360. Utility Theory

A framework for decisions under risk aversion. Instead of maximizing expected payoff (EMV), maximize expected utility — where the utility function encodes the decision-maker’s preferences over wealth.

360.1. Why EMV is incomplete

Consider a fair coin flip: win M or lose M. EMV . But most people refuse this gamble — the M downside is worse in subjective terms than the M upside is better.

This is risk aversion: declining marginal utility of wealth.

360.2. Utility function

maps wealth to utility. The decision-maker maximizes:

instead of .

For a risk-averse decision-maker, is concave: .

360.3. Common forms

Linear — risk-neutral: . Reduces to EMV.

Logarithmic (Bernoulli 1738): . Strongly risk-averse for losses; resolves the St. Petersburg paradox.

Exponential: for some risk parameter . Constant absolute risk aversion (CARA).

Power: for . Constant relative risk aversion (CRRA).

Quadratic: . Mean-variance — used implicitly in Markowitz portfolio theory.

360.4. Risk aversion measures

Arrow-Pratt absolute risk aversion:

Arrow-Pratt relative risk aversion:

Used to parameterize and measure risk aversion empirically. Typical values for stock-market investors: to .

360.5. Certainty equivalent

The certainty equivalent (CE) of a gamble is the certain amount with the same utility:

The risk premium is the amount the decision-maker would pay to avoid the risk:

For a risk-averse DM ( concave): → . The DM is willing to give up expected return to reduce variance.

360.6. Insurance application

You face a risk of losing with probability .

This is why insurance exists — risk-averse buyers transfer risk to risk-pooling insurers.

360.7. Calibration: lottery method

To estimate someone’s utility function:

  1. Anchor: ,
  2. For intermediate : ask “you’re indifferent between guaranteed and a 50/50 lottery between and — but at what ?”
  3. Result: at the indifference point
  4. Repeat to fill in the curve

Subjective and biased — but operationally usable.

360.8. See also