 # Math - Elo Rating System

In this lecture we looked at the formulas that make up the Elo rating system, which is one of the most well known and widely used systems for rating player skills in 1v1 games.

The Elo rating system was originally designed for ranking players in competitive chess but, due to its simplicity and popularity, it can now be found in all manner of games.

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Great work @valyfox!

As a bit of a time saver, you don’t really need to work out all the math for the second player.
The Elo rating system is transactional, so the rating of the second player will change by the opposite amount to the first player.
e.g. If one player gains 5 points, you automatically know that the other player will lose 5 points.

Still, it’s a good way to double check your answer - you know you made a mistake somewhere if the transfer between players isn’t equal.
I also like how you used the complement rule to speed things up when finding the second players expected score!

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The score for player A is 1270 and for player B is 1295

Ea = 0.6958
R’A = 1251 + 32 * (1-0.89)
1251 + 32 * 0.11
1251 + 3.52
1254.52

EB = 0.5896
R’B = 1314 + 32 * (0 - 0.5896)
1314 + (32 * -0.58)
1314 - 18.56
1295.44

A 1270
B 1295

The probability that the player A has to win is 41%

If we consider 1 point for the win the new score of player A will be approximately 1270

A 1269.87
B 1295.13

Player A rating = 1251
Player B rating = 1314

Ea = 1 / (1 + 10^((Rb - Ra) / 400)) = 1 / (1 + 10^(0,1575)) = 0,41 = 41%
R*a = 1251 + 32(1 - 0,41) = 1251 + 32 * 0,59 = 1251 + 18.88 = 1270

Eb = 1 / (1 + 10^((Ra - Rb) / 400)) = 59%
R*b = 1314 + 32(0 - 0.59) = 1314 + 32 * (-0.59) = 1295

New Ratings
Player A = 1270
Player B = 1295

EA = 1/(1 + 1.44) = 1/2.44 = 0.41 = 41%
R′A = 1251 + 32 * (1 - 0.41)
R′A = 1251 + 32 * (0.59)
R′A = 1251 + 18.88
R′A = 1269.88
R′B = 1314 - 18.88 = 1295.12