Stackelberg type dynamic symmetric three-players zero-sum game with a leader and two followers

We study a Stackelberg type symmetric dynamic three-players zero-sum game. One player is the leader and two players are followers. All players have symmetric payoff functions. The game is a two-stages game. In the first stage the leader determines the value of its strategic variable. In the second stage the followers determine the values of their strategic variables given the value of the leader's strategic variable. On the other hand, in the static game all players simultaneously determine the values of their strategic variables. We show that if and only if the game is fully symmetric, the Stackelberg equilibrium and the static equilibrium are equivalent.