Vertex π-Floodlights

How many $\pi$-floodlights are always sufficient to illuminate any polygon of $n$ vertices, with at most one floodlight placed at each vertex? An $\alpha$-floodlight is a light of aperture $\alpha$. (We consider here \textquotedblleft{}inward-facing\textquotedblright{} floodlights, whose defining halfspace lies inside the polygon, locally in the neighborhood of the vertex. Other models of the problem allow general orientations of floodlights or restricted orientations (e.g., \textquotedblleft{}edge-aligned\textquotedblright{}).)