Concave Rationalization with an Ideal Point: An Afriat Theorem and an Application to Survey Design

This paper develops an Afriat-type characterization of concave rationalization with an unknown ideal point. We show that, for each candidate peak, a finite system of linear inequalities is necessary and sufficient for the existence of a continuous concave utility with an ideal point that rationalizes choices from linear budget sets anchored at different corners of the choice space. A stronger characterization adds the requirement that supergradients at observed choices point coordinatewise toward the peak, a necessary condition for single-peaked rationalizability. The resulting peak-oriented system has a transparent geometry - budgets anchored at different corners triangulate the ideal point - and yields a nonparametric set of candidate ideal points. This provides the theoretical foundation for the Priced Survey Methodology (PSM), in which respondents complete the same survey under different linear constraints. We apply the PSM to study political preferences in a sample of French respondents.