## Stochastic Dominance

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# Stochastic Dominance

Stochastic dominance is based on the prevalent issue and debates as to whether a random variable can be judged as riskier than the other, irrespective of who is the judge but given that the utility functions applied have shared properties. Economists always have though and used variance and mean as measures of comparative risk. This is not accurate. This is because variance and mean can only be accurately used as risk measures if;

- The utility functions of the agents applied are quadratic
- The probability distributions are all normal distributions. However, the normal distributions are considered very restrictive.

#### Types of Stochastic Dominance

There are two rankings of the stochastic dominance, namely;

**First-order stochastic dominance**

Given X˜ A and X˜ B are the random variables, X˜ A is said to first-order stochastically dominate X˜ B if X˜ A is preferred to X˜ B by every vN-M expected utility maximizer.

In other words; Random variable A is said to have first-order stochastic dominance over random variable B if, for any outcome *x*, A has a high probability of having at the very least *x* as B does, and for some *x*, A has a higher probability of having at least *x.*

**Second-order Stochastic dominance**

If every the expected utility maximizer of risk-averse vN-M prefers X˜ A to X˜ B, then the random variable X˜ A is said to be second-order stochastically dominate X˜ B random variable.

The decision makers with concave or increasing utility functions, also known as the risk-averse expected utility maximizers, have a preference of the second-order stochastically over the dominated one.