Forum Archive : Ratings

No limit to ratings

From:   David desJardins
Address:   desj@math.berkeley.edu
Date:   19 December 1998
Subject:   Re: Average FIBS rating
Forum:   rec.games.backgammon
Google:   vohd85gm5wk.fsf@bosco.berkeley.edu

David desJardins writes:
> THEOREM.  Suppose that player A has rating ra, and player B has rating
> rb.  Set a "target rating" rt for player A.  Then there is a finite
> number N, depending on ra, rb, and rt, such that, if player A wins N
> consecutive matches from player B, player A's rating will exceed rt.

It occurs to me that there's another interesting theorem that can be

THEOREM 2.  Suppose that Player A and Player B play an infinite sequence
of matches, with the winner of each match determined at random (A wins
with probability P, where 0 < P < 1).  Then, for any "target rating" rt,
A's rating will at some point exceed rt, with probability one.

What this implies is that arbitrarily high ratings will eventually be
achieved, with probability one, even without any collusion of any sort.
Of course, this will take a *really* long time, for high target ratings.

This result still only relies on basic properties of the FIBS rating
system.  We need the following:

  1. The winner of each match gains a positive number of rating points
     which depends only on the ratings difference between the players.
     The loser loses the same number of rating points.

  2. The rating point gain for the winner is a strictly decreasing
     function of the rating difference between the winner and the loser,
     and goes to zero as the rating difference goes to infinity.

PROOF of THEOREM 2.  (I'll only give a sketch, I can fill in more
details if anyone cares.  All of this follows from properties 1 and 2.)
There is a pair of "equilibrium ratings" (ra,rb), such that if Player A
has rating ra and Player B has rating rb, then the expected change in
A's rating from the next match is zero; i.e.,

  P * (A's gain for winning) - (1-P) * (B's gain for winning) = 0.

If A's rating exceeds ra, then the expected change in A's rating from
the next match is negative.  If A's rating is below ra, then the
expected change in A's rating from the next match is positive.

It follows that, infinitely often, A's rating will be at or above ra.
[This follows from standard results about random walks; it's
intuititively obvious, but also the hardest part of the proof.]

Consider the target rating rt, which corresponds to the rating
difference 2*rt-ra-rb.  As in the proof of Theorem 1, let rg be the
rating point gain for A for winning when the rating difference is
2*rt-ra-rb.  Then, for any smaller rating difference, A's gain for
winning exceeds rg.  Let N = floor((rt-ra)/rg) + 1.  Then if A wins N
consecutive games, starting with a rating at or above ra, A's rating
after the N wins will exceed rt.

The probability of such a sequence of wins is P^N, which is greater than
zero.  Given an infinite sequence of occasions (times when A's rating is
at least ra) when such a sequence of wins might occur, with a positive
probability of occurrence, with probability one it will eventually
occur.  QED.

David desJardins

Tom Keith  writes:

Douglas Zare and Adam Stocks comment on this posting in their article
"Ratings: A Mathematical Study"
Did you find the information in this article useful?          

Do you have any comments you'd like to add?     



Constructing a ratings system  (Matti Rinta-Nikkola, Dec 1998) 
Converting to points-per-game  (David Montgomery, Aug 1998)  [Recommended reading]
Cube error rates  (Joe Russell+, July 2009)  [Long message]
Different length matches  (Jim Williams+, Oct 1998) 
Different length matches  (Tom Keith, May 1998)  [Recommended reading]
ELO system  (seeker, Nov 1995) 
Effect of droppers on ratings  (Gary Wong+, Feb 1998) 
Emperical analysis  (Gary Wong, Oct 1998) 
Error rates  (David Levy, July 2009) 
Experience required for accurate rating  (Jon Brown+, Nov 2002) 
FIBS rating distribution  (Gary Wong, Nov 2000) 
FIBS rating formula  (Patti Beadles, Dec 2003) 
FIBS vs. GamesGrid ratings  (Raccoon+, Mar 2006)  [GammOnLine forum]
Fastest way to improve your rating  (Backgammon Man+, May 2004) 
Field size and ratings spread  (Daniel Murphy+, June 2000)  [Long message]
Improving the rating system  (Matti Rinta-Nikkola, Nov 2000)  [Long message]
KG rating list  (Daniel Murphy, Feb 2006)  [GammOnLine forum]
KG rating list  (Tapio Palmroth, Oct 2002) 
MSN Zone ratings flaw  (Hank Youngerman, May 2004) 
No limit to ratings  (David desJardins+, Dec 1998) 
On different sites  (Bob Newell+, Apr 2004) 
Opponent's strength  (William Hill+, Apr 1998) 
Possible adjustments  (Christopher Yep+, Oct 1998) 
Rating versus error rate  (Douglas Zare, July 2006)  [GammOnLine forum]
Ratings and rankings  (Chuck Bower, Dec 1997)  [Long message]
Ratings and rankings  (Jim Wallace, Nov 1997) 
Ratings on Gamesgrid  (Gregg Cattanach, Dec 2001) 
Ratings variation  (Kevin Bastian+, Feb 1999) 
Ratings variation  (FLMaster39+, Aug 1997) 
Ratings variation  (Ed Rybak+, Sept 1994) 
Strange behavior with large rating difference  (Ron Karr, May 1996) 
Table of ratings changes  (Patti Beadles, Aug 1994) 
Table of win rates  (William C. Bitting, Aug 1995) 
Unbounded rating theorem  (David desJardins+, Dec 1998) 
What are rating points?  (Lou Poppler, Apr 1995) 
Why high ratings for one-point matches?  (David Montgomery, Sept 1995) 

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