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Learning in Bayesian Stackelberg Games With Unknown Follower's Types

Matteo Bollini, Francesco Bacchiocchi, Samuel Coutts, Matteo Castiglioni, Alberto Marchesi

TL;DR

The paper tackles online learning in Bayesian Stackelberg games where the leader lacks knowledge of follower payoffs and the type-distribution. It proves a strong impossibility result for action feedback and introduces a no-regret algorithm under type feedback that operates in epochs, learning follower-type probabilities, best-response polytopes, and pruning the leader’s feasible space. The method combines Find-Types, Find-Partition, and Prune procedures to achieve a sublinear regret bound $\tilde{O}(\sqrt{T})$ when the number of leader actions $m$ is fixed, with polynomial dependence on problem size otherwise. This work significantly broadens the practical applicability of Stackelberg-learning methods by removing strong a priori knowledge assumptions and providing rigorous regret guarantees with explicit dependence on the instance parameters. Its results have potential impact on security, contract design, and any domain where leaders must learn robust commitments against unknown follower profiles in sequential settings.

Abstract

We study online learning in Bayesian Stackelberg games, where a leader repeatedly interacts with a follower whose unknown private type is independently drawn at each round from an unknown probability distribution. The goal is to design algorithms that minimize the leader's regret with respect to always playing an optimal commitment computed with knowledge of the game. We consider, for the first time to the best of our knowledge, the most realistic case in which the leader does not know anything about the follower's types, i.e., the possible follower payoffs. This raises considerable additional challenges compared to the commonly studied case in which the payoffs of follower types are known. First, we prove a strong negative result: no-regret is unattainable under action feedback, i.e., when the leader only observes the follower's best response at the end of each round. Thus, we focus on the easier type feedback model, where the follower's type is also revealed. In such a setting, we propose a no-regret algorithm that achieves a regret of $\widetilde{O}(\sqrt{T})$, when ignoring the dependence on other parameters.

Learning in Bayesian Stackelberg Games With Unknown Follower's Types

TL;DR

The paper tackles online learning in Bayesian Stackelberg games where the leader lacks knowledge of follower payoffs and the type-distribution. It proves a strong impossibility result for action feedback and introduces a no-regret algorithm under type feedback that operates in epochs, learning follower-type probabilities, best-response polytopes, and pruning the leader’s feasible space. The method combines Find-Types, Find-Partition, and Prune procedures to achieve a sublinear regret bound when the number of leader actions is fixed, with polynomial dependence on problem size otherwise. This work significantly broadens the practical applicability of Stackelberg-learning methods by removing strong a priori knowledge assumptions and providing rigorous regret guarantees with explicit dependence on the instance parameters. Its results have potential impact on security, contract design, and any domain where leaders must learn robust commitments against unknown follower profiles in sequential settings.

Abstract

We study online learning in Bayesian Stackelberg games, where a leader repeatedly interacts with a follower whose unknown private type is independently drawn at each round from an unknown probability distribution. The goal is to design algorithms that minimize the leader's regret with respect to always playing an optimal commitment computed with knowledge of the game. We consider, for the first time to the best of our knowledge, the most realistic case in which the leader does not know anything about the follower's types, i.e., the possible follower payoffs. This raises considerable additional challenges compared to the commonly studied case in which the payoffs of follower types are known. First, we prove a strong negative result: no-regret is unattainable under action feedback, i.e., when the leader only observes the follower's best response at the end of each round. Thus, we focus on the easier type feedback model, where the follower's type is also revealed. In such a setting, we propose a no-regret algorithm that achieves a regret of , when ignoring the dependence on other parameters.
Paper Structure (24 sections, 28 theorems, 115 equations, 1 figure, 4 algorithms)

This paper contains 24 sections, 28 theorems, 115 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Original Contributions
  3. Related Works
  4. Preliminaries
  5. Bayesian Stackelberg Games
  6. Follower’s Best-Response Regions
  7. On the Representation of Numbers
  8. Online Learning in Bayesian Stackelberg Games
  9. A Negative Result With Action Feedback
  10. No-Regret With Type Feedback
  11. Structure of the Leader's Decision Space
  12. The $\texttt{Find-Types}$ Procedure
  13. The $\texttt{Find-Partition}$ Procedure
  14. The $\texttt{Prune}$ Procedure
  15. Theoretical Guarantees
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $L \in \mathbb{N}$ be the bit-complexity of the follower’s payoffs. Under action feedback, for any learning algorithm, there exists a BSG instance with constant-sized leader/follower action sets and set of follower types, and $T=\Theta(2^{2L})$ rounds, such that $R_T \ge \Omega(2^{2L})$.

Figures (1)

  • Figure 1: Each subsimplex represented in the figure is associated with a single instance of the lower bound.

Theorems & Definitions (48)

  • Theorem 3.1
  • Lemma 4.0
  • Lemma 4.0
  • Lemma 4.0
  • Lemma 4.0
  • Lemma 4.0
  • Lemma 4.0
  • Theorem 4.1
  • Theorem A.1
  • proof
  • ...and 38 more