On Round-Robin Tournaments with a Unique Maximum Score

Richard Arnold Epstein (1927-2016) published the first edition of"The Theory of Gambling and Statistical Logic"in 1967. He introduced some material on round-robin tournaments (complete oriented graphs) with n labeled vertices in Chapter 9; in particular, he stated, without proof, that the probability that there is a unique vertex with the maximum score tends to one as n tends to infinity. Our object here is to give a proof of this result along with some historical remarks and comments.

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