hgm wrote:
Empirical data has shown that draws are a stronger indication for equality of the player strength than equally scoring series of wins and losses. It could just as well have been the other way around; it all depends on the win, loss and draw probability as a function of rating difference. These probabilities happen to be such that draws are more significant.
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So one win and one loss should count as something less than two draws as assumed by the way tournaments are scored. Since I have a lot of respect for your expertise in such mathematical issues, I would like your opinion as to what number of draws should have the same weight as one win and one loss based on all evidence of which you are aware. My own feeling is that the value of 1.5, as assumed by the middle of the three models, is close to correct, but I would defer to your opinion on this.
Well, the evidence I have seen is based purely on what Adam Hair has posted here some time ago, for computer-computer games. He had plotted the fractions of wins, draws and losses as a function of the Elo difference of the players. As I remember it, the fit through the wins and the losses (which are nearly straight lines around Delta_Elo = 0), when multiplied with each other, gave a parabolic peak that, after vertical scaling, fitted the points for the fraction of draws quite well.
If the width at half-height of this Win x Loss product would be the same, it would mean that 1 draw is indeed as significant as 1 win + 1 loss. For 2 draws to be equivalent to 1 win and 1 loss, the draw fraction would have to be fitted by the square root of that, which is not a parabola, but a circle. The width of a parabola at half height is 0.5*sqrt(2) = 0.71 times its width at the base, for a (semi-)circle it is 0.5*sqrt(3) = 0.87. This is sqrt(3/2) = 1.22 times wider.
From what I remember the data points seemed closer to the parabola than to the circle. The calculation shows that the difference between the two is not very large, however, and it would be difficult to sayby merely visual inspection (which is what I did) whether W+L equals 1.9 D or 1.8 D. But doing a curve fitting should most likely be able to do that.
You have to take into account that in the Davidson model 2 draws=1 win + 1 loss is only true if maximum likelihood estimation is used for elo.
FIDE uses a much more naive scoring scheme (win=1 point, draw=0.5 point, loss=0 point) and then applies the inverse of the logistic function to this.
The thus computed elo is quite different from the one computed by maximum likelihood estimation. One may correct this to some extent with scaling
but this does not work so well for large elo differences.
The best thing would be for FIDE to switch to maximum likelihood estimation, irrespective of the draw model being used.
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Thanks. One consequence of the notion that one win plus one loss equals less than two draws is that it gives theoretical validation to using number of wins as a tiebreak, I think. The current popular practice of scoring draws as 1 but wins as 3 might also be justified with this logic,although it seems way too severe to me (and also works in reverse for the players in the bottom half of the table). Can you suggest a better scoring system for tournaments that captures this idea? The best I can come up with is something like this: wins 3 out of 3, draws 2 out of 4, losses zero out of 3, scoring by percentage of total possible points. So one win and three draws gets 9 out of 15 for 60%, while two wins, one draw, and one loss (the same by standard count) gets 8 out of 13 for 61.5%. Practical effect would be the same as just making number of wins the tiebreak with standard count for the top half, but it gets the bottom half right too. Comments welcome.
lkaufman wrote:Thanks. One consequence of the notion that one win plus one loss equals less than two draws is that it gives theoretical validation to using number of wins as a tiebreak, I think. The current popular practice of scoring draws as 1 but wins as 3 might also be justified with this logic,although it seems way too severe to me (and also works in reverse for the players in the bottom half of the table). Can you suggest a better scoring system for tournaments that captures this idea? The best I can come up with is something like this: wins 3 out of 3, draws 2 out of 4, losses zero out of 3, scoring by percentage of total possible points. So one win and three draws gets 9 out of 15 for 60%, while two wins, one draw, and one loss (the same by standard count) gets 8 out of 13 for 61.5%. Practical effect would be the same as just making number of wins the tiebreak with standard count for the top half, but it gets the bottom half right too. Comments welcome.
The whole idea is ridiculous to me, though I haven't been following the thread carefully. Could you summarize why in the world one would want to use anything other than a standard {0, 0.5, 1} scoring metric?
lkaufman wrote:Thanks. One consequence of the notion that one win plus one loss equals less than two draws is that it gives theoretical validation to using number of wins as a tiebreak, I think. The current popular practice of scoring draws as 1 but wins as 3 might also be justified with this logic,although it seems way too severe to me (and also works in reverse for the players in the bottom half of the table). Can you suggest a better scoring system for tournaments that captures this idea? The best I can come up with is something like this: wins 3 out of 3, draws 2 out of 4, losses zero out of 3, scoring by percentage of total possible points. So one win and three draws gets 9 out of 15 for 60%, while two wins, one draw, and one loss (the same by standard count) gets 8 out of 13 for 61.5%. Practical effect would be the same as just making number of wins the tiebreak with standard count for the top half, but it gets the bottom half right too. Comments welcome.
The whole idea is ridiculous to me, though I haven't been following the thread carefully. Could you summarize why in the world one would want to use anything other than a standard {0, 0.5, 1} scoring metric?
-Sam
There are two ways to answer this. The theoreticians would say that since empirical evidence indicates that two draws should count for more than one win plus one loss, the scoring system should reflect this. The practical reason is that the standard scoring produces too many ties, and some way to break ties is needed. But as I pointed out, there is not much difference between using my system and using the normal one with number of wins as a tiebreak. Personally I am opposed to the fad of using the "football" system (0,1,3), I'd like to see something far less drastic to break ties while still favoring a win and a loss over two draws (for the players in the top half at least) slightly.
lkaufman wrote: The theoreticians would say that since empirical evidence indicates that two draws should count for more than one win plus one loss, the scoring system should reflect this.
In what sense? In terms of producing a rating that predicts future results more accurately? If so, are these future results measured using current scoring systems (loss 0, draw 0.5, win 1)?
lkaufman wrote:Thanks. One consequence of the notion that one win plus one loss equals less than two draws is that it gives theoretical validation to using number of wins as a tiebreak, I think. The current popular practice of scoring draws as 1 but wins as 3 might also be justified with this logic,although it seems way too severe to me (and also works in reverse for the players in the bottom half of the table). Can you suggest a better scoring system for tournaments that captures this idea? The best I can come up with is something like this: wins 3 out of 3, draws 2 out of 4, losses zero out of 3, scoring by percentage of total possible points. So one win and three draws gets 9 out of 15 for 60%, while two wins, one draw, and one loss (the same by standard count) gets 8 out of 13 for 61.5%. Practical effect would be the same as just making number of wins the tiebreak with standard count for the top half, but it gets the bottom half right too. Comments welcome.
The whole idea is ridiculous to me, though I haven't been following the thread carefully. Could you summarize why in the world one would want to use anything other than a standard {0, 0.5, 1} scoring metric?
-Sam
The guy is seriously confused you are advised to ignore him. The winning percentages are the same for both 1W+3D and 2W+1D namely 2.5 points out of 4 => 62.5%. The goal is not to change to soccer scoring system by changing this percentage but the _ratings_ assigned to the players. For example if I draw with white for a 50% score, I would have less rating than if I did it with black. All draw models give exactly the same winning percentages for every player, even though the players are assigned different ELOs based on home advantage (white or black) or draw ratio (handled via drawElo by bayeselo)...
So Glenn-David, Rao-Kupper, and Davidson give the same winning percentages but different elos, so someone please stop this guy from spreading his misinformed crap.
Last edited by Daniel Shawul on Sun Sep 22, 2013 3:15 pm, edited 2 times in total.
lkaufman wrote: The theoreticians would say that since empirical evidence indicates that two draws should count for more than one win plus one loss, the scoring system should reflect this.
In what sense? In terms of producing a rating that predicts future results more accurately? If so, are these future results measured using current scoring systems (loss 0, draw 0.5, win 1)?
-Sam
The experts here say that weighting draws more heavily in the rating formula improves the predictability of the system. It would seem like common sense that if the rating system were changed to reflect this, the scoring system should match it as much as possible. Theoretically this should increase the chance that the best player will win the tournament, or to put it another way it should decrease the luck factor in the final standings.
