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Characterising Simulation-Based Program Equilibria

Emery Cooper, Caspar Oesterheld, Vincent Conitzer

TL;DR

This work investigates simulation-based program equilibria where players submit programs that reason about opponents’ code. It generalizes the ε-GroundedπBot approach to both correlated (shared randomness) and uncorrelated settings, proving a folk theorem in the correlated case and characterizing what equilibria are possible without shared randomness. It introduces correlated and uncorrelated variants, the use of apply^* to screen private randomness, and a repeated-game interpretation with screening and correlating signals to establish broad sets of achievable payoffs. It further shows intrinsic limits: without shared randomness, the full Tennenholtz folk theorem cannot be reached in general, though additively separable utilities admit a stronger result, and it analyzes how general simulation-based programs face fundamental limits even with advanced schemes.

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) $ε$Grounded$π$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) $ε$Grounded$π$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) $ε$Grounded$π$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.

Characterising Simulation-Based Program Equilibria

TL;DR

This work investigates simulation-based program equilibria where players submit programs that reason about opponents’ code. It generalizes the ε-GroundedπBot approach to both correlated (shared randomness) and uncorrelated settings, proving a folk theorem in the correlated case and characterizing what equilibria are possible without shared randomness. It introduces correlated and uncorrelated variants, the use of apply^* to screen private randomness, and a repeated-game interpretation with screening and correlating signals to establish broad sets of achievable payoffs. It further shows intrinsic limits: without shared randomness, the full Tennenholtz folk theorem cannot be reached in general, though additively separable utilities admit a stronger result, and it analyzes how general simulation-based programs face fundamental limits even with advanced schemes.

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) GroundedBot 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) GroundedBot. 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) GroundedBot. 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.

Paper Structure

This paper contains 37 sections, 42 theorems, 35 equations, 1 figure, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Background and Preliminaries
  4. Normal-form games
  5. Program games
  6. Correlated Program Games
  7. Our algorithm
  8. Halting results
  9. Expected runtime
  10. Repeated games with correlating signals and screening
  11. Relating the program game to the repeated game
  12. Epsilon-Grounded Simulation without Shared Randomness
  13. Halting Behaviour
  14. No full folk theorem
  15. Characterising the equilibria
  16. ...and 22 more sections

Key Result

Theorem 1

Payoffs $v_{1:n}$ are achievable in an equilibrium of the uncorrelated program game of $G$ (given aforementioned restrictions on the programs available to each player) if and only if they are feasible (without correlation) and individually rational.

Figures (1)

  • Figure 1: The simulation tree in a game between $n$ correlated $\epsilon$Grounded$\pi$Bots with common $\epsilon$ in the case $T= 3$ (i.e., when $r_0,r_1,r_2\geq \epsilon$ and $r_3<\epsilon$). An arrow from a group of programs with one input, to a second group of programs with a second input, indicates that the first group with the first input all run all of the second group of programs with the second input (in this case, via an $\mathit{apply}^*$ call). We omit the part of the input that is simply the list of other programs.

Theorems & Definitions (89)

  • Example 1
  • Definition 1: Program games
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 79 more