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

Optimal protocols for the most difficult repeated coordination games