The Powers of Two

Visitors to the printable brackets page will see that all of the brackets there are designed for tourneys with a number of entrants that’s an even power of two: 16, 32, 64, or 128. There are reasons for that, relating to the basic fact that an individual match is between two players or teams. When the number of entrants is not a power of two, some measure of unfairness is usually unavoidable in any elimination-style format.

In this post, I’ll discuss the common practice of using the results of a preliminary competition, usually some sort of round-robin regular season, to seed a championship playoff that observes, at least in its later rounds, the power of two constraint. In the next, I’ll consider some other ways of dealing with the awkward situation of running a tourney with a number of entrants that’s not a power of two.

In the case of a knock-out, or single elimination, tourney, the unfairness stems from the fact that the path to victory is longer for some entrants than for others. If there are 32 entrants, for example, each team in a full, balanced bracket needs to have five wins to win the tourney. But if there are 24 entrants, eight of the entrants will draw byes in the first round, to that those eight need only four wins, while the other 16 need five. Things are a little more complicated in a double elimination, but the same basic pattern obtains.

In many contexts, then, tourneys are organized so that they begin with a number of entrants that is an even power of two. The playoffs in the NBA and the NHL each start with 16 teams that qualify from among the 30 (or 31 in next year’s NHL) that play the regular season.

Other sports embrace the imbalance inherent in a number of entrants that’s not a power of two to enhance the importance of the regular season. The NFL playoffs involve twelve of the leagues 24 teams, four of which earn a first-round bye as a reward for a superior record in the regular season.

Major League Baseball qualifies ten of its 30 teams for the playoffs, but requires the four of these that did not win their divisions in the regular season to play a chancy single-game play-in round to arrive at a bracket of eight.

All 14 of the “Big 10” basketball teams are allowed to compete in the conference championship playoffs, but they are seeded by their regular season results into three groups: four that play in the first round, six with a one-round bye, and four that get a double bye into the quarter finals.

Whether these various arrangements are fair or not depends on how you define the competition. As far as the playoffs themselves are concerned, they are distinctly unfair. But if you consider the playoffs together with the regular season as part of a single large competition, they seem eminently fair – not as fair, perhaps, as simply awarding the championship to the winner of the regular season, as happens in the English Premier League, but more fair than just choosing a winner with a short knock-out format.

For my money, the enthusiasm of organizers for creating dramatic loser-goes-home matches with a knockout tourney has gotten out of hand in professional sports. The most recent Cricket World Cup, for example, had 42 matches that reduced the field from 14 teams to eight. Given that there were eight teams widely regarded as clearly superior to the others, these 42 matches were expected to resolve very little, leaving it to the seven playoff games to select the overall winner. This format was apparently designed to ensure that all of the teams that represented major television markets would make the playoffs, though the ineptness of the English side thwarted that plan, allowing Bangladesh to advance instead.

The value of seeding the playoff in such a way as to add significance to a playoff is compelling enough that you sometimes see it when the number of qualifying teams is already a power of two. The Indian Premier League (which plays cricket) uses the page playoff system:

page playoff

This not only creates another playoff match, but is also adds significance to the regular season by giving a big advantage to the top two qualifying teams. This and similar techniques are used in some other competitions, including softball and curling, and should be more widely adopted.

The most extreme use of differential seeding for a playoff is the ladder format used in many professional bowling tourneys:

bowling ladder

Any of the top five qualifiers can win the tourney, but none can lose more than a single place with respect to their qualifying performance. I’d argue that this format, despite the extreme imbalance of the bracket, is the fairest of the lot. It manufactures dramatic, televisable matches at the end, but uses the much fairer and more balanced qualifying competition to determine a very large part of the outcome.

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