Houjun Liu

rational preference


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:


universal comparability

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


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


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 \(C\)


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.