Power-matching
Power-matching is a tournament format building block in which teams' schedules depend on their performance in previous matches. Teams generally play matches against other teams of the same or similar records, resulting in many more meaningful (i.e, close) matches than round robins provide, but at the expense of it becoming impossible to meaningfully compare teams' strengths of schedule.
Power matching is used at many NAQT national championships, Scobol Solo, and some other local events. Although he was not the first to use it, power matching is most prominently associated with David Reinstein, whose innovations in pairing algorithms led NAQT to ask him to write the power-matched schedules for several years, though he has since been voluntarily replaced by a computer program (that he helped create).
Reinstein and NAQT independently started using power-matching during the 2001-2002 season. Neither had any idea the other was doing it until New Trier attended the 2005 HSNCT.
Teams are generally seeded going into power-matching, although this is not strictly required. It is possible to obfuscate the seeds such that teams cannot tell who actually has what seed, and that is done at NAQT Tournaments and Scobol Solo.
Power-matching is essentially equivalent to Swiss pairing based only on wins rather than other statistics, and done in advance rather than adapting during the tournament to avoid repeats.
Card System
A card system is by far the most common way to implement power-matching. (It can also be implemented using individual instructions given out after each match.) Teams are given cards with a prominent number and a schedule of when the team holding the card at the time will play the team holding a specified other card in each round (as well as other auxiliary information). Game officials have a "room card" indicating what cards should appear in what rounds. (You can see a sample team card and a sample room card from the 2017 HSNCT.)
After each round, the winning team takes the card whose number is closer to 1 and the losing team takes the card whose large is closer to n, where n is the number of teams playing on the card system. It is important to explain this carefully; in particular, do not say or write "the winning team takes the lower-numbered card", since although this is mathematically correct, lower-numbered cards correspond to higher rankings within the tournament, which may result in confusion.
Examples
Clean Example
Power-matching works out very cleanly if the number of teams is a power of two and the number of rounds is the exponent of that power of two. For example, if there are 16 teams that play 4 round, then there will be one undefeated team and no chance of a team playing the same opponent twice or a match in which the two teams have different records. In such a case, the number of teams with each final record can be found using either Pascal's triangle or combinations, so there will be one 4-0 team, four 3-1 teams, six 2-2 teams, four 1-3 teams, and one 0-4 team.
To design such a system, start by figuring out who the first round opponents will be and who the 1 card will play over the course of the four rounds. The first round opponents should be set so that all pairings add up to the same number, which is one plus the number of teams. With 16 teams, the first round pairings should be 1 vs 16, 2 vs 15, 3 vs 14, 4 vs 13, 5 vs 12, 6 vs 11, 7 vs 10, and 8 vs 9. The opponent of the 1 card each round should be the previous round's opponent divided by 2, so over the course of four rounds the 1 card should play the 16 card, then the 8 card, then the 4 card, and then the 2 card.
For the second round opponents, pair up first round matches. The winner of 1 vs 16 from the first round will play the winner of 8 vs 9, so those matches are paired up with winner vs winner, and loser vs loser. Likewise, 2 vs 15 is paired with 7 vs 10, 3 vs 14 is paired with 6 vs 11, and 4 vs 13 is paired with 5 vs 12. The resulting pairings are 1 vs 8, 9 vs 16, 2 vs 7, 10 vs 15, 3 vs 6, 11 vs 14, 4 vs 5, and 12 vs 13.
For the third round opponents, pair up the pairs of second round matches. Because you want 1 vs 4 in the third round, take the grouping that contains cards 1, 8, 9, and 16, and pair it up with the grouping that contains cards 4, 5, 12, and 13. Each group has one 2-0 team, and those teams should play each other. Each group has two 1-1 teams, and those teams should play each other so that they cross groups and so that the stronger 1-1 team from one group plays the weaker 1-1 team from the other group. The 0-2 teams should also be paired up. The resulting matches are 1 vs 4, 5 vs 9, 8 vs 12, and 13 vs 16. The other third round matches, which are set up the same way, are 2 vs 3, 6 vs 10, 7 vs 11, and 14 vs 15.
In the fourth round, each match should feature one team from the group 1, 4, 5, 8, 9, 12, 13, and 16. The opponents should be from the group 2, 3, 6, 7, 10, 11, 14, and 15. That way, there is no chance of two teams playing each other a second time, because teams will only have played against opponents in their own group. The 3-0 teams will have cards 1 and 2, and they should play each other. The 2-1 matchups should be 3 vs 8, 4 vs 7, and 5 vs 6. The 1-2 matchups should be 9 vs 14, 10 vs 13, and 11 vs 12. The 0-3 matchup will be 15 vs 16.
Messy Example
If there is a situation such as 12 teams playing 4 matches, then the schedule will not work out cleanly. In such cases, there will be some matchups in which teams have different records, and there will also be a chance of teams playing each other twice.
With 12 teams, it makes sense to start with 1 vs 12, 2 vs 11, 3 vs 10, 4 vs 9, 5 vs 8, and 6 vs 7. After the first round, there will be six 1-0 teams and six 0-1 teams. For the second round, you can make three pairs of the original six matches, and get the pairings 1 vs 6, 2 vs 5, 3 vs 4, 7 vs 12, 8 vs 11, and 9 vs 10. However, after two rounds there will be three 2-0 teams, six 1-1 teams, and three 0-2 teams. You cannot just have the 2-0 teams play each other, because three is an odd number. In addition to that, you want to consider the cards 1, 6, 7, and 12 to be in a group that should not play each other because of a possible repeat match, and the same goes for the cards 2, 5, 8, and 11, and also for the cards 3, 4, 9, and 10.
In such a case, you can have the pairings of 2 vs 3, 1 vs 9, 5 vs 7, 6 vs 8, 4 vs 12, and 10 vs 11. The winner of 2 vs 3 will have the 2 card and be 3-0, and the loser will have the 3 card and be 2-1. The winner of 1 vs 9 will have the 1 card and be either 3-0 or 2-1 depending on which team wins, and the loser will have the 9 card and be 2-1 or 1-2. The winners of 5 vs 7 and 6 vs 8 will have the 5 and 6 cards and be 2-1, and the losers will have the 7 and 8 cards and be 1-2. The winner of 4 vs 12 will have the 4 card and be either 2-1 or 1-2 depending on which team wins, and the loser will have the 12 card and be 1-2 or 0-3. The winner of 10 vs 11 will have the 10 card and be 1-2, and the loser will have the 11 card and be 0-3.
In the fourth round, you want the teams that could be undefeated playing each other, which is 1 vs 2. Similarly, you want the teams that could be winless playing each other, which is 11 vs 12. The 3, 5, and 6 cards are definitely 2-1, and the 4 card is likely to be 2-1, so those 4 teams should be paired up in the last round. Similarly, the 7, 8, and 10 cards are definitely 1-2, and the 9 card is likely to be 1-2, so those teams should be paired up. When it is all done, there will be 0 or 1 4-0 teams, anywhere from 1 to 5 3-1 teams, anywhere from 2 to 8 2-2 teams, anywhere from 1 to 5 1-3 teams, and 0 or 1 0-4 teams.