Optimal protocols for the most difficult repeated coordination games

TL;DR

This paper analyzes optimal protocols for repeated win-lose coordination games, especially Choice Matching Games, providing a complete classification of optimal times and demonstrating their significance for all two-player WLC-games.

Contribution

It offers a complete classification of optimal expected and guaranteed coordination times in CM-games and proves their role as the most difficult among two-player WLC-games.

Findings

Unique optimal protocols for most CM-games identified.

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

CM-games are the most challenging among 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 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 also 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.