# How to break ties in a round robin?

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).

• I know that one standard method is to break the tie in favor of whoever won the head-to-head match between the tied players, but I haven't seen a satisfactory justification for that - sure that player beat the most formidable opponent, but they also racked up more losses against less formidable opponents so it's not obvious to me that they're the better player. – Benjamin Cosman Mar 22 '17 at 6:29
• How about ranking players using some method similar to ELO in chess? – mlc Mar 22 '17 at 6:31
• You certainly could, but please convince me that the player with the higher ELO has a greater than 50% chance of being the better player. Also for ELO, wouldn't it matter what order you face your opponents in? That's quite inelegant for a round robin, though I'll take it if it works. – Benjamin Cosman Mar 22 '17 at 6:34
• This book reviews several ranking schemes (but has no proofs that one is more reliable than a coin flip): Who's #1? The Science of Rating and Ranking, Amy N. Langville and Carl D. Meyer, 2013. – mlc Mar 22 '17 at 7:19
• @BenjaminCosman I decided to Monte Carlo the head-to-head tiebreaker, and my results were that the success rate at picking the player I had given a higher probability of winning with it was 50% – Arcanist Lupus Sep 13 '18 at 22:32