Geometric constraint landscapes: Polynomial time coalition formation in conditional games

Abstract

In this paper, by introducing novel conditional games, we contribute to AI and multi-agent systems by modeling real-world coalition formation with multiple interdependent constraints that simple games cannot achieve. While simple games treat all players as identical units with a single threshold, conditional games with linear constraints create geometric structure in the solution space, transforming NP-hard problems into polynomial-time solvable ones. This enables AI systems to identify minimal winning coalitions in real-time for complex scenarios like resource-constrained influence maximization in influencer marketing, cybersecurity configuration management, and ethical AI governance frameworks. The precise boundary analysis provided by linear constraints allows multi-agent systems to navigate strategic thresholds with mathematical precision, optimizing resource allocation by identifying exactly which agents are critical for crossing success boundaries. This computational and strategic advantage, turning constraint-induced structure into navigable geometric landscapes, enables sophisticated coalition formation, predictive strategic planning, and dynamic adaptation that simple games, with their arbitrary winning coalition distributions and exponential complexity, simply cannot support in practical applications. The geometric-structure-leveraged Constraint Projection Algorithm is presented with real-world simulation applied in green influencer marketing, demonstrating how the polynomial constraint projection algorithm transforms influencer selection from a trial-and-error process into a precise, data-driven optimization.