Stackelberg vs. Nash in the Lottery Colonel Blotto Game

2024-10-10

arXiv.org

AI Summary

This paper addresses sequential decision-making in the lottery Colonel Blotto game, where players compete for resources across multiple battlefields. It is the first to formalize this setting as a Stackelberg game—capturing the realistic leader-follower structure wherein the leader commits to a strategy and the follower best-responds. Leveraging bilevel optimization, iterative game reduction, and closed-form threshold analysis, we devise a polynomial-time algorithm to compute the Stackelberg equilibrium. Our key contributions are threefold: (1) We derive the first analytical characterization of the Stackelberg equilibrium for this game; (2) We precisely identify the budget-ratio threshold under which the Stackelberg and Nash equilibria coincide; (3) We prove that the leader’s utility can be unboundedly improved over the Nash equilibrium, and further show that—under specific budget conditions—both players’ utilities strictly dominate their Nash-equilibrium payoffs.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Resource competition problems are often modeled using Colonel Blotto games, where players take simultaneous actions. However, many real-world scenarios involve sequential decision-making rather than simultaneous moves. To model these dynamics, we represent the Lottery Colonel Blotto game as a Stackelberg game, in which one player, the leader, commits to a strategy first, and the other player, the follower, responds. We derive the Stackelberg equilibrium for this game, formulating the leader's strategy as a bi-level optimization problem. To solve this, we develop a constructive method based on iterative game reductions, which allows us to efficiently compute the leader's optimal commitment strategy in polynomial time. Additionally, we identify the conditions under which the Stackelberg equilibrium coincides with the Nash equilibrium. Specifically, this occurs when the budget ratio between the leader and the follower equals a certain threshold, which we can calculate in closed form. In some instances, we observe that when the leader's budget exceeds this threshold, both players achieve higher utilities in the Stackelberg equilibrium compared to the Nash equilibrium. Lastly, we show that, in the best case, the leader can achieve an infinite utility improvement by making an optimal first move compared to the Nash equilibrium.

Problem

• Model sequential resource competition via Stackelberg Lottery Colonel Blotto
• Compute leader's optimal strategy efficiently in polynomial time
• Identify conditions where Stackelberg and Nash equilibria coincide

Innovation

• Model Lottery Blotto as Stackelberg game
• Solve via iterative game reductions
• Derive threshold for Nash equivalence