Summary
The doubleelimination draw leading to a singlegame final has become a standard of sorts at USCA tournaments, but the versions in common use can result in an unfair outcome. By reconfiguring the “losers” bracket, it is possible to construct a better and fairer draw of this type.
Background
The doubleelimination draw has been a mainstay of USCA tournaments since the first National Championship in 1977. In those early days croquet courts in the US were few and far between. This left few practical options for how to format a tournament. Doubleelimination was chosen as a way to guarantee every player at least two games without making the tournament too long.
Since then we have seen the development of new and larger tournament venues. Players now expect to play many games at a tournament, usually in a block (round robin) format followed by a playoff. With the block rounds guaranteeing each player a good number of games, singleelimination playoffs have become more acceptable. But at many tournaments, and certainly at the National Championship, doubleelimination remains the playoff format of choice.
Diagram 1:

Double trouble
Except that somewhere along the way, an insidious innovation crept in: the “if needed” game was dropped. Apparently one reason for this is a compulsive desire to know exactly when the tournament will end. At any rate, the doubleelimination playoff with singlegame final has become the standard playoff format. Indeed, in seven years of tournament play I have never seen a doubleelimination draw of the “classic” type, as shown in Diagram 1.
Obviously, a “doubleelimination” draw that ends with a singlegame final is not a true doubleelimination draw. (I’ll call this type of format “pseudodoubleelimination”.) While, theoretically, the loss of the “if needed” game should make little difference to any player’s overall chances of winning the tournament ^{(1)}, it can easily lead to a dubious outcome.
Referring to Diagram 1, if L7 wins his next game, he will immediately earn a rematch with the player who beat him in Game 7. If L7 then wins Game 8, these two players will have each beaten the other once. It is hardly logical to declare L7 the winner at this point, and yet that is exactly what happens in the pseudodoubleelimination format.
The usual reply to this criticism is to point out that L7 finishes with a better overall record than L8. While this is true (barely), it falls flat as a justification. One player reached Game 8 by means of a 30 record (i.e., three wins and no losses), the other with a 31 record. In other words, these two players achieved an equal position even though one of them had a patently inferior record. This is simply unfair.
The obvious solution is to restore the “if needed” game. But general opinion seems to be firmly in favor of the singlegame final, even in light of its inherent unfairness.
The “faceoff” draw
One variation that has been introduced in an effort to solve this disparity is the faceoff draw. This is now the standard playoff format at the USCA National Championship.
The faceoff simply divides the field in half, each half comprising a separate doubleelimination ladder, complete with “if needed” games. (There is some crossover from one half to the other to avoid immediate rematches in the losers brackets.) The tournament final is a singlegame playoff between the two ladder winners.
The problem with the faceoff format is that while it decreases the likelihood of the unfairness described above, it doesn’t eliminate it. There is still the chance that one finalist will have a decidedly inferior record to the other (e.g., 41 vs. 40).
A better compromise
The pseudodoubleelimination format can be reasonably fair, so long as winning the losers bracket (the lower bracket) always requires more wins than winning the winners bracket (the upper bracket).
I devised a draw ^{(2)} that accomplishes this by a rearrangement of the final rounds of the losers bracket. Note Diagram 2:
Diagram 2:

Because L15 has to win two games to get to the final, this draw remedies the main unfairness of the standard pseudodoubleelimination draw.
The asymmetrical construction of the losers bracket has two interesting side effects. First, it shortens the entire draw by one round. The number of games remains the same, so this is an unqualified bonus. The Diagram 2 draw can be completed in seven rounds, versus eight rounds for either the standard pseudodoubleelimination draw or the faceoff draw for the same number of players.
Second, all paths to the final that pass through the bottom bracket are of the same length. In the Diagram 2 draw, a player who loses a game at any stage and then goes on to win the final will have won six games. (In the standard draw, such a player might have won five, six, or seven games, depending on which game he lost.) This is also a benefit, because it means that within each bracket, all wins have equal value.
Of course, this losers bracket configuration can also be used in a true doubleelimination draw, with some of the same advantages over the standard doubleelimination draw. Not that I ever expect to see true doubleelimination making a comeback as a USCA tournament format.
A limitation of my draw is that it doesn’t work with smaller numbers of players. Sixteen is the minimum if all players are to start from equal positions. Twelveplayer versions are also possible, either by giving four players byes, or by starting four players directly in the bottom bracket. The latter is a more attractive format, and is shown in Diagram 3.
Diagram 3:

Conclusion
The “standard” form of the pseudodoubleelimination draw can easily result in an unfair outcome, and for this reason alone it should not be used. The faceoff draw is somewhat better, but still suffers from the same basic problem (albeit to a lesser extent).
In terms of fairness, the best choice is to use a true doubleelimination format. But the strong demand for formats with singlegame finals calls for a different solution.
My modified draw eliminates the principal unfairness of the other pseudodoubleelimination draws. It is also more compact, requiring one less time period to complete. Either point alone is reason enough to use this draw in preference to the others.
Notes
 I made this determination after doing some computer analysis of various doubleelimination formats. (This was a computer simulation using the Association Croquet World Ranking formula to predict outcomes, as suggested by Louis Nel’s tournament simulations.) I was actually expecting to find that doing away with the “if needed” game had a significant effect on outcome. I was then going to use that evidence to argue that pseudodoubleelimination formats are decidedly inferior to true doubleelimination formats. But I found only a slight effect, and so have had to draw a different conclusion.
 I say that “I devised” this draw because I arrived at it independently. However, I expect that this is yet another example of reinventing the wheel. Doubleelimination is used in many different sports, and it seems likely enough that my discovery is not an original one.
Copyright notice
Copyright 2001–2008 by Jeff Soo.
You may make and distribute paper copies of this article, under the following conditions:
 each copy must include the complete text of the article, unmodified, including this copyright notice.
 the copies must be distributed free of charge.
Electronic distribution or republication is prohibited, unless permission has been granted by the author. However, you are
welcome to publish links to this article via email or WWW. Link to
http://ipsedixit.net/croquet/articles/formats/
8 December 2001