Stackelberg Equilibrium Set in 2x2 Mixed Extended Games

Consider a bimatrix () mixed extended Stackelberg game. Player 1 has the payoff matrix . Player 2 has the payoff matrix . Player 1 is the leader and he moves first. Player 2 is the follower and he moves second. The leader knows ex ante (beforehand) that the follower observes his action. The set of Stackelberg equilibria (red) in a particular game is determined as the solution-of-optimization problem on the graph-of-best-response mapping (blue) of the player 2 (follower); its vertices are given at the bottom. Green points are not equilibrium, but have the same value of the cost function of the leader on the interior vertex of the set of Stackelberg equilibria.

Sliders define the elements of the matrix. First, mixed strategies of both the players and ) are used for the graphic representation of the set of Stackelberg equilibria. The values of the second strategy of the players are simply and .

Reference:

H. von Stackelberg, Marktform und Gleichgewicht (Market Structure and Equilibrium), Wien/Berlin: Springer, 1934.