Defining Decision Theory - Behavioral Mathematics for Game AI - page 49

For example, I've heard it said that “four out of five dentists surveyed recom-
mend sugarless gum to their patients who chew gum.� That is a description of the
aggregate of the dentists surveyed. Literally, when the data was summarized, 80% of
them recommended sugarless gum. The other 20% just wanted more repeat business.
There are a few things that we can't assume based on this data. First, we cannot
make the case that dentists should recommend sugarless gum four-fifths of the time
and recommend the dental time bomb the rest of the time. It also doesn't mean that
any one dentist recommends sugarless gum 80% of the time and regular gum to the
rest of his patients. Descriptive decision theory doesn't deal with the specifics of
individual behaviors like that. However, from the description provided by this data,
we can pull a random dentist off the street, ask his opinion, and feel reasonably cer-
tain (80% so) that he is going to recommend sugarless gum to us. (Come to think
of it, is there any gum that has sugar in it anymore?)
Past Performance May Not Be an Indication
So, the problem with theory is that it's… well… theoretical . It is not really what “is�
in a concrete sense but what “seems to be� if you back up far enough to blur out all
the detail. If I can draw from a prior life of mine, when I was a musician studying
theory and composition, my mother bestowed upon me her copy of the topic
Harmony by Walter Piston. While reading about music theory is almost as dry as
reading about game AI, there was a line in the preface that caught my eye. Walter
said, “Theory must follow practice… theory is not a set of directions for composing
music. It tells us not how music will be written in the future , but how music has
been written in the past � (emphasis mine). What Mr. Piston was pointing out is that
music theory is descriptive in nature rather than prescriptive . It gives us a window
into what has become accepted practice over time rather than what should be done
by anyone writing music today. (One could make the point that today's musicians
seem to take Piston's edict the wrong way around… and, accordingly, that the
creative variety of today's music suffers in merciless homogeneity.)
The only way we can make a connection between what has been done (descriptive
decision theory) and what should be done (normative decision theory) is to assume
that what has come before had a valid, reasonable rationale. That is, the decisions
that the collection of dentists made were for a reason. We may not know what that
reason is, but, if we trust the dentists, we can decide that their collective opinion is
reasonably correct.
One problem with using these assumptions is that we may be misled by what
has been collected. In investment program commercials, we hear this as “past per-
formance may not be an indication of future results.� The same could be said for