When two people are together and they both know the relationship has no future, but both refuse to take the step and end it. The situation might entail "upping the ante" to make the other one end it, such as suggesting marriage or kids. Another tactic in this game could be intentionally provoking each other.
Marshall: If they can’t stand each other, why don’t they end it?
Ted: Neither one wants to be first. They’re playing relationship chicken