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

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.