Algebraic Properties of Blackwell's Order and A Cardinal Measure of Informativeness

TL;DR

This paper introduces a new cardinal measure of experiment informativeness based on algebraic properties of Blackwell's order, which is invariant, complete, and computationally simple, extending Blackwell's framework for economic applications.

Contribution

It establishes a translation invariance property of Blackwell's order, defines a new measure of informativeness using the infinity norm, and shows this measure extends Blackwell's order with desirable properties.

Findings

The measure coincides with Blackwell's order when applicable.

Experiment closeness to perfect information is captured by the inf-norm distance.

The measure is prior-independent and computationally simple.

Abstract

I establish a translation invariance property of the Blackwell order over experiments, show that garbling experiments bring them closer together, and use these facts to define a cardinal measure of informativeness. Experiment $A$ is inf-norm more informative (INMI) than experiment $B$ if the infinity norm of the difference between a perfectly informative structure and $A$ is less than the corresponding difference for $B$. The better experiment is "closer" to the fully revealing experiment; distance from the identity matrix is interpreted as a measure of informativeness. This measure coincides with Blackwell's order whenever possible, is complete, order invariant, and prior-independent, making it an attractive and computationally simple extension of the Blackwell order to economic contexts.