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The main page states that on purpose of tournaments is ordering players by playing strength.
A tournament produces a set of results based upon games played by pair of entrants. The results are the performances (wins and losses) of each player. The entrants, being people, do not produce repeatable results of the type that we know from Mathematics or Physics (macroscopic, non-quantum). Rather an entrant performance in a tournament is a point in time representation of that person playing strength. That strength may change over time. One of the original ideas behind Elo Ratings was to track deterioration of playing ability vs age.
The statistical model is that an entrant has a playing strength distribution. Typically a Gaussian distribution is used. There is good reason to believe that the Logistical distribution would be more accurate at the cost of significantly more difficult calcuations. A game consists of each entrant sampling his own playing strength distribution to produce a single numeric value that is compared with the result of sampling the other player's playing strength distribution. The larger value wins the game. For the basis of these statements, please see H A David's book [The Method of Paired Comparisons | Review: H. A. David, The Method of Paired Comparisons]. (You should be able to find it in your local University library.)
Therefore, a tournament can not provide a ordering players by playing strength. It can provide a ordering players by performance that may or may not accurate represent each player's playing strength distribution.
Let's reword the page to say ordering players by performance.
What a tournament attempts is to order players by strength. What is does is order players by performance. This is not its purpose, it is simply what it does.
If you state that the purpose of the tournament is to order players by performance, then there is no difference at all between the tournament systems. All of them order players by performance equally well.
One of the reasons the purpose is listed as ordering people by strength is the fact that we can then, further down, make statements about how well different tournament systems fulfill this purpose.
Hmm...not sure that all tournaments order the players by performance equally well. A single knockout produces a single unique winner, but does a poorer job of ordering the rest of the field. A Swiss or McMahon does an equally good/bad job of ordering all players in the field. A round-robin provides a much richer set of data points upon which to build and ordering.
Choosing the tournament format (knockout, swiss, mcmahon, round robin) is one choice the tournament director must make.
Which tie breakers are chosen is a second choice. The ability of various tie breakers to better estimate strength based upon performance is a method of valuing the tiebreakers.
Nonetheless, all that a combination of tournament format and tiebreakers can do is produce an approximation of strength at a particular point in time based upon a sparse matrix (except for round robin) of results.
Hence, it measures preformance and attempts to infer strength.
Exactly. I have never disagreed with that conclusion. What is does is measure performance (and order by it), what it attempts is to measure strength (and order by it). So the purpose (what is attempted) of all the systems is to order the players by strength. The means (what it does) is ordering by performance.
To evaluate tournament systems, we can then consider how well the means approach the purpose.
I think the evaluations on the page of how well different systems do in estimating strength by tournament performance are largely fair, ie: