Comparing Information in General Monotone Decision Problems

I study the value of information in monotone decision problems where the action spaces are potentially multidimensional. In these general environments, there exist sensible partial orders on actions and higher actions are optimal for higher signal realizations. As a criterion for comparing information structures, I develop a condition called monotone quasi-garbling meaning that an information structure is obtained by adding reversely monotone noise (more noisy in a lower state and less noisy in a higher state) to another. It is shown that monotone quasi-garbling is a sufficient condition for decision makers to get a higher ex-ante expected payoff. This new criterion generally permits more comparisons than the garbling condition by Blackwell (1951, 1953) and the effectiveness condition by Lehmann (1988). To illustrate, I apply the result to an optimal insurance problem and a nonlinear monopoly pricing problem.