Howard wrote:The reason the '40 point' rule is not a '50 point' rule is because a '50 point' rule would provide no grading benefit to the stronger player and the weaker player would win or draw the occasional game. Statistically (i.e. looking at actual game results) the stronger player only scores something like 98% against someone graded 50 points lower.
According to Condition 1...
Condition 1: The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores or higher. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher.
...which is equivalent to...
Condition 1: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). If the players play a match it is expected (according to their grades) that player A performance 'p' (in percent) will be p = 50+a-b, or equivalently if performance of player A is 'p', p = 50+a-b, the players performed as expected (according to their grades) and their grades would remain unchanged after the match. For example, if a=130 and b=100 and A performs 80% in the match the grades will not change after the match (A's expected performance is p = 50+a-b = 50+130-100 = 80%).
...the stronger player is expected to score 100% against someone graded 50
or more points lower (the actual percentage of 98% does not look to me far from the theoretical 100%, or is this treated as 'far').
I see no reason why would one choose to obey Condition 1 if the difference in grades is 40 points or less and break it in all other cases. The number '50' in '50 point' rule, as opposed to '40 point' rule, comes form the formula 'p = 50+a-b', which is a mathematical form of Condition 1, and in my opinion should not arbitrarily be changed into '40', if one wishes that
the difference in ratings is half the difference in percentage scores.
Puttying it simple, if one uses
'40 point' rule rather than '50 point' rule I think he or she
breaks Condition 1. If you think I'm wrong, could you please elaborate where does '40' come from? Thanks.
Howard wrote:Zero and negative grades are allowed for and used in the calculations - the only difference is that negative grades are published as zero rather than the true figure.
I see, I think that is how it should be (negative grades used in calculation).
Howard wrote:You then go on at great length regarding problems grading players who only play each other. Your suggested improvement may solve the two-player problem but does very little for the real world case.
If there are two grading systems: the first (GS and CGS) which fails on simple examples (with two players who play only each other in the matches) and the second (AGS) which does not, one would expect the second to perform in the real world better than the first.
Howard wrote:If a player grade 100 scores 50% against 30 players with an average grade of 120, then his grade under current ECF rules would be 120.
Correct (both GS and CGS give that result, GS would give a wrong result if the difference in the grades was more than 40).
Howard wrote:Similarly, if a player graded 140 scored 50% against the same 30 players he would also have a grade of 120.
Correct (both GS and CGS give that result, GS would give a wrong result if the difference in the grades was more than 40).
Howard wrote:Under your suggested changes the 100 player would have a grade of 110 and the 140 player would have a grade of 130
Correct (AGS gives that result).
Howard wrote:surely as the two players have had an identical performance they should also have an identical grade?
In my opinion
they should not have an identical grade. Let me explain...
According to AGS, after the games with the 100 player, an average grade of the 30 players would be 110 (assuming that in the other games the 30 players performed as expected according to their average grade of 120), Now, if the 100 player scores 50% against the same 30 players again, say in the next season, (assuming that in the other games the 30 players performed as expected according to their average grade which is now 110) neither the 100 player's grade (which is now 110) nor the average grade of the 30 players (which is now 110) would change (it will remain 110). That is the consequence of the fact that AGS satisfies Condition 2.
Similarly, according to AGS, after the games with the 140 player, an average grade of the 30 players would be 130 (assuming that in the other games the 30 players performed as expected according to their average grade of 140), Now, if the 140 player scores 50% against the same 30 players again, say in the next season, (assuming that in the other games the 30 players performed as expected according to their average grade which is now 130) neither the 140 player's grade (which is now 130) nor the average grade of the 30 players (which is now 130) would change (it will remain 130). That is the consequence of the fact that AGS satisfies Condition 2.
You may also look at this as follows...
The 100 player's new grades should be 110 (rather than 120) as you do not assume that the 100 player performance is only due to increase in his or her chess ability, but also due to possible decrease in chess ability of the 30 players (you in fact do not know if the 100 player improved or the 30 players worsen, that is why you take to be both). Similar reasoning holds for why the 140 player's new grade should be 130 (rather than 120).
If the 100 player had played 30 games against a 120 player scoring 50% (in a match), then I think you would agree that you do not know if the 100 player has improved or the 120 player has worsened, and, in my opinion, a new grade of 110 for both players would be fair (as if they play again and score the same, 50%, their grade of 110 will remain unchanged). I think nothing should change if, instead, the 100 player had played against 30 players with an average grade of 120, as the fact that they are 30 players should not, in my opinion, imply that they are an absolute measure and that we know that the score is solely due to increase in the 100 player's chess ability.
Code...
Code: Select all
(*Amended Grading System*)
ClearAll[a, b, d, d0, gw, gd, gl, g, a0, b0, nw, nd, nl, nt];
a = 120; b = 100;
nw = 15; nd = 0; nl = 15;
nt = nw + nd + nl;
d = a - b;
g = 50;
If[d > g, a0 = b + g, a0 = a];
If[d > g, b0 = a - g, b0 = b];
If[d > g, d0 = g, d0 = d];
gw = (g + d0)/2; gd = d0/2; gl = (g - d0)/2;
a2 = (nw*(b0 + gw) + nd*(b0 + gd) + nl*(b0 - gl))/nt;
b2 = (nw*(a0 - gw) + nd*(a0 - gd) + nl*(a0 + gl))/nt;
a2
Print[];
b2
110
110
Code: Select all
(*Amended Grading System*)
ClearAll[a, b, d, d0, gw, gd, gl, g, a0, b0, nw, nd, nl, nt];
a = 140; b = 120;
nw = 15; nd = 0; nl = 15;
nt = nw + nd + nl;
d = a - b;
g = 50;
If[d > g, a0 = b + g, a0 = a];
If[d > g, b0 = a - g, b0 = b];
If[d > g, d0 = g, d0 = d];
gw = (g + d0)/2; gd = d0/2; gl = (g - d0)/2;
a2 = (nw*(b0 + gw) + nd*(b0 + gd) + nl*(b0 - gl))/nt;
b2 = (nw*(a0 - gw) + nd*(a0 - gd) + nl*(a0 + gl))/nt;
a2
Print[];
b2
130
130
If I require for AGS to obey your condition above, which I may call Condition 3, I get...
Code: Select all
(*Amended Grading System*)
ClearAll[a, b, d, d0, g0, g, a0, b0, nw, nd, nl, nt];
a = 140; b = 120;
nw = 15; nd = 0; nl = 15;
nt = nw + nd + nl;
d = a - b;
g = 50;
If[d > g, a0 = b + g, a0 = a];
If[d > g, b0 = a - g, b0 = b];
If[d > g, d0 = g, d0 = d];
a2 = 120; b2 = 140;
Solve[{
a2 == (nw*(b0 + gw) + nl*(b0 - gl))/nt,
b2 == (nw*(a0 - gw) + nl*(a0 + gl))/nt}, {gw}]
{{gw -> gl}}
...and as AGS has to obey Condition 1 too, I get...
Code: Select all
(*Amended Grading System*)
ClearAll[a, b, d, d0, g0, g, a0, b0, nw, nd, nl, nt];
a = 130; b = 100;
nw = 8; nd = 0; nl = 2;
nt = nw + nd + nl;
d = a - b;
g = 50;
If[d > g, a0 = b + g, a0 = a];
If[d > g, b0 = a - g, b0 = b];
If[d > g, d0 = g, d0 = d];
a2 = 130; b2 = 100;
Solve[{
a2 == (nw*(b0 + g0) + nl*(b0 - g0))/nt,
b2 == (nw*(a0 - g0) + nl*(a0 + g0))/nt}, {g0}]
{{g0 -> 50}}
...which implies that the only possible grading system which would obey both Conditions 1 and 3 is CGS...
Code: Select all
(*Corrected Grading System*)
ClearAll[a, b, d, g, a0, b0, nw, nd, nl, nt];
a = 140; b = 120;
nw = 15; nd = 0; nl = 15;
nt = nw + nd + nl;
d = a - b;
g = 50;
If[d > g, a0 = b + g, a0 = a];
If[d > g, b0 = a - g, b0 = b];
a2 = (nw*(b0 + g) + nd*(b0) + nl*(b0 - g))/nt;
b2 = (nw*(a0 - g) + nd*(a0) + nl*(a0 + g))/nt;
a2
Print[];
b2
Nevertheless, as Condition 3 needs not (or better, must not) be satisfied by a good grading system, AGS appears to be the best of the three.
