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Quasi-Concavity, Convexity of Optimal Actions, and the Local Single-Crossing Property

Kailin Chen

TL;DR

Consider a decision problem with payoff u and Bayesian updating to a belief p; the paper proves that quasi-concavity (unimodality of u in a for each p) is equivalent to convexity of the optimal-action correspondence A*(p), with separate proofs for finite and continuous action spaces. It then shows that QCC implies a local single-crossing property after relabeling states by the ordering of a*(θ), the left-endpoint of the argmax. The results rely on Berge's maximum theorem and first-order condition analysis, linking payoff shape to the structure of optimal actions and enabling a clearer understanding of information design under uncertainty. Overall, the findings provide a structural bridge between unimodality and state-wise single-crossing, aiding theoretical and applied analyses of Bayesian decision problems and related economic applications.

Abstract

This note presents two results. First, it shows that under mild conditions, a decision problem is quasi-concave if the set of optimal actions is convex under every belief. Second, it shows that if a decision problem is quasi-concave, then it satisfies the local single crossing property after relabeling the states.

Quasi-Concavity, Convexity of Optimal Actions, and the Local Single-Crossing Property

TL;DR

Consider a decision problem with payoff u and Bayesian updating to a belief p; the paper proves that quasi-concavity (unimodality of u in a for each p) is equivalent to convexity of the optimal-action correspondence A*(p), with separate proofs for finite and continuous action spaces. It then shows that QCC implies a local single-crossing property after relabeling states by the ordering of a*(θ), the left-endpoint of the argmax. The results rely on Berge's maximum theorem and first-order condition analysis, linking payoff shape to the structure of optimal actions and enabling a clearer understanding of information design under uncertainty. Overall, the findings provide a structural bridge between unimodality and state-wise single-crossing, aiding theoretical and applied analyses of Bayesian decision problems and related economic applications.

Abstract

This note presents two results. First, it shows that under mild conditions, a decision problem is quasi-concave if the set of optimal actions is convex under every belief. Second, it shows that if a decision problem is quasi-concave, then it satisfies the local single crossing property after relabeling the states.
Paper Structure (5 sections, 4 theorems, 16 equations)

This paper contains 5 sections, 4 theorems, 16 equations.

Table of Contents

  1. Basic Setting
  2. Equivalence between Quasi-concavity and Convexity
  3. Finite Action Space
  4. Continuous Action Space
  5. Relation to Local Single Crossing Property

Key Result

Proposition 1

If eqn:38 holds, then $\{A,u\}$ is QCC if $A^*(\boldsymbol{p})$ is convex relative to $A$ for each $\boldsymbol{p}\in\widehat{\Delta}_n$.

Theorems & Definitions (8)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 1
  • proof
  • Proposition 3
  • proof