Estimates for the Branching Factors of Atari Games

TL;DR

This paper estimates the average branching factors of 103 Atari games using state enumeration, revealing that actual branching is often much lower than the maximum available actions, which informs AI research in videogames.

Contribution

It provides the first comprehensive estimates of branching factors for Atari games, clarifying state distinctions and offering insights into game complexity for AI development.

Findings

Average branching factors are often just above 1.

Maximum actions per frame range from 3 to 18.

State enumeration clarifies what constitutes a distinct state.

Abstract

The branching factor of a game is the average number of new states reachable from a given state. It is a widely used metric in AI research on board games, but less often computed or discussed for videogames. This paper provides estimates for the branching factors of 103 Atari 2600 games, as implemented in the Arcade Learning Environment (ALE). Depending on the game, ALE exposes between 3 and 18 available actions per frame of gameplay, which is an upper bound on branching factor. This paper shows, based on an enumeration of the first 1 million distinct states reachable in each game, that the average branching factor is usually much lower, in many games barely above 1. In addition to reporting the branching factors, this paper aims to clarify what constitutes a distinct state in ALE.