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.

翻译：在Tennenholtz的程序均衡中，博弈参与者提交程序代表其进行博弈。每个程序接收其他程序的源代码并输出一个行动。这可以用于建模涉及智能体、相互透明的机构或承诺的交互。Tennenholtz (2004) 证明了程序博弈的民间定理，但所构建的均衡极其脆弱。因此，我们考虑基于模拟的程序——即通过运行对手程序来运作的程序。这类程序具有相对鲁棒性（特别是，行为相同的两个程序会被同等对待），并且比基于证明的方法更具实用性。Oesterheld (2019) 的 $\epsilon$Grounded$\pi$Bot 即为此类方法。遗憾的是，该方法通常不适用于三人或更多参与者的博弈，且在双人博弈中仅允许有限的均衡范围。本文提出了对Oesterheld (2019) $\epsilon$Grounded$\pi$Bot 的推广。我们在可访问共享随机源的设定下，证明了我们程序的民间定理。随后，我们在无共享随机源的设定下刻画了其均衡特性。无论是否存在共享随机源，我们都实现了比Oesterheld (2019) $\epsilon$Grounded$\pi$Bot 更广泛的均衡范围。最后，我们探讨了基于模拟的程序均衡的局限性，证明在无法访问共享随机源的情况下，基于模拟的程序无法达到Tennenholtz民间定理所描述的均衡状态。