An example showing that A-lower semi-continuity is essential for minimax continuity theorems

TL;DR

This paper demonstrates that A-lower semi-continuity is a necessary condition for certain minimax continuity theorems, showing that weaker lower semi-continuity assumptions are insufficient in general.

Contribution

The paper provides a counterexample illustrating that A-lower semi-continuity cannot be replaced by lower semi-continuity in minimax continuity results.

Findings

A-lower semi-continuity is essential for minimax continuity theorems.

Lower semi-continuity alone does not guarantee the same results.

Counterexample shows the necessity of the stronger assumption.

Abstract

Recently Feinberg et al. [arXiv:1609.03990] established results on continuity properties of minimax values and solution sets for a function of two variables depending on a parameter. Such minimax problems appear in games with perfect information, when the second player knows the move of the first one, in turn-based games, and in robust optimization. Some of the results in [arXiv:1609.03990] are proved under the assumption that the multifunction, defining the domains of the second variable, is $A$-lower semi-continuous. The $A$-lower semi-continuity property is stronger than lower semi-continuity, but in several important cases these properties coincide. This note provides an example demonstrating that in general the $A$-lower semi-continuity assumption cannot be relaxed to lower semi-continuity.