Designing Tourneys for the Riddler

One of my favorite web sites,, runs a weekly feature by Oliver Roeder called the Riddler, which each week poses a couple of interesting (and usually difficult) questions in math, logic, and probability.

This week’s “Riddler Classic” question concerns tournament design. Given a competition in which the better player wins two-thirds of the time, and that you only care about maximizing the probability that that best player wins, how should your construct a blind-draw tournament with these rather severe constraints: four entrants and four total games; and five entrants and five total games.

(Note that this success criterion is the same as tourneygeek’s earliest measure of fairness (C). So we can relate this challenge to the frequent discussions of fairness (C) elsewhere on this site.)

Now, I’m unable to imagine any competition in which the better player wins two-thirds of the time regardless of the size of the skill differential between the two players. The best player ought to beat the worst player rather more often than they beat the next best. I can see why the Riddler doesn’t want to put its readers to the trouble of using a more realistic match model. But since I have one ready to use, I’m going to seek a solution using my simulator, in which skill levels are handled rather better. In deference to the question as posed, however, I’ll use a skill parameter of 2.4, which is about what’s required to give the better of any two randomly-chosen players a two-thirds chance to prevail.

My solution is below the fold – before looking at it, you might want to give it a shot yourself.

Continue reading “Designing Tourneys for the Riddler”

Do Two Singles Make a Double?

An interesting design problem from a reader.

The tourney is for “Cold War”, a two-handed version of the famous board game Diplomacy. As many as 64 players are expected. The chief limitation is time – apparently even in its two-handed form, the game takes a long time to play. So minimizing the number of rounds is crucial.

A single elimination 64 bracket takes six rounds to play. A full double elimination can take as many as 13 rounds, but since time is at a premium, that can be pared down to 10 rounds. One round is saved by not having a recharge, and two more by shifting the lower bracket.

But an unusual feature of this particular game offers a third possible bracket structure. Each player can easily play two games at the same time! So perhaps we can give players a second chance by simply playing two separate 64 brackets at the same time, with a playoff between the two bracket winners as a seventh round if the two brackets are not won by the same person.

This is marvelously efficient. Each player gets to play until they lose twice, but you save at least three rounds, and maybe four if you would otherwise insist on a recharge, because a recharge is never needed.

How should such a bracket be drawn, and how does it compare with conventional single and double elimination brackets in the item of fairness (C)?

Continue reading “Do Two Singles Make a Double?”

TGT: Fairness

Here is another draft from my slowly-growing manuscript, Tourneygeek’s Guide to Tournaments. TGT Fairness

Only the first five pages, discussing fairness in general and fairness (A), are new, but because this now completes a first full chapter I’m also including the previously posted parts of the chapter than discuss fairness (B) and fairness (C).

As before, comments and criticisms are welcome.

Setting the Luck Parameter

One interesting bit of NCAA tourney trivia is that this year, apparently for the first time ever, one of the tens of millions of brackets submitted to online bracket challenge games was entirely correct through the first two rounds, or 48 games, of the tourney.

The chance that this bracket will remain perfect through the remaining 15 games is very small, but there will be many eyes on it. One year, there was a $1,000,000,000 prize offered for a perfect bracket (though, apparently for legal reasons, that prize is no longer on offer).

In a related discussion, I ran across a tidbit of expert opinion that may be useful in helping calibrate tourneygeek’s simulator.

Continue reading “Setting the Luck Parameter”

Learning from Ottawa, part III

In the last post, I compared the double-elimination portion of the Ottawa Men’s Bonspiel curling bracket against a slightly revised version of itself. But the more important comparison to make is with the a more conventional version of a double-elimination for 91 teams. And, as I’ll show, comparing the Ottawa bracket to a more conventional approach unexpectedly seems to open new avenues of inquiry for bye management.

As discussed in BBBR: 128s, there are a number of suitably-sized brackets from which to choose a comparator. In deciding which to use, I came upon another surprising virtue of the Ottawa-adapted format: it runs in just ten rounds, and there are almost no long waits except those caused by byes. The quickest of the 128s is the “128supershift“, which takes 11 rounds. So, to capture as much of the virtue of the Ottawa adaptation as possible in a (not very) conventional format, I chose the supershift.

Continue reading “Learning from Ottawa, part III”

Learning from Ottawa, part II

In the previous post, I suggested that the first parts of the Ottawa Men’s Bonspiel design could be adapted to a large double-elimination tourney in other contexts. It’s time to put it to the test.

In the last post, I linked to brackets for the Ottawa design, both (nearly) as it was played in Ottawa, and with a few technical adjustments intended to improve fairness. In this post, I’ll look at the effect of those technical adjustments.

Continue reading “Learning from Ottawa, part II”