## Information in Games and Decision-Making

*10 Apr 2009 17:40*

The sense in which "information" is used in decision-theory and game-theory,
and so in economics, seems to be quite different than the way it's used in
information theory as such. In the latter, "information" is a numerical
property of a random variable, or a relationship between random variables,
related to coding or forecasting --- how much could the value of one variable
be used to shorten the encoding of another, or tighten the best-achievable
prediction interval for it. In decisions, information is used in a sense
closer to what someone knows --- which, of several possible alternatives, does
the state of the world fall in to? This seems to be most generally formalized
not in terms of entropy or related quantities, but rather in terms of a sigma
algebra. (I am *not* going to explain sigma algebras here.) That is,
the agent is taken as conditioning the value of all random variables on some
sigma algebra or another, and if the sigma algebra increases, then the agent
knows more. Thus the agent knows exactly the value of any function which is
measurable w.r.t. that algebra, and has a certain conditional distribution for
the others. Since not all algebras are comparable, it may not be possible to
say which of two agents knows more.

All well and good, but I wonder what the relationships are between this sort
of algebraic information and entropic information. After all, quantities like
the Kullback divergence (relative entropy) play an important role in problems
like hypothesis testing, which is equally a decision-theoretic problem. I'd
also like to know what we could intelligbly say about the *value* of
information in a decision problem. Given a change between one conditioning
algebra and the other, we can imagine computing the relative entropy of any
given random variable. Could we somehow bound this over all variables, and use
that to give a more quantitative idea of *how much* information an agent
has acquired? Again, a natural way to talk about the value of information, in
a decision problem, would be to examine the distribution of future rewards
conditional on the informational algebra. Could one show that there is always
some strategy such that the expected reward is non-decreasing as the algebra
grows? Could the change in expected reward be related to the KL divergence of
the reward distribution?

These have the sound of questions people worked out the answers to a long time ago...

- To read:
- Larry Samuelson, "Modeling Knowledge in Economic Analysis",
Journal of Economic Literature
**62**(2004): 367--403