Center for Economic Research, Department of Economics, University of Minnesota
Although its specification in economic models with uncertainty is
critical to the results obtained, agents' information has not been defined
in a way allowing isolation of its effects on other variables. In addition,
information needs to be allowed to be optimally chosen by agents
when, for example, endogenous shifts in information are a major part of
Both problems require a space of information with some appropriate
mathematical structure depending only on the uncertainty in the model.
Given a probability space, the space of information is the set of all sub-
fields of events. The structure defined here is a metric derived from
the topology defined to be the weakest one such that for any integrable
random variable, the function mapping information to the expected value
of the random variable conditional on that information is continuous.
This metric is shown to be complete and separable with two additional useful
properties. First, set-theoretic convergent (monotone) sequences of information
converge, so martingale convergence is modelled by this metric.
Second, the set of finite partitions of the state space is dense, so any
information can be finitely approximated. The topology derived from this
metric is more tractible than the one used by Allen (1982). A generalization
of the topology is provided to handle special cases such as information
consisting of state plus noise.
Cotter, K.D., (1983), "An Information Metric of Similarity of Expectations", Discussion Paper No. 186, Center for Economic Research, Department of Economics, University of Minnesota.
Cotter, Kevin D..
An Information Metric of Similarity of Expectations.
Center for Economic Research, Department of Economics, University of Minnesota.
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