DEFINITION. A (FIBS/Elo-style) rating system is one in which:
1. Each player has a rating which is a real number;
2. If Player A with rating RA defeats Player B with rating RB, then
Player A gains G(RA-RB) rating points, and Player B loses the same;
3. The function G(x) is strictly positive: G(x) > 0 for all real x;
4. The function G(x) is nonincreasing: G(x) <= G(y) if x > y.
THEOREM. In a (FIBS/Elo-style) rating system, suppose that Player A has
rating RA, and Player B has rating RB. Set a "target rating" RT. Then,
there is an integer N such that if Player A wins N times in succession
against Player B, Player A's rating will exceed RT after the N wins.
PROOF of THEOREM. By conservation of rating points [property 2], when
Player A has rating X, player B has rating Y such that X+Y = RA+RB.
Therefore the rating difference at that time is:
X-Y = 2*X-(X+Y) = 2*X-RA-RB.
If X < RT, then X-Y = 2*X-RA-RB < 2*RT-RA-RB. Since G is nonincreasing
[property 4]:
G(X-Y) >= G(2*RT-RA-RB).
Also, because G is strictly positive [property 3]:
G(2*RT-RA-RB) > 0.
Choose N such that N >= (RT-RA) / G(2*RT-RA-RB), which is finite by the
previous inequality. Assume that Player A's rating after N games
remains below RT. Then, for each of the first N wins, Player A's rating
X is less than RT, so Player A gains at least G(2*RT-RA-RB) rating
points for that win. Therefore, Player A's rating after N wins is at
least
RA + N * G(2*RT-RA-RB) >= RA + (RT-RA) = RT.
contradicting the assumption. Therefore N wins are sufficient. QED.
David desJardins