On the equivalence of Stackelberg equilibrium and static equilibrium of symmetric multi-players zero-sum game

We study a Stackelberg type symmetric dynamic multi-players zero-sum game. One player is the leader and other 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.