Learning in Conjectural Stackelberg Games

We extend the formalism of Conjectural Variations games to Stackelberg games involving multiple leaders and a single follower. To solve these nonconvex games, a common assumption is that the leaders compute their strategies having perfect knowledge of the follower's best response. However, in practice, the leaders may have little to no knowledge about the other players' reactions. To deal with this lack of knowledge, we assume that each leader can form conjectures about the other players' best responses, and update its strategy relying on these conjectures. Our contributions are twofold: (i) On the theoretical side, we introduce the concept of Conjectural Stackelberg Equilibrium -- keeping our formalism conjecture agnostic -- with Stackelberg Equilibrium being a refinement of it. (ii) On the algorithmic side, we introduce a two-stage algorithm with guarantees of convergence, which allows the leaders to first learn conjectures on a training data set, and then update their strategies. Theoretical results are illustrated numerically.