The quickest possible introduction to Bayesian decision theory, pt. 2 : Expected Utility

More of what they should have taught you in high school

Okay. I’m sorry. The quickest possible introduction to Bayesian decision theory is ending up longer than I expected. Part 2 will itself be a two-parter.

In my last post in this series, we covered some basic probability theory, including probability functions, updating & Bayes’ theorem, and “Dutch book” arguments for the main rules of Bayesianism. Today, we move from belief to action. I hope I made a good case that degrees of belief aren’t fake, and today I hope I’ll make a good case that utilities aren’t totally fake. Of particular importance is the idea of a representation theorem, which is a theorem that says that if your preferences among choices under uncertainty satisfy certain axioms (such as transitivity), then we can come up with an assignment of utilities (real numbers) to outcomes such that you prefer one action to another if and only if it has greater expected utility. Some philosophers, then, argue that the axioms are constraints on rational choice, from which it would follow that insofar as you are a rational agent, you will be an expected utility maximizer for some utility function.

The outline for §2 is as follows. Today we’ll cover §2.1-§2.4.

§2 Expected Utility

§2.1 The formula and some simple calculations

§2.2 Preferences

§2.3 The von Neumann-Morgenstern Representation Theorem

§2.4 Money-pump arguments

§2.5 A brief overview of Savage’s representation theorem

§2.6 Problems concerning expected utility theory

Let us begin!