The Structure of N-Player Games when Influence and Independence Collide

TL;DR

This paper explores the complex interplay between influence and independence in multi-player probabilistic causal models, revealing conditions under which effects can be correlated or independent despite causal influences.

Contribution

It provides a complete classification of symmetric processes where influence and independence coexist in n-player games, extending previous understanding.

Findings

Characterization of symmetric processes for all n ≥ 3

Description of the geometry and topology of the probability space

Identification of conditions allowing influence and independence to coexist

Abstract

We study the mathematical properties of probabilistic processes in which the independent actions of $n$ players (`causes') can influence the outcome of each player (`effects'). In such a setting, each pair of outcomes will generally be statistically correlated, even if the actions of all the players provide a complete causal description of the players' outcomes, and even if we condition on the outcome of any one player's action. This correlation always holds when $n=2$, but when $n=3$ there exists a highly symmetric process, recently studied, in which each cause can influence each effect, and yet each pair of effects is probabilistically independent (even upon conditioning on any one cause). We study such symmetric processes in more detail, obtaining a complete classification for all $n \geq 3$. Using a variety of mathematical techniques, we describe the geometry and topology of the underlying probability space that allows independence and influence to coexist.