363. Utility Theory

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

363.1. Why EMV is incomplete

Consider a fair coin flip: win $1$M or lose $1$M. EMV $=0$. But most people refuse this gamble — the $1$M downside is worse in subjective terms than the $1$M upside is better.

This is risk aversion: declining marginal utility of wealth.

363.2. Utility function

$𝑢:ℝ\to ℝ$ maps wealth to utility. The decision-maker maximizes:

instead of $𝐸\left[𝑊\right]={\sum }_{𝑗}{𝑝}_{𝑗}{𝑊}_{𝑗}$.

For a risk-averse decision-maker, $𝑢$ is concave: ${𝑢}^{″}\left(𝑥\right)\le 0$.

363.3. Common forms

Linear — risk-neutral: $𝑢\left(𝑥\right)=𝑥$. Reduces to EMV.

Logarithmic (Bernoulli 1738): $𝑢\left(𝑥\right)=log\left(𝑥\right)$. Strongly risk-averse for losses; resolves the St. Petersburg paradox.

Exponential: $𝑢\left(𝑥\right)=-{𝑒}^{-\frac{𝑥}{𝑅}}$ for some risk parameter $𝑅>0$. Constant absolute risk aversion (CARA).

Power: $𝑢\left(𝑥\right)=\frac{{𝑥}^{1-𝑟}-1}{1-𝑟}$ for $𝑟>0$. Constant relative risk aversion (CRRA).

Quadratic: $𝑢\left(𝑥\right)=𝑥-𝑎{𝑥}^2$. Mean-variance — used implicitly in Markowitz portfolio theory.

363.4. 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: $𝑅\approx 1$ to $5$.

363.5. Certainty equivalent

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

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

For a risk-averse DM ($𝑢$ concave): $𝑐<𝐸\left[𝑋\right]$ → $𝜋>0$. The DM is willing to give up expected return to reduce variance.

363.6. Insurance application

You face a risk of losing $𝐿$ with probability $𝑝$.

• Expected loss: $𝑝𝐿$
• Certainty equivalent of loss: $𝑐$ where $𝑢\left(𝑊-𝑐\right)=\left(1-𝑝\right)𝑢\left(𝑊\right)+𝑝𝑢\left(𝑊-𝐿\right)$
• Maximum premium you’d pay: $𝑐$ — could be much greater than $𝑝𝐿$ if you’re risk-averse

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

363.7. Calibration: lottery method

To estimate someone’s utility function:

1. Anchor: $𝑢\left({𝑊}_{\text{min}}\right)=0$, $𝑢\left({𝑊}_{\text{max}}\right)=1$
2. For intermediate $𝑊$: ask “you’re indifferent between $𝑊$ guaranteed and a 50/50 lottery between ${𝑊}_{\text{min}}$ and ${𝑊}_{\text{max}}$ — but at what $𝑊$?”
3. Result: $𝑢\left(𝑊\right)=0.5$ at the indifference point
4. Repeat to fill in the curve

Subjective and biased — but operationally usable.

363.8. See also

• EMV — risk-neutral baseline
• Decision Trees — apply utility to payoffs at leaves
• Economics: Utility — classical economic foundation
• Decision Criteria — non-probabilistic alternative