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The Elo Rating System for Chess and Beyond

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