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Swim till You Sink: Computing the Limit of a Game

Rashida Hakim, Jason Milionis, Christos Papadimitriou, Georgios Piliouras

TL;DR

This work studies the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game, and proves that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

Abstract

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

Swim till You Sink: Computing the Limit of a Game

TL;DR

This work studies the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game, and proves that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

Abstract

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.
Paper Structure (19 sections, 8 theorems, 20 equations, 8 figures)

This paper contains 19 sections, 8 theorems, 20 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Justification.
  4. A Combinatorial Algorithm for the Hitting Probabilities
  5. Outline of the Algorithm
  6. Definitions
  7. Algorithm Correctness
  8. Algorithm Progression
  9. Pseudosink Collapse
  10. Deletion of $\epsilon$-Edges at the Final Step
  11. Running Time of the Algorithm
  12. Experiments
  13. Some interesting games and their better-response graphs
  14. $3 \times 3$ Game
  15. Game with Order 1 Profile
  16. ...and 4 more sections

Key Result

theorem thmcountertheorem

The sink equilibria can be computed in time near-linear in the description of the game presented in normal form, whereas computing them in a graphical game is PSPACE-complete.

Figures (8)

  • Figure 1: The better-response graph of the $3 \times 3 \times 3$ game depicting the hitting probabilities of the pure profiles as pie charts.
  • Figure 2: $3\times 3$ game. Left: the better-response graph. Right: the game utilities.
  • Figure 3: Tie game. The profile $(3,3)$ is order 1. Left: the better-response graph. Right: the game utilities.
  • Figure 4: The $3 \times 3 \times 3$ game.
  • Figure 5: Convergence example. 40 independent runs of noisy RD were used for each sample.
  • ...and 3 more figures

Theorems & Definitions (21)

  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • ...and 11 more