[Paper Review] Be a Leader or Become a Follower: The Strategy to Commit to with Multiple Leaders (Extended Version)

Matteo Castiglioni, Alberto Marchesi+1 more|arXiv (Cornell University)|May 30, 2019

Complex Systems and Decision MakingDecision Sciences32 references3 citations

TL;DR

This paper introduces a novel framework for multi-leader Stackelberg games where leaders first negotiate a correlated strategy commitment via a sequential agreement stage, choosing to lead or defect to follower status. It proposes three solution concepts—SCE, SCE-PA, and SCE-PAPE—guaranteeing existence and enabling efficient computation via a stability oracle, with polynomial-time solvability for compact games like anonymous, symmetric, and bounded-treewidth graphical games.

ABSTRACT

We study the problem of computing correlated strategies to commit to in games with multiple leaders and followers. To the best of our knowledge, this problem is widely unexplored so far, as the majority of the works in the literature focus on games with a single leader and one or more followers. The fundamental ingredient of our model is that a leader can decide whether to participate in the commitment or to defect from it by taking on the role of follower. This introduces a preliminary stage where, before the underlying game is played, the leaders make their decisions to reach an agreement on the correlated strategy to commit to. We distinguish three solution concepts on the basis of the constraints that they enforce on the agreement reached by the leaders. Then, we provide a comprehensive study of the properties of our solution concepts, in terms of existence, relation with other solution concepts, and computational complexity.

Motivation & Objective

• To address the lack of theoretical and computational frameworks for Stackelberg games with multiple leaders and followers.
• To model a preliminary agreement stage where leaders decide whether to commit to a correlated strategy or defect to follower status.
• To define and analyze three solution concepts—SCE, SCE-PA, and SCE-PAPE—based on stability, perfect stability, and efficiency properties.
• To establish the existence and computational tractability of these solution concepts in general and compact game classes.
• To design a general computational framework relying on a game-independent stability oracle for scalable solution computation.

Proposed method

• Models the agreement stage as a sequential game where leaders decide in turn whether to commit or defect, forming a factorial-sized game tree.
• Introduces three solution concepts: SCE (stability and efficiency), SCE-PA (perfect stability and efficiency), and SCE-PAPE (perfect stability and perfect efficiency).
• Uses a stability oracle to verify whether leaders have incentive to defect, enabling iterative computation of optimal correlated strategies.
• Reduces the exponential state space by focusing only on the set of defecting leaders and the last defector, achieving exponential compression.
• Applies the ellipsoid method to solve the separation problem in polynomial time for compact games, linking it to optimal correlated equilibrium computation.
• Leverages existing algorithms for optimal correlated equilibria (e.g., Jiang and Leyton-Brown, 2011) to construct a polynomial-time stability oracle for classes like anonymous, symmetric, and bounded-treewidth games.

Experimental results

Research questions

• RQ1 Can a correlated strategy commitment be meaningfully extended to games with multiple leaders who can choose to lead or become followers?
• RQ2 What solution concepts ensure that leaders have no incentive to defect from a proposed correlated commitment?
• RQ3 How can the computational complexity of finding optimal commitments be reduced in multi-leader settings?
• RQ4 For which classes of games can the stability oracle be computed in polynomial time?
• RQ5 Can the state space of agreement outcomes be compressed without losing optimality or stability guarantees?

Key findings

• SCEs and SCE-PAs are guaranteed to exist in any finite game, ensuring theoretical robustness of the proposed solution concepts.
• The number of agreement states can be reduced from factorial to exponential in the number of leaders by focusing on the set of defectors and the last defector.
• The stability oracle can be implemented in polynomial time for compact games, including anonymous, symmetric, and bounded-treewidth graphical and polymatrix games.
• The optimal correlated strategy to commit to under SCE-PA can be computed with O(|L|2^{|L|-1} + 1) queries to the stability oracle.
• The stability oracle is tightly connected to the problem of computing optimal correlated equilibria, enabling reuse of existing efficient algorithms.
• The framework supports general games and extends beyond security-specific models, offering a game-theoretically grounded and computationally scalable solution.