Pure or Unstable: A Generic Dichotomy for Strong Stackelberg Commitments
We study the robustness of the Strong Stackelberg Equilibrium (SSE) in finite leader--follower games when the follower's best-response correspondence is set-valued. While optimistic tie-breaking (in the leader's favor) is commonly adopted, it can hinge on knife-edge indifferences. We formalize a stability notion: an SSE is unstable if, at the leader's committed strategy, the follower has an alternative best response that strictly reduces the leader's payoff. Our main results establish a sharp generic dichotomy. Fixing the follower's utility and sampling the leader's utility from any continuous distribution, with probability one the optimal Stackelberg commitment is unique and is either (i) pure, or (ii) mixed and unstable. When both players' utilities are sampled generically, this strengthens to: with probability one, the unique optimal commitment is either pure and stable or mixed and unstable. These theorems complement the classic generic-value result of von Stengel and Zamir by showing that even when optimistic and pessimistic leader values coincide generically, the strategy-level SSE prediction is generically fragile whenever optimality requires genuine randomization. We further apply this perspective to Stackelberg satisfaction games, disproving a conjecture from prior work via counterexamples and identifying conditions under which it nonetheless holds.