The Value of Ambiguous Commitments in Multi-Follower Games (2409.05608v1)

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.

• The paper demonstrates that ambiguous commitments in Stackelberg games can yield superior outcomes compared to precise strategies through a novel model.
• It introduces distinct classes of ambiguity and develops algorithms that expose NP-hard challenges in computing optimal commitment strategies.
• The findings emphasize that ambiguity helps secure higher payoffs and offers robust decision-making tools under uncertain conditions.

Commitments with Ambiguity

In the given academic paper, the authors investigate the concept of "ambiguous commitments" within the context of Stackelberg games with multiple followers. This essay will provide an expert overview, elucidating the formal model, key results, and implications of this research.

Overview and Modeling

The paper extends the classical Stackelberg game framework by introducing ambiguity into the leader's strategy. In a classical Stackelberg game, a leader commits to a strategy first, and the followers subsequently choose their best responses. However, in this work, the leader's strategy is not a single distribution but a set of distributions, introducing a layer of uncertainty about the leader's actions.

The authors define several classes of ambiguous commitments:

• Precise Multi-follower Stackelberg: Classical model where the leader commits to a precise mixed strategy.
• Full Ambiguous Stackelberg: The leader's strategy is any non-singleton set of distributions.
• Pure Ambiguous Stackelberg: The leader's strategy is a specific non-singleton set of distributions.
• Ambiguous Stackelberg: Any set of distributions that introduces ambiguity.

Given these categories, the concept of "ambiguous Stackelberg equilibria" is explored, where followers respond to the set of distributions in a maximin fashion, considering the worst-case scenario.

Algorithms and Hardness

The paper details algorithms for various classes of ambiguous Stackelberg games. Each commitment class has a corresponding algorithm designed to compute the optimal strategy within that class, with specific focus on:

• Optimal ambiguous sets which provide the best guaranteed worst-case value.
• The difficulty and computational complexity of finding these sets.

The computational hardness of finding the optimal ambiguous commitment is a central theme. The authors demonstrate that determining the optimal pure ambiguous commitment is NP-hard, underpinning the computational challenges inherent in these models.

Key Results

The paper presents several important numerical and theoretical results:

1. Comparative Advantages: It is shown that full ambiguity can be strictly better than the precise multi-follower Stackelberg commitment. Similarly, pure ambiguity can be better than full ambiguity, and further, general ambiguity can outperform pure ambiguity.
2. Correlating Ambiguity Across Games: The research finds that correlating ambiguous strategies across unrelated games can be beneficial. Specifically, it is discussed whether ambiguity can lead to better cumulative payoffs compared to classical strategies.
3. Ambiguity Gap: The concept of the "ambiguity gap" is introduced, quantifying the advantage gained by a leader employing ambiguous strategies over precise ones. This gap highlights scenarios where ambiguity provides measurable benefits.

Practical and Theoretical Implications

Practically, this research provides insights into strategic decision-making in environments where uncertainty is inevitable. For fields such as economics, military strategy, and competitive business environments, where leaders must make decisions under uncertainty and ambiguity, these models offer robust strategies that can provide better outcomes than traditional precise commitments.

Theoretically, the work challenges the classical notion of commitment in game theory by introducing and formalizing the benefits of ambiguity. The findings align with existing literature on robust decision-making under uncertainty and extend it by providing a grounded algorithmic and computational perspective.

Future Directions

The authors suggest several avenues for future research:

1. Characterizing Games with Ambiguity Advantage: Further research is needed to understand which types of games and scenarios benefit the most from ambiguous commitments.
2. Algorithmic Improvements: Developing more efficient algorithms for computing optimal ambiguous strategies to address the NP-hardness issue.
3. Experimental Validation: Applying these theoretical models to real-world scenarios to validate their practical utility and robustness.

By establishing a formal framework for ambiguous commitments in Stackelberg games, this research paves the way for more nuanced and effective strategies in decision-making processes where uncertainty is a significant factor. The implications extend beyond theoretical interest, offering practical tools and insights for complex strategic environments.