Parametric Open Source Games

Traditional open-source game theory relies on symbolic or proof-theoretic programs to model how agents inspect each other's decision procedures. This paper asks whether these strategic insights can be translated into a continuous, differentiable framework compatible with modern machine learning, allowing for the analysis of how parameter-based policies influence cooperation and equilibrium structure.

The authors define a parametric open-source game where each player chooses a parameter vector. A semantics map then translates the full profile of parameters into mixed actions for a finite base game. This setup allows the researchers to apply standard tools from continuous optimization and gradient-based learning. They analyze two specific semantics: a sigmoid-based model, which provides an explicit threshold for cooperation, and a neural network-based model, which generalizes the findings to more complex, non-linear dependencies.

• Cooperation Threshold: In symmetric 2x2 games, there exists a critical coupling threshold (gamma) for cross-player sensitivity. When this sensitivity exceeds the threshold, selfish gradient ascent naturally shifts from defection toward cooperation.
• Neural Generalization: The framework extends to neural semantics, where the ratio of cross-player to self-player sensitivity serves as the primary driver for cooperative behavior.
• Optimization Barriers: While the model can represent cooperative equilibria, the authors find that gradient-based learning may fail to discover these outcomes unless the system is initialized with sufficient cross-player coupling, suggesting that the threshold acts as both a theoretical condition and an optimization barrier.

This work bridges the gap between symbolic program equilibrium and differentiable multi-agent learning. By showing that internal parameter access can reshape learning dynamics, the authors provide a foundation for designing more cooperative AI systems that can account for their own transparency and the transparency of others.