Discuss the cardinal utility theory and its assumptions.

Utility can be thought of as "goodness" or "desirability"; if we are rational, we do things in order to maximize utility.

There are two basic concepts of utility:

Ordinal utility, in which utility can only be determined in relative terms; we can know that A is better than B, but not by how much. Ordinal utility can be thought of as measuring preferences--we prefer A to B.

Cardinal utility, in which utility is actually a real numerical measure of something, so that we can say that A is 5 units better than B or 7 units better than B. Cardinal utility can be thought of as measuring value--A is 5 units more valuable than B.

The core assumption of cardinal utility theory is that there is actually a cardinal utility we can measure; many economists believe that this is simply not possible, and all we can do is measure preferences.

Some define cardinal utility as "defined up to a linear transformation," but all they're really talking about is a unit of measurement. Length is "defined up to a linear transformation" in the same sense because you can measure it in meters or in feet.

The challenge then becomes figuring out what our units of measurement are, and whether we can actually measure them usefully; perhaps they are "happy seconds" or "quality-adjusted life years" (QALY, which are actually in widespread use in public health).

If we can't find a usable unit of measurement, then cardinal utility theory falls apart; if all we know is that some things are better than others but not by how much, then we can no longer make judgments under risk, because we can't say how many Bs we should be willing to give up for a given chance to get an A. The necessity of cardinal utility for rational judgment under risk is proven by the Von Neumann-Morgenstern Expected Utility Theorem.