Existence of share equilibrium in symmetric local public good economies

Share equilibrium was introduced in van den Nouweland and Wooders (2011) as an extension of ratio equilibrium to local public good economies. In that paper, we took an axiomatic approach to motivate share equilibrium. In the current paper we consider questions related to the existence of share equilibrium and we derive a necessary and sufficient condition for the existence of share equilibrium in symmetric economies. Along the way, we develop a deeper understanding of the possible variation in share equilibrium by considering when symmetric players necessarily have the same share indices in equilibrium.