## Motivation

Suppose we would like to say that “we prefer all to well \(A\) more than bad blood \(B\)”

\begin{equation} A \succ B \end{equation}

No right or wrong answers in this statement by itself, but we can check whether or not your preferences are **inconsistent** with itself.

## von Neumann and Morgenstern Axioms

Axioms for checking if a set of preferences are rational. The axioms allow you to check if a set of decisions are Rational Preferences.

For three conditions \(A, B, C\), we have:

### completeness

either \(A \succ B\), \(A \prec B\), \(A \sim B\) (you have to like either better, or be indifferent)

### transitivity

If \(A \succeq B\), \(B \succeq C\), then \(A \succeq C\)

### continuity

If \(A \succeq C \succeq B\), then there exists some probability \(p\) such that we can form a lottery of shape \([A:p; B:1-p] \sim C\)

That is, if \(C\) is between \(A, B\), then we can create a situation where we mix the chance of \(A\) and \(B\) happening such that selecting from that situation feels equally as good as selecting from \(C\)

### independence

for \(A \succ B\), then for any \(C\) and probability \(b\) and any probability \(p\), then the lotteries \([A:p; c:1-p] \geq [B:p; C:1-p]\)

As in, if you swap out a component of a lottery with something less desirable, your new lottery should be more undesirable as well.