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Recursive Joint Simulation in Games

Vojtech Kovarik, Caspar Oesterheld, Vincent Conitzer

TL;DR

It is shown that the resulting interaction between AI agents is strategically equivalent to an infinitely repeated version of the original game, allowing a direct transfer of existing results such as the various folk theorems.

Abstract

Game-theoretic dynamics between AI agents could differ from traditional human-human interactions in various ways. One such difference is that it may be possible to accurately simulate an AI agent, for example because its source code is known. Our aim is to explore ways of leveraging this possibility to achieve more cooperative outcomes in strategic settings. In this paper, we study an interaction between AI agents where the agents run a recursive joint simulation. That is, the agents first jointly observe a simulation of the situation they face. This simulation in turn recursively includes additional simulations (with a small chance of failure, to avoid infinite recursion), and the results of all these nested simulations are observed before an action is chosen. We show that the resulting interaction is strategically equivalent to an infinitely repeated version of the original game, allowing a direct transfer of existing results such as the various folk theorems.

Recursive Joint Simulation in Games

TL;DR

It is shown that the resulting interaction between AI agents is strategically equivalent to an infinitely repeated version of the original game, allowing a direct transfer of existing results such as the various folk theorems.

Abstract

Game-theoretic dynamics between AI agents could differ from traditional human-human interactions in various ways. One such difference is that it may be possible to accurately simulate an AI agent, for example because its source code is known. Our aim is to explore ways of leveraging this possibility to achieve more cooperative outcomes in strategic settings. In this paper, we study an interaction between AI agents where the agents run a recursive joint simulation. That is, the agents first jointly observe a simulation of the situation they face. This simulation in turn recursively includes additional simulations (with a small chance of failure, to avoid infinite recursion), and the results of all these nested simulations are observed before an action is chosen. We show that the resulting interaction is strategically equivalent to an infinitely repeated version of the original game, allowing a direct transfer of existing results such as the various folk theorems.
Paper Structure (17 sections, 14 theorems, 8 equations, 2 figures)

This paper contains 17 sections, 14 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Normal-Form Games
  5. Repeated Games
  6. Recursive Joint Simulation Games and Equivalence to Repeated Games
  7. Extensions of the Formalism
  8. Voluntary Simulation
  9. Variable Simulation Probability
  10. Limited Simulation Budget
  11. Self-locating Beliefs
  12. Two Objections to the Simulation Framework
  13. Could indistinguishability be impossible in principle?
  14. Could indistinguishability be impossible in practice?
  15. Conclusion
  16. ...and 2 more sections

Key Result

Proposition 1

Let ${\mathcal{G}}$ be any NFG. Let $v$ be a feasible payoff vector s.t. for all players $i$ we have that $v_i > \bar{v}_i$. Then there exists $p_0 \in [0, 1)$ s.t. for every $p \in [p_0, 1)$, there is some $\pi \in \textnormal{NE}(\textnormal{Rep}_\omega({\mathcal{G}}, p))$ s.t. $\forall i : u_i(\p

Figures (2)

  • Figure 1: A game with a single (non-recursive) joint simulation.
  • Figure 2: A description of the main setting of this paper in pseudocode.

Theorems & Definitions (26)

  • Proposition 1: Folk theorem for repeated games; e.g., Osborne2004, Prop. 454.1
  • Theorem 1: Strategic equivalence to infinitely-repeated games
  • proof
  • Lemma 1
  • proof : Proof sketch
  • Lemma 1
  • proof : Proof sketch
  • Corollary 2: Folk theorem for recursive joint simulation
  • Proposition 2
  • proof : Proof sketch
  • ...and 16 more