The Value of Ambiguous Commitments in Multi-Follower Games
Natalie Collina, Rabanus Derr, Aaron Roth
TL;DR
The paper investigates ambiguous leader commitments in multi-follower Stackelberg games, showing that coupling ambiguity across followers can yield unbounded utility gains for the leader in nonzero-sum settings, while no such coupling advantage exists in zero-sum games. It introduces a hierarchical taxonomy of commitment classes (precise, full ambiguous, pure ambiguous) and develops algorithms and hardness results across these classes, including polynomial-time Solutions for 2-action leaders and NP-hardness in the general case. The work also explores correlations of ambiguity across unrelated games and correlations between follower responses and leader action sets, providing finite-time approximation methods and concrete constructions such as the Parasol-Umbrella games to illustrate when ambiguity is advantageous. Overall, the results illuminate the potential and limitations of ambiguity as a robust-commitment device in strategic interactions with multiple, heterogeneous followers and point to rich avenues for characterization and computation. The framework unifies imprecision, robustness, and commitment design, with implications for mechanism design, security, and multi-agent decision-making under uncertainty.
Abstract
We study games in which a leader makes a single commitment, and then multiple followers (each with a different utility function) respond. In particular, we study ambiguous commitment strategies in these games, in which the leader may commit to a set of mixed strategies, and ambiguity-averse followers respond to maximize their worst-case utility over the set of leader strategies. Special cases of this setting have previously been studied when there is a single follower: in these cases, it is known that the leader can increase her utility by making an ambiguous commitment if the follower is restricted to playing a pure strategy, but that no gain can be had from ambiguity if the follower may mix. We confirm that this result continues to hold in the setting of general Stackelberg games. We then develop a theory of ambiguous commitment in games with multiple followers. We begin by considering the case where the leader must make the same commitment against each follower. We establish that -- unlike the case of a single follower -- ambiguous commitment can improve the leader's utility by an unboundedly large factor, even when followers are permitted to respond with mixed strategies and even. We go on to show an advantage for the leader coupling the same commitment across all followers, even when she has the ability to make a separate commitment to each follower. In particular, there exist general sum games in which the leader can enjoy an unboundedly large advantage by coupling her ambiguous commitment across multiple followers rather than committing against each individually. In zero-sum games we show there can be no such coupling advantage. Finally, we give a polynomial time algorithm for computing the optimal leader commitment strategy in the special case in which the leader has 2 actions (and k followers may have m actions), and prove that in the general case, the problem is NP-hard.