Lower Semicontinuity of Intersections of Set-Valued Maps and Applications on Bilevel Games

We discuss about the lower semicontinuity of set-valued maps, which is a crucial concept in parametric optimization and game theory. The focus is on the intersection of set-valued maps and the preservation of lower semicontinuity under this operation. We present new results based on some minimal properties that ensure the lower semicontinuity of the intersection with other lower semicontinuous maps. Additionally, the lower semicontinuity of the intersection of an infinite family of set-valued maps is considered, which has not been studied in the literature. We provide two examples to illustrate the applications of this concept in bilevel games, including a single-leader multi-follower game with robust followers and a secured multi-leader-follower game.