Self-stabilizing uncoupled dynamics

TL;DR

This paper investigates self-stabilizing uncoupled dynamics in distributed systems, analyzing the minimal recall needed for stochastic and deterministic players to reach stable states in various game settings.

Contribution

It characterizes the minimum recall periods required for self-stabilization in uncoupled game dynamics, providing protocols for deterministic players with specific action constraints.

Findings

Stochastic players need a bounded number of recall periods for self-stabilization.

Deterministic players require at least two steps of recall in certain game classes.

Single-step recall is insufficient for deterministic self-stabilization regardless of actions.

Abstract

Dynamics in a distributed system are self-stabilizing if they are guaranteed to reach a stable state regardless of how the system is initialized. Game dynamics are uncoupled if each player's behavior is independent of the other players' preferences. Recognizing an equilibrium in this setting is a distributed computational task. Self-stabilizing uncoupled dynamics, then, have both resilience to arbitrary initial states and distribution of knowledge. We study these dynamics by analyzing their behavior in a bounded-recall synchronous environment. We determine, for every "size" of game, the minimum number of periods of play that stochastic (randomized) players must recall in order for uncoupled dynamics to be self-stabilizing. We also do this for the special case when the game is guaranteed to have unique best replies. For deterministic players, we demonstrate two self-stabilizing uncoupled protocols. One applies to all games and uses three steps of recall. The other uses two steps of recall and applies to games where each player has at least four available actions. For uncoupled deterministic players, we prove that a single step of recall is insufficient to achieve self-stabilization, regardless of the number of available actions.