Ctp Pbem Rankings 12th April 2002

[Kralj Matjaz]

*deleted*

[blackice]

The example shown is missleading no one will have the same score or rarely. The small amount of points given to the second and rarely third place person is to reflect the fact they did beat at least one opponent in the game. This does not happen on every ranking it depends on the rank of the players in the game and the points wagered. This is to reflect the fact that the score is calculated every 9th turn. It should not be that "the winner only" scores points every ninth turn. That would simply be unfair and would not promote players playing or beiing ranked. I hope that helps.

It is also incentive to play and be ranked. By doing this their was no need to impose upon people the rule of having to be ranked.

Quinn post the win loss so people can get on with this why the delay?

[J Bytheway]

Well, after some research I find that the actual formula is 1/(2+d^2) where d is the difference and ^ denotes exponentation - which makes more sense.

*Here begins the heavy maths *

But it's still a very strange formula - I can't imagine what probabalistic assumtions would give rise to it. Basically if A and B have scores a and b and d=a-b this formula claims that the probability of B beating A is:

1/(2+d^2) for d>=0
and 1-1/(2+d^2) for d<0

this has zero gradient at d=0 which seems exceptionally peculiar. I would have expected a formula more along the lines of:

(1+tanh(d))/2

or:

(pi/2 + atan(d))/pi

Since these have maximal gradient at the origin, but again, these are unlikely to be derived from theory.

Probably you'd actually end up with some formula involving the phi function (phi(d) being the integral from -infinity to d of 1/(sqr(2pi))*e^((d^2)/2), which unfortunately cannot be expressed in closed form) but that's of little use to us.

* Here ends the heavy maths *

But, after some thought I do understand and approve of the application of the probabilities to the manipulation of the scores as described (scores move by half the probability of the observed outcome) since this should award/penalise most for play where the low-ranked beats the high-ranked.

However, the introduction of probabilities is an unnecessary obfuscation of the relevant process, except insofar as it explains the otherwise arbitrary introduction of the '1-' to the defenition of the function used. It might be more sensible, perhaps, to manipulate the function and use it directly to deduce the change made to the scores, rather than including an intermediate step. In this case a function such as that used has some benefits which perhaps explain its use rather than any probability theory.

I can see the Klair-syndrome hidden in this system - it arises from the fact that results on consucutive *9 turns are not independant. Without further study I can't place a cure, but my instict says that it might be a good idea to use a reduced power (change 2 to 1.5 or 1.6), as I suggested above, so that the function decays less quickly - but that might in fact exacerbate the problem. I believe Gav recommended diminishing returns (i.e. you get less for leading if you've been leading for a long time), and that might also help if applied judiciously but my gut warns me off, if only because of the increased administrative burden. Perhaps if a suitable program were written to store the information then it would be plausible.

All in all it's a knotty problem - I'll give it thought as I lie in bed this evening and get back to you tomorrow with any insights I have.

[blackice]

1. Kralj Matjaz 23.2532 10% wagered=2.3-3.9= 1.6 new ranking 24.8532
2. Faded Glory 22.2950 10% wagered=2.2-2.619= 0.419 new ranking 22.714
3. Darth Viper 10.6760 10% wagered=1.0-1.649= 0.649 new ranking 11.325
4. Nuke Boy 15.0358 10% wagered=1.5-0.97= -0.53 new ranking 14.5058
5. TheBirdMan 27.8815 10% wagered=2.7-0.485= -2.6515 new ranking 25.23

9.7 points wagered total divided by the forumula below:

5..1ST..41%..2nd..27%..3rd..17%..4th..10%..5th..5%

This game is waiting to be ranked so I used it as an example. 1st 2nd and 3rd receive some points be it token points for 2nd and 3rd. 4th and 5th negative points. As you can see award/penalise most for play where the low-ranked beats the high-ranked. Is built into this system also. But unlike the old system the new one may or may not award points to the bottom ranked players. They did not win.

Again here we go making new rules to make a system work maybe. The new system works and awards points for winning. No new rules are needed and you do not have to have a math degree to understand the formula.

John the old system also awards points for second and third and forth if they beat a higher ranked player. I guess that means it is flawed too even more so...Bottom line is I don't get your flawed reasoning since both do it the old system more so...In a five player game with the bottom position being the highest ranked player. Every person above them will get points it guess I am missing the logic here? You dismissed the new system for something the old one does more of?
Maybe you could explain why?

It also awards way more points for doing so and way more negative points for the loser. Look at the blackice example.

The difference is the ease of the new system and the built in no need for rules to create balance and fairness.

[J Bytheway]

Originally posted by blackice
But unlike the old system the new one may or may not award points to the bottom ranked players. They did not win.

Eh? The old system would never have awarded points to the bottom player, but the new one might if they were ranked sufficiently lowly.

The new system works and awards points for winning.

Unless the winner is far ahead already in which case they lose points for winning...

John the old system also awards points for second and third and forth if they beat a higher ranked player. I guess that means it is flawed too even more so...

But that's a good thing, isn't it?

It also awards way more points for doing so and way more negative points for the loser. Look at the blackice example.

Is that the one where he wins 2 games, loses 1 and drops in score? That is primarily due to the falloff speed of the formula used (I presume). If that situation continued then an equilibrium would have been reached with Blackice at some lower score, but still higher than all his competitors, just as it should be. The details of the formula will dictate the position of this equilibrium. I don't see the problem.

The difference is the ease of the new system and the built in no need for rules to create balance and fairness.

Ease is nice, but I'd rather have fairness (Naturally, I would say that). I'm fairly sure an appropriate system of a similar sort can be devised without recourse to additional rules like a 3-game minimum.

[blackice]

Eh? The old system would never have awarded points to the bottom player, but the new one might if they were ranked sufficiently lowly.

Oh so it does not award players for beating a higher ranked player?
Or is it only the first placed player that beats a higher ranked player that gets points? If the higher ranked player is last of five players. fourth place gets no points for beating them? they do in fact get points. J. They are awarded one win over the last place player and then three losses for being in fourth. So in each and every 9th ranked turn someone takes 3 losses and one win for fouth place. They also lose points for beating a higher ranked player. The reality is 1st and 2nd get all the awards for beating the Higher ranked player. In the new system 1,2 maybe three and maybe four will get points for beating the higher ranked player. More fair.

Unless the winner is far ahead already in which case they lose points for winning

Are you refering to LUNG? if so then yes you have top spot you have to play games to keep it. on top of that you have to play at least one higher ranked player to gain points. In Lungs case his remaining game he is way ahead of his opponents so no he will not be rewarded big for beating far lower ranked players.

But that's a good thing, isn't it?

You mean it is flawed if the new system does it but not if the old one does? Makes no sense still...

Is that the one where he wins 2 games, loses 1 and drops in score? That is primarily due to the falloff speed of the formula used (I presume). If that situation continued then an equilibrium would have been reached with Blackice at some lower score, but still higher than all his competitors, just as it should be. The details of the formula will dictate the position of this equilibrium. I don't see the problem.

Really the other players would be going up and he down so what you are saying is they meet and all stay the same? How would blackice stay ahead if he losses points for one loss and two wins? Makes no sense.

Ease is nice, but I'd rather have fairness

The new system is fair the winner gets points as do second and third sometimes. The losers lose points(sometimes depends on the rank of the players)what is not fair about that? makes no sense again.

Quinn where is that complete example? You are not going to post it or what? Do you not think the people should see the two systems side by side in an equal and fair comparision?

This is all talk until we see a clear example of the old system J. could you take the example I just gave and use the old system to show how it works to the other players here? We can see them side by side with the same players as an example. Would that be possible?

[quinns]

Thanks for the very reasoned response J Bytheway! You are right about the above formula... I will fix it in the above post (i.e. it is the differential alone squared, without the +2). Regarding "probablity", it was a stretch to use that term because it is an imperical number, but I thought it might help to understand what the formula was doing. Also, I had considered using an exponent of 1.5 or 1.6 instead of 2, but thought it would make the formula too messy. But maybe your right. It would level out the curve somewhat.

Blackice, I would be happy to post the "loss ratio" but I don't know exactly what you mean by that. For example? You mean the number of ratings points that the losers lose? Or the ratio of winner gain points to loser loss points? That ratio is always 1. That is, if Albert gains .02 then Bob loses .02. The ratio = .02/.02 = 1. This is always true. I don't think that is what you meant though when you said "loss ratio". Edit: Ice, I added what I think you meant by "loss ratio". Take a look at my previous post where I show the ratings of Bob, Charlie, and Dallas after the first *9 turn.

Obviously, JBTW has much more knowledge about this subject than I. I await his thoughts on this tomorrow.

[blackice]

Thanks Quinn

What I mean is take the birdman game example I gave above:

http://apolyton.net/forums/showthread.php?s=&threadid=47344&perpage=10&pagenumber=26

and with the old system do the calculations. That way we have a real example of the two systems rankings side by side. That it easy really then people can see in a real game scenerio how they both would work.

In this example given the losser is the highest ranked player.
This has been the complaint in the past and the real difference between the two systems. The fact that Albert gains so little playing lower ranked players is one very good reason NOT to play lower ranked players.

The new system promotes playing with lower ranked players, primarily because in 9 out of ten games you play lower ranked players here. The loss for playing lower ranked players is even more important to show as that is the big difference between the systems.

Quite frankly most can not do the math required to make you calculations.

If we can do that we can talk apples and apples go to a vote and offer the third solution a new ranking system all together. Should not take you more than a few minutes. Thanks :d:

[quinns]

Ice, it isn't really apples to apples to use the Birdman example you mentioned. At this point, the PBEM ratings are already way out of skew as compared to the old system. In my opinion, there is no possible way that Birdman should be 17 points higher than Darth Viper. I have played both players and know both of their relative playing abilities. At most Darth Viper and Birdman should be within 1 to 3 points of each other.... not 17! See, I would have to use an example BEFORE the GameLeague based system took over in order to compare apples to apples. However, if you really feel it would clarify things for you and everyone else, I can do the calculations if you like. But as I said, I would be using the old system to calculate the already heavily skewed GameLeague based rating numbers. It would not be apples to apples.

[blackice]

Old
BLACKICE 21.342

New
Blackice 26.301

Thanks it will do an number is a number looking at my old score compare to the new one? 5.041 difference. I do not get the skewing part but lets move on and use the example and compare the difference ok?

This will show a lower ranked person winning over a higher ranked. That is the whole point here.

Darth has lost more games than Birdy has won and played far less.

Thanks Quinn

[J Bytheway]

Originally posted by blackice

Eh? The old system would never have awarded points to the bottom player, but the new one might if they were ranked sufficiently lowly.

Oh so it does not award players for beating a higher ranked player?
Or is it only the first placed player that beats a higher ranked player that gets points? If the higher ranked player is last of five players. fourth place gets no points for beating them? they do in fact get points. J. They are awarded one win over the last place player and then three losses for being in fourth. So in each and every 9th ranked turn someone takes 3 losses and one win for fouth place. They also lose points for beating a higher ranked player. The reality is 1st and 2nd get all the awards for beating the Higher ranked player. In the new system 1,2 maybe three and maybe four will get points for beating the higher ranked player. More fair.

Sorry, I misunderstood you - I thought you meant the bottom player, not just one near the bottom. It is indeed true that If a few low-ranked players beat a sufficiently highly ranked one then all of them will gain, and the highly ranked player will lose a lot. Surely the highly ranked player should lose a lot in this situation, and since the system is conservative (total rank of all players constant) it is inevitable that the other players will rise. Is your concern that too much of this rise is allocated to the lower amongst those lowly ranked players - I can see that this might be a problem.

Unless the winner is far ahead already in which case they lose points for winning

Are you refering to LUNG? if so then yes you have top spot you have to play games to keep it. on top of that you have to play at least one higher ranked player to gain points. In Lungs case his remaining game he is way ahead of his opponents so no he will not be rewarded big for beating far lower ranked players.

This I view as a bad thing - I think you should be rewarded for every victory, no matter how apparently trivial, but that those rewards should be small. If this is what you want then certainly the new system (or something like it) is appropriate.

But that's a good thing, isn't it?

You mean it is flawed if the new system does it but not if the old one does? Makes no sense still...

No, the flaw in the new system to which I was referring in this context was that the very last player (5th of 5, or whatever) can gain points if losing to sufficiently highly ranked players, not the awards to those near the bottom for beating a player who is highly ranked and last.

Is that the one where he wins 2 games, loses 1 and drops in score? That is primarily due to the falloff speed of the formula used (I presume). If that situation continued then an equilibrium would have been reached with Blackice at some lower score, but still higher than all his competitors, just as it should be. The details of the formula will dictate the position of this equilibrium. I don't see the problem.

Really the other players would be going up and he down so what you are saying is they meet and all stay the same? How would blackice stay ahead if he losses points for one loss and two wins? Makes no sense.

No, I mean that he will drop and they will rise until they are a certain distance apart appropriate to this win/loss ratio. The fact that he is losing points for 2 wins/1 loss is essentially because - according to the system - someone with a 2/3 chance of victory should not be so far ahead of his opponents as he is. As his score drops the aqmount by which it drops will slow.

Ease is nice, but I'd rather have fairness

The new system is fair the winner gets points as do second and third sometimes. The losers lose points(sometimes depends on the rank of the players)what is not fair about that? makes no sense again.

If you don't mind situations like that in which Lung found himself (and the converse for very lowly ranked players) then there is nothing wrong with that. That is the only obvious flaw to me (but that doesn't mean there aren't others). If you can live with that then the ease of use almost certainly outweighs any other problems. I'm certainly not going to volunteer to maintain the rankings and so I don't think my opinion carries very much weight, but it was requested and so I give it.

This is all talk until we see a clear example of the old system J. could you take the example I just gave and use the old system to show how it works to the other players here? We can see them side by side with the same players as an example. Would that be possible?

Yes, but I won't because I don't have a spreadsheet to hand and my graphical calculator is out of batteries . This system does need a calculational aid in excess of Windows calculator to do in any sensible time, which I'll be the first to admit.

The best recommendation I can come up with so far is this:

Use a differential system as before, so if A beats B and d=a-b then increase a by f(d) and decrease b by f(d) where f is the function:


f(d) = { (1/(2+d))/n if d>=0
.......{ (1-1/(2-d))/n if d<0


where n could be 2 for simplicity, but if you wish to avoid Klair syndrome then it might be better to use n equal to the number of times A has beaten B in this game so far (including this one) - but that does of course require storing additional data and I don't think it would be worth it.

I reduced the power all the way to 1, rather than stopping at 1.5 since this avoids the 'messiness' quinns mentions, and simplifies things further.

Though it's purely gut instinct, I think that you should indeed calculate all changes before applying any of them as someone (blackice?) recommended a couple of pages ago - but it probably won't make that much difference so if it would be simpler not to then I wouldn't worry.

In any case, I believe that this is a fascinating subject (ranking theory?) and I think I'll do some additional research into it, though probably not in time to be of any use to you.

I've noticed a mistake in the above post - I confused the probability of A beating B with that of B beating A - I'll fix that.

[blackice]

2..1ST..67%..2nd..33%
3..1ST..54%..2nd..31%..3rd..15%
4..1ST..46%..2nd..29%..3rd..17%..4th..8%
5..1ST..41%..2nd..27%..3rd..17%..4th..10%..5th..5%
6..1ST..36%..2nd..25%..3rd..17%..4th..11%..5th..7% ..6th..4%
7..1ST..33%..2nd..23%..3rd..16%..4th..12%..5th..8% ..6th..5%..7th..3%
8..1ST..30%..2nd..22%..3rd..16%..4th..12%..5th..8% ..6th..6%..7th..4%..8th..2%

No, the flaw in the new system to which I was referring in this context was that the very last player (5th of 5, or whatever) can gain points if losing to sufficiently highly ranked players, not the awards to those near the bottom for beating a player who is highly ranked and last.

As you can see the odds of that happening beyond a two player game are extremely rare. you put 10% of your points into the pot and gain 8% or less. We have not allowed a two player game to be ranked as yet nor should we. The two and three player points are to reflect the fact that a 5 or 6 player game is down to 3 or 2 players. People should be awarded point for sticking out a game and especially for beating out the other players. Mind you like the old system if the last two people where one is high ranked and the other way lower points will be small.
IMHO if a 5 player game has only 2 or 3 players that want to be ranked. The game should still be ranked out as a 5 or 6 player game. Until there are 3 or 2 players left.

Is your concern that too much of this rise is allocated to the lower amongst those lowly ranked players - I can see that this might be a problem.

Mine and a pile of others too. this was the biggest problem with the old system. We are just waiting for quinn to post this senerio so we can move on.

I think you should be rewarded for every victory, no matter how apparently trivial, but that those rewards should be small.

Lung senerio is the fact he has one game and the opponent is lower ranked. If you have first place you have to work to keep it. Lung will also win points when he wins this game. no system should allow a 1st ranked player to play but one game against a weaker opponent and remain in the top spot while others have several and against better opponents.
The Klair factor will not happen in this new system.

As his score drops the aqmount by which it drops will slow.

Correct be if I am wrong but that depends on his opponents winning. if the opponents are losing games at a faster rate this player will continue to drop in the rankings by large amounts. This also would depend on how many games the player is playing. If only one game it could put them down huge. It also depends on the other games they are in. If they are winning other games they are gaining less than they are losing. Back to the 2 wins mean less than one loss.

I don't think my opinion carries very much weight, but it was requested and so I give it.

Everyone's opinion should count here I hope! Thanks for your input we need more people to speak up.

This system does need a calculational aid in excess of Windows calculator to do in any sensible time, which I'll be the first to admit.

The average person does not have the calculator or the knowlegde to make these formulas work. That is a nessessity in order to maximize ones wins with the old system. That was the other problem because of the huge wins vs loss difference. With the old system you would want to play a game with less or no players way down the rankings to maximize your win vs loss on the ranking board.

The new system does the same thing but with less of a hit both ways, winning or losing. This is more friendly to PBEM as the chances of playing lower ranked players are great. The fact less damage is done promotes games with lower ranked players. This is needed without the need for rules. This also eases the "bad land" scenario that is a huge factor. You are playing lessor ranked opponents you have tundra they have rivers and mountains. Like gary said we should take the time each and every game to balance the land factors.
Any volunteers?

[quinns]

Alright Blackice, here is the old way of calculating your example (rounding to two decimal places). It took me about an hour of manual work, but in the past the program did all of the work once the *9 finish orders were entered. I hope this clarifies things.

Differential Based Calculation Example

*9 Finish Order:

1. Kralj Matjaz 23.30 (K)
2. Faded Glory 22.30 (F)
3. Darth Viper 10.70 (D)
4. Nuke Boy 15.00 (N)
5. BirdMan 27.90 (B)

K d. F: Prob = 1/(2 + 1.0**2) = .33 --- K is higher than F so Change = .33/2 = 0.17.
K d. D: Prob = 1/(2 + 12.6**2) = .01 --- K is higher than D so Change = .01/2 = 0.01.
K d. N: Prob = 1/(2 + 8.3**2) = . 01 --- K is higher than N so Change = .01/2 = 0.01.
K d. B: Prob = 1/(2 + 4.6**2) = .04 --- K is lower than B so Change = (1 - .04)/2 = 0.48.
F d. D: Prob = 1/(2 + 11.6**2) = . 01 --- F is higher than D so Change = .01/2 = 0.01.
F d. N: Prob = 1/(2 + 7.3**2) = . 02 --- F is higher than N so Change = .02/2 = 0.01.
F d. B: Prob = 1/(2 + 5.6**2) = .03 --- F is lower than B so Change = (1 - .03)/2 = 0.49.
D d. N: Prob = 1/(2 + 4.3**2) = . 05 --- D is lower than N so Change = (1 - .05)/2 = 0.48.
D d. B: Prob = 1/(2 + 17.2**2) = . 01 --- D is lower than B so Change = (1 - .01)/2 = 0.49.
N d. B: Prob = 1/(2 + 12.9**2) = . 01 --- N is lower than B so Change = (1 - .01)/2 = 0.48.

Net Change:

Kralj Matjaz: +.17 + .01 + .01 + .48 = +0.67
Faded Glory: -.17 + .01 + .01 + .49 = +0.34
Darth Viper: -.01 - .01 + .48 + .49 = +0.95
Nuke Boy: -.01 - .01 - .48 + .48 = -0.02
Bird Man: -.48 - .49 - .49 - .48 = -1.96

New Ratings (Differential Based):

1. Kralj Matjaz 23.30 (K) +0.67 = 23.97
2. Faded Glory 22.30 (F) +0.34 = 22.64
3. Darth Viper 10.70 (D) + 0.95 = 11.65
4. Nuke Boy 15.00 (N) - 0.02 = 14.98
5. Bird Man 27.90 (B) -1.96 = 25.94

[TheBirdMan]

In my opinion, there is no possible way that Birdman should be 17 points higher than Darth Viper. I have played both players and know both of their relative playing abilities.

Just because you have all the Gods, the luck, the terrain and all the babarians (so I feel it ) on your side in our challenge-game .

You might even have ME on your site. It took ages before I noticed what you were doing . Outnumbering me in populationsize/number of cities at a rate of 1:3 or even 1:4. And now it's TOO LATE - soon your PG will be more 4 times mine and I am without doubt going to LOOSE .

[quinns]

Originally posted by TheBirdMan


Just because you have all the Gods, the luck, the terrain and all the babarians (so I feel it ) on your side in our challenge-game .

You might even have ME on your site. It took ages before I noticed what you were doing . Outnumbering me in populationsize/number of cities at a rate of 1:3 or even 1:4. And now it's TOO LATE - soon your PG will be more 4 times mine and I am without doubt going to LOOSE .

Sorry Bird, I didn't mean it that way. I think you are a very good player, but Darth is good too. A 17 point relative difference essentially means that you are "Beyond Grand Master of CTP" and Darth is "Way Below Beginner Level". That is not exactly true!

Regarding our current PBEM challenge game, I would be happy to grant you a rematch! You are right about your terrain and bad luck with Barbarians. Let's start afresh. PBEM is the most reasonable way to play CTP IMHO. We play when we can without having to commit many hours at designated times and places. It is the best! Internet play is for those who have GADS of free time. Lucky them.