Maxitive functions with respect to general orders
M. Kupper, J. M. Zapata
TL;DR
This work generalizes maxitive aggregation from the classical pointwise order to general preorders, providing a unifying representation framework for monotone, countably maxitive functionals via a penalty function $\alpha$ and a duality with sublevel sets. The main theorem shows that, on a countably complete preorder induced by a Polish-indexed family $\Phi$, a maxitive functional $\psi$ admits $\psi(x) = \inf\{s : \sup_{\phi\in\Phi} (\phi(x) - \alpha(s,\phi)) \le 0\}$, with countable-stability of $\alpha$ ensuring the representation. The paper then specializes the result to several stochastic orders—usual, increasing convex, convex, increasing concave, and dispersive—deriving order-specific representations in terms of quantile, integrated-quantile, and dispersion measures, and highlighting connections to risk measures such as Value-at-Risk and Expected Shortfall. Overall, the framework extends maxitivity to a broad class of decision-theoretic and probabilistic orders, enabling robust worst/best-case analysis beyond the pointwise setting and providing new tools for risk aggregation and decision under uncertainty.
Abstract
In decision-making, maxitive functions are used for worst-case and best-case evaluations. Maxitivity gives rise to a rich structure that is well-studied in the context of the pointwise order. In this article, we investigate maxitivity with respect to general preorders and provide a representation theorem for such functions. The results are illustrated for different stochastic orders in the literature, including the usual stochastic order, the increasing convex/concave order, and the dispersive order.