Super Perfect 4-4, 8-8, 4-3, 8-7

Debates and discussions on the various race scheduling methods that can be used and their fairness and accuracy in determining the winners.
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Super Perfect 4-4, 8-8, 4-3, 8-7

Post by MathGuy »

Studying Race Charts and I stumbled some rather super efficient race for the final top 4 racers on either a 4 or 3 lane track. It satisfies the Complementary Perfect N criteria. (Even Lanes, Even Competitors, Competitors face each other on each lane.)

I believe this is going only to be possible for 4,8, 16, etc. (who has a 16 lane track?).


Super Perfect 4-3
Race #1 #2 #3 #4
Lane1 1 2 3 4
Lane2 2 1 4 3
Lane3 3 4 1 2

Super Perfect 4-4
Race #1 #2 #3 #4
Lane1 1 2 3 4
Lane2 2 1 4 3
Lane3 3 4 1 2
Lane4 4 3 2 1

Super Perfect 8-8. (Remove Lane 8 for Perfect 8-7)
Race #1 #2 #3 #4 #5 #6 #7 #8
Lane1 1 2 3 4 5 6 7 8
Lane2 2 1 4 3 6 5 8 7
Lane3 3 4 1 2 7 8 5 6
Lane4 4 3 2 1 8 7 6 5
Lane5 5 6 7 8 1 2 3 4
Lane6 6 5 8 7 2 1 4 3
Lane7 7 8 5 6 3 4 1 2
Lane8 8 7 6 5 4 3 2 1
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Cory »

MathGuy wrote:Super Perfect 4-3
Race #1 #2 #3 #4
Lane1 1 2 3 4
Lane2 2 1 4 3
Lane3 3 4 1 2
This one is documented at Stan's site (http://members.aol.com/StanDCmr/cyoung2.html, bottom of the page), and I was aware of the 4-4 one, also. The 4-4 and the 4-3 are basically the same chart.
MathGuy wrote:I believe this is going only to be possible for 4,8, 16, etc. (who has a 16 lane track?).
This is a MUCH better conjecture than the one we posted years ago.

MathGuy, I've been trying to find new 1-round CPNs for years with no luck, but I never went past 6 lanes. (Who has an eight lane track?)

Congratulations on finding the "key"!!! Very cool!
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by MathGuy »

Cory,

http://members.aol.com/StanDCmr/cyoung2.html

I remember seeing this a while ago, and getting rather tired very quickly at the formal math. (I ussually do) Now I am going to have to review the entire document.

:mrgreen: Boy, I know that I have something better to be doing. :lol: :mrgreen:
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Cory »

MathGuy wrote: :mrgreen: Boy, I know that I have something better to be doing. :lol: :mrgreen:
There is nothing earth-shattering there. Stan and I just wanted to convince ourselves that all the things that seemed to be "obviously true" were, in fact, true.

The Super Perfect 4-3 was also in my email to Stan -- web page cyoung.html at Stan's site. We just threw it onto the theory page as an oddity. We figured that someone like you would eventually come along and figure it out for us!!! ;)
Last edited by Cory on Thu Mar 11, 2004 12:55 pm, edited 1 time in total.
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Stan Pope »

Yes! Very cool, indeed!

I was trying to find that 3-lane chart, myself, because it looked familiar ... something that Cory mentioned ... couldn't find it anywhere on Cory's Pack site! (Something to do with "creeping senility", perhaps, though, now-a-days, it seems to be "thundering" rather than "creeping.")

The generalization to 2^n and (2^n)-1 is good work. Were you a number theorist in a past life?

I'd expect this chart to be somewhat more accurate than the "ppngen" generated PN chart of the same size. How would it's accuracy compare to the generated CPN chart? Intuition is easily tested, I guess ... just need to flip this chart about its diagonal into the form needed by Cory's simulation program and chug it through with a few different variances.
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by MathGuy »

For the 6 lane track (we have one) the best I can do is this chart. Adjacent Car numbers have perfect complementary lane assignments. Thus 1-2,2-3,3-4,4-5,5-6,(6-1).

This would be useful for a Finals round, if the preliminary rounds rank is used to sort the The Cars. Thus assuming the preliminary rounds comes close, this would make the rankings more exact.

6 Lane
Race #1 #2 #3 #4 #5 #6
Lane1 1 2 3 4 5 6
Lane2 2 1 6 5 4 3
Lane3 3 4 5 6 1 2
Lane4 4 3 2 1 6 5
Lane5 5 6 1 2 3 4
Lane6 6 5 4 3 2 1

We have a 6 lane track, this year for the initial rounds used the sorting method, we only used 3 or 4 lanes depending on the # of boys in the rank. And for the finals, we used a simple lane rotation for the top 4 (3 lane configuration) cars and top 5 cars (4 lane configuration).

Next year, I will probably want to fill all 6 lanes for the trophy round, and using the above chart makes me believe that we are probably going to be ok. The more boys in the finals, the more excitement. This chart will obviously work for a 5 lane track by ommission of lane 6
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Stan Pope »

Here's DerbySim comparisons:
Key Parameters:
Sigmas:
Car Speed: 2
Lane Speed: 1
Random Elem: 1
Timed Tie Break
5000 Trials

Since the proposed chart usage is "Finals", the cars variance is low relative to lane and random variance, and comparison is based on 3-trophy, i.e. First, Second, and Third Place trophies are awarded correctly.

4-3:
PN: 77.59%
SPN: 78.35%
CPN: 79.73%

4-4:
PN: 78.13%
SPN: 78.31%
CPN: 81.41%

6-6:
PN: 72.35%
SPN: 72.73%
CPN: 78.69%

Even averaging results over 5000 trials, the percentages flopped around in a narrow +/- 1% band, so replication will not be exact, but probably close.

Conclusions:
1. For those cases in which SPN and CPN charts coexist,
SPN are slightly more accurate than PN, and CPN are more accurate than SPN.
2. Complying with CPN criteria help accuracy. Adding heats while maintaining CPN compliance helps accuracy more.

Cory, I love that little chunk of software! :)
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by MathGuy »

Thanks for the analysis, interesting. Let me challenge you though:
Key Parameters:
Sigmas:
Car Speed: 2
Lane Speed: 1
Random Elem: 1
Timed Tie Break
5000 Trials
I know that a car "unique" run will generate a Random Elem, but so does any performance of any competition. If a car#1 runs true to form with an expected time of 2.500, and assume no lane bias. If car#2 runs true to form, it will go 2.505, If Car 1 and Car 2 race, with Random Elem, let's assume these are the times:

Race CAR1 CAR2 Winner
#1 2.490 2.510 Car1
#2 2.505 2.500 Car2
#3 2.510 2.495 Car2

Overall winner is Car2. The question is which one deserves the trophy?? Car2. I don't think you should count this as an "Error". Even though Car1 might (theoretically) have a superiour potential, it did not perform well enough to win. The sports equivalent, the best team doesn't always win. If a superior car has a few bad races, and losses, so be it.

The key question when analyzing schedule peformance is how often lane or schedule biases interfere with the car overall finish, not the random element effect of results modeling.

Thus I think a better test is to set Random Elem to 0.
Also, for 4-4 and 6-6, it would also be relavent to see a single heat results. Thus if 1st through 3rd was determined by a single race.
Last edited by MathGuy on Fri Mar 12, 2004 11:16 am, edited 2 times in total.
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by MathGuy »

Stan, I am sure it is going to be difficult to test the SP6-6 as I had envisioned it's use. The Car#'s are set based on a preliminary races PPN scoring. Thus the PPN rank#1 would be car 1 and PPN rank#2 would be car 2, etc. Thus it takes advantage of the preliminary rounds sorting, and will be complementary perfect to correct that initial ranking that might be off by one.

But a simple Random assignment will be interesting to see regardless.
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Stan Pope »

Here's DerbySim comparisons:
Key Parameters:
Sigmas:
Car Speed: 2
Lane Speed: 1
Random Elem: 0 <---
Timed Tie Break
5000 Trials

Since the proposed chart usage is "Finals", the cars variance is low relative to lane and random variance, and comparison is based on 3-trophy, i.e. First, Second, and Third Place trophies are awarded correctly.

4-3:
PN: 90.53%
SPN: 91.52%
CPN: 93.89%

4-4:
PN: 92.85%
SPN: 92.91%
CPN: 95.33%

6-6:
PN: 88.82%
SPN: 89.74%
CPN: 92.40%

Even averaging results over 5000 trials, the percentages flopped around in a narrow +/- 0.5% band, so replication will not be exact, but probably close.

Conclusions:
1. For those cases in which SPN and CPN charts coexist,
SPN are slightly more accurate than PN, and CPN are more accurate than SPN.
2. Complying with CPN criteria help accuracy. Adding heats while maintaining CPN compliance helps accuracy more.

The relationships between results for variance for the random element at 0 and 1 are consistent.
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by MathGuy »

Good stuff. I was surprised at the error rate in the 4-4 SPN and CPN. I would think the error would have been 0%, if there was no randomness left in a race. Do I miss understand the model?

I will have to modify my spreadsheet to confirm this result in my mind. (You do, You understand)
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Stan Pope »

MathGuy wrote:Good stuff. I was surprised at the error rate in the 4-4 SPN and CPN. I would think the error would have been 0%, if there was no randomness left in a race. Do I miss understand the model?
Cory can probably explain this better, but it boils down to a few possible combinations of lane differences and car differences that can fool the system. They are few, but they are present. This is why the cars should be randomized into the chart, rather than someone deciding which car gets which chart number/position.
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Cory »

MathGuy wrote:I was surprised at the error rate in the 4-4 SPN and CPN.
There are very rare cases like the one documented at http://pack146.nova.org/pinewood/track.html, which may be that to which Stan was referring.

In this case, though, I think the error rates are so high because lane bias is half the size of car bias. A trial having two cars which are more closely matched than the lanes becomes fairly likely, and they'll end up tied after you run the chart. Then you run a timed tie-breaker and whoever gets the better lane wins -- you might as well have flipped a coin.

Since Stan did 3-trophy accuracy -- which is the same as 4-trophy accuracy when there are only 4 cars -- this means lots of errors.
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Re: Super Perfect 4-4, 8-8, 4-3, 8-7

Post by Cory »

Oops, sorry...I went back and checked the code. Timed tie-break uses the objective car speeds only -- it is not affected by lane bias.

My re-assessment is that the "very rare" condition I mentioned above is not so rare when cars and lanes have similar variance.
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