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):
Player |
Odds |
Betsson |
True |
Anand |
2.8 |
35.7% |
34.8% |
Topalov |
4.2 |
23.8% |
23.2% |
Leko |
5.2 |
19.2% |
18.8% |
Svidler |
16 |
6.3% |
6.1% |
Morozevich |
18 |
5.6% |
5.4% |
Polgar |
20 |
5.0% |
4.9% |
Adams |
20 |
5.0% |
4.9% |
Kasimdzhanov |
50 |
2.0% |
2.0% |
|
|
102.6% |
100.0% |
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."