Global Ranking is here to stay; the whole Championship Point system rests on it. The best that can be done short of revamping the whole system is to tweak GlobalR.  Scaling to a perceived equilibrium is a good start.

Well, a lot has been said but I would still like one question answered:

How often does it occur that a player on a board play more than 75 games and has a H rating higher than 80%.  This is when a 40 point cap would effect equilibrium.  If this number is very low, which I expect it is, you can still implement a 40 point cap.   

SG - Look at the stats of some boards yourself. I did.  Suffice it to say 80% is not likely to happen on standard boards like WarGear Warfare.

But as far as all boards are concerned, I'd still say it's too early to tell. In games where luck is less a factor (Like Go-Geared), I think it's much more likely for a few dominant players to emerge.   If they play each other, that'll keep their H-ratings down. But if they avoid each other, they could maintain a high H-Rating.  Regardless, their Global ratings will not move when they play each other, so if they dominate the players below them, their GRatings could fly.

Hey M,

Yes I agree with what you are saying, however, before giving up on a possible solution I would like to have this proven.  Hugh came up with one example, 75 games weren't played, but it was close.  If if only happens once or twice it stay still be worth trying to implement a 40 point cap.  If we have the stats to show it happens more often then we can cross it off the list and move on.  I just see too often where things are left uproven and 'up int the air' so they are constantly revisited.  There has been a lot of good information presented here so it would be great if we could come to a solid conclusion.

We can change the cap and watch the stats ..or we can just watch the stats and change the cap later.  Either way, the stats won't be any different.

Which way is prudent? If we change it and it doesn't work, and then change it again, that could be confusing to a lot of players, not to mention a bit of wheel spinning on the programming side.

On the other hand, if there's consensus that something needs to be done sooner than later..

And, finally.. on the other (third) hand.  We should probably consider some other options.  It seems to me there should be alternatives to a hard cap.

For instance We could modify your idea to create a more progressive system..

Hard: all points over 40 are reduced by a 3:1 ratio.

Soft: all points over 40 are reduced by a ratio determined by a curved function to be determined.

If we are not able to get an answer to the question above, then I would recommend implementing a filter based on rate ratio.  For example, only allow players to filter those with a rate 1/2 of theirs or lower.  So for example you could select a filter from a list:

.50 of your rate and higher

.45 of your rate and higher

.40 of your rate and higher

etc

So, this, of course would mean you would allow people to avoid playing opponents who have half their rate and lower.

I think this is reasonable and would not be considered a method of protecting your rate - others may see it differently.  Maybe to start you would only implement it for 1 v 1 games to see how it goes.

Instead of a simple ratio of rankings times 20 to determine point transfers, how about some kind of asymptotic function, to get rid of the hard cap.

For example, given S1 and S2 (Score of two players).  In the case of a win by 1,

you have a formula P = f(S1,S2), that tells you the # of points at stake when 1 beats 2.

Or if we assume it is a function of the ratio of their score (i.e. R = S1/S2), we can simplify and say P = f(R).

Right now the formula is P = 1/R*20.

Maybe we can attack it by listing sets of R & P that we think make the ideal, and then try to fit a curve to them.

So 1st point (R,P) should be (1,20).  That keeps things consistent with current play where evenly scored players risk 20 points per game.

Next is to choose a 'soft cap' that would never be approachable.  Say max 50 points lost, min 5 points lost.  We can represent those as (R,P) points (+infinity,5), (0,50).  So now every game risks between 5 & 50 points.

Now 3 points is not enough to get a very nice curve, so lets pick two more points.  How about if assume that up between a ratio of 1/2 to 2 (i.e games between players within half as many score as their opponents).  We keep the original 20*R formula, so that gives us two more (R,P) points: (.5,40) and (2,10). 

So now we have 5 points:

(0,50), (.5,40), (1,20), (2,10), (+infinity,5)

Can someone fit a curve to that?

Something like this?

P=-(30+3 R)/(-6+R)

Edit - way off -- kind of inverted..

P= 3.5(4.5+R)/(R)

This should be pretty close.

I like the idea Ozy, I put the points in Excel and used the trendline function to try to get a match.  The closest is an exponentail equation but it is not a great match and is a bit unwieldy.

P= 3.5(4.5+R/(R) yields approximately

(0.5, 35) (1,19) (2,11) (3,9) (5,6.5) ...

(10.5,5) hits your suggested threshold and  (+infinity,3.5)

This solution does not satisfy (0,50) because 0 is the vertical asymptote, but that's not a problem because the ratio of 0 should not be possible.

When R approaches 0 the solution looks to work pretty nicely

For instance,

1/5 Ratio:  (5000 Loses a game to 1000) = -82 (Currently 100)

1/10 Ratio: (5000 loses a game to 500) = -161  (Currently 200)

1/50 Ratio (10,000 loses a game to 200) = -796 (Currently 1000)

As you can see it pretty much puts a light 4/5 damper on the amount the better player can lose compared to the current system.  This looks to nicely balance the gains the better player makes on the other side. 

Is this what you had in mind, Ozy?  I'm thinking you wanted stronger limits on the top.  This is a pretty soft solution.  I'm sure the equation could be tweaked to come down harder on the top end (when R approaches 0).

SquintGnome wrote:

Hugh came up with one example, 75 games weren't played, but it was close.  If if only happens once or twice it stay still be worth trying to implement a 40 point cap.  If we have the stats to show it happens more often then we can cross it off the list and move on.  I just see too often where things are left unproven and 'up int the air' so they are constantly revisited.  There has been a lot of good information presented here so it would be great if we could come to a solid conclusion.

I gave an example that satisfied >75 games and > 80% H-rating:

http://www.wargear.net/boards/view/Axes+and+Allies/Rankings

Now, I know 75 games and 80% H-Rating were based on a 1000 opponent. The 1000 opponent is common, but there is some variation here. So, in the spirit of your data-based approach, a fine approach, a good question to ask is what a re-run on WGWF would yield:

http://www.wargear.net/boards/view/WarGear+Warfare/Rankings

We've got some post-2000 with under 80% H-Rating (way over 75 games, and clearly they played a lot of post-1000 players). If we had direct access to the won/lost data, we could see how a 40-cap would have played out, which would be very interesting.

Getting back to the "worry about the future" approach, I gave a board likely to cause problems. I had others in my quiver, they just weren't as convincing :) Once a player is post-1500, they can keep playing these boards with no risk to their CPs. Boards where the leader has a high win rate could cause problems in the future, but it would require the leader playing more. Or another motivated player to play at a similar win rate. Here are some:

http://www.wargear.net/boards/view/Pangaea%3A+Rise+of+the+Chimp/Rankings

http://www.wargear.net/boards/view/Gear+Wars%3A+The+Force+Unleashed/Rankings

http://www.wargear.net/boards/view/Rockem+Sockem/Rankings

http://www.wargear.net/boards/view/Five/Rankings

http://www.wargear.net/boards/view/Seven/Rankings

http://www.wargear.net/boards/view/Ten+-+Propagate/Rankings

http://www.wargear.net/boards/view/StarGear

http://www.wargear.net/boards/view/Octagons

Anyway, I think you're right that a 40-cap wouldn't have affected much to date. But, a real problem arises when a player reaches a point where the 40-cap regularly applies. Equilibrium does go up. My understanding of your argument is that it wouldn't happen too often in the common 1000 matchup situation, and it would protect players against the 500 rated player. It probably wouldn't have happened too often so far, except in global ratings. But, it _can_, and the potential is there for it to happen more often, because once a player is past 2000, it always applies against a 1000. High win rates already exist on some maps, and on others there is reason to believe the rates could be higher, or that the cap would apply anyway, as in WGWF.

RiskyBack wrote:

The amount gained or lost depends on how favored the winner was to win.

This is my favorite property of Elo. It's an excellent innovation, and the formula is only slightly more complicated than ones like ours.

If they are serious about re-vamping their system, they should also consider the more modern Glicko:

http://en.wikipedia.org/wiki/Glicko_rating_system

Thanks for all the hard work Hugh.  I can't disagree with your point that a 40 point cap will affect equilibrium at times.  I suppose the last point I would like to make on the topic is this. 

If you have the perspective that losing more than 40 points is unfair then although some inflation would occur with a cap it would help more than hurt. Up till now we have discounted the fact that the current system may be unfair at times.  If you accept that it can be unfair at times you do not have to throw away any proposed solution that may have an adverse affect.  It may be that a solution has a bad side effect but is still better than the current system.  No sytem is perfect.

I am not saying a cap is the best solution, maybe it's the filter or something else.

Do any of these other Elo system use caps?

Elo is automatically capped. Glicko is similar, but the cap moves depending on your "ranking reliability", a stat kept in addition to the ranking itself. (More reliable ratings go up/down less than less reliable.)

On the one hand, an Elo player can feel great about never losing more than, say, 32 points in a game. But, the higher ranked players have a different thing to complain about: For some matchups, they gain 1 point if they win, and lose 31 if they lose. (Though, to be fair to the system, this only occurs when the probabilities really are that high for the winner. If the edge really is only 4 to 1, the system shouldn't produce that kind of rating match-up.)

The fairness issue is one I'll have to think about. I can see an argument for unfair if playing a low rated player doesn't give the corresponding edge. I certainly don't think the 2000 versus 500 thing is "unfair" if the 2000 player actually has the proper edge to have it balance over time. Frustrating, terrifying, etc, would be the adjectives I'd use instead of "unfair" ;)

M57 wrote:

P= 3.5(4.5+R)/(R)

This should be pretty close.

That's pretty close.  I came up with:

P=5+100/(1+e^(1.6*R))

Which is also pretty close.  It has pretty similar values to your equation in the typical range, but is better behaved at the extreme points.  I think something of the form A+B/(C+e^(D*r) would give us the right shape.  Mine ranges from 5 to 55 as we approach our limits.

I stuck these all in a online google doc in case anyone is curious:

https://docs.google.com/spreadsheet/ccc?key=0AjtIsJkyopekdGVfa2VUdHlvMGZoSUtRSFJGOVUtdHc

Hugh wrote:

The fairness issue is one I'll have to think about. I can see an argument for unfair if playing a low rated player doesn't give the corresponding edge. I certainly don't think the 2000 versus 500 thing is "unfair" if the 2000 player actually has the proper edge to have it balance over time. Frustrating, terrifying, etc, would be the adjectives I'd use instead of "unfair" ;)

I guess the judgement of 'fairness' would be board specific.  You have shown case of very high H ratings for some boards, a small percentage I think.  For Wargear Warfare, which I play mostly, I think the data supports that losing 80 points on this board is unfair since it requires a 94% win rate for equilibrium when the rate ratio is 4:1.  The highest rate for players with a substantial amount of games is 82% after 40 games.  IF I used the criteria of a minimum of 75 games, the highest rate is 72%.

But, of course, the system must be used for all boards, so it is difficult to have a 'one size fits' all system.  Which would be difficult, but better, would be to set an equilibrium win rate for each board and base the point cap on this.  This would prevent runaway scores.  For Wargear Warfare theh cap would be about 40, but for Ocatagons the cap might be 80.

Squint -

(Maybe I'm not understanding the math properly, and if so, then you can ignore my comment, but I think my response would be as follows)

I think that the basis for your evaluation of the "fairness" of the scoring system assumes that 2000PointPlayer deserves a ranking of 2000 points on a given board/GR, based on his/her skill.
Maybe your perception of the point-to-skill ratio is weighted incorrectly.
Maybe everyone who has 2000 points has an artificially high score, and requirement of a high win rate is the current scoring system's way of trying to bring those artificially high scores back into equilibrium.
The loss of more than the 40 point cap equates to the normalizing modifier that should be bringing their score back to the range that our scoring system would have them normalize at (based on the point-to-skill ratio that is included within it)
If you remove the preconceived ratio of points-to-skill from your evaluation of the current scoring system, you'd see that the only flaw you've pointed out is that it (as Hugh mentioned) "frustrates/terrifies" those few high ranked players, but it is not unfair (ie: benefits one tier of player over another).

Put another way:
I propose instead that you would have less of a problem with the scoring system if you adjusted your perception of what 2000PointPlayer's score SHOULD be. 
The current theory is only "unfair" if you are insisting that a player with skill X needs to be able to normalize at 2000 points.
Even if you don't like it, I think that the scoring system is dictating that a player with 2000 points should be able to beat a player with 500 points 95% of the time after 75 games, because that's what the points-to-skill ratio requires.
If so, then "2000PointPlayer" is only good enough to normalize to 1600 (which requires a smaller win percentage to maintain, after your arbitrarily chosen 75 game minimum).
And in this case, there is nothing unfair about him losing that 40+ points to the 500 point player.

Final thought:  I think that Hugh has shown that no matter what it is set at, a point cap does give an unfair advantage to one group of players (those with high enough rank). 
It shouldn't matter how unlikely it is that this unfairness could be capitalized on, it's still a hole in the system, and as such, that system needs to be unilaterally discarded.

Let's assume we're designing this site from the ground up:
If Tom informed us that he was going to implement a dice system that had a mathematical pitfall in it, where, in some very unlikely circumstances, a tiny group of the best players would have an unfair advantage over the lower ranked players on the site, we'd be erecting the gallows right now.

I VERY much admire your persistence in finding a fix for a problem that you see...but a point cap, no matter how it is implemented, is NOT a fair solution.