, they adopt a cautious strategy.

2.3 Determining the Utility Curve

Utility can be expressed by a value u ranging between 0 and 1. In a decision problem, the maximum possible gain is assigned u = 1, and the worst possible loss is assigned u = 0, so 0 ≤ u ≤ 1. Plotting monetary outcomes on the horizontal axis and utility values on the vertical axis yields the decision‑maker’s utility curve, which varies from person to person. The utility curve can be identified using the von Neumann‑Morgenstern (VNM) psychological‑test method, developed in 1944 by John von Neumann and Oskar Morgenstern, also known as the standard measurement method.

A decision‑maker faces a project with a maximum possible profit of 200,000 yuan or a maximum loss of 100,000 yuan. Determine the decision‑maker’s utility curve.

Step 1: Set the utility of the best outcome u(200 k) = 1 and the worst outcome u(‑100 k) = 0.

Step 2: Present two options: (a) a 50 % chance of gaining 200 k and a 50 % chance of losing 100 k; (b) a certain gain of 50 k (the expected value of option a).

If the decision‑maker chooses option (b), it indicates that the utility of (b) exceeds that of (a), and the test proceeds.

Step 3: Replace the certain 50 k in option (b) with 20 k and ask the decision‑maker to choose again. If they still prefer the certain gain, the utility of the certain gain remains higher.

Step 4: Ask the decision‑maker what they would do if refusing option (a) required paying 10 k. If they now prefer option (a) to avoid the payment, it shows that paying 10 k has lower utility than option (a).

Repeating such psychological tests narrows down the utility values. Eventually the decision‑maker may become indifferent between paying nothing and choosing option (a), implying that the utility of a monetary amount of 0 equals the utility of option (a), which is 0.5 (0.5 × 1 + 0.5 × 0). This process can be applied across the range 0–200 k and –100 k–0 to construct the full utility curve.

Reference

Guo Wenqiang, Sun Shixun, Guo Lifu. Decision Theory and Methods (3rd Edition). 2020. Beijing: Higher Education Press.