Ok I say EMin = 4387, EMax = 4594 => always solvable (and unbreakable) for these 12
EDIT: EMin = 4429, EMax = 4528 if we add the "half" rule
note that AbsoluteMin = 4387, AbsoluteMax = 4594; So there is only ONE solution in this case (the full legal interval)
Can we find a tighter interval than [4387 .. 4594] for these 12 that is unbreakable, with or without "half" rule?
Maybe also a tighter one than [4429 .. 4528]?
Sven
If my implementation is correct (exhaustive search), then I fear no, it's impossible
So it seems that, without the "half" rule, all intervals tighter than the biggest possible one are breakable, at least for your "12 players" example. If that turned out to be the essential outcome then this would imply that
1) Graham (or any other TD in his place) would have to explicitly include the "half" rule, and
2) no other restriction which can remain constant throughout the whole selection process can be made without making the configuration "breakable".
The only consistent rule (that I have found) that guarantees selection from both halves and is unbreakable is the half rule.
A consistent range is (min Elo + median Elo, max Elo + median Elo).
As Martin found, this can be improved by substituting median Elo in the lower limit with the greatest Elo less than the median and substituting the median Elo in the upper limit with the least Elo greater than the median.
I've tried to work with quartiles (engine from top quartile is paired with engine from bottom quartile; engine from upper middle paired with engine from lower middle quartile), but the range based off of quartiles does not prevent breaks.
Adam Hair wrote:The only consistent rule (that I have found) that guarantees selection from both halves and is unbreakable is the half rule.
A consistent range is (min Elo + median Elo, max Elo + median Elo).
As Martin found, this can be improved by substituting median Elo in the lower limit with the greatest Elo less than the median and substituting the median Elo in the upper limit with the least Elo greater than the median.
I've tried to work with quartiles (engine from top quartile is paired with engine from bottom quartile; engine from upper middle paired with engine from lower middle quartile), but the range based off of quartiles does not prevent breaks.
Disregard what I said about a consistent range. The pairing of Chesley and Parrot (4498 Elo) is in that range, but that can lead to Carballo and Bubble being the only engines left.
Sven Schüle wrote:So it seems that, without the "half" rule, all intervals tighter than the biggest possible one are breakable, at least for your "12 players" example. If that turned out to be the essential outcome then this would imply that
1) Graham (or any other TD in his place) would have to explicitly include the "half" rule, and
2) no other restriction which can remain constant throughout the whole selection process can be made without making the configuration "breakable".
Sven
Hi Sven,
1) I agree that explicitly including the "half" rule would be reasonable, without breaking the fun factor
2) If you mean EMin..EMax then yes, it can be the full legal interval in the case without the "half" rule. With half rule a tighter interval can be found but still too large.
So my conclusion (if my computations were correct) is that these unbreakability rules don't limit selection much (intervals still not tight enough). Besides, adding more restrictions => more annoying for us humans => less fun So I'd say that our experiment was a failure.
Adam Hair wrote:As Martin found, this can be improved by substituting median Elo in the lower limit with the greatest Elo less than the median and substituting the median Elo in the upper limit with the least Elo greater than the median.
I didn't work with quantiles at all. Just tried all combinations. But it's an interesting idea and I'll try to compare median to the minimum solvable bound that I obtained by using that and see it they match.
