Mixed Stackelberg strategies in continuous-kernel games

It is shown that continuous-kernel nonzero sum games with compact strategy spaces could admit both pure and mixed Stackelberg equilibrium solutions, if the cost function of each player is either nonquadratic or nonconvex in his own decision variable. In such a case, the mixed Stackelberg strategy will yield a lower average cost for the leader than the pure Stackelberg strategy. It is also verified that, if the cost function of the players are quadratic and strictly convex, then only pure Stackelberg strategies can exist.