Multiplayer General Lotto game

2024-01-26

AI Summary

This paper investigates the existence, uniqueness, and structural properties of Nash equilibria in generalized multi-battlefield, multi-player Lotto games under general asymmetric settings—encompassing both heterogeneous battlefield valuations and asymmetric budget constraints. Methodologically, it employs game-theoretic modeling with probabilistic resource allocation strategies, supported by support-set analysis, existence proofs under expectation constraints, and derivation of upper bounds on the expected number of contested battlefields. The work establishes, for the first time, the existence of Nash equilibria in fully asymmetric generalized Lotto games; characterizes endpoint consistency of equilibrium supports in single-battlefield subgames and shows that the smallest positive support value scales inversely with player budgets; demonstrates pervasive non-uniqueness in multi-battlefield equilibria and constructs explicit counterexamples; and, in the symmetric case, derives closed-form equilibrium solutions while establishing tight upper bounds on support size. These results collectively advance the theoretical foundations of generalized Lotto games.

Technology Category

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

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, we investigate the multiplayer General Lotto game across multiple battlefields, a significant variant of the Colonel Blotto game. In this version, each player employs a probability distribution for resource allocation, ensuring that their expected expenditure does not exceed their budget. We first establish the existence of the Nash equilibrium in a general setting, where players' budgets are asymmetric and the values of the battlefields are heterogeneous and asymmetric among players. Next, we provide a detailed characterization of the Nash equilibrium for multiple players on a single battlefield. In this characterization, we observe that the upper endpoints of the supports of players' equilibrium strategies coincide, and that the minimum value of a player's support above zero inversely correlates with his budget. We demonstrate the uniqueness of Nash equilibrium over a single battlefield in some scenarios. In the multi-battlefield setting, we prove that there is an upper bound on the average number of battlefields each player participates in. Additionally, we provide an example demonstrating the non-uniqueness of the Nash equilibrium in the context of multiple battlefields with multiple players. Finally, we present a solution for the Nash equilibrium in a symmetric case.

Problem

• Establishes Nash equilibrium existence in asymmetric multiplayer General Lotto games
• Characterizes equilibrium strategies for multiple players on single battlefields
• Proves bounds on battlefield participation in multi-battlefield scenarios

Innovation

• Probability distribution for resource allocation
• Nash equilibrium in asymmetric multiplayer setting
• Upper bound on battlefield participation rate