Coming up with a Ranking formula for KotZ

[eMarkM]

Hi, probability and math gurus. I've recently taken over the role as Tournament Director for the King of the Zoo series of tournaments. For those not familiar, it will be a series of 18 events on PokerStars open to any 2+2 poster with at least 25 posts. Mostly HE, but featuring every game Stars has touneys for. See here (http://tinyurl.com/2eprc) for more info on the series.

Anyway, much of the fun in playing in this is for bragging rights of being the best, or King of the Zoo. For that we had a formula for determining the best player over the course of all events. That formula, devised by the previous TD went as follows:

[ QUOTE ]
After each tournament, the participants are awarded points based on their finishing position, relevant to the field. The percentage of the field they beat is multiplied by 10 to derive a points total. For example - If a player finishes 13th in a 56 runner tournament, they will have beaten 43 of their 55 opponents = 78.2%. Thus, their points total is 782 for that event. The winner receives 1000 points, and the last placed competitor scores 0.


[/ QUOTE ]

The problem I have with this is that it led to a situation where a player who was consistently in the top 20 would get a high ranking without ever making the money. In fact, last year 2nd place went to someone who only made one final table and went out 9th. This leaves a bad taste in my mouth, doesn't seem right. It also gives those who play the most events an automatic pop in the rankings by just surviving for awhile in each event. We'll have many more events this year, so this will get more pronounced. This leaves me wanting a better formula.

On one hand, I don't want a system where someone plays two events and takes first or second in each and hops to the top. I want players to be somewhat rewarded for participating. But I think the formula above gives too much reward for just showing up a lot. I'd like to at least weight it for players that make the money so we don't have situtions where players are in the top ten that never moneyed. Would also like a formula where everyone earns points somehow, so not like Stars where the majority of participants get 0 because they don't finish in the top 15%. A lot to ask of a formula I guess and not sure the best way to go about it.

Any input, ideas, suggestions would be appreciated. And of course, every 2+2 poster with at least 25 posts is invited to play in the series. Just give me an email address and I will email passwords to the events as they come up. Thanks for your help.

[Wenona]

Hi Mark

This is my first ever 2+2 post, so I'll quickly try and get a few more in so I try to become King of the Zoo.

I would suggest changing the points allocation from a linear equation to a curve. This will decrease the points differences between the lower placings and increase the gap as peoples placings improve. To do this you simply put the result of your previous equation to a power greater than 1.

Eg
No of entrants = E
Finish Position = F

Current formula = (((E-F)/(E-1))*1000

Suggest formula = (((E-F)/(E-1))^X)*1000 where X = input variable

Results - 50 player tourney

Current X=1.5 X=2 X=3
1st 1000 1000 1000 1000
10th 816 738 666 544
25th 510 364 260 133
40th 204 92 42 8

Suggest you just set it up for say a 50 player tourney and play around with "X" until you get a spread of points that you like.

Also suggest perhaps using say best 9 finishes only (if 18 tourneys) in total points. People who have played lots can still improve by beating their previous 9th best finish, and people who only play say 12 tourneys are not at a huge dissdvantage against those that have average finishes in all 18.

Hope this is some assistance.

Wenona

[Bozeman]

Why not just use money won?

This is an already existing non-linear weighting that happens to naturally correspond to what we think of as winning.

Craig

[bigpooch]

Even though there might be other methods that are arguably
superior, I would have to agree with this post. Simply look
at the net amount of total dollars won. Sure, someone might
get lucky, but it's tournament play!

[irchans]

Net amount of money won is great, but it does favor frequent players.

Another formula you might consider is

(average percentage of top pay out -
entrance fee/top payout)*Sqrt( number of games).

For example: Assume a player plays in four tournaments with an entrance fee of $1 and top pay out of $100. Assume that he win $10 in one tournament. Then his rating would be

( 2.5% - 1% )*Sqrt(4) = 3%.

[BruceZ]

When Mike Haven approached me about coming up with a ranking for team competition, my first suggestion was to use money, but scaled to equalize teams with different numbers of players. My thinking was that since the player's strategy is based on the payout structure anyway, it didn't make sense to have an alternate reward system that conflicts with that. It seems that others now agree with this. Mike didn't want this because he wanted all players to contribute some points, just as now you seem to want everyone to get points for participating. My next suggestion was to have a linear reward system, but add bonus points for placing in the money, and more bonus points for winning. My suggestion was (for 40 players) 50, 44, 43, 42, 41, 35, 34, 33, 32, 31, 30, ... which is the linear system 40, 39, 38, ... with 10 added to the winner, and 5 added for each place in the money. I believe we ended up deciding on a "grand prix" system of 50, 45, 41, 38, 36, 35, 34, ...

[Bozeman]

Personally, I think that you should reward greater number, since there are only 18 a year or so. However, it is reasonable to take somewhere between the (linear in # played) scoring of total money and the (flat in # played, and a bad choice by most reckonings) average money. Money won should be net, not gross.

If you want all places to receive something, a couple of alternatives are Sqrt(players/place) and (number of players beat)^n (n~2-5 depending on how much you would like to reward top finishers).

The "grand prix" system uses quadratic (n=2) scaling in the money, and then linear scaling for the rest.

All these scoring systems could be used with a non-linear relation to number of events played, using (AveragePts)*(events played)^x, where 0<x<1 (x=1 and x=0 are the linear and average methods mentioned earlier).

Craig

[BruceZ]

Personally, I think that you should reward greater number, since there are only 18 a year or so.

The problem is that this can conflict with correct monetary strategy, for example, by causing a player to pass on opportunities to double-through for fear of busting and ending up in a low place. I'm assuming that the players are concerned with actual money as much or more so than their ranking. Normally if you are out of the money, it makes no difference how far out of the money you are. Low placement does not necessarily correlate with low skill, and can be more associated with bad luck. Most of the battle of skill takes place between those on the bubble, those who just make the cut, and those who don't. This would suggest a fairly sharp cutoff after the money places.

[Simon Diamond]

How glad I am that I quit mathematics when I was 16! For the record I devised the original system, and cannot believe how simple it is in comparison to all these complex suggestions.

My wish was to reward loyalty and overall consistency - however I did compile a prize money list too, so people could see who fared well in that category. Strangely enough the same person topped both the official rankings and the prize money charts, though I do see Ed's point about the second ranked player only making one final and finishing 9th. I finished 5th in the rankings with 2 finals and 4 bad results - so maybe that proves my system is flawed.

Interesting debate guys, good luck in finding something that the honorable TD eMarkM can work with in the KotZ series.

Simon

[Lori]

I haven't tested this, or even thought too hard, but you might want to use Simon's 1000 point scale, but weight it according to the first prize.

Eg: First prize is $250, the scale runs from 0 to 250, but with 0 being awarded for one behind last place (so everyone scores)

After applying the linear scale, add the number of dollars actually won, so the winner would get 500 points.

In a 10 player $11 sng for example, the points would be as follows:

1st 100 (50+50)
2nd 75 (30+45)
3rd 60 (20+40)
4th 35
5th 30
6th 25
7th 20
8th 15
9th 10
10th 5 (Afterall, he turned up)

I like this because it is simple, rewards the money but also gives points for bothering to enter.

Lori

[dogsballs]

I like Wenona's distribution here, with the X=2. I'm sure there wouldn't be many complaints with or something like this.

Some of the tourneys have different entry fees, and I think we should make sure they are weighted equally despite this. The distribution of points should only be according to the no of entrants, not whether it was $22, $33 or $55 to enter. The arbitrary entry fee shouldn't impact the points distribution; only the no of competitors in the field and your finish position should do that. Hence, since entry fees differ, I vote not to use the prize distributions in the calculations.

I vote Wenona.

I'd say at least the top 10 finishes out of the 18, or better 12, to reward a bit of dedication to the series.

dogs

[eMarkM]

Thanks for everyone's input so far. This is great stuff. So far the idea I like is some combination of the linear ranking already in place, but with some added weight for money. I haven't back tested all the ideas here as it would take quite a bit of time doing a recompiling of standings on my part. What I did do was to take the data that was readily available and came up with what I have below. Please poke holes in this math gurus.

On the KotZ website that Simon manages he has a variety of statistics including Stars rankings and total money won. All I did was combine two of them, the linear ranking used last year and the Stars ranking system that he also compiled. I divided Simon's final result by 10 and just added to it the person's Stars points. So instead of getting 1000 pts for winning an event, you get 100. Combined with Stars formula that pretty much only gives points to those in the money gives making the money a lot more weight. But you can still crawl up the rankings by just participating a lot and making decent finishes. Money is not so much weighted that you could win one event and be in the top 10. Here's the results of a backtest:

First the top 20 using the original ranking formula. Here I show how they were ranked, how many events they played in (there were six total), total points using old system and how much money they actually won.

<font class="small">Code:</font><hr /><pre>
rank name played $ won pts
1 Lorinda 6 $1,348 5,423
2 Glenn 6 3,637
3 Myrtle 5 $456 3,575
4 Havanabanana 6 $633 3,564
5 Simon Diamond 6 $427 3,221
6 David 6 3,219
7 William 4 $182 3,150
8 eMarkM 5 $134 2,920
9 Inthacup 5 $468 2,637
10 jasonHoldEm 3 $367 2,572
11 C M Burns 4 $156 2,299
12 Fmonti 4 2,294
13 ZeeJustin 4 $70 2,174
14 ohkanada 6 2,113
15 KurnsonofMogh 5 2,090
16 PlayerA 3 $256 2,036
17 MS Sunshine 3 2,019
18 Acesover8s 4 $189 1,946
19 duxDelux 2 $408 1,870
20 mrbaseball 5 1,854
</pre><hr />

Here I think you can see my problem with the approach used last year. Sure, the best player in terms of money won, but the other rankings do not correlate with money as much as I'd like. Glenn played all six events and never moneyed, but was very consistent. I want to reward this, but not with 2nd, which in my mind should clearly be Havana. So I figured if I just add the Stars points to this while giving the linear ranking less weight, it would correlate more with the money actually won. Here's what it looks like:

<font class="small">Code:</font><hr /><pre>
new rank name played total $ won
1 Lorinda 6 1047.82 $1,348
2 Havanabanana 6 678.89 $634
3 Myrtle 5 631.52 $456
4 Simon Diamond 6 521.76 $427
5 Inthacup 5 442.59 $468
6 jasonHoldEm 3 437.84 $367
7 William 4 403.29 $182
8 C M Burns 4 379.26 $156
9 David 6 375.49
10 duxDelux 2 370.85 $408
11 eMarkM 5 367.79 $134
12 Glenn 6 363.7
13 PlayerA 3 358.67 $256
14 yct 2 351.3 $338
15 sdplayerb 2 316.1 $192
16 Acesover8s 4 287.42 $189
17 curtains 2 286.5 $79
18 ZeeJustin 4 280.65 $70
19 ohkanada 6 273.73
20 KurnsonofMogh 5 269.77
</pre><hr />

Here, Lorinda's dominance is even more clearly shown. She played every event and was a staple of the final tables, winning twice. She has twice as much money as the next best, and almost twice as many points. The other top five also correlate very nicely with total money won. We have one player, duxDeluxe, who only played two events, but won once. So he's in the top ten and has won more money than some of those ranked higher, but can't get higher because he hasn't played enough. And we still see those players who are playing every event, but can't crack the money, rewarded in the rankings for their perseverance. There are several others who moneyed that are ranked lower, but in every case they only played 2 events or less. This new system points correlating to money won = 0.92, which is good enough for me. I think this approach is a good combination of rewarding participation and making money. All input appreciated.

[Bozeman]

"Hence, since entry fees differ, I vote not to use the prize distributions in the calculations."

Since the few I played last year had the same entry fee, I assumed this was not an issue. If it is, I think you should use normalized (by entry fee) instead of absolute money. Don't overreward the big money winner, the money itself is its own reward.

Craig

[dogsballs]

I agree.

[Lori]

The one thing I would change if done that way is that Simon only awards points for the top 20% using the stars System.

As the reason behind this is to reward money finishes anyway, I think that every player should get their stars points just for turning up.
(They tail off pretty dramatically towards the bottom anyway)

Lori

[Lori]

Please note, my above post is timestamped before the start of the tourney where I just came 14/65.

Have to love irony /images/graemlins/laugh.gif

Lori