why does a battle of the sexes game best response curve for two players in game theory have three points in common?

Question

Why does the mixed strategy best response curve shown in the image linked above have three common points and what does it signify?

Answer 1

The graph superimposes the map for the best reply of Player 1 and the map for the best reply of Player 2. Each map reports the probabilities with which Player $i$ should pick his first strategy as a function of the probability with which the opponent picks her first strategies. The three Nash equilibria for the game correspond to three intersection points: two are equilibria in pure strategies (corner points) and one is in completely mixed strategies (interior point).