What is stochastic dominance?

Stochastic dominance is a concept in economics and finance that relates to the comparison of different probability distributions or random variables. It is used to compare risk and return characteristics of different investment options or gambles.

There are three main types of stochastic dominance:

1. First-order stochastic dominance: A random variable X is said to first-order stochastically dominate another random variable Y if the cumulative distribution function of X is always below that of Y, and there is at least one point where X has a strictly higher probability of being realized than Y.

2. Second-order stochastic dominance: A random variable X is said to second-order stochastically dominate another random variable Y if the expected value of any increasing function of X is greater than the expected value of the same function of Y for all increasing and concave functions.

3. Third-order stochastic dominance: A random variable X is said to third-order stochastically dominate another random variable Y if the expected value of any increasing and convex function of X is greater than the expected value of the same function of Y for all increasing and convex functions.

Stochastic dominance is often used in decision-making under uncertainty, as it provides a way to rank different alternatives based on their risk-return profile. It is a key concept in the field of financial economics, particularly in the evaluation of investment portfolios and asset pricing models.