Suppose that there are two incumbent icecream vendors, but that there is a possibility of entry of a third vendor. Specifically, at any location on the beach a third vendor can, after observing the locations of the two incumbents, op en at a cost of $c=0.05$ and will serve all clients who are located closest to him.
Question : Find the Subgame Perfect Nash Equilibria of the game. Does there exist an equilibrium where all three firms enter?
Hint: Draw the unit square in the $(x,y)$ plane, and let each vector represent a subgame where firm 1 has chosen location $x$ and firm 2 location $y$. By symmetry, it suffices to analyze the case where $x \leq y$, since other equilibria are easily derived from the equilibria for this case (why?). Partition the region $0 \leq x \leq y$, according to the best response of the firm*. Compute (and draw) the reduced (given the best response of the third firm) best response correspondence of the incumbents.
* In a knife edge case that $x=y$, a best response does not exist. If $z$ is the location of the third firm, any $z\neq x$ is not optimal, since moving closer to $x$ increases market share but taking the location $z=x$ is also not optimal since the market is then shared with two incumbents, and moving slightly away in the right direction leads to an improvement. To avoid this technical difficulty, you may assume that $|z-x| \geq \epsilon$ for $\epsilon >0$ a small number
I don't have a clue about how I should approach this question. I already spent hours thinking about it. Could anyone please help?