A Modest Proposal

The previous post discussed a way in which the game of tennis is broken. The advantage accruing to the server has become so great that it is no longer sensible to play and score tennis in the traditional way.

The response of the tennis establishment has been the invention of a new type of game, the tiebreak, in which the advantages of serving are shared more equally because in a tiebreak both players get a chance to serve. And this solution is probably, now, a permanent part of tennis – it’s hard to imagine going back to the days before the tiebreaker.

There is still a need, however, to address the underlying problem lest the imbalance between serve and return grow so large that all games except the tiebreaks become meaningless. And this should probably be done in a way that requires the smallest possible alteration of the existing rules of tennis.

I have a modest proposal.

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The Broken Serve

Tourneygeek is enjoying its annual pilgrimage to Mason, Ohio, for the Western and Southern Open, one of the leading tennis tournaments leading up to the U.S. Open.

In past years, I’ve used the opportunity to explore such things as the effect of the draw on the expectations of individual players, and the effect that tennis’s distinctive tiered seeding system has on the way a tournament plays out.

This year, I’ve mostly just sat in my implausibly comfortable seats and enjoyed the tennis. But on the second day of the tourney I watched a match between John Isner and Dusan Lajovic, and this put me in mind, again, of the fact that one element of tennis, the serve, is broken.

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The Last Chance Salon

The last post discussed the two alternatives for the Consolation bracket needed for an upcoming tournament. ┬áBut in addition to a Consolation, my friend also needs a Last Chance bracket. Third brackets are almost always rather difficult brackets to build, and they depend on the second brackets they’re attached to in the same way that second brackets relate to first brackets.

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Be Squirrelly Early

In the last post, I started showing how to build a bracket for the consolation in my friend’s tourney, which he expects to draw up to 48 entrants. In showing the method, I digressed to show how it generated the more familiar 64 brackets. Now let’s return to the problem of building the 48.

Along the way, I’ll introduce a new half maxim of tourney design: be squirrelly early.

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Building Lower Brackets

A friend asked me for a 48 bracket for a tournament he’ll be running next month. The tourney will have a main flight, a consolation, and a last chance. He wants to pay two places in the main, and two more in the consolation.

Rather than just giving you the bracket I came up with, let me show you the process by which the bracket was built. The will take a few posts. For the first one, I’ll begin the process, and then digress to show how second brackets are built for a 64 main.

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Designing Tourneys for the Riddler, Corrected!

One of my favorite web sites, FiveThirtyEight.com, 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.

I was dead wrong! For the correct answer (or at least a better one), scroll down to the comment of Donald the Potholer. His “Page Ladder” brackets improve on the ones I found. For the 4/4 case he succeeds 49% of the time, and his 5/5 bracket succeeds 44.8%.

I’ll rewrite this post with new analysis after the official results have been announced.

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