Friendly frogs, stable marriage, and the magic of invariance

TL;DR

This paper explores a two-player game involving token movements on fixed sets, revealing deep connections to Gale-Shapley stable marriage and employing advanced probabilistic tools to analyze outcomes on random infinite sets.

Contribution

It introduces a novel game model linked to stable marriage theory and applies invariance and ergodic methods to analyze game outcomes on random infinite sets.

Findings

Game outcomes are characterized using invariance and ergodicity.

Connections established between the game and Gale-Shapley stable marriage.

Analytical framework applicable to random infinite point sets.

Abstract

We introduce a two-player game involving two tokens located at points of a fixed set. The players take turns to move a token to an unoccupied point in such a way that the distance between the two tokens is decreased. Optimal strategies for this game and its variants are intimately tied to Gale-Shapley stable marriage. We focus particularly on the case of random infinite sets, where we use invariance, ergodicity, mass transport, and deletion-tolerance to determine game outcomes.