TITLE: Algorithm for least required matches to rank players in tournament
QUESTION [8 upvotes]: Assuming the following conditions:

A higher skill level always beats a lower skill level.
Given n players, each have a distinct skill level compared to the other (n-1).
If player A has beat player B, and player B has beat player C, then player A is better than player C and no match need to occur.
The skill level of each player can only be used to determine if it is complete (It cannot use the skill level to help the algorithm, it must only use the match history)
The algorithm is only considered complete when each player is correctly ranked according to their skill level

What matching algorithm would rank the n players in the least number of matches?
Notice I state matches, and not rounds. So concurrent matches occurring does not help. Although I am curious of the algorithm that would do it in the least number of rounds as well.
If there is no clear obvious answer, what methods/techniques are worth considering?
The answer may (probably?) be a well known tournament style.
If not I have a feeling the answer is very simple and is related to some graph traversal problem, or even related to sorting algorithms on random integers.

As an example, I will use a basic algorithm for 4 players:
Player A (skill 4)
Player B (skill 3)
Player C (skill 2)
Player D (skill 1)

Matches
Round 1 (Match-making is random in round 1 as per condition 4)
A vs C: A wins
B vs D: B wins

Known: (A > C), (B > D)

Round 2
A vs B: A wins
C vs D: C wins

Known: (A > BCD), (B > D), (C > D)

Round 3
B vs C: B wins

Known: (A > BCD), (B > CD), (C > D)

So given 4 players, I was able to find the rank for all players in 5 matches.

REPLY [2 votes]: The merge insertion sort is optimal if the number of players is less than 15 or between 20 and 22. For larger numbers of players there exists a sorting algorithm with fewer comparisons than the merge insertion sort.
In general, this is an open problem.