The Basics of the Decision Model: Acts, States, Losses and Utilities

In decision theory, there is an almost unanimously agreement on how we should view a decision procedure. I believe a simple example is the best way to set things straight. Suppose you must decide whether to go to a park, to go to the movies, or to stay home, and two things can happen: it may be sunny or it may be cloudy.

act park in a sunny day =

To make things manageable mathematically, we need to represent the consequences by numbers. Let us say that going to the park in a sunny day is the best, evaluated 10. Now we can state that

act park in a sunny day = 10

Let us say going to the park in a cloudy day is worst; you get -10. Going to the movie is fun but not very exciting and requires a lot of energy; if it is cloudy, you get 4, but if it is sunny, then you get -5. Staying home is boring either way, so it is worthy zero. Now we reduced the problem to a table with numbers:

	sunny	cloudy
park	10	-10
market	-5	4
home	0	0

Each possible act is a row in the table. So, an act determines a function from the states (sunny or cloudy) to the consequences. This is the definition of act used in Quasi-Bayesian theory; note that it has nothing to do with actual ``action'' or ``movement''. It is an statement of what could happen to you in the various states of nature.

The trick above, reducing consequences to numbers, is the whole point of utility theory. The theory gives axioms [4,9] that guarantee the existence of numbers that represent the value of consequences. The theory is of course subjectivist in that any agent has a ``personal'' scale of values. The scale of values is called the utility function, or the loss function (utility with a minus sign).

Now I hope the model is clear. The agent specifies the states of the world, and the acts are functions from the states to utility values (which measure the value of the consequences). Decision theory starts when the states and acts are defined.

A Digression: The Mathematics Behind the Model

More mathematically, the numeric utility scale is unique up to a positive affine transformation, i.e., if u(x) is the utility, then a u(x) + b is equivalent for any constant a > 0 and any constant b. The axioms of utility theory, which were initially proposed in the theory of games of von Neumann-Morgenstein [29], do not constrain the utility scale to a single function: any positive affine transformation is accepted.

To obtain this result given a finite number of consequences, three axioms suffice.

The first axiom says that every pair of acts can be compared.

The second axiom is best understood through an example. Suppose you decide that going to the park is better than going to the movie. The axiom says that you should not change your opinion if you have the same chance of receiving an extra popcorn bag both in the park or during the movie. More formally, the axioms says that if you have the chance to obtain something no matter what you choose, then this new something cannot affect your preferences.

The third axiom says that there cannot be a consequence that is infinitely better, or infinitely worse, than any given consequence. It is called the ``no-heavens, no-hell'' axiom.

Here is an important point about classical decision theory. To be able to formalize such axioms, von Neumann-Morgenstein theory invokes the use of lotteries (see one of Peter Fishburn's works for a mathematical definition [9]). A lottery is something that gives you the chance of obtaining one of a number of consequences. So, in its heart, von Neumann-Morgentein theory assume the existence of some chance-generating mechanism, where chance has the properties defined by Kolmogorov's axioms (chance is a positive measure which adds to 1). But chance does not define the behavior of the agent; chance is not a representation for any beliefs; chance is simply a tool used by von Neumann-Morgenstein axioms.

For sets of consequences with infinitely many members, the conclusions of finite utility theory can be reproduced using a fourth axiom. The particular form of this fourth axiom varies, but in general it is a statement of dominance. In words, the axiom usually says that if the consequences of an act are always better than the consequences of a second act, no matter which state of the world, then the first act should be considered better than the second act. With such a fourth axiom, the utility function is bounded [8]. It is possible, using a different set of axioms, to obtain unbounded utility [4,9], but the preference relation will not be complete for all lotteries, and the theory becomes very involved.