Elo rating - Revision history

Selinger: /* Definition */ Added 125 to table

2024-10-03T00:44:17Z

‎Definition: Added 125 to table

Selinger: /* Details of specific Hex sites that use Elo ratings */ Fixed Elo rules for BGA.

2022-11-10T23:33:24Z

‎Details of specific Hex sites that use Elo ratings: Fixed Elo rules for BGA.

Selinger: /* Details of specific Hex sites that use Elo ratings */ Updated strongest players rating for BGA

2021-07-26T03:37:02Z

‎Details of specific Hex sites that use Elo ratings: Updated strongest players rating for BGA

Selinger: Use 25 instead of 20 in the rating table.

2021-07-26T03:35:59Z

Use 25 instead of 20 in the rating table.

Selinger: Punctuation

2021-07-26T03:25:41Z

Punctuation

Selinger: Added category

2020-12-28T20:55:08Z

Added category

Selinger: Typo and minor copy-edit

2020-11-29T04:10:01Z

Typo and minor copy-edit

Selinger: Reformatted a formula.

2020-10-30T14:05:59Z

Reformatted a formula.

Selinger: /* Motivation */ Minor wording

2020-10-30T13:17:33Z

‎Motivation: Minor wording

Selinger: /* Definition */ Improved formatting of a formula.

2020-10-30T13:13:53Z

‎Definition: Improved formatting of a formula.

@@ Line 31: / Line 31: @@
 |-
 | 100 || 1.7782 ≈ 7 : 4 || 64%
 |-
 | 150 || 2.3714 ≈ 7 : 3 || 70%

@@ Line 84: / Line 84: @@
 == Details of specific Hex sites that use Elo ratings ==
-* [http://en.boardgamearena.com Board Game Arena]: K-factor is ''k'' = 60 for the first 10 games, ''k'' = 40 for the next 10 games, and ''k'' = 20 thereafter. There are special rules for ratings under 100: Players rated below 100 can never lose Elo points, and win at least 1 Elo point for each game. Once a player has reached Elo 100, they cannot go below 100 again, so 100 points is basically the baseline rating. The strongest players are rated in the 900s.
+* [http://en.boardgamearena.com Board Game Arena]: The K-factor is ''k'' = 60 for the first 10 games, ''k'' = 40 for the next 10 games, and ''k'' = 20 thereafter. There are special rules for ratings under 100: Players rated below 100 can never lose Elo points, and win at least 1 Elo point for their first game. Once a player has reached Elo 100, they cannot go below 100 again, so 100 points is basically the baseline rating. The strongest players are rated in the 900s.
 [[category: Rules and Conventions]]

@@ Line 84: / Line 84: @@
 == Details of specific Hex sites that use Elo ratings ==
-* [http://en.boardgamearena.com Board Game Arena]: K-factor is ''k'' = 60 for the first 10 games, ''k'' = 40 for the next 10 games, and ''k'' = 20 thereafter. There are special rules for ratings under 100: Players rated below 100 can never lose Elo points, and win at least 1 Elo point for each game. Once a player has reached Elo 100, they cannot go below 100 again, so 100 points is basically the baseline rating. The strongest players are rated in the 800s.
+* [http://en.boardgamearena.com Board Game Arena]: K-factor is ''k'' = 60 for the first 10 games, ''k'' = 40 for the next 10 games, and ''k'' = 20 thereafter. There are special rules for ratings under 100: Players rated below 100 can never lose Elo points, and win at least 1 Elo point for each game. Once a player has reached Elo 100, they cannot go below 100 again, so 100 points is basically the baseline rating. The strongest players are rated in the 900s.
 [[category: Rules and Conventions]]

@@ Line 24: / Line 24: @@
 | 0 || 1.0000 = 1 : 1 || 50%
 |-
-| 20 || 1.1220 ≈ 9 : 8 || 53%
+| 25 || 1.1548 ≈ 7 : 6 || 54%
 |-
 | 50 || 1.3335 ≈ 4 : 3 || 57%

@@ Line 9: / Line 9: @@
 Of course this assumption is not strictly true in practice, for a number of reasons. For example, if player X has a particular weakness that Y knows how to exploit, and player Y has a weakness that Z knows how to exploit, and player Z has a weakness that X knows how to exploit, it is quite possible that in matchups between two players, X on average loses to Y, Y on average loses to Z, and Z on average loses to X. Nevertheless, the assumption that odds are multiplicative seems to hold at least approximately in many practical situations.
-The Elo rating is a ''relative'' rating. Specifically, the odds of X winning against Y are meant to be reflected in the ''difference'' between X's and Y's Elo rating. By design, Elo ratings do not have an absolute meaning, as the strengths of players are always calibrated in relation to that of other players they have played against. For example, there is no absolute definition of how good a player must be to have an Elo rating of 800 --- and indeed, this can differ from game to game (e.g. Hex vs. chess), organization to organization, website to website, and may also change over time. It would be very difficult to objectively compare the strength of a player from 1942 to that of a player from 2020, since it is not likely that they have ever played against each other or against a common opponent. Moreover, players' strength changes over time. However, the statement "player X's Elo rating is 100 points higher than that of player Y" theoretically has a well-defined meaning (given the assumption on odds, above).
+The Elo rating is a ''relative'' rating. Specifically, the odds of X winning against Y are meant to be reflected in the ''difference'' between X's and Y's Elo rating. By design, Elo ratings do not have an absolute meaning, as the strengths of players are always calibrated in relation to that of other players they have played against. For example, there is no absolute definition of how good a player must be to have an Elo rating of 800 — and indeed, this can differ from game to game (e.g. Hex vs. chess), organization to organization, website to website, and may also change over time. It would be very difficult to objectively compare the strength of a player from 1942 to that of a player from 2020, since it is not likely that they have ever played against each other or against a common opponent. Moreover, players' strength changes over time. However, the statement "player X's Elo rating is 100 points higher than that of player Y" theoretically has a well-defined meaning (given the assumption on odds, above).
 == Definition ==

← Older revision		Revision as of 20:55, 28 December 2020
Line 85:		Line 85:

	* [http://en.boardgamearena.com Board Game Arena]: K-factor is ''k'' = 60 for the first 10 games, ''k'' = 40 for the next 10 games, and ''k'' = 20 thereafter. There are special rules for ratings under 100: Players rated below 100 can never lose Elo points, and win at least 1 Elo point for each game. Once a player has reached Elo 100, they cannot go below 100 again, so 100 points is basically the baseline rating. The strongest players are rated in the 800s.		* [http://en.boardgamearena.com Board Game Arena]: K-factor is ''k'' = 60 for the first 10 games, ''k'' = 40 for the next 10 games, and ''k'' = 20 thereafter. There are special rules for ratings under 100: Players rated below 100 can never lose Elo points, and win at least 1 Elo point for each game. Once a player has reached Elo 100, they cannot go below 100 again, so 100 points is basically the baseline rating. The strongest players are rated in the 800s.
		+
		+	[[category: Rules and Conventions]]

@@ Line 61: / Line 61: @@
 To explain how the method works, consider a player X playing against an opponent Y. Let us assume that initially, X's Elo rating is 100 points higher than that of Y. According to the above formula or table, this means that X is expected to win against Y about 64% of the time. Now suppose the players play a number of games. How should X's Elo rating be updated? This should be done in such a way that on average: (1) if X actually wins 64% of the time as predicted, then X's Elo rating should not change; (2) if X wins more than 64% of the time, X's Elo rating should increase; and (3) if X wins less than 64% of the time, X's Elo rating should decrease.
-We achieve this by wagering some of X's Elo points on the outcome of each game. In each game, X will have the chance to win a certain number of points, say 10 points. If X's winning probability is truly 64%, this means that X will win 6.4 points on average. To compensate for this, X must pay 6.4 points at the beginning of each game. Now if X actually wins 64% of the games on average, then X wins the same number of Elo points that X paid, and X's rating does not change. But if X wins more than 64% of the games, then X wins on average more points than X paid, and X's rating increases. Similarly, if X wins less than 64% of the games, X's rating decreases.
+We achieve this by wagering some of X's Elo points on the outcome of each game. In each game, X will have the chance to win a certain number of points, say 10 points. If X's winning probability is truly 64%, this means that X will win 6.4 points on average. To compensate for this, X must pay 6.4 points at the beginning of each game. Now if X actually wins 64% of the games, then X wins on average the same number of Elo points that X paid, and X's rating does not change. But if X wins more than 64% of the games, then X wins on average more points than X paid, and X's rating increases. Similarly, if X wins less than 64% of the games, X's rating decreases.
 The number of points that X can win in each game is called the ''K-factor'' of player X. In the above example, X's K-factor was 10. A larger K-factor causes ratings to change more quickly, but it also causes larger random fluctuations in ratings. A smaller K-factor causes ratings to change more slowly, but it also causes smaller random fluctuations. For this reason, it is common to give different K-factors to different players. Typically, new players will be given larger K-factors so that they have a chance to reach their "true" rating quickly. Established players will be given smaller K-factors, because it is likely that they are already in the vicinity of their true ratings, so it is better to minimize random ups and downs.
@@ Line 74: / Line 74: @@
 * Let ''N'' be the difference of X's and Y's Elo ratings prior to the game (i.e., positive if X's rating was higher, and negative if Y's rating was higher).
 * Calculate X's predicted odds of winnings: ''odds'' = 10<sup>''N''/400</sup>
-* Calculate X's predicated winning probability: ''p'' = ''odds'' / (''odds'' + 1)
+* Calculate X's predicted winning probability: ''p'' = ''odds'' / (''odds'' + 1)
 * Take ''pk'' Elo points from X.
 * If X wins, give ''k'' Elo points to X.

@@ Line 15: / Line 15: @@
 By definition, the Elo rating is calibrated in such a way that a difference of 400 Elo points corresponds to odds of 10 : 1.
-In formulas, this means that the difference between two players' Elo ratings is 400 times the 10-based logarithm of the odds. Thus, if the odds of player X winning against player Y are ''a'' : ''b'', then player X's true Elo rating exceeds that of player Y by 400 log₁₀ (''a''/''b''). Conversely, if player X's Elo rating exceeds that of player Y by ''N'' points, the odds of X winning against Y are 10^(''N''/400) : 1.
+In formulas, this means that the difference between two players' Elo ratings is 400 times the 10-based logarithm of the odds. Thus, if the odds of player X winning against player Y are ''a'' : ''b'', then player X's true Elo rating exceeds that of player Y by 400 log₁₀ (''a''/''b''). Conversely, if player X's Elo rating exceeds that of player Y by ''N'' points, the odds of X winning against Y are 10<sup>''N''/400</sup> : 1.
 In table form, the following table translates various relative Elo ratings to the corresponding odds and probabilities of winning: