Information order in monotone decision problems under uncertainty

This paper examines the robustness of Lehmann's ranking of experiments (Lehmann, 1988) for decisionmakers who are uncertainty averse à la Cerreia-Vioglio et al. (2011). We show that, assuming commitment, for all uncertainty-averse indices satisfying some mild assumptions, Lehmann's informativeness ranking is equivalent to the induced uncertainty-averse value ranking of experiments for all agents with single-crossing vNM utility indices (Theorem 1). Moreover, Lehmann's ranking can also be detected by varying the uncertainty-averse indices for a fixed finite collection of vNM utility indices (Proposition 1). Our findings suggest that Lehmann's ranking can be a useful enrichment of Blackwell's ranking for monotone decision problems even if ambiguity is present. We apply our results to problems that include social aggregation of information preferences and investment decisions.