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Is there any literature on the order structure coming from round-robin tournaments? I play games on littlegolem and the tournaments are mostly 5x5 round-robins. I noticed at the end, after sorting for wins and "sum of scores" all the matrices look the same. $$ \begin{array}{c|ccccc|cc} 1 & & 2 & 2 & 2 & 2 & 8 & 24\\\\ 2 & 0 & & 2 & 2 & 2 & 6 & 12 \\\\ 3 & 0 & 0 & & 2 & 2 & 4 & 4 \\\\ 4 & 0 & 0 & 0 & & 2 & 2 & 0 \\\\ 5 & 0 & 0 & 0 & 0 & & 0 & 0 \end{array}$$

Can all tournament grids be ordered this way even for 5x5? Is there literature on partial orders or well-orders induced from tournaments?

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    $\begingroup$ I can’t say I understand what the last column means, but it is perfectly possible that player A beats player B, player B beats player C, and player C beats player A, in which case the tournament does not induce an order, and the matrix looks different. $\endgroup$
    – Emil Jeřábek
    Sep 23, 2011 at 18:14
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    $\begingroup$ And of course, if it does not contain a cycle like this, the tournament induces a total order. A total order with a given finite number of elements is unique up to isomorphism, hence the tournament matrix can be reordered to the triangular form. Either I am missing something, or this is nowhere near a research-level question. $\endgroup$
    – Emil Jeřábek
    Sep 23, 2011 at 18:22
  • $\begingroup$ In a Martin Gardner column, probably the 1960's or 1970's, he gave diagrams for four ordinary cubical dice with altered numbers, call them A,B,C,D. I think one of them had all numbers the same, each side had the same number (4?). Anyway, he called them "nontransitive," as, when rolled against each other competitively, A beats B 2/3 of the time (by easy counting/finite probability arguments), B beats C 2/3 of the time, then C beats D, finally D beats A, all 2/3 of the time. $\endgroup$
    – Will Jagy
    Sep 23, 2011 at 19:42
  • $\begingroup$ The last column is "Sum of Opponents' Scores". It can be calculated from the table, and contains no new information. It is used on littlegolem for ordering the players. $\endgroup$
    – maproom
    Sep 24, 2011 at 9:26
  • $\begingroup$ @maproom: Thanks for the information. @Will: The number of nonisomorphic tournaments is given in oeis.org/A000568 (and it’s easy to check by hand for these small numbers), there are indeed 4 tournaments with 4 players, and there are 12 with 5 players. Diagrams of all tournaments of up to 6 players are presented in oeis.org/A000568/a000568.pdf . $\endgroup$
    – Emil Jeřábek
    Sep 26, 2011 at 10:26

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