Transposition Limits

Running Chess Tournaments
1

I agree. Having just three limits (80 points for alternation, 200 points for equalization, and 9999 points for three of the same color in a row) is too simplistic.

Some transpositions, even within the same category (alternation or equalization), are worse than others. In the alternation department, for example, in round 5 I would rather give black to BWWB than to WBWB. In fact, the 1st (Morrison) edition of the rulebook actually said that a player with BWWB was due black. (This was changed in the 2nd, also Morrison, edition.)

There needs to be some sort of sliding scale as to how large the transposition limit should be to avoid various degrees of bad colors. Something along these lines:

Giving W to WB could be worth 80 points.
Giving W to WBWB could be worth 80 points.
Giving W to BWWB could be worth 50 points.
Giving W to WBWBWB could be worth 80 points.
Giving W to BWWBWB could be worth 60 points.
Giving W to WBBWWB could be worth 40 points.

Or, in the equalization department:

Giving W to B (after one round) could be worth 200 points.
Giving W to BWB could be worth 200 points.
Giving W to BBW could be worth 160 points.
Giving W to BWBWB could be worth 200 points.
Giving W to BWBBW could be worth 170 points.
Giving W to BBWBW could be worth 140 points.

Still further, in the “extremely bad colors” department:

Giving W to WBB could be worth 1200 points.
Giving W to WWBB could be worth 800 points.
Giving W to WBWWBB could be worth 800 points.
Giving W to WWBBWB could be worth 600 points.

No specific science was used to arrive at these numbers. There’s nothing exact here, I’m just throwing out some ideas to begin a discussion. Have at it!

Bill Smythe

2

Why would that be 600 points? In the 5th edition rulebook wouldn’t that be under the 80 point rule for alternation?

3

A player with WWBBWB is in a situation similar to a player with WWBB. Both of these involve only alternation, not equalization. Yet, assigning 3 blacks in a row is still regarded as Very Bad Colors.

If a player with WWBBWB is assigned black, he will have had four blacks in his last five games. In my book, that’s almost as bad as having three blacks in a row. But your mileage may vary.

Bill Smythe

4

I’ll have to think about it. The player did get White in his first two games. On the other hand if those games were against much lower rated players perhaps they shouldn’t count as much as games played in a later rounds. It might make a difference whether we’re pairing a player in one of the top score groups or a player in the middle or at the bottom.

Your suggestions would make the pairing algorithm more complicated. This might not matter very much since most pairings these days are made by computers, as long as we trust the computers. On the other hand it makes it harder for players to work out their pairings, so the new rules might generate more complaints.

5

So did the player with WWBB.

That consideration would apply equally (or nearly so) to both cases, WWBB and WWBBWB.

True. Perhaps, for manual pairings, a simple statement of preference (maybe a TD Tip) would suffice. Such as: “Other things being more or less equal, it may be preferable to assign due color to a player more strongly due his color than to one less strongly due. For example, assigning W to WBWB is preferable to assigning W to BWWB.”

And there could be an example:

Raw pairings going into round 5:

2030 WBWB vs 1700 BWWB
1940 BWBW vs 1650 BWBW
1850 WBWB vs 1580 WBWB

Here colors are bad in all three pairings. Switching 1700 with 1650 would fix two of them, with a 50-point transposition. Switching 1650 with 1580 would also fix two of them, but with a 70-point transposition. Nevertheless, the latter switch might be preferable, because the only player left with bad colors would be 1700, who is less strongly due white than any other.

Bill Smythe

6

Bill, if your color history was WWBBWB which pairing would you rather get in round 7: white vs. a player rated 2200 or black vs. a player rated 1600? According to your 600 point rule giving you your due color is more important than the 600 point rating difference, so the TD would pair you against the 2200 instead of your natural 1600-rated opponent in order to improve colors. Would you be happy about that?

I’m skeptical about the rule against giving a player the same color three times in a row unless there’s no other way to pair the score group, although I obey it because it’s in the rulebook. It could mean making a 1000+ point transposition or interchange in order to improve colors. Is color really that important?

According to Tim Redman’s introduction to the 3rd edition rulebook, “George Cunningham, Professor Emeritus of Mathematics at the University of Maine in Orono, provided valuable assistance in defining the relation between the value of proper color allocation and rating-point spread”. The original 3rd edition rule was (rule II.6.H on page 55):

The rule against assigning the same color to a player three times in a row was introduced between the 3rd and 4th editions. Was there any statistical justitification for this or was it a knee-jerk reaction to the complaint of a player who got black three times in a row (GM Walter Browne, if I recall correctly)? In the 4th edition the limit was changed to 200 points for equalization of colors and 80 points for alternation. It’s not clear whether this was the result of a statistical analysis beyond what was done for the 3rd edition. Maybe instead of 200/80 it should have been something like 100/50.

In any case, it seems to me that there should be an upper limit to the rating point difference for making any change to improve colors, including getting the same color three times in a row or four times in the last five rounds, such that a player would have equal changes before and after a maximum switch. Say the limit is 200 points. If a player’s color history is WWBB he should be equally happy being paired with white against a player rated 2200 or black against a player rated 2000. Personally I think I’d have better chances with black against the 2000, and I wouldn’t mind getting my third black in a row if I knew that the alternative was to be white vs. a 2200.

I agree with that, but I think it should be quantified in some way.

7

According to 29E5f both the 80 point and 200 point rules are out the window if it would give someone the same color three consecutive times. In effect, instead of an 80 or 200 point rule there’s a 9999 point rule.

For example:

2000 BB vs. 1300 BB
1700 WB vs. 1000 WB

If it weren’t for 29E5f the 1300 / 1000 transposition to equalize colors would be illegal because it would violate the 200 point rule.

8

If you are arguing that it is better to pair two players out of their score group than to assign anybody 3 in a row of the same color, then your argument is certainly not without merit (though it would not be according to current USCF rules). Maybe that’s why FIDE (apparently) does it differently.

If you are following George Cunningham’s logic, then it seems to me you are forced to the conclusion that there should not be multiple transposition limits at all (like 80 and 200). Instead, both equalizations and alternations should be worth the same (both 80, or both 100, or whatever).

Apparently, though, the 80- and 200-point limits are not based on any mathematical theory of how many extra points the white pieces are worth. Instead, they seem to be based merely on the relative undesirability of large transpositions vs large color problems. Failure to equalize is worse than failure to alternate, so the argument would go, so a greater stretch should be allowed.

Personally, I’m more sympathetic to the second approach, but I can certainly see the merit in the first as well.

That’s a loaded question. Or rather, it’s a question to which I’m going to give you a loaded answer. I would prefer to play the 2200. Recently I’ve had better luck with white against 2200s (I beat one a couple weeks ago) than with black against 1600s (sometimes I struggle to get a draw). :slight_smile:

It doesn’t always play out like that. Sometimes it simply means that a 2200 with two blacks in a row, instead of being paired against a 1900 also with two blacks in a row, would instead play a 2000 whose colors have alternated.

Bill Smythe

9

You are correct, of course. Actually both the 80- and 200-point rules are out the window when they would result in three straight blacks or whites.

Bill Smythe

10

If you come east to explain this system to the first chess lawyer at our mostly laid-back club who complains about it I will pay half your travel costs. I like your idea in principle, but I find “the simpler the better” is best practice in re Swiss pairings.

What I would like to see in print is a reflection of how things work in practice: TDs can and do have wiggle room to ‘massage’ the 200- and 80-point limits, if that works best for their particular events. Note that there is a variation at the end of 29E (I think) that allows TDs to disregard all rating restrictions to improve colors.

I believe that’s called the “FIDE” option in pairing programs; apparently FIDE SOP is to improve colors within score groups as much as possible, regardless of ratings. This might be due to both the relatively closer rating spread in most international events compared to USCF Swisses, as well as higher sensitivity to Swiss colors in the chess world outside the USA.

Anyway, I do not want to completely disregard ratings when pairing Swiss score groups—I want the leeway to say that a 203-point swap might work in a given situation to equalize colors, while a 198-point swap seems less necessary in another given situation. Based on posts here from experienced TDs who say they have tweaked the 80- and/or 200-point thresholds in their pairing software at times, it seems this is how things work in the field.

Discussion of how many rating points it’s worth to maintain ‘proper’ colors could be a TD tip, with the rule itself allowing for some discretion.

I wonder if Professor Cunningham’s statistical analysis is still extant somewhere?

11

Actually I’m arguing the other way: that the rules should be changed so that a switch to avoid giving a player three colors in a row should only be made if it’s within the 200 point limit, or possibly less than that if Professor Cunningham’s calculations which were used in the 3rd edition rulebook are still valid.

Instead of going from 100/100 to 200/80 they could have gone from 100/100 to, say, 100/50. From a mathematical point of view Cunningham’s numbers, assuming they’re correct, should be the maximum rating difference for any switch to correct colors. Apparently Cunningham didn’t consider that there might be one limit for switches to equalize colors and a smaller limit for alternation.

On the other hand there’s also a psychological effect to getting multiple blacks in a row. Even if the player getting multiple blacks is getting easier opponents than he would have if the colors were balanced he might feel that he’s being treated unfairly.

I do agree with your basic point that if getting three blacks in a row is bad getting four blacks in the last five rounds is also bad.

That would be a 100 point switch and therefore acceptable (although according to Cunningham the switch shouldn’t be more than 80 points for players rated under 2100).

It would be interesting to look at current playing statistics and see whether George Cunningham’s calculations of the effect of color on playing strength still hold up today.

12

They probably would. Chess hasn’t changed much (except for faster time controls, which I guess could have an effect).

I wonder where Cunningham got his data. Crosstables with color information are hard to come by, even today.

There are other reasons to transpose, besides the theoretical equating of color difference with X number of rating points. If too many players have bad colors in early rounds, there will be even worse color problems in later rounds (e.g. too many with two consecutive blacks). Late-round pairings are difficult enough as it is. On this basis, transpositions larger than the theoretical 100 points seem justified.

Besides, if you follow the Cunningham logic completely, it would seem to require that a transposed player that was set to get white with the raw pairings, but is now getting black with the transposed pairings, must be transposed downward only. And, in reverse, a player originally set to get black but now getting white should be transposed upward only. Somehow, I don’t think this was the idea behind the transposition rules, so I tend to reject the Cunningham considerations as a basis for making transpositions.

Bill Smythe

13

Two players may be switched in the natural pairing so as to alleviate color problems. For example, in a 10-player score group, the “natural” pairings should be 1-6,2-7,3-8,4-9,5-10 (by ranking order). The 80-pt rule says that we can switch 6 with 10 to correct an alternation problem provided 6 and 10 are within 80 rating points of one another. (Sometimes 6 and 10 cannot be switched consistent with the 80 pt rule, but 1 and 5 can be, and the 1-5 switch and the 6-10 switch have the same effect.) Similarly we can switch 6 and 10 to solve an equalization problem if they are within 200 rating points of one another. This is the “200-pt rule”.

The idea behind these rules seems to be that, on top of the “right” to be paired against someone in the same score group, a player has a “right” to be paired against the specific person in the other half of the score group that the “natural” pairings give him. The idea seems to be that you have a right to the opponent given you by natural pairings, and you should only be compelled for color allocation reasons to swap opponents with someone who is close to you in rating. You can’t be required to swap with someone who is very different to you in rating. However, in addition to you being swapped, your opponent can be swapped with someone. By the weird logic of the 80-pt and 200-pt rules, in that case your rating is then completely irrelevant, because you aren’t being swapped; it is your opponent who is being swapped (even though the result is just the same as you being swapped.)

How does this make sense? Where did this come from? Why are rating differences even relevant here? Isn’t the relevant thing how “far” you and the three other players involved in the swap are being moved from the natural pairing, rather than how close in rating the swapped players are, and which two of the four affected players are decreed to be the “swapped” players?

For example, it seems to me in the above score group, switching 8 and 9, and switching 8 with 7 are equal with respect to the natural pairings. They are both “1 rank” swaps. Both of these switches is a bigger change from natural pairings than switching 8 with 6, which is “2 rank” swap. Switching 1 with 5, or 6 with 10 (which are equivalent) is a “4 rank” swap. Both the number and “size” of the swaps is relevant. Two “1 rank” swaps might be preferable to one big “4 rank” swap. One big “4 rank” swap might be preferable to two “3 rank” medium swaps, and equal in preference to two “2 rank” swaps.

It seems to me that rather than the 80-pt and 200-pt rules we need a measure like this of the distance of a pairing from the “natural” pairings and a metric to compare two pairings so that we can say which is “better” as regards color allocation. Then the rules are: (1) use the pairing of the score group which gives the “best color allocation”, and (2) between two pairings that give equally good color allocation, use the one that is the least “distant” from the natural pairings.

I don’t think the “distance” measure from natural pairings need be ratings-based at all. It could be based just on the ranking of the players within the score group. (Ratings would play a part in rank, of course, but they would be relative ratings. Rank within the score group would be the important thing.)

14

Of course, you are largely describing FIDE pairing rules. Not precisely, but the 80/200 point rules do not apply.

15

If what I described is the FIDE approach to adjustments for color allocation, we should adopt them. Because the 80/200 point rules are, not to put too fine point on it, stupid.

16

Mine was not a comment on the validity of the appraoch, simply pointing out that such rules largely already exist so there is a framework to follow.

17

This might be a decent idea. Let’s look only at part (1) for the moment. I’d like to run through an example, for my own (and others’) edification.

Let’s say we’re pairing round 3. There are nine possible color sequences players may have experienced in rounds 1 and 2. I have listed these nine sequences below, in order from most strongly due white to most strongly due black. x denotes no color (bye, forfeit, late entry, etc):

(a) BB
(b) xB
(c) Bx
(d) WB
(e) xx
(f) BW
(g) Wx
(h) xW
(j) WW

Let’s say there are 40 players in the score group, divided as follows:

(a) BB – 2 players
(b) xB – 5 players
(c) Bx – 4 players
(d) WB – 11 players
(e) xx – 1 player
(f) BW – 9 players
(g) Wx – 3 players
(h) xW – 4 players
(j) WW – 1 player

Thus we have 22 players due white, 17 due black, and 1 due neither. Therefore, we throw the due-neither player into the due-black group, to try to even out the two groups.

Now we have 22 due white, 18 due black. It follows that 2 players due white will get black. These two should be selected from subgroup (d), since they are “less” due white than anybody else.

Therefore, any set of pairings which assigns black to two of the players in subgroup (d) and the one player in subgroup (e), and due colors to everybody else, is preferable to any other set of pairings.

Now let’s go to round 4, and say there are 30 in a score group. There are now 27 possible combinations of W, B, and x. For simplicity let’s assume everybody is in one of the following ten:

(a) WBB – 2 players
(b) BWB – 9 players
(c) BBW – 3 players
(d) WxB – 2 players
(e) xWB – 1 player
(f) xBW – 1 player
(g) BxW – 1 player
(h) WWB – 2 players
(j) WBW – 8 players
(k) BWW – 1 player

So there are 17 due white, 13 due black. Two will necessarily get the wrong color. These should be the one player in subgroup (e) and one of the two in subgroup (d).

Am I understanding your (or FIDE’s) idea correctly, so far?

Bill Smythe

18

FIDE’s color equalization system is very simple to understand.

Each player gets +1 for each white they get and -1 for each black and everyone starts at zero. No player should go above +2, below -2 or get the same color three times in a row.

This means there are 6 possible “color equalization” scores:

-2, -1, -0, +0, +1, +2

(+0 is given to players who were on -1 and received white and -0 to players who were on +1 and received black. This is to maximize alternation.)

Players in the negative scores are due white, either absolutely (-2), strongly (-1) or mildly (-0) according to FIDE’s wording. Players in the positive scores are due black.

FIDE pairings will maximize those “due” white against those “due” black. If there is a pairing where both players must have the same color then you use the “color equalization” score to determine who is “more” due that color. If that is equal then you grant the color preference of the higher ranked player.

There are no rating point limits on transpositions and interchanges but FIDE does provide a priority order in which you should do them. Also note that there are always exceptions to the above rules, such as the special rules for the last round for score groups over 50%.

19

So, according to FIDE, a player with WBWB is equally due white as a player with BWWB? Both seem to have a FIDE color score of +0.

If so, I don’t like it. I would much rather assign black to BWWB than to WBWB. Assigning black to BWWB at least equalizes his colors in the most recent four rounds.

Bill Smythe

20

According to FIDE both would be -0 and “mildly” due white. Both have had two whites and two blacks and FIDE would prefer them to alternate from their previous color. (Note: WWBB also gives a score of -0 but the player must get white in the next round to avoid the same color three games in a row.) Fairly simple to understand and not overly complicated, which is what I think they were aiming for.