Houjun Liu

utility theory

utility theory is a set of theories that deals with rational decision making through maximizing the expected utility.

utility theory can be leveraged to choose the right actions in the observe-act cycle in a graphical network via decision networks

additional information

never have a utility function that’s infinite

If something has infinite utility, doing two of the good things is the same as doing one good thing, which is wrong.

Say going to a Taylor concert has \(+\infty\) utility. Then, you would be indifferent to the difference between Taylor + Harry vs. Taylor only. However, the former case clearly has higher utility as long as Harry concert doesn’t have negative utility.

utility elicitation

see utility elicitation

expected utility

expected utility is the utility we expect from taking an action \(a\) at a state \(o\). To compute it based on transition probabilities:

\begin{equation} EU(a|o) = \sum_{s’} p(s’ | a,o) U(s’) \end{equation}

the expected utility of taking some action \(a\) at an observation \(o\) is the probability of any given next state \(s’\) happening times the utility of being in that state \(U(s’)\).

See also expected utility of wealth.

maximum expected utility principle

MEU states that a rational agent should choose an action which maximizes expected utility. That is,

\begin{equation} a^{*} = \arg\max_{a} EU(a|o) \end{equation}

Notably, this is not always the best action. This action maximizes utility NOT outcome.

utility of Rational Preference

For rational values, for two situations, \(A, B\), we have, with utility function \(U\):

\begin{equation} U(A) > U(B) \iff A \succ B \end{equation}

\begin{equation} U(A) = U(B) \iff A \sim B \end{equation}

and this \(U\) is unique up to the same affine transformation

risk aversion

see risk aversion

common utility functions

see utility function