Decision theory

Beliefs alone are not sufficient for making decisions. Preferences are also needed. Decision theory points out how the beliefs and preferences should be combined when making decisions. We shall denote by U(A) the utility of proposition A. By definition, A is preferred over B if U(A) > U(B).

Decision theory can be summarised in a single rule:

$$U(A) = P(B \vert A) U(AB) + P(\neg B \vert A) U(A \neg B) \, .$$ (4)

We shall call it the rule of expected utility. In case of mutually exclusive and exhaustive propositions $B_1, B_2, \ldots$, it generalises into

$$U(A) = \sum_i P(B_i \vert A) U(AB_i) \, .$$ (5)

The significance of the rule becomes apparent if one considers A to be an action and B_i to be the possible consequences. Basically the rule indicates that the utility of A depends on the utilities of the possible consequences of A, weighted by the probabilities of the consequences.

The sum and product rules of probability theory fix the scale by which degrees of beliefs are measured to be the canonical scale of probabilities. The rule of expected utility does the same for utilities, up to linear scaling and an additive constant. This is because the beliefs have clear limits in absolute belief and disbelief while there are no absolutely worst or best possible states of the world, or if there are, the difference of their utility is probably infinitely greater than the difference between any other states of the world.