On the Maximization of Menu Dependent Interval Orders

We study the behavior of a decision maker who prefers alternative x to alternative y in menu A if the utility of x exceeds that of y by at least a threshold associated with y and A. Hence the decision maker's preferences are given by menudependent interval orders. In every menu, her choice set comprises of undominated alternatives according to this preference. We axiomatize this broad model when thresholds are monotone, i.e., at least as large in larger menus. We also obtain novel characterizations in two special cases that have appeared in the literature: the maximization of a fixed interval order where the thresholds depend on the alternative and not on the menu, and the maximization of monotone semiorders where the thresholds are independent of the alternatives but monotonic in menus