Develop a behavioral characterization of genuine rational inattention

Develop a full behavioral characterization of the menu-based genuine Rational Inattention value $\mathcal{RI}(q;F,\theta)$, in the spirit of the paper's Assumptions E1--E3, to clarify the behavioral conditions generating the expected-log value and its relationship to the robust-control bounds.

Background

For a single fixed act, the paper proves that Bayes-plausible information acquisition is degenerate: information cannot improve the payoff and only adds cost. With a nontrivial menu of acts, genuine Rational Inattention becomes meaningful and has an expected-log closed form involving a stochastic choice rule and mutual information.

The paper establishes that the Exponential SOEU and robust-control values bound the genuine menu-level Rational Inattention value from below and above, respectively. It does not provide a complete behavioral axiomatization of the genuine menu-level value analogous to the entropic assumptions used for the main characterization.