The cheating coach redux

A few of you might remember a post I wrote last year about a really cool math problem that a friend sent to me. For those who don’t remember, here it is again:

Two chess teams are going to play a chess match. The best players on each team are supposed to play on board 1, the second best players on board 2, etc. However, the coach of team B thinks he can win more games by putting his players in a different order. Is he right? If so, what is the best order for the cheating coach to put his players in?

Fortunately, shenanigans like this are not possible in the actual chess world because we have a rating system that keeps the coach from intentionally putting the players in the wrong order. However, in a world with no rating system this would be a real problem. If there are 3 players on each team, it turns out that the best order to put them in is 3-1-2. If there are 4 players, the best order is 4-1-2-3. In each case the idea is to “throw” the game on the first board in order to improve your odds on the other boards. For example, in the four-player match, the cheating team will win on average by a score of 2.34-1.66 if they adopt this strategy.

(Technical note: The above calculation is only true in a world where upsets never happen, i.e., the better player always wins. If upsets do happen, I suspect the effect would be to reduce but not eliminate the cheating team’s advantage.)

Now let me get to my news. Recently I wrote an article about the cheating coach problem for a science website called Nautilus. If you’re interested, the article — which also explains the optimal strategy for teams of N players — is now online.

But that’s not all! The problem is also going to be featured in the New York Times “Numberplay” blog, written by Gary Antonick. Even if you’ve read my Nautilus article, I think you will find Antonick’s post to be interesting because we pose the problem in a different way. Gary thought that the original version was a little bit too complicated for readers to understand, and I kind of agree with him. The fine print about no upsets is, unfortunately, a necessary assumption to get a unique solution, and that assumption is both unrealistic and detracts from the simplicity of the problem.

So we worked out a different way to pose the problem, in terms of a card game, which I think is simple and elegant and really the best way to do it. It distills the problem to its essence and also, I think, greatly enhances the surprise factor. The card game seems as if it must be fair, when in fact it is biased in favor of one player. It should be a hustler’s delight!

Anyway, I won’t say any more about Gary’s article now, because I don’t want to scoop him, but it think it will be really, really cool. I’ll let you know when it comes out.

P.S. As an unrelated aside, this blog got 305 hits on Tuesday, the day of my “Anatomy of a Meltdown” post. That’s the first time that I have topped 300 hits in one day except during the 2010 and 2012 world championship matches. (During those matches I was translating GM Sergey Shipov’s commentary, and got as many as 2143 visitors in a single day.) Thanks to all 305 of you! In general there seems to be a subtle but unmistakable increasing trend in the number of readers of “dana blogs chess,” so I must be doing something right.