Drop the lowest, or drop the middle?

Running Chess Tournaments
1

Discussion of this topic began in the conversation “Tournament Exercise Form AC Confuses Me” in this Tournament Direction thread. The issue was, if there are an odd number of players in a score group, should the lowest player, or the middle player, be dropped and paired against the highest player in the next lower score group?

This is a matter of opinion. The current standard (ever since the Morrison rulebook editions, at least) is drop-the-lowest, but some TDs still use the drop-the-middle variation.

One thing to keep in mind is that a player near the top of his score group “should” face weaker opposition in the next round, while a player near the bottom of his score group “should” face stronger opposition. (Top half vs bottom half.)

Normally, there is ratings overlap in the score groups. For example, due to upsets and draws, the lowest 2.0 will probably NOT be higher-rated than the highest 1.5. In this case, pairing the lowest 2.0 against the highest 1.5 will mean that the former will face a stronger opponent, and the latter a weaker one, just as “should” be the case. If, instead, the middle player is dropped, the pairing is much less likely to be as it “should” be. The middle 2.0, being in the middle, can reasonably go either up or down, so he has no complaint either way, BUT the highest 1.5 is quite likely to be facing a player stronger than himself, which is not as it “should” be.

So it seems to me that drop-the-lowest is preferable, usually.

Somebody who prefers drop-the-middle ought to step in here and tell us why their way is better, but I think the argument goes something like this: Suppose you have a large, single-section tournament like the U.S. Open. Then in round 1 the masters will be paired against 1700 players, while the 1800 players are playing 700 opponents. With such a large field, there will almost certainly be some upset draws, so that in round 2 a 700 (the lowest rated 1.0) will then be paired against a master (the highest 0.5). The huge rating difference in this pairing, it is argued, seems absurd. It would, however, result in a sure knock-out, which is exactly what the pairing is supposed to accomplish.

Bill Smythe

2

Thanks, Bill. I don’t find the US Open illustration terribly convincing. The odds are that round 2 in such a tournament is irrelevant to the final score of both the 700 upset winner and the master. And that is probably true for a 30 player 5 rounder too.

3

Last Thursday at the club, I got a tiny bit of flak for using the USCF method (actually SwissSys implemented the USCF method for me). Here is the situation before round 3.

2200 2.0
2000 2.0
1940 2.0
1894 1.5

SwissSys using USCF’s drop the lowest 2.0 rule gave
2200-2000
1894-1940

The master questioned why in round 3 he had to play the 2000 player, and our National TD who overhead his question answered, “because he’s using the inferior USCF rules.” The NTD makes it a policy of bringing up the highest 1.5 and pairs the group top vs. bottom, in effect dropping the middle 2.0 (the 2000) to play the top 1.5 (the 1894). It occurred to me that if you had a three-round tournament with only a few people and the situation was like:

2200 2.0
2100 2.0
1600 2.0
1500 1.5

and SwissSys came up with
2200-2100
1600-1500

If I were the 2100, I would feel a bit gypped that I wasn’t playing the 1500. To add insult to injury, if I lost to the 2200 finishing at 2.0, no matter what the result on board 2, I would finish in third (1600 wins=3.0, 1600 draws=2.5, 1600 loses=1500 at 2.5!). This was only a club tourney with no prizes anyway, but perhaps if money got involved (like Maurice Ashley’s $500,000 HB Global Challenge tournament next year), I might be at a loss to justify the pairings. Stated briefly, the USCF rules have the effect of favoring the bottom member of an odd score group to the detriment of the middle member.

4

The Swiss pairing rules are a real-world approximation of a mathematical model, and they are never going to be a perfect fit. (Consider the use of fluid dynamics formulas for traffic analysis – they work perfectly for water pipes, but only fairly well for automobiles.)

One of the points where the reality doesn’t match the model is the odd man. In very large score groups, the distortions will cancel out. In small groups (either small tournaments or small scoregroups), someone will often get lucky. Suppose, in your example, that the top player in the next scoregroup had been the top player in the tournament, rated 2300, who had lost in the previous round by blundering a piece. Would the middle man in the top group still argue that he “ought” to play him?

There are still a few TDs around who like to drop the middle man. They tend to be ones who learned to direct back in the 60’s, from the Harkness blue book. (Kashdan used to pair that way at Lone Pine; it did not reduce the number of pairing dispute.) When the USCF revised the rulebook around 1972, they had to pick one, and, for whatever reason, they chose “drop the low man.” Arguing that one or the other of these is “better” may be a fine way to spend time at the chess club, but it’s really pretty pointless. If you prefer dropping the middle man, announce it as a variation and pair that way.

5

On the other hand, if it was done the other way (drop-the-middle) and you were the 1600, you would almost certainly feel cheated that YOU weren’t playing the 1500.

In your example, the bottom 2.0 is higher-rated than the top 1.5. In such a situation, there is NO good way to do it.

But in a more typical situation, where the lowest 2.0 is LOWER-rated than the top 1.5, drop-the-lowest is more likely to produce the desired result, i.e. that players near the bottom of their score groups face stronger opposition, while those near the top face weaker opposition.

So, since drop-the-lowest works better in the majority of cases, and since there is no good solution anyway in the other cases, drop-the-lowest has been adopted as the standard.

I still wish somebody who actually favors drop-the-middle (as standard procedure) would jump in here with his rationale, so that the discussion can become truly interesting and double-edged.

Bill Smythe

6

Well, I can’t really debate it from his side, but the NTD who gave me a hard time in my previous example gave me more flak last Thursday. I believe I began to see his point.

After round 1, we had:

2015 1.0 W
2000 1.0 B
1934 1.0 W
1916 1.0 B
1602 1.0 W
1192 0.5 (bye)

SwissSys gave:

1916-2015 result (1-0)
2000-1934 result (1/2-1/2)
1602-1192 result (1-0)

The NTD asked me if game 2 had been decisive, would I have dropped the 1602 again according to USCF rules?

I went back and changed the result of 2000-1934 to 1-0 and then paired for round 3.

SwissSys gave
1916-2000
1934-1602

So yes, using SwissSys, I would have dropped the 1602 again, but he doesn’t seem to get an “easy” pairing. I think the NTD’s point is that if the players in the top score group is repeatedly odd (in this case 5 then 3), the lowest player might enjoy a string of “easy” pairings. A drop-the-bottom advocate could respond that the pairing for the 1602 would likely be a strong player who had fallen on hard times in an earlier round. That gets back to Bill’s point that different score groups will likely have ratings overlaps. Getting back to the drop-the-middle advocates, perhaps it is “better” for the 1602 to have gotten a hard pairing in round 2 instead of round 3. I realize it’s possible to construct examples that illustrate the opposite events.

7

Perhaps it’s getting off topic, but it concerns a follow-up question for the tournament in my last post:

In round 1,
2015-1578 (1-0)
1543-2000 (0-1)
1934-1430 (1-0)
1429-1916 (0-1)
1907-1356 (1-0)
1602-1334 (0-1)
1192 half-point bye

In round 2, the 1907 withdrew and the 1543, and 1578 took half-point byes. There were five 1.0’s
1916-2015 (1-0)
2000-1934 (1/2-1/2)
1602-1192 (1-0)
1430-1334 (1-0)
1356-1429 (1/2-1/2)

For round 3, SwissSys proposes
1602-1916 (2.0 vs 2.0)

2015-2000 (matching 1.0 vs 1.5)
1934-1578 (matching 1.5 vs 0.5 because 1934 already played 1430)
1429-1430 (matching highest 0.5 who is due black with 1.0)

1192-1543
1334-1356

My question concerns the pairings for boards 2-4. Instead of having the large group skip between 1.5 and 0.5, isn’t

2015-1934 (matching 1.0 vs 1.5)
1430-2000 (matching 1.0 vs 1.5)

“better”?

8

When I first did Swiss pairings dropping the middle seemed to give the most natural pairings. I do admit that I started doing pairings as a TD assistant and it wasn’t until I finally opted to go for certification that I read the rulebook and found that I had been doing the Harkness pairing variation.

By pairing the 1/2 AS the lowest 1 instead of AGAINST the lowest one (equivalent to dropping the middle), the rating difference on that board would often be similar to the rating differences on the boards above and below.

If the first round cut was just below 1500 and there were no upsets then you might have either:

Dropping the middle
2150 (1) - 1700 (1)
2025 (1) - 1630 (1)
1975 (1) - 1500 (1)
1900 (1) - 1475 (0)

or

dropping the bottom
2150 (1) - 1900 (1)
2025 (1) - 1700 (1)
1975 (1) - 1630 (1)
1500 (1) - 1475 (0)

Dropping the middle often avoids having one board in particular having a rating difference way out of whack with the boards around it.

One advantage of dropping the bottom here is that the close matches may be more likely to have draws and reduce the number of perfect scores (or the 1476 is more likely to score an upset against the 1500).

9

You could have responded to that rude and annoying question from the holier-than-thou NTD by throwing the question back in his face. If drop-the-middle had been used, round 2 pairings would have been:

1916-2015
2000-1602
1192-1934

– with the following possible results:

1916-2015 1-0 (as in the actual tournament)
2000-1602 1-0 (the most probable result)
1192-1934 0-1 (by far the most probable result)

Now the 1934 is, for the second round in a row, the middle player in his score group. Ask your TD acquaintance, would he drop the 1934 again in round 3?

So, the question of dropping a player twice in a row may be a legitimate issue, but it has nothing to do with drop-the-bottom vs drop-the-middle. The question comes up about equally often either way.

If you want to avoid dropping the same player twice, just put an up-arrow and a down-arrow on the two pairing cards, next to the round number, whenever two players from different score groups are paired. That way, next round you’ll notice, and can make a judgment call as to whether to let it affect the pairings.

In your drop-the-bottom example (with the 2000 beating the 1934 in round 1), when the 1602 is dropped the 2nd time, he is “dropped” against a player rated 332 points higher than himself, so (as you pointed out) it may be OK not to worry about it.

But in the above drop-the-middle case, when the 1934 is dropped the 2nd time, he plays an opponent only 81 points higher than himself. So he could skate from 1-0 to 3-0 by beating a player much lower than himself and another player barely higher than himself. THIS is much more worrisome to a TD who doesn’t like seeing a dropped player slide through.

So here we have a case where drop-the-middle causes more problems than drop-the-bottom, if cakewalks for dropped players are considered undesirable.

Bill Smythe

10

Yes, I agree. Reversing the order of which 1.5 plays which 1.0 is FAR preferable to pairing one 1.5 and one 1.0 each against 0.5’s. The former involves two half-point score differences (total 1.0), whereas the latter involves a half-point score difference, a full-point score difference, and another half-point score difference (total 2.0).

Transposing the two 1.0’s is legal, since the 80- and 200-point limits do not apply when pairings are transposed to prevent two players from playing each other again.

It looks as though Swis-Sys didn’t “look ahead” when it paired the first 1.5 against the first 1.0. Maybe the program doesn’t “look ahead” at all when pairings across score groups are involved.

Bill Smythe

11

That may seem “natural” to the players with 1 point, but it is ANYTHING but that to the 1475, who is paired against an opponent 425 points above himself despite being the highest-rated in his score group!

I’ve never understood why the debate is always between drop-the-bottom and drop-the-middle, or, more precisely:

(1) drop the lowest, raise the highest
(2) drop the middle, raise the highest

The following are equally plausible:

(3) drop the lowest, raise the middle (mirror image of (2))
(4) drop the middle, raise the middle (why not both?)
(5) drop the player at the 3/4 mark, raise the player at the 1/4 mark

– or, I suppose, even

(6) drop the highest, raise the lowest

– or any of a number of other possible schemes.

If the goal is, as jwiewel suggests, to keep inter-group rating differences reasonably small, then wouldn’t the following make the most sense of all?

(7) drop the Nth lowest, raise the Nth highest, where N is chosen to keep inter-group rating differences similar to intra-group rating differences.

For example:

Seven players at 2.0:
2200
2100
2000
1900
1800
1700
1600

Seven players at 1.5:
2100
2000
1900
1800
1700
1600
1500

Here you could choose N=2 (second-lowest vs second-highest) and get the following pairings:

2200-1900 (2.0)
2100-1800 (2.0)
2000-1600 (2.0)

1700-2000 (2.0 vs 1.5)

2100-1700 (1.5)
1900-1600 (1.5)
1800-1500 (1.5)

– which maintains a fairly constant rating difference of 300-400 throughout the pairings.

Bill Smythe

12

I learned drop-the-middle (indeed, we were using papyrus to make the pairing cards at the time and had I heard stories of when they used to chisel results into bits of stone) and it seemed less natural when drop-the-bottom came into effect. It is a compromise as Bill S. just said, and while his idea to select N to optimize scoring is great, it’s too complicated to expect humans to implement in the real world and opens the TD up to lots of questions. So we need a simple answer for what N should be. Middle and Bottom are two simple answers.

In later rounds of a tournament, drop-the-bottom will generally cause the dropped player to be paired well up in rating. This makes the Swiss Gambit a realistic strategy. Drop-the-middle seemed to produce a more even pairing (in rating) for the dropped player.

Dropping into the middle of the next-lower score group (“raise the middle”) is a novelty I’ve never thought of. I always thought of the top score groups as being the most important (that’s why we focus on them in the language, “drop” but not “raise”) but I realize that’s now Politically Incorrect.