Assume that each player in a round robin has some unknown skill level, and the probability a player wins a match is determined by the difference in skill of the two players. Two players end the round robin with the same number of wins, tied for first place. Does there exist a tiebreaking scheme that picks the more skilled player more reliably than a coin flip?
It is ok if your tiebreaker only works under some reasonable restriction of this model (e.g. you may choose a monotonic function for determining win chances from skill differential). The tiebreaker may not rely on anything other than who won which matches (e.g. do not assume matches have point differentials).