FEAT
EATURE ARTICLE
ANANT V. PATKI
As Viswanathan
Anand gets
ready to defend
his World title
in November at
Chennai, here’s
a look at how
chess players
are rated at the
international
level.
S
CIENCE and its methods allow us
to study and understand nature.
Mathematics stretches it further, helping
us to build simple analytical models that
describe how some of these patterns,
processes or the outcomes can be
predicted. With such an extraordinary
ability, science and mathematics have
become essential and inseparable parts of
all human endeavour.
Sir Isaac Newton was the first
to revolutionize our thinking power
and bring out the amazing power of
mathematics in natural processes. His
famous gravitational law, so simple and
elegant in terms of mathematics, is a key
to many of our activities. And though
nobody has been able to prove it as yet,
it works; we use it so extensively and
had our successful mission to Moon,
Chandrayaan, based on it. This power of
mathematics is also exploited in sports,
hobbies and games. The Elo rating system
in Chess is one such useful product.
SCIENCE REPORTER, NOVEMBER 2013
Chess is a popular game, played all
across India for hundreds of years. It is a
good entertainer where many onlookers
join the players, suggesting better moves,
in effect converting it into virtually a team
game. This popularity is probably due to
its inherent low resource requirement. All
it needs is a small desk-top board, chess
coins and two players. Often when the
coins are missing, they are substituted by
small household objects, sometimes even
by stone pieces and sand granules.
The game is played between two
opponents and replicates the battlefield
situation prevailing in war. The field
is simulated by board with 64 squares
coloured black and white alternately.
Each player has an army of 16 pieces, a
king, a queen, 2 rooks, 2 bishops, 2 knights
and 8 pawns, placed on opposite sides
of the board to start with. Each piece has
to follow certain pre-determined ways
of movement. The aim of the game is to
capture the opponent’s king, which results
26
in a win. The game can often result in
a draw offering equal scores for both
players, by certain rules or by mutual
agreements.
A quick game can end in about 25
moves but some games could even extend
to over 80 moves. Though the numbers
FEATURE ARTICLE
The names of the pieces are now
Rook, Bishop and Queen rather than
Elephant (Hathi), Camel (Oont) and
Minister (Wazir). The original boards
of plain squares are now changed to
chequered ones, making it easy for the
bishop’s movement. To accelerate the
game time limits are set. On average, a
player needs to make 20 moves in an hour.
The Britishers took keen interest
in chess, promoted it and had the first
international (actually European) event
when we were fighting our first war of
independence in 1857. It is the national
sport in Russia and is more popular
than football. Russian chess players have
dominated world chess after the Second
World War for over 60 years.
of pieces are limited and have restricted
ways of advancement, the game produces
huge numbers of possible alternatives.
In fact, within the first few moves itself,
the board position can result in one of
thousands of possible situations.
There are many claims on the origins
of chess, but now it is widely believed
to be of Indian origin, where its early
form in the 6th century was chaturanga,
which translates as “four divisions of the
military” – infantry, cavalry, elephants,
and chariots, represented respectively
by pawn, knight, bishop, and rook.
The earliest reference is found in some
versions of Ramayana where King Ravana
was said to be good at the game of chess.
There are stories where kings (real ones)
used to sponsor chess-fights with another
kingdom with many people witnessing
the game real-time. Often, even clashes
between kings were resolved based on
chess matches between their nominees.
The game was taken to Persia
thousands of years ago (acquiring the
name ‘shatranj’), from where it slowly
moved to Europe and then to Russia. Over
the years, traditional chess has become an
international game.
The number of pieces, their power
and movement laws have essentially
remained the same over centuries.
Nevertheless, rules for the international
games
have
undergone
many
modifications over the game played in
traditional India. The king’s position,
the double move option for pawn, the
castling and choice of wide option on
pawn reaching the eighth rank are the
major differences.
The game of chess combines elements
of technique, theory and intuition. Today,
it is one of the world’s most frequently
played indoor games. It is also the one
practiced most widely, has the mostdocumented results, most written books
and theories to help practitioners. One can
also find records of games played over a
century ago. This popularity is due to its
simplicity and low resource requirements
and also intellectual excitement and
stimulation that it offers. The game has no
place for luck unlike card games.
Not only for players, even for a
follower, analyzing a good chess game
is very stimulating and is an exercise
in logic. Chess is said to improve one’s
intelligence. So much so that many a
computer algorithms are tried on chess
routine first. In fact, chess programs are
used to incorporate artificial intelligence
and to test their robustness. Finally, it
is a mind game and helps one to keep
the mind active, especially in old age,
lowering risks of memory degeneration
and Alzheimer.
Today, chess is a well organized game
at the state, national and international
level. At the international level, the
World Chess Federation (Federation
Internationale des Echecs, known as FIDE
from its French acronym) organises chess
championships. FIDE also maintains
a record of all events, gives ratings to
all players and updates these ratings
regularly. This rating, known as the Elo
rating, helps organisers to sort players in
different categories or classes and arrange
the events accordingly.
FIDE implemented the Elo system
in 1970, to track the relative strength
of all chess masters worldwide.
27
27
SCIENCE REPORTER, NOVEMBER 2013
FEATURE ARTICLE
a player with rating of 200 points higher
than his opponent has 75% probability
of winning a game or a set of games.
The third premise was on the maximum
weightage per game, called K value,
which was set to 32.
Let us take a simple example to
understand the concept. Say, Two players
A and B play six matches and A wins five
of them. Now the current rating for each
player, based on his past performances,
should be known at the beginning of
the tournament. Assume they are new
entrants, and the committee has allotted
them rating of 1400 each. This means they
are expected to win three games each.
Now we use the rating formula,
For A, Da=(Aw-Ae) x 32, where
Fig. 1: Win probability for a given difference in rating
This revolutionary rating system was
devised by Prof. Arpad Emrick Elo. He
was born in Hungary, but moved to the
United States in his early days. A professor
of physics, he was a devoted chess
player and won the State Championship
several times. As the chairman of the
rating committee, he proposed a system
of rating, which was adopted by the US
Chess Federation in 1960.
Further, he applied this rating
system to tournament results dating back
to the last century, making it possible to
compare the relative strength of any two
chess players operating over different
time periods. Now this rating system is
also practiced in many other competitive
games.
Elo made very simple assumptions
in the model proposed, which made the
calculations leading to rating changes
very easy, one of the greatest assets of this
system. A chess participant can himself
calculate what his next officially published
rating will be, with a pocket calculator
based on the result of tournaments he
has played. This brought transparency to
the rating system thereby making it quite
popular.
Elo’s system is based on probabilistic
approach. It is assumed that the mean of
the performances of a player is stable but
varies from game to game as a normally
distributed variable. Performance is
inferred from results of the games he
played and the ratings of the opponents.
Assume two new players are playing
a chess game. When a player wins, he
gets a higher rating, if he loses his rating
SCIENCE REPORTER, NOVEMBER 2013
is lowered and if the game is a draw,
the two players would have the same
rating. The expectancy of win depends
on the difference between the ratings of
the two players and is given by normal
distribution (Figure 1).
As far as the actual number scale
is concerned, the Elo system has three
basic premises. First is about the initial
ratings presumed. A good beginner
would start with a rating of 1000 and a
regular, competitive player can have a
rating up to 1800. The second is about the
class or category concept. Here 200 point
difference is defined as a class, meaning
Sir Isaac
Newton was
the first to
revolutionize
our thinking
power and
bring out the
amazing power
of mathematics
in natural
processes.
His famous
gravitational
law, so simple
and elegant
in terms of
mathematics, is
a key to many
of our activities.
28
Da is the change in rating of A
Aw is the number of wins by A and
Ae is the expected wins of A
And then, New Rating = Old Rating +Da
This gives us Da= (5-3)*32=64
So after this event the rating of A
becomes 1464 and that of B becomes 1336.
Based on this simple theory, a
detailed procedure is followed by FIDE
to update the ratings after every major
tournament: start with individual’s
ratings, find his expected score against
each opponent, input the actual score, use
FEATURE ARTICLE
TABLE 1: WIN PROBABILITY FOR A GIVEN DIFFERENCE IN RATING
diff
-640
-600
-560
-520
-480
-440
-400
-360
-320
-280
-240
p
0.015
0.022
0.029
0.040
0.053
0.069
0.089
0.112
0.140
0.173
0.209
diff
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
p
0.250
0.272
0.295
0.318
0.343
0.368
0.394
0.420
0.46
0.473
0.500
diff
0
20
40
60
80
100
120
140
160
180
200
p
0.500
0.527
0.554
0.580
0.606
0.632
0.657
0.682
0.705
0.728
0.750
appropriate K value and derive the new
ratings. The expectancy of winning, based
on the normal distribution assumed,
depends on the difference in the rating
and can be calculated easily (Table 1). The
actual K value used is rather dynamic,
ranging from 32 for beginners to 16 for
masters and 10 for grandmasters, making
it harder and harder to gain points as you
climb up.
Today, chess is
a well organized
game at the
state, national
and international
level.
A further example may help to
clarify more complex situations. Suppose
three players A, B and C with ratings of
2000, 2050, 2100 respectively, play a tworound tournament. And each scores a win
and loss against the other two, scoring
two points each. K value assigned is 10.
For A, actual score is 2.
though it is known to have distinct
advantage in the real world. The system
also has a problem of tackling rating
inflation taking place due to increased
knowledge, documentations and use of
computers. The main weakness arises
from the fact that the ratings are relative
to your competitors and can be boosted
by tournaments within a controlled group
or weaker players. However, with all
these, the system on the whole has stood
the test of time for the last 50 years and is
considered the best rating system so far.
From the table,
With rating difference of -50, p = 0.43 and
With rating difference of -100, p = 0.36
His expected score = 2(0.43+0.36) = 1.58
Change in the rating, Da = 10 (2- 1.58) =
4.8 ~ 5
Therefore his new rating will be 2005.
Here rating of A has improved since
he played better than expected; he got
equal scores with higher rated players.
On the same account, rating of C will go
down by 5 points and that of B will remain
same as a result of this tournament.
The Elo rating system does have its
weaknesses. The values in the numerical
scales are assigned ones and hence
differ from organization to organization.
Further, it gives no advantage to white,
The primary goal of giving Elo
ratings is to rate, rank and classify the
players. For top players, this rating is very
important. FIDE updates the ratings list
four times a year. The categories include
World
Championship
Contenders,
Grandmasters, International Masters,
National Masters, Candidate Masters,
Experts, Amateurs and Novices. There
are about 20 players in the top most
category (rating above 2700), 150 playergrandmasters above rating of 2600 and
about 20000 rated as candidate masters
(rating above 2200). Only four players,
Garry Kasparov, Vladimir Kramnik,
Veselin Topalov and Viswanathan Anand
29
diff
240
280
320
360
400
440
480
520
560
600
640
p
0.791
0.827
0.860
0.888
0.911
0.931
0.947
0.960
0.971
0.978
0.985
have exceeded a rating of 2800 in the
history of chess.
In India, Viswanathan Anand’s
records and achievements are simply
extraordinary. He was India’s first
Grandmaster (1987), he was the youngest
Grandmaster in the world, he was the
World Junior title holder, he is the only
person in the history of chess to become
the winner in all formats of the game, and
to top it all that he is the World Chess
Champion since 2007. His achievements
have brought attention, recognition and
prestige to the game in India. Today, India
has over 30 Grandmasters, and about 100
International Masters waiting to become
Grandmasters..
Prof. A.V. Patki Retired as ‘Outstanding Scientist’
from the Indian Space Research Organisation
(ISRO). He led a technical group towards the
design and development of systems of Aryabhata,
Bhaskara, APPLE, ROHINI and INSAT spacecrafts.
He was also the founder editor of the Journal of
Spacecraft Technology. Address: Flat No B 2/202,
‘Dahanukar Regency’, Plot No 1, Dahanukar
Colony, Kothrud, Pune-411029; Email: patki.
anant@gmail.com
SCIENCE REPORTER, NOVEMBER 2013