Random tie-breaker

A possible tie-breaker, typically used as a last resort and when prizes are indivisible, is to break the tie randomly such that each tied player has an equal chance of winning.

For two players, a simple tossing of a coin can be used for such a tie-breaker. For more players, their relative placement may be determined by giving each player a lot and then drawing lots in order.

Random tie-breakers are independent of pairing and as long as the procedure remains randomized not easily manipulated (such as by intentional losses by third players which would often affect SOS, or in rare cases including multiple tied players even Direct Comparison).