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The Elo Rating System for Chess and Beyond
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hi everyone today I want to talk about
the elo rating system which is a system
they're using competitive chess to rate
how good you are so if you are playing a
tournament then each player would have a
rating which is a number a new player
would have a rating of 1000 and then if
you win games your rating goes up and if
you lose games your rating goes down and
it's useful in a tournament because it
means you can match players of similar
ability together
now it's turned out to be really popular
in chess and then it's spread to other
competitive games like baseball and
basketball and online games and
competitive Scrabble and all these
things and it's really by the clever but
when you look it up online or they give
you is two formulas without explanation
so the first formula tells you how to
predict the outcome of the game so
whether you're going to win or lose and
the second formula tells you how to
update your rating depending on whether
you won or loss and then no explanation
is given so what I want to do is show
you where these formulas come from what
the idea behind it all is this system
was devised for chess in the 1960s it
was devised by an American physicist
called Arpad elo and his idea was to
assume that each player's ability forms
a kind of bell curve so if we forget
chess for the moment imagine each player
brings with them a box of numbers now
each player pulls a number from their
box and whoever has the highest number
wins well those numbers represent the
player's ability so each player has the
potential at playing at a range of
different abilities so some days you're
playing really well some days you're
picking a high number some days you've
got a cold you just had an unhappy
breakup you're pulling a low number but
most of the time you're picking a number
from somewhere in the middle
but a strong player will be picking
numbers from a box of generally higher
numbers so they're more likely to win
but not always if you look at the
frequencies of the numbers you see that
they form bell curves and those curves
overlap now it's assumed that those
curves are exactly the same with the
only difference being the center
of those curves down the center of the
curves is the average and the average is
that players rating now I want to derive
the first of those two formulas which is
the probability of winning the game so
if you want to know this you need to
look at the frequency of the differences
between the two players numbers if you
do that you also get a bell curve it's
called a logistic curve and the idea is
that the more of this curve that is to
the right of zero the more likely that
player is to win so the elo rating
system is designed so that if a player
has a rating that is 400 points more
than another player they are 10 times
more likely to win so on the curve the
area to the right of zero would be 10
times the area to the left of zero if a
player has a rating that is 800 points
more than another player there'd be 100
times more likely to win so if we turned
that into a formula this is the formula
that you would get so this is the
probability that player a wins with RA
and RB being the ratings of player a and
player B now we can tighten it up a
little bit more because the probability
that player B wins is just 1 minus the
probability player a wins we can put
that into the formula we can now tidy
this up and we can get the probability
that player a wins in its final form and
you can see it's just based on the
ratings of the two players and using
this you can predict who's going to win
the game if the probability of winning
is one you're definitely going to win
the game if the probability of winning
is zero you're definitely going to lose
if the two players have an equal rating
then the probability of winning is 1/2
so you're gonna win half the games and
lose half the game this is a kind of a
draw so let's say winning a game is
one-point losing a game is 0 and a draw
is half then the probability can be used
as the expected score even when it's
something weird like if you have the
probability of winning being 0.75 that
means you're going to win 75 percent of
the games or you might win 50 percent of
the games and draw
50% of the games or somewhere in between
either way we would say that the
expected score is point seven five so we
can use the probability to predict the
outcome of the game but what happens if
we do better or worse than the predicted
outcome
whenever player does better than
expected their rating will increase and
the more surprising that their win is
the more points they'll get up to a
maximum of 32 points and there's nothing
special about 32 that's just a choice
that they made in the same way if a
player does worse than expected their
rating will decrease up to 32 points so
after a game or tournament a player's
rating is updated using something called
an update formula this is the second
formula that was going to talk about and
you can see it's based on the difference
between their expected score and their
actual score so we just do a quick
example of this let's say we've got
player a he's the weaker player he's
expected to lose the game his expected
win is 0.35 and then in an upset he
actually wins the game and scores one
point if you put those figures into the
update formula you can see that player
age rating increases it actually
increases by 21 points in this example
which is an equal transaction from the
losers rating to the winner but that's
the idea so if a player does better than
expected their rating will increase if a
player does worse than expected their
rating will decrease and if a player
does about as expected there is little
difference so there's little benefit to
strong players picking on weak players
but this is really clever because that
means that eventually and over time a
players rating becomes a true reflection
of their ability the e low rating system
has become very popular with players as
well because it allows you to monitor
your own progress so you can see
yourself getting better or worse through
your rating the only problem with that
is that your rating is a measure of your
ability relative to the population so a
rating of 1800 this year may not be the
same as a rating of
800 next year or five years from now
even if your ability stays the same your
rating can go up and can go down it's
kind of like a currency in that way so
there are other more sophisticated
methods that solve those kind of
problems but ultimately all those
methods are based on ellos original idea
and I think that's all for me for now so
if you have been thanks for watching
