On the Generic Structure and Stability of Stackelberg Equilibria

We consider a noncooperative Stackelberg game, where the two players choose their strategies within domains $$X\subseteq {{\mathbb {R}}}^m$$ X ⊆ R m and $$Y\subseteq {{\mathbb {R}}}^n$$ Y ⊆ R n . Assuming that the cost functions F, G for the two players are sufficiently smooth, we study the structure of the best reply map for the follower and the optimal strategy for the leader. Two main cases are considered: either $$X=Y=[0,1]$$ X = Y = [ 0 , 1 ] , or $$X={{\mathbb {R}}}, Y={{\mathbb {R}}}^n$$ X = R , Y = R n with $$n\ge 1$$ n ≥ 1 . Using techniques from differential geometry, including a multi-jet version of Thom’s transversality theorem, we prove that, for an open dense set of cost functions $$F\in {{\mathcal {C}}}^2$$ F ∈ C 2 and $$G\in {{\mathcal {C}}}^3$$ G ∈ C 3 , the Stackelberg equilibrium is unique and is stable w.r.t. small perturbations of the two cost functions.