Characterising Simulation-Based Program Equilibria (2412.14570v2)

Abstract: In Tennenholtz's program equilibrium, players of a game submit programs to play on their behalf. Each program receives the other programs' source code and outputs an action. This can model interactions involving AI agents, mutually transparent institutions, or commitments. Tennenholtz (2004) proves a folk theorem for program games, but the equilibria constructed are very brittle. We therefore consider simulation-based programs -- i.e., programs that work by running opponents' programs. These are relatively robust (in particular, two programs that act the same are treated the same) and are more practical than proof-based approaches. Oesterheld's (2019) $\epsilon$Grounded$\pi$Bot is such an approach. Unfortunately, it is not generally applicable to games of three or more players, and only allows for a limited range of equilibria in two player games. In this paper, we propose a generalisation to Oesterheld's (2019) $\epsilon$Grounded$\pi$Bot. We prove a folk theorem for our programs in a setting with access to a shared source of randomness. We then characterise their equilibria in a setting without shared randomness. Both with and without shared randomness, we achieve a much wider range of equilibria than Oesterheld's (2019) $\epsilon$Grounded$\pi$Bot. Finally, we explore the limits of simulation-based program equilibrium, showing that the Tennenholtz folk theorem cannot be attained by simulation-based programs without access to shared randomness.

• The paper introduces a simulation-based program framework that interprets opponent code, generalising simulation methods for multi-player games.
• The paper proves a folk theorem demonstrating that feasible and individually rational payoffs are achievable in correlated program games using these simulation methods.
• The research has profound implications for developing AI systems in multi-agent cooperative and competitive environments by enabling reliable equilibrium concepts.

Analysis of "Characterising Simulation-Based Program Equilibria"

The paper "Characterising Simulation-Based Program Equilibria" by Emery Cooper, Caspar Oesterheld, and Vincent Conitzer offers an extensive exploration into refining simulation-based approaches for program equilibria in normal-form games. The authors address the limitations of previous works, notably the brittleness of equilibria in \citeauthor{tennenholtz}'s program equilibrium framework and the challenges posed by \citeauthor{oesterheld2019robust}'s Grounded$\pi$Bot in extending to games with more than two players. Here, the focus is articulated on generalizing simulation-based approaches to handle wider classes of games and achieve more robust equilibria by leveraging new computational techniques.

Key Contributions

• Simulation-Based Program Framework: The authors introduce a framework leveraging simulation-based programs, which are distinguished by their ability to interpret opponents' program code comprehensively. This approach contrasts substantially with proof-based tactics, which are often impractical for complex program structures like neural networks.
• Generalized Simulation Approach: They extend the capabilities of the Grounded$\pi$Bot to accommodate more than two players and integrate a shared randomness feature, which allows for a broader span of strategic equilibria. This development aims to solve previously infeasible cases such as certain three-player game structures.
• Theoretical Insights and Folk Theorem: A significant result is the proof of a folk theorem under the correlated program game setting, demonstrating that feasible and individually rational payoffs can be achieved by these generalized simulation approaches. This theorem is an essential aspect of establishing the efficacy and flexibility of the proposed models across different game-theoretic scenarios.
• Limitations without Shared Randomness: The research provides a detailed examination of the limitations inherent in simulation-based programs that lack shared randomness, noting that such programs cannot achieve the complete set of equilibria characterized by Tennenholtz's folk theorem.

Implications for AI Development

The research carries profound implications for the development of AI systems in cooperative and competitive environments. In AI domains where systems need to autonomously interact—consider multi-agent systems, smart contracts, or autonomous economic agents—such simulation-based equilibrium concepts become crucial. The ability to simulate opponent strategies and achieve reliable equilibria paves the way for AI that can effectively negotiate, cooperate, and compete with other AI entities programmed by diverse principals.

Future Directions

Looking forward, several avenues can be pursued:

• Exploration of Decentralized Simulation: Extending the notion of shared randomness more practically in decentralized AI environments remains an open question. Although theoretically demonstrated, practical implementation poses significant challenges.
• Integration with Large-Scale AI Models: The integration of these equilibrium concepts with contemporary AI models like neural networks would be invaluable, offering more robust frameworks for AI interactions where complete transparency of internal structures is not feasible.
• Expanding Scope Beyond Game Theory: Beyond game-theoretic applications, exploring these methodologies in other fields where strategic program execution is required could reveal novel applications and further theoretical insights.

In summary, this paper advances the understanding of simulation-based equilibria in program games, providing a solid theoretical foundation for more complex multi-agent systems while addressing critical limitations of preceding models. The paper is an important step towards realizing game-theoretic equilibria in applied AI, setting the stage for future research to explore and harness its practical potentials.