Ordering information content using the quantile function.

The significance of the information conveyed by prices and the actions of economic agents is fundamental in various fields of economics. In 1953, Blackwell introduced a comprehensive concept of information content, stating that a random variable X holds more information than another random variable Y if, regardless of the decision problem, an observer would prefer to know X over Y. However, this Blackwell ordering doesn't effectively rank numerous relevant cases. For instance, Lehmann (1988) reveals that Blackwell's ordering surprisingly fails to rank the amount of information about a variable θ, conveyed by the set of random variables Xκ = θ + νκ, where ν follows a uniform distribution between [-1, 1].

Lehmann (1988) proposes an alternative concept of information content that surpasses Blackwell's ordering by encompassing more scenarios, including the one mentioned. Lehmann's approach is designed to resonate with economists by focusing on monotone decision problems, where a fully informed decision-maker's choice would consistently follow the underlying state variable's monotonicity. Despite its potential, Lehmann's ordering hasn't gained substantial traction among economists. Moreover, Lehmann's concept seems somewhat complex to apply at first glance.

The primary finding of this brief paper is that Lehmann's ordering is equivalent to a specific characteristic of the quantile function known as the single-crossing property. This equivalence, while straightforward, hasn't been recognized before. This equivalence significantly facilitates the practical use of Lehmann's ordering, particularly in differentiable scenarios. By utilizing the Spence-Mirrlees formulation of single-crossing, it becomes feasible to compare the information content of prices and actions without needing a full solution for prices or optimal actions.