Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Selection pressure/Noise driven cooperative behaviour in the thermodynamic limit of repeated games

Rajdeep Tah, Colin Benjamin

TL;DR

This work investigates whether cooperation can emerge when an infinite number of players repeatedly engage in a Prisoner's Dilemma under selection/noise. It combines analytical Nash equilibrium mapping to a 1D Ising chain with numerical agent-based modelling, using game magnetization as the order parameter to study the thermodynamic limit. Analyzing five strategy pairings (All-D, TFT, GRIM, GRIM*, WSLS) reveals first-order phase transitions in several cases, with transition points depending on the discount factor and selection intensity. The results demonstrate substantial agreement between ABM and NEM in the thermodynamic limit and illuminate how cooperation can arise or fail to emerge depending on strategy dynamics and game horizon in infinite populations.

Abstract

Consider the scenario where an infinite number of players (i.e., the \textit{thermodynamic} limit) find themselves in a Prisoner's dilemma type situation, in a \textit{repeated} setting. Is it reasonable to anticipate that, in these circumstances, cooperation will emerge? This paper addresses this question by examining the emergence of cooperative behaviour, in the presence of \textit{noise} (or, under \textit{selection pressure}), in repeated Prisoner's Dilemma games, involving strategies such as \textit{Tit-for-Tat}, \textit{Always Defect}, \textit{GRIM}, \textit{Win-Stay, Lose-Shift}, and others. To analyze these games, we employ a numerical Agent-Based Model (ABM) and compare it with the analytical Nash Equilibrium Mapping (NEM) technique, both based on the \textit{1D}-Ising chain. We use \textit{game magnetization} as an indicator of cooperative behaviour. A significant finding is that for some repeated games, a discontinuity in the game magnetization indicates a \textit{first}-order \textit{selection pressure/noise}-driven phase transition. The phase transition is particular to strategies where players do not severely punish a single defection. We also observe that in these particular cases, the phase transition critically depends on the number of \textit{rounds} the game is played in the thermodynamic limit. For all five games, we find that both ABM and NEM, in conjunction with game magnetization, provide crucial inputs on how cooperative behaviour can emerge in an infinite-player repeated Prisoner's dilemma game.

Selection pressure/Noise driven cooperative behaviour in the thermodynamic limit of repeated games

TL;DR

This work investigates whether cooperation can emerge when an infinite number of players repeatedly engage in a Prisoner's Dilemma under selection/noise. It combines analytical Nash equilibrium mapping to a 1D Ising chain with numerical agent-based modelling, using game magnetization as the order parameter to study the thermodynamic limit. Analyzing five strategy pairings (All-D, TFT, GRIM, GRIM*, WSLS) reveals first-order phase transitions in several cases, with transition points depending on the discount factor and selection intensity. The results demonstrate substantial agreement between ABM and NEM in the thermodynamic limit and illuminate how cooperation can arise or fail to emerge depending on strategy dynamics and game horizon in infinite populations.

Abstract

Consider the scenario where an infinite number of players (i.e., the \textit{thermodynamic} limit) find themselves in a Prisoner's dilemma type situation, in a \textit{repeated} setting. Is it reasonable to anticipate that, in these circumstances, cooperation will emerge? This paper addresses this question by examining the emergence of cooperative behaviour, in the presence of \textit{noise} (or, under \textit{selection pressure}), in repeated Prisoner's Dilemma games, involving strategies such as \textit{Tit-for-Tat}, \textit{Always Defect}, \textit{GRIM}, \textit{Win-Stay, Lose-Shift}, and others. To analyze these games, we employ a numerical Agent-Based Model (ABM) and compare it with the analytical Nash Equilibrium Mapping (NEM) technique, both based on the \textit{1D}-Ising chain. We use \textit{game magnetization} as an indicator of cooperative behaviour. A significant finding is that for some repeated games, a discontinuity in the game magnetization indicates a \textit{first}-order \textit{selection pressure/noise}-driven phase transition. The phase transition is particular to strategies where players do not severely punish a single defection. We also observe that in these particular cases, the phase transition critically depends on the number of \textit{rounds} the game is played in the thermodynamic limit. For all five games, we find that both ABM and NEM, in conjunction with game magnetization, provide crucial inputs on how cooperative behaviour can emerge in an infinite-player repeated Prisoner's dilemma game.
Paper Structure (21 sections, 44 equations, 14 figures, 1 table)

This paper contains 21 sections, 44 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory
  3. Prisoner's Dilemma in the one-shot settings
  4. The discount factor
  5. Nash equilibrium for 2-player repeated games
  6. TFT vs. All-D
  7. GRIM vs. All-D
  8. GRIM vs. GRIM*
  9. WSLS vs. TFT
  10. WSLS vs. GRIM
  11. Thermodynamic limit of Repeated games
  12. Motivation: Why do we study games in the thermodynamic limit?
  13. Introduction to the two methods
  14. Nash equilibrium mapping (NEM)
  15. Agent-based Modelling (ABM)
  16. ...and 6 more sections

Figures (14)

  • Figure 1: TFT vs. All-D in the 2-player limit.
  • Figure 2: GRIM vs. All-D in the 2-player limit.
  • Figure 3: GRIM vs. GRIM* in the 2-player limit.
  • Figure 4: WSLS vs. TFT in the 2-player limit.
  • Figure 5: Both $\mathfrak{P}_1$ and $\mathfrak{P}_2$ are playing WSLS in the 2-player repeated prisoner's dilemma.
  • ...and 9 more figures