Best Response Games on Regular Graphs

TL;DR

This paper investigates the complex dynamics of best response games on regular graphs with multiple strategies, revealing geometric and combinatorial properties that determine when such games correspond to cellular automata.

Contribution

It characterizes the convex geometric structures underlying best response games and links them to graph circuit theory, providing criteria for when these games are equivalent to cellular automata.

Findings

Almost every convex division in high-dimensional space has a single point where all regions meet.

Connections between convex geometry and graph circuits are established.

Conditions for best response games to be equivalent to cellular automata are identified.

Abstract

With the growth of the internet it is becoming increasingly important to understand how the behaviour of players is affected by the topology of the network interconnecting them. Many models which involve networks of interacting players have been proposed and best response games are amongst the simplest. In best response games each vertex simultaneously updates to employ the best response to their current surroundings. We concentrate upon trying to understand the dynamics of best response games on regular graphs with many strategies. When more than two strategies are present highly complex dynamics can ensue. We focus upon trying to understand exactly how best response games on regular graphs sample from the space of possible cellular automata. To understand this issue we investigate convex divisions in high dimensional space and we prove that almost every division of $k-1$ dimensional space into $k$ convex regions includes a single point where all regions meet. We then find connections between the convex geometry of best response games and the theory of alternating circuits on graphs. Exploiting these unexpected connections allows us to gain an interesting answer to our question of when cellular automata are best response games.