The Rating Principle (Rating Update January 3, 2016)

Suppose Michel Lamote, BIC 2015 champion, and Guy Van Middelem, play a 13-point match. What effect will the outcome of that match have on both players’ ratings? The winner’s rating should go up, of course, and the loser’s rating should go down. But by how much? What is the rating principle?

Let us admit that Michel is the better player. Then Michel is more likely to win the match than Guy. (In the recent rating update, Michel broke his own rating record again: 1875.48, taking into account matches played up to January 3, 2016. Michel is currently more than 200 rating points up to Guy, the second ranked player on the list.)

Fair or not?

Suppose Michel and Guy would agree to the following: no matter who wins, the winner’s rating goes up by 1 point, and the loser’s rating goes down by 1 point. Sounds fair, right? Well, actually, it isn’t.

Enjoying life and quite a couple of beers, Michel and Guy play 100 matches. Michel is likely to win about 60 matches, say, and Guy about 40. Michel’s rating will thus go up by 20 points, and Guy’s will go down by 20 points.

The more they play, the more Michel’s rating will go up and Guy’s will go down. This isn’t fun for Guy. Guy will not want to play with Michel for the rating anymore. Too bad, really, for the two Brugge comrades.

Handicap

Then Guy, a passionate golf player, gets an idea.

“Perhaps I should get a handicap,” proposes Guy to Michel.

“All right, if that can convince you to play me again, why not,” replies Michel reluctantly. He has been enjoying himself quite a lot, cashing in rating points while nipping his Brugse Zot. But something needs to change, he admits. “What do you propose?”

Guy searches in his wallet, takes out a small piece of paper (it’s the receipt of last night’s dinner at the Brasseries George — hmm, that salmon was really delicious). He finds a pen and starts to scribble a few numbers. After a few minutes,  he looks up and smiles.

“Look here,” says Guy. “Of the 100 recent matches, you won 60 and I only 40. Suppose that for each match you win, you get 2 points and I lose 2 points. But for each of my victories, I win 3 points and you lose 3 points. Then after 100 matches, our ratings wouldn’t have changed.”

Michel can see the logic of that. He feels like saying okay, but then an objection crosses his mind.

The odds are changing

“Why not,” says Michel. “But what if you start to play better? I mean, you’re filming your matches, learning from your errors, reading books…. I’m even teaching you things once in a while. That cube you just took, for instance. How many times didn’t I tell you already that when you’re leading 3-away 4-away, you should be very careful with gammonish cubes? Anyway, perhaps that in six months from now, you’ll play better than you do now. You may then win 45 matches out of 100, rather than 40. Then you would get rewarded too much for your victories, wouldn’t you?”

“I see,” says Guy. Two middle-aged white-skirted lady tennis players just enter the Brasseries du Longchamp, and for a moment, Guy’s focus is elsewhere.

“A-hum.” Michel clears his throat discreetly, and Guy wakes up from his reverie.

“I’m sorry? — Right, what if I start to play better.” Guy looks at his envelope again.

“Well, then the odds should change too.” Guy explains: “As soon as I start to win more matches, my handicap should get smaller. And if you, Michel, would be winning even more matches than you do now, my handicap should become even bigger.”

Guy continues: “In fact, the rating points I can win or lose should be according to my odds of winning the match. Same thing for you. On average, once our rating difference is a good indicator of our relative playing strengths, neither you or I should win or lose rating points.”

The Rating Principle

What Guy has just discovered can be called the rating principle. Let us say that Guy’s chance of winning a match to Michel is p. Michel’s chance of winning is then q = 1-p. Let us further assume that when underdog Guy wins the match, he should get G points, and Michel would lose G points. If Michel, the favourite, would win, then Michel’s rating should go up by M points, and Guy’s should go down by the same amount. What can we say about G and M?

Guy’s explanation amounts to saying that G and M should be chosen such that the average rating update should be equal to zero. From Guy’s perspective, this update will be p*G – q*M on average: a proportion p of the matches, he wins G, and a proportion q of the matches, he loses M. We should thus have that p*G – q*M = 0. This is true provided G/M = q/p.

“Algebra was never my cup of tea, and my college days are longer ago than I’d like to admit,” says Michel. “But if you understand you correctly, you’re saying the following. The amount of points that you can win relative to those you can lose are proportional to your odds of losing the match. The more you are the underdog, the less a defeat will cost you and the more a victory will profit you.”

“That’s it, my friend,” answers Guy. “Another beer, perhaps?”

“Thank you, no, I’m driving,” says Michel. “By the way, you seem to know the lady on the left. Could you perhaps introduce me to her?”