Adding Decision Problem Makes Information More Valuable

We consider decision-making under incomplete information about an unknown state of nature. We show that a decision problem yields a higher value of information than another, uniformly across information structures, if and only if it is obtained by adding an independent, parallel decision problem.

💡 Research Summary

The paper investigates the comparative value of information (VoI) across decision problems when a decision‑maker faces incomplete information about an unknown state of nature. The authors formalize a decision problem as a utility function (U: A \times \Omega \to \mathbb{R}) where (A) is a non‑empty set of actions and (\Omega) is a finite set of states. For any belief (probability distribution) (p \in \Delta(\Omega)) the decision‑maker selects an action that maximizes the expected utility, yielding the (convex) value function (V_U(p)=\sup_{a\in A}\sum_{\omega\in\Omega}p(\omega)U(a,\omega)).

Information is modeled as a distribution (Q) over posterior beliefs, i.e., (Q\in\Delta(\Delta(\Omega))). The value of information for a given decision problem (U) and information structure (Q) is defined as the difference between the expected value under the posterior distribution and the value under the prior (the barycenter of (Q)):
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