Pareto rationalizability by two single-peaked preferences

We study, in a finite setting, the problem of Pareto rationalizability of choice functions by means of a preference profile that is single-peaked with respect to an exogenously given linear order over the alternatives. This problem requires a new condition to be added to those that characterize Pareto rationalizability in the general domain of orders (Moulin (1985)). This new condition appeals to the existence of a central range of options such that the choice function excludes alternatives which are distant from that range.