Short challenges Kasparov: who will win?

9/23/2005 – In less than a week the FIDE World Championship will take place in San Luis. Garry Kasparov has said that there is a 95% chance that one of the trio Anand, Topalov or Leko will take the title. Nigel Short, who is the official commentator at the event, would like to wager money on one of the other five – at 17:1 odds. What are the true chances?

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The Chances of Winning

By Nigel Short

Prognostication is an art fraught with error. If we are to be the least bit scientific when attempting to assess the likely victor of the World Chess Championship in San Luis, Argentina, it makes sense, as a starting point, to defer to the opinion of the highest authority – that of the recently retired Garry Kasparov. In his New In Chess column, (issue 6/2005, page 105) his eminence forthrightly states “I’d put the chances of the San Luis winner coming from the trio of Anand, Leko and Topalov as high as 95%. Two of the top three spots will most likely be occupied by this trio. It's hardly going out on a limb to call Anand a slight favourite, while the length of the tournament improves Topalov's chances.”

Fair enough. I have learned, through long and very bitter experience, never to underestimate the opinion of the greatest chess player of all time. However I believe, on this occasion, that the big Russian has got it quite wrong. Put another way – do such immense talents as Svidler, Morozevich, Adams, Polgar and Kasimjanov together really have no more than a 5% of winning? Rephrased again, does Kasparov seriously believe that is nineteen times more likely that Anand, Leko and Topalov will win than the other five?

World Championship 1993. Before the qualification was over Kasparov
famously predicted: "It will be Short and it will be short!"

If so, Garry will no doubt be willing to put his money where his mouth is. I am by no means a betting man, but would be quite happy to publicly wager a modest $100 at 17-1 odds that one of my five players triumph. Probably I will lose, but the prospect of collecting $1,700 is sufficiently enticing, should I prevail.

Don’t get me wrong – like Kasparov, I believe Vishwanathan Anand to be the favourite. I also believe that Peter Leko and the ever-combative Veselin Topalov have very good chances. But please, please, do not dismiss the others too lightly. It is only one tournament, of a mere 14 rounds, and anything can happen. Who rated the almost unknown Harry Nelson Pillsbury at the start of Hastings 1895? That is too long ago to be relevant you may say: but one need only remember the astonishing recent triumph, in Dortmund this year, of Arkady Naiditsch – who is quite frankly nowhere near the calibre of the four-times Russian Champion Peter Svidler or the brilliant, although erratic, Alexander Morozevich – to see what is possible. Michael Adams has good nerves. He will fear no silicon tormentors in San Luis. Judit Polgar, the one woman in the field, will undoubtedly command a devoted following. She is a fascinating outsider. And Rustam Kasimjanov – the World Champion, lest we forget – should certainly not be ignored.

Ladies and gentlemen, we are in for a treat. Theoretical novelties, prepared in home laboratories, will play their part in determining the outcome, to be sure. But nerves and strength of character will count for more. Enjoy the spectacle!

The probability game in San Luis

By Frederic Friedel

Probability is a tricky business. There are a lot of cases in which reality contradict intuition, something that shown up in numerous probability paradoxes that are circulated in mathematics classes and social gatherings. One of the most famous is the Birthday Paradox. How big must a group of people be in order for you to be reasonably certain that two of them will share the same birthday?

The intuitive answer often given is 183, i.e. 365 divided by two. The correct number is not something people easily arrive at, certainly not by intuition. The surprising fact is that if you have a gathering of 23 people the chances of two of them having the same birthday is better than 50%.

The solution does not seem quite as surprising if you put the question in a different way: if 23 people are gathered in a room, what are the chances that no two of them will have the same birthday? And this is the simplest way to approach the mathematics of the paradox. We calculate what the chances are of not sharing birthdays.

Assume you are alone in a room. The chances that every person in the room has a different birthday is obviously 100% – or in the language of probabilities: 1. Now a second person enters. The chances that he or she will have a different birthday to you is 364/365 (we are going to ignore leap years in this calculation), or 0.9973, which is the same as 99.73%. A third person enters. The chances that this person has a different birthday from both you and the second person is 363/365. The chances that all three have different birthdays is 364/365 times 363/365, or 0.9918.

So the chances of 23 people having different birthdays is 364/365 * 363/365 * 362/365 * 361/365 ... 343/365, which comes out to 0.493. This means there is a 49.3% chance of everyone in the room having different birthdays, and conversely a 50.7% chance of at least two sharing the same birthday.

Numerically and graphically the chances for different numbers of people sharing a birthday are as follows:

No. prob   No. prob
5 0.027 30 0.706
10 0.117 35 0.814
15 0.253 40 0.891
18 0.347 50 0.970
20 0.411 60 0.9951
23 0.507 70 0.99916
25 0.569 80 0.99991
27 0.627 90 0.99999

The conclusion of the birthday paradox is: if you have 23 random people at a party, go ahead and bet that two have the same birthday. If there are 50 people present then you could easily give 30:1 odds that two share a birthday.

Which brings us back to chess as a betting person's game. Let us take a look at what the professional wagering odds for the San Luis tournament are. The site Betsson allows you to place money on all types of sporting events, so too on the FIDE world championship. Note that here the odds are not calculated on the basis of scientific data, like Elo points, or even by Betsson itself. They simply reflect the opinions of the people who are placing the bets. If someone places a large bet on one of the players the odds automatically sink. Currently the odds are as follows (but note that they are changing all the time):


The three columns are to be interpreted as follows: the first gives us the odds and tells us what you will get if you bet on a specific player and win. Betting $100 on Anand will net you $280 if he actually wins; you get $1600 if you bet on Svidler and he wins; and there is $5000 waiting for you if you put the money on Kasimdzhanov.

The next two columns tells you what the probability of each player's winning the title is. Note that the Betsson probabilities add up to 102.6%, because they include the commission the company takes. If they gave you the exact percentage, as shown in the third column, they would not make any money. Incidentally the Betsson commission (or betting margin) is very small, certainly far less than most bookmakers on High Street.

According to Betsson – or more precisely the people who are placing wagers there – the odds of one of the trio of Anand, Topalov and Leko winning the event is 78.7% (with the betting margin). You cannot place such a wager, but they would give you odds of 1.3 if you could. Conversely the combined odds of one of the other five winning in the opinion of bettors is 23.9%, and they would give you odds of 4.2. Which means that betting $100 would net you $420 – and that Kasparov would be giving Nigel Short unreasonably favorable odds, if he would indeed accept the wager that Nigel is offering him.

Post Scriptum

Paul Prescott of Callander, Scotland, draws attention to an inaccuracy in the final paragraph. "You say that one cannot place a wager on one of the trio Anand, Topalov or Leko winning, and that such a wager would give you odds of 1.3 if you could. But you can in fact place such a wager. Simply bet $45.35 on Anand, $30.23 on Topalov and $24.42 on Leko, for a total bet of $100. Whoever wins, you get fractionally less than $127. More generally, you can make any compound bet you like by dividing your stake between the players in proportion to their chances as shown in the odds."

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