Colonel Blotto with Battlefield Games

2025-11-09

AI Summary

This paper studies a generalized two-player zero-sum Colonel Blotto game: players allocate finite resources across multiple battlefields, where each battlefield hosts an independent (potentially heterogeneous) normal-form game whose payoff function explicitly depends on both players’ resource allocations—extending the classical Blotto model that relies solely on relative resource magnitudes. For discrete and continuous resource allocation, we formulate two global objectives: linear aggregate payoff and worst-case battlefield payoff. Our key methodological innovation is a strategic space reformulation that restores convex–concave structure, enabling the otherwise non-convex equilibrium problem to be recast as a tractable optimization problem. We prove existence of Nash equilibria and design polynomial-time algorithms for their computation. Empirical evaluation on synthetic benchmarks and real-world security adversarial scenarios demonstrates the method’s effectiveness, scalability, and practical applicability.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents

Application Category

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

Abstract

We study a class of two-player zero-sum Colonel Blotto games in which, after allocating soldiers across battlefields, players engage in (possibly distinct) normal-form games on each battlefield. Per-battlefield payoffs are parameterized by the soldier allocations. This generalizes the classical Blotto setting, where outcomes depend only on relative soldier allocations. We consider both discrete and continuous allocation models and examine two types of aggregate objectives: linear aggregation and worst-case battlefield value. For each setting, we analyze the existence and computability of Nash equilibrium. The general problem is not convex-concave, which limits the applicability of standard convex optimization techniques. However, we show that in several settings it is possible to reformulate the strategy space in a way where convex-concave structure is recovered. We evaluate the proposed methods on synthetic and real-world instances inspired by security applications, suggesting that our approaches scale well in practice.

Problem

• Generalizes Colonel Blotto with battlefield normal-form games
• Analyzes Nash equilibrium existence and computability
• Reformulates strategy space to recover convex-concave structure

Innovation

• Reformulates strategy space to recover convex-concave structure
• Analyzes Nash equilibrium for discrete and continuous allocations
• Generalizes Blotto games with battlefield-specific normal-form interactions