From: hau4@ellis.uchicago.edu (sven  hauptfeld)
Subject: Re: Inflation in ELO's system
Reply-To: hau4@midway.uchicago.edu
Organization: University of Chicago Computing Organizations
Date: Tue, 29 Sep 1992 01:42:42 GMT

In article <1992Sep28.181643.16032@cis.uab.edu> sloan@cis.uab.edu (Kenneth Sloan) writes:
>In article <1992Sep28.171802.1@ulys.unil.ch> jboschun@ulys.unil.ch writes:
>>In the various questions about Fischer's strength somebody talked about
>>inflations in ELO ratings...
>>Is somebody able to give a quantitativ evaluation of it from 1972 and so
>>to give, for instance, the value that Kasparov would have had at that time
>>(or the value of the Fischer of '72 now) ?
>
>No.
>
But it is possible to do. Just nobody has tried.

>>More can somebody explain the exact origin of the ELO's inflation (is it
>>coming, like in the case of money, from the increasing number of chess
>>players or is there other reasons to that fact, I once heard about a bad
>>management of the system ?) ?
>
>Neither.  There is an *de*-flationary pressure - which can be managed.
>[...]
>The deflationary pressure comes from the (observed) fact that most
>players leave the rating pool with a higher rating than they had when
>they entered the pool.  When rating points are managed as a conserved
>quantity, these retiring players slowly but surely will remove all of
>the rating points from the pool.  The remaining players compete for
>fewer and fewer rating points, and the average rating must (of course?)
>go down.

This sounds very fishy. First of all, there is no obvious correlation 
between average rating of all players and the points for which players
compete. Secondly, people leave the pool higher rated than they enter
because they, on the average, gain points as they play. Therefore, when a
player leaves the pool, 'taking away' points, players still in the pool
gain points and make up for the loss. 

Let's be more precise (math haters skip this paragraph): There are N(t)
players in the pool at time t. Their average rating is r(t) and the sum
of their ratings is R(t)=N(t)*r(t). In time dt, dN[-] players leave the
pool and dN[+] enter. Their average ratings are respectively r[-] and r[+].
So the number of players changes by dN = dN[+] - dN[-], the rating sum by
dR = r[+]dN[+] - r[-]dN[-] and average rating, to first order, by dr=dR/N.
However, this is not the whole change. All players in the pool 'improve' in
time (for generality, allow improvement to be negative as well - all this
is valid whether they improve or deteriorate). If their rate of improvement
is I, defined as I(t) := dr[improvement](t)/dt, where r[imp.] is change due
only to individual changes of players in the pool (i.e. not due to players
leaving and entering the pool), then TOTAL change in rating sum is
	dR = r[+]dN[+] - r[-]dN[-] + NI.
Obviously, if I>0, players will leave with higher ratings than they enter,
so the sum of first two terms will be negative (deflation) while the third
will be positive (inflation). The opposite is true if I<0 (players deteriorate)

Now note that this says nothing about the sign of dR. It depends on how the 
rating of the players who enter the pool is determined.

Also note that all this is oversimplified because AVERAGE rating is taken as
unweighted (simple) mean of all players' ratings, while it should be weighted
by the number of games they play during period dt, as well as by some other
factors ('K factor' of their opponents!). So instead of players not in the
pool and those in the pool, we have a continuous distribution of players by
their frequency of play, with players not-yet-in-the-pool and those-who-left-
the-pool simply belonging to frequency zero.

Further, and MUCH MORE SIGNIFICANT complication is that the pool is not 
uncorrelated, i.e. each players' rating is not determined homogeneously by
all others' ratings, but there are 'sub-pools' of players who play more
frequently against each other... (Well, OK, this is taken care of if the
above mentioned average K factors of opponents are accounted for.)

In any case, it should now be clear that the difference in rating between
people who leave the pool and those wo enter it does not by itself cause
either inflation or deflation of ratings.

For the explanation of how inflation (which exists) really works, see below.


>What to do?  Everyone is unhappy when it becomes increasingly difficult
>to garner rating points.  Players especially feel that their strength
>has gone up, but their rating has stayed the same (or fallen!).  The
>solution is to print more money (oops, I mean rating points)!  Some
>mechanism must be used to insert rating points into the system to
>balance out the ones disappearing with retiring players.

Could you give an example of an intentional inflation (or anti-deflation)
mechanism being in use? I haven't heard of any.

Bonus points (of yesteryear) and rating floors (of today), which indeed cause
inflation, were established for quite different purposes, so inflation is their
side-effect rather than desired effect. Floors actually cause large inflation
of USCF (where they exist) ratings. The way they work is obvious: if the 
player at his floor would lose points by rating formula, the sum change of the
two players' ratings is positive instead of zero, which is not ballanced by
any 'opposite' situation (which doesn't exist).

But there is also a more subtle inflation mechanism, which plagues FIDE as well
as USCF ratings. It is caused by the K factor (maximum up or down change in one
game) being dependent on the player's rating. For example, USCF K factor is 32
for players under 2100, 24 for players between 2100 and 2400 and 16 for players
above 2400. (Maybe the proper definition actually gives only half of these
numbers, but this is true: if I beat Kamsky, my rating would go up 32 points,
but his would fall only 16.)

Actually, in most cases everything works fine: even though the sum change in
individual games is not zero, the stronger player will win just as many more
games as necessary to pull the sum back to zero. (This is because ratings are
based on probability of the outcome.) However, if there exists any other
inflation mechanism, this will amplify it, because the TRUE differences 
between players will be smaller than their ratings predict and thus the 
scores with positive sum will appear more often than those with negative sum.

There is also one way in which this rating-dependence of K actually generates
(not just amplifies) inflation. That happens when players reach the borderline
rating. Namely, few players improve their ratings continuously; most oscillate
with a slight overall drift upwards. So when a player reaches, say, 2100 pts,
he will stay in the vicinity of 2100 for a while, crossing that line many
times. But, due to the fact that K is determined by OLD rating, every time he
crosses 2100 going up, he gets more than when he crosses it going down, thus
generating inflation.

There is a solution, which has not been used, as far as I know: do BOTH of the
following:
(1) Make K a smooth function of rating (actually, continuity of dK/dr is
sufficient) instead of abruptly changing at arbitrary values (like 2100);
(2) Evaluate K at the midpoint between old and new rating. (Pedestrian approach
requires iteration, but redefining K(r) with different values for win, draw
and loss - in effect, making the rating formula apparently asymmetric -
will have the same effect, saving computational time.)

Note that (2) is necessary, while (1) can be avoided, but the players crossing
the line must be treated as though (1) were effective.


	Sven

P.S. I just remembered that there is/was a FIDE rating rule with inflation as
its side-effect, analogous to floors: the rule that the winner of a tournament
cannot lose points. (This rule has been in and out of use and I don't know if
it is effective now.) Again, its purpose is quite different.


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