The Optimal Way to Play the Most Difficult Repeated Coordination Games

TL;DR

This paper characterizes optimal protocols for repeated coordination games, especially Choice Matching Games, providing a complete classification of their expected and guaranteed coordination times, and establishing their difficulty relative to other WLC-games.

Contribution

It offers a complete analysis of optimal coordination protocols in CM-games and extends these results to all two-player WLC-games, identifying the most challenging cases.

Findings

Optimal protocols are unique except in the four-choice case.

CM-games have the highest expected coordination times among WLC-games.

Complete bounds for coordination times are established for all two-player WLC-games.

Abstract

This paper investigates repeated win-lose coordination games (WLC-games). We analyse which protocols are optimal for these games, covering both the worst case and average case scenarios, i,e., optimizing the guaranteed and expected coordination times. We begin by analysing Choice Matching Games (CM-games) which are a simple yet fundamental type of WLC-games, where the goal of the players is to pick the same choice from a finite set of initially indistinguishable choices. We give a fully complete classification of optimal expected and guaranteed coordination times in two-player CM-games and show that the corresponding optimal protocols are unique in every case - except in the CM-game with four choices, which we analyse separately. Our results on CM-games are essential for proving a more general result on the difficulty of all WLC-games: we provide a complete analysis of least upper bounds for optimal expected coordination times in all two-player WLC-games as a function of game size. We also show that CM-games can be seen as the most difficult games among all two-player WLC-games, as they turn out to have the greatest optimal expected coordination times.