What merit is there to the so-called “Swiss gambit?”

Question

The "Swiss gambit" refers to the idea of intentionally lowering one's performance at the beginning of a Swiss-system tournament, to one degree or another, in order to have an easier path through the later stages of the tournament. E.g. a highly-rated player in a field could take a half-point bye in the first round (instead of a likely win) in order to likely be paired against weaker competition for the next few rounds than would have happened otherwise.

It certainly shouldn't be the case that it's always a good idea, or Swiss-system tournaments would be very poorly conceived. But depending on where a given player sits ratings-wise in the field, and the exact distribution of the ratings of players in the tournament, it is not inconceivable that such a Swiss gambit could increase a player's expected final score. Of course, intentionally giving points away at any point could obviously decrease the expected score as well. This is all I am asking for here:

Has any serious research been done concerning scenarios in which a Swiss gambit is/isn't effective, probabilistically speaking, in the sense of raising/lowering the gambiteer's expected tournament score?

Pointers to any legitimately mathematical (and mathematically legitimate) discussions of the matter would be most appreciated, as would the creation of such in an answer of course.

Answer 1

I tried writing a crude pairings/results simulator, to see whether taking a bye could increase a top player's score. When generating pairings, the program ignored pairing history and color (which I realize can matter, but I didn't want to have to program it to go back and redo pairings if there was a conflict - this is a crude simulator, not an actual pairings engine!) But it was able to handle pairing people in a scoring category with the usual top half/bottom half method, as well as the all-important "pairing up" of the top person from the lower category with the bottom person from the higher category if there was an odd number of players in the higher score category.

I assumed a uniform ratings distribution through whatever ratings range I selected. I used the "Standard winning expectancy" formula at the bottom of page 11 in this document. I did not account for fatigue. I assumed a draw probability of half of the probability that the higher-rated player would lose (for example, if the expected score was 0.75 according to the formula, I assumed a win would happen 70%, a draw 10%, and a loss 20%. For even matches with 0.5 expected score it would be 40% - 20% - 40%.) I set the program to run 100000 tournaments at a time, to get a good average.

The Swiss gambit pretty much always decreased a high-rated player's overall score regardless of the number of players, rounds, or rating spread (unless I set the draw probability parameter to zero, which is unrealistic.) At best it had only a small negative effect on the final score. Although the player's performance in later rounds was indeed better due to weaker opponents, that performance did not quite overcome the nearly half-point lost. The top players were better off playing in all the rounds.

For example, in a simulation of an 8 round 200 player tournament, with player ratings ranging from 200 to 2000, the 2000-rated player had an average score of about 6.35 if they did not take a bye. If they took a first-round bye, the average was only about 6.24.

However, for some small tournaments with large point spreads and a certain number of players, although the average score dropped, the probability of placing first actually increased. For example, in a 5 round 32 player tournament with players rated from 200 to 2000, taking a first round bye decreased the average score from 4.23 to 3.95, but increased the probability of taking clear first from 33.2% to 34.7%. I'm not sure if these are artifacts of an imperfect pairing engine, though; the exact pairings matter more in such situations. In most of my simulations the decrease in score corresponded with a lower probability of taking first (and the decrease was somewhat larger than the increase shown here.)

Interestingly, although it also wasn't effective compared to playing all the rounds, it seems that taking a half-point in the second or even third round often gave a slightly better score than taking one in the first round, especially when the ratings spread was large (in the 8 round 200 player example, they scored about 6.26 by taking the bye in the second or third round, as opposed to 6.24 by taking it in the first round.) The first round has a top half player against an easy opponent; why skip a game you'll almost certainly win, instead of skipping the next one where your opponent may have something of a chance?

So, overall: The average score decreases when using the Swiss gambit. The odds of winning the tournament might go up in certain specific scenarios, but I'd need a better program to say for sure, and if there is any such effect it's sensitive to the exact number of players.

Answer 2

It would help to know the motivation of the gambiteer. It never occurred to me to game the Swiss system, or that it was even gameable. Does the gambiteer want prize money? Rating points?

I've seen far too many Swiss systems at the class level where the winner had a perfect score. It's hard to believe that the goal is to win the tournament when the gambiteer voluntarily loses 1/2 point.

Let's start off by agreeing that surely, it is possible to get an easy 2nd and 3rd round if a player honestly loses or draws in the first round. The question becomes, can a cheater manipulate the system in a meaningful way.

So, let's assume for the SG (Swiss Gambit) to work:

1. it's a class-level tournament (my world.)      
2. all players have the exact same rating.
3. the ratings are accurate.

I don't believe the SG would yield a positive result in this case; at best, the gambiteer would play people who might be off their games. However, it's much more likely he'd just be playing someone who lost. At the class level the games are almost always decisive.

Thus, I draw the conclusion that the SG only works dependably if there's a wide range of rated players. In large tournaments where players are grouped by class (D and under, C, B, A, Expert) I can't imagine a measurable result; the maximum difference between ratings is 200 points.

So, I posit:

1. it's a class-level tournament (my world.)      
2. the brackets must include players of wildly different ratings
3. the ratings are accurate
4. the point of cheating is to get prize money

#1 implies draws are rare. This becomes important once the ratings are widely distributed. If the gambiteer secures the only draw, in round 1, he'll almost certainly end up playing in the "0-wins" bracket since the only other player with a draw would be his opponent and you can't be paired twice in a Swiss. And because of #2, the "0-wins" bracket will contain mostly lower rated players.

#2 implies a small tournament where there aren't enough players to fill out class-specific brackets.

#3 is a dicey assumption since I'd expect a cheater to sandbag his rating. I'd also expect a cheater to also play with a cheesy style designed to knock out less experienced players. For example I've seen players talk during play, make really fast moves in an attempt to psychologically rush the opponent, etc. This is probably not germane to the discussion, however.

#4 is my assumption of the motivation. This means the gambiteer wants to win the remainder of his games and be alone at the top. It doesn't do much good to get a cut of 3rd place with 5 other people. Because it's probably a small tournament (else #2 may not be true) the gambiteer needs very good score.

As I work through this, I begin to understand the SG. The SG exploits the Swiss method of

a. pairing people with the same scores
b. not allowing duplicate pairing, and
c. splitting the brackets in half by rating and pair the top of the top with the top of the bottom.

So the gambiteer scores a factional point in rd 1 in the hope he's always paired with someone in the group with the lesser score. So in round 2, he's paired with the "0/1" group. Further, he'll be paired with a player whose rating puts him in the middle of that group.

Consider the last round of a 5-round Swiss: The gambiteer, at 3.5 points, will be playing a middling 3.0 scorer. Compare this to the others at the top - two 4's battling it out. The Gambiteer is going to probably come out on top of one of them. The worst case scenario is that the 4's draw and all three of them share 1st, 2nd, and 3rd places.

Conclusion #1: I'm convinced it is easily possible to materially manipulate the 2nd round of tournaments that meet criteria #1 and #2.

Conclusion #2 - the SG is real in theory if dicey in practice. Draws, drop-outs, and the leeway allowed to the TDs can ruin the gambiteer's day.

Solution - group draws with the category above them, not below them. This will stop the SG in its tracks. That is, in rd 2, the gambiteer would be playing winners, not losers. Further, because of their scores, the gambiteer would be playing someone at the bottom of the top half of the group. Probably not the intent and certainly not a path to a prize-by-cheating. In fact the first-round draw works against him now as he always gets paired up. This may be too harsh. It could be in rds 2 and 4 the fractional scores would get paired up and in rd 3 they would get paired down.