An Extension of Landau's Theorem on Tournaments
نویسنده
چکیده
In an ordinary (round-robin) tournament there are n people, Pi, , Pn, each of whom plays one game against each of the other n — 1 people. No game is permitted to end in a tie, and the score of Pi is the total number Si of games won by pim By the score sequence of a given tournament is meant the set S = (slf •••,«»), where it may be assumed, with no loss of generality, that sx ^ ^ sn. Landau [3] has given necessary and sufficient conditions for a set of integers to be the score sequence of some tournament. The object of this note is to show that these conditions are also necessary and sufficient for a set of real numbers to be the score sequence of a generalized tournament; a generalized tournament differs from an ordinary tournament in that as a result of the game between Pi and Pj> i Φ J, the amounts aiό and aH = 1 — aiS are credited to Pi and pjf respectively, subject only to the condition that 0 ^ aiά S 1. The score of Pi is given by
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