Author: Stephen A. Boak
Date: 17:31:51 01/02/00
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Hi Jim, I agree that individual players may inflate their ratings by various artificial means. [Let's set that aside for a moment.] The ELO system by which all players in a rating pool are measured involves an overall scaling aspect that is the same for all players within the same K-factor band in the pool. If a player has a lucky streak (natural variation at work--the same player will on occasion have a bad streak) in a match of say, 6 wins in a row against equally rated opponent, the player's rating will rise 6*(K-factor/2). By the same token, the opponent's rating will drop 6*(K-factor/2). It is true that if the K-factor is doubled, the player's rating rise will rise by twice as much as otherwise, and the opponent's rating will drop by twice as much as otherwise. These rating gains/losses are not properly called 'inflation' or 'deflation'. Why? An ELO system expects that *all* players will oscillate in rating above and below their 'true' strength. This oscillation is totally natural variation in measured rating versus 'true' strength. The oscillation is not dependent (for its existence) on the K-factor. The K-factor merely sets the scale of the oscillation--equally so for natural variations above and below 'true' playing strength. If the K-factor were doubled, the typical oscillations would be larger in terms of absolute rating points (for both rating increases and decreases above/below 'true' strength). But since the same K-factor applies to the similarly rated opponent(s), the *relative* rating of the player will be in the same relationship to the other similar players in his rating range. A lucky streak of 6 wins against equally rated opponents might result in twice the gain of rating points if the K-factor were doubled. And twice the loss of rating points for the opponent. However, among a band of similarly rated players, the relative ratings would be identical, simply the scaling across the band would be wider (if higher K-factor used) or narrower (if lower K-factor used. Now, back to the player with an assumed artificially inflated rating. If such a player contrives to win 6 games in a row against an equally rated opponent, his rating will rise by 6*(K-factor/2). Relative to a player who merely has natural variation and a similar but lucky win streak, both the lucky player and the contriving player have the same rating changes. Let's double the K-factor and re-analyze the relative rating changes for the above two kinds of players. The contriving player will gain 6*(2*K-factor/2) points. The lucky player will also gain 6*(2*K-factor/2) points. Here again, the relative gain between the contriving player and the lucky player is the same (in this example, it is 1 for 1). My point is that the relative amount of inflation under a doubled K-factor system between contriving players (defined as players who artifically inflate their ratings) and normal players (defined as players who do not artifically inflate their ratings, but simply have natural variance in results), is the *same* regardless of the specific K-factor that is used. This is because the K-factor also determines the scale (or typical breadth) of rating oscillation for normal players, as well as the scale of rating oscillation for contriving players who may somehow artifically win games. The fact that the absolute point difference between a contriving player and a normal player is larger under a doubled K-factor system is offset by the fact that the doubled K-factor also increases the scale of natural oscillation (it also doubles) for normal players, such that the relative amount of inflation is between contriving players and normal players is the same under both K-factor scenarios. In a simplistic example, assume a normal player of strength X has a rating of 3 on a scale of 1 to 5, and a contriving player of same strength X has a rating of 4 on a scale of 1 to 5. After doubling the scale from 1-5 to 1-10 (for example by doubling the K-factor), the normal player will have a rating of 6 on the scale of 1-10, and the contriving player will have a rating of 8 on the scale of 1-10. Yet the relative rating ratio ContrivedRating/Normal Rating is the same in both cases: 4/3=1.333 and 8/6=1.333, and the amount of inflation is the same (33.3%). These are my thoughts about the mathematics of 'inflation' under varying K-factor ELO systems. --Steve Boak
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