In Appendix A: Optimal Choice, the author nicely illustrates the framework of decision theory and its key concepts such as axioms and utility function. The following might be especially helpful for those who are against or suspicious about the fundamental tool of economics, utility maximization.
A fundamental of optimal choice theory is the distinction between feasibility and desirability. A choice is feasible if it is possible for the decision maker, that is, one of the things that she can do. An outcome is desirable if the decision maker wishes to bring it about. Typically, feasibility is considered to be a dichotomous concept, while desirability is continuous: a choice is either feasible or not, with no shades in between; by contrast, an outcome is desirable to a certain degree, and different outcomes can be ranked according to their desirability.
We typically assume that desirability is measured by a utility function u, such that the higher the utility of a choice, the better will the decision maker like it. This might appear odd, as many people do not know what functions are and almost no one can be observed walking around with a calculator and finding the alternative with the highest utility. But it turns out that very mild assumptions on choice are sufficient to determine that the decision maker behaves as if she had a utility function that she was attempting to maximize. If the number of choice is finite, the assumptions (often called axioms) are the following:
1. Completeness: for every two choices, the decision maker can say that she prefers the first to the second, the second to the first, or that she is indifferent between them.
2. Transitivity: for every three choices a, b, c, if a is at least as good as b, and b is at least good as c, then a is at least as good as c.
It turns out that these assumptions are equivalent to the claim that there exists a function u such that, for every two alternatives a and b, a is at least as good as b if and only if u(a) ≧ u(b). (...) Any other algorithm that guarantees adherence to these axioms has to be equivalent to maximization of a certain function, and therefore the decision maker might well specify the function explicitly.