I subscribe to The Cricketer, a glossy cricket magazine published in England. My copy arrives in Indiana by way of Budapest after about two months transit time. But it’s worth waiting for. In the last issue I received, there’s an interesting tournament design problem relating to the scheduling of round robins.
Here’s a small extract from a Cricketer article by Wasim Kahn, the head of county cricket for Leicestershire:
The idea behind the seeding system was that when you have an asymmetrical system such as 10 teams and 14 matches, if you are at the top end, you might not play some of the other top teams, and others will. There needed to be a fairer, robust seeding system. They are using experts who do this at top-level sport, so everybody buys into that. [The Cricketer, v. 16, no. 4, p. 13.]
Who are these experts? Apparently there’s a cadre of tourneygeeks in England who command universal respect. I’m in the wrong country. But wait. On the same spread there’s another article by BBC cricket expert David Townsend:
Now I have no problem with the invention of a ‘science’ that enables the thick but athletic sons and daughters of middle-class parents to accumulate as much student debt as their peers, but not when it comes to messing with the cricket season. [Ibid p. 12].
So maybe the tourneygeeks are not so universally respected, but at least they’re acknowledged to exist.
I’m a little hurt that the England and Wales Cricket Board did not seek me out for my expert advice. But I’m not one to hold a grudge, so I’ll solve their problem for them anyway. What’s a special relationship for if I can’t reach out across “the pond” to come to the aid of my thick but athletic colleagues.
Here’s the problem. Cricket is a Summer game, and the English Summer simply isn’t long enough to accommodate all of the cricket that people want to play. Back in the day, there was enough time, but that’s before the ECB decided that it really needed to run a full season for each of the three forms of the game.
Now there are 18 counties that play first class cricket. For many years, these were split into two divisions of nine teams each. There were 18 rounds in the championship, so the season could be run on a scrupulously fair pattern, with each team playing each other team twice, once at home, and once away, and drawing two byes.
But starting in 2017, the ECB decided that there was only time enough for fourteen rounds of first class cricket if there was also going to be time to play both T20 cricket and one-day cricket. The top division was limited to eight teams, so that it could continue to play a full double round robin. But the lower division, with ten teams and only the same 14 rounds would have to settle for less than a complete round robin.
But people are unhappy. Some of the teams that have fallen out of the top flight are old stalwarts of English County Cricket like Middlesex, who play their home matches at Lord’s (FN1). The powers that be want to expand the top division to ten teams, shrinking the second division to eight. And, while they’ve had a couple of years to work out how truncate the round robin for the ten-team division, they’re only worrying about it being really fair now that it’s the best teams that might suffer an inequity.
So, how do you decide which four rounds of the double round robin should get the axe?
The trick is not to include, in the second set of pairings, any in which the seeds are both even or both odd. Thus, seed 1 will have repeat fixtures against 2, 4, 6, 8, and 10, but not against 3, 5, 7, or 9. Seed 2 gets a second chance at 1, 3, 5, 7, and 9, but not at 4, 6, 8, or 10. And so forth.
Here’s how it works out. Start with a single, full round robin of the ten teams:
Since you want every team to play every other team at least once, you can use these nine sets of pairings directly. Now you need to generate another five rounds.
Start with the same nine rounds, but this time mark every match in which the seed numbers are both even or both odd:
These are the matches you don’t want. Reassemble the bits left over into five new rounds, which will be added to the first nine to make the full, 14-round schedule:
The point is that observing the even/odd rule means that the repeated matches will be distributed as evenly as possible through the seeds. It’s true that odd-numbered seeds have a small advantage. They always, for example, get to play the worst seed twice and never have to play the best seed twice. But absolute equality is not possible.
That’s it. I hope it helps.
We’re worried about you, over here – worried that about the time balls start crashing into stumps this year Britain will also be crashing out of the European Union.
On the bright side, if that happens my copy of The Cricketer may not be routed through Budapest.
FN1: Lord’s is probably the most iconic cricket ground in England. It is so named not because it is frequented by male peers, but because it was originally developed by a man named “Lord”. Still, it is frequented by peers, and perhaps the name makes them feel a little more at home there.