A Note on Ordinally Concave Functions

TL;DR

This paper explores properties of ordinally (weak-)concave functions, introduces a weaker notion, characterizes these functions, and provides an efficient algorithm for their maximization, with applications in economics and optimization.

Contribution

It introduces ordinal weak-concavity, characterizes these functions, and develops an efficient maximization algorithm, expanding the theoretical framework of ordinal concavity.

Findings

Characterization of ordinally weak-concave functions

Development of an efficient maximization algorithm

Analysis of duality and lexicographic composition in these functions

Abstract

The notion of ordinal concavity of utility functions has recently been considered by Hafalir, Kojima, Yenmez, and Yokote in economics while there exist earlier related works in discrete optimization and operations research. In the present note we consider functions satisfying ordinal concavity and introduce a weaker notion of ordinal weak-concavity as well. We also investigate useful behaviors of ordinally (weak-)concave functions and related choice correspondences, show a characterization of ordinally weak-concave functions, and give an efficient algorithm for maximizing ordinally concave functions. We further examine a duality in ordinally (weak-)concave functions and introduce the lexicographic composition of ordinally weak-concave functions.