Monday, May 21, 2007

The Mathematics of Rake/Bonuses

The mathematics of rake is pretty poorly understood, even though on the surface it seems so simple. I rattled some numbers off the top of my head during my last bonus challenge that probably should have been questioned, but were not. So I will try to formalize it a bit here. The results may be more interesting then you imagine.

I will use FullTilt as an example here, but the rake structure is pretty much the same at most sites. FullTilt rakes 5% of the pot in 5 cent increments up to $3 maximum. There is no rake if there is not a flop (the “no flop no drop” rule), and no rake after $3 or after the pot reaches $60 for the mathematically challenged.

The question is, how much of the rake do I get credit for (when working a bonus), and how much did I actually pay? How much did I get credit for will determine how much bonus that I earned, or how much rake back I will get. How much did I actually pay determines what it is actually costing me to play. To simplify things, I will assume that every hand has a flop, and no pot exceeds $60 (all pots are $20 for simplicity). If you are playing at a preflop aggressive table, or at stakes where the pots go over $60 routinely your rake will be less than what this analysis shows. The first term is "Personal Rake Credit" which means how much credit does the site give me for the rake generated at my table.

Personal Rake Credit = Pot Size * Rake Percentage / Number of Players Dealt In

PRC = PS *(0.05) / # of Players

Note that number of players = number of players dealt into the hand, and not the number of seats at the table. Now for an example let’s assume that there was a $20 pot at a full-ring table with 8 players and a $20 pot at a 6-handed table with 5 players.

PRC = $1 / 8, PRC =$1 / 5

For the full ring case you get credit for 12.5c or 1/8th of the rake, for 6-handed you get credit for 20c or 1/5th of the rake. This is the credit you get, not the actual amount that you paid. It represents the average rake paid for everyone at the table for that hand. So based on this you can see that your rake expense on average is a lot more for 6-handed than full ring. If on average 60h/hr are played full ring and 90h/hr at 6-max, you pay about $7.50 per table hour at full ring, and about $18 per table hour at 6-handed.

But people do not pay the average rake. Some pay more, and others pay less. I claim that I pay about ½ as much as the average player. How do I claim this? Let’s look at actual rake paid, and not just what you get credit for. I now add to the equation your Personal Rake Factor (PRF) which is based on how much on average that you contribute to the pot. For a perfectly average person this would be equal to 1. For looser players it is greater than one, for tighter players it is less than 1. For any given table it always averages to 1. If you contribute mainly to pots that you win, you keep this number small. If you fire at a lot of pots, including a bunch that you don’t win, your number will be greater than 1. For the tightest possible play where you always fold preflop, you get a factor of about 0.075. For the loosest possible play where you play every pot heads up you contribute 1/2 of the table rake for a factor of # players / 2 (4.25 for the 8.5 player case) Most people should be in the 0.5 to 1.5 range with a mean of 1.

So let’s look again at the full ring case for someone like me who does not play a lot of pots, and does not contribute much to pots that I do not win. This time I will assume 8.5 players at a full ring (which is about average long term).

Actual Rake = Personal Rake Factor * Pot Size * 0.05 / # of players

AR = (0.5) * (20) * (.05) / 8.5 = 5.9c/hand

Now for a 6-max expert who plays an aggressive style with a personal rake factor of 1.25, assuming 5.5 player average.

AR = (1.25) * (20) * (.05) / 5.5 = 22.7c/hand

So the actual rake paid can be 4x as much for an aggressive 6-max player vs. a tight full-ring player. That is a pretty huge difference, and may be hard to overcome with good play. The 6-max tables are clearly great for the website as they take in much more rake, but is it good for the players?

Now, I will try to pull things together from a reload or deposit bonus perspective. When you are working a bonus on FullTilt you get 6c credit per dollar raked at your table. For a 9 player full table FullTilt would pay 54c per dollar raked in bonus. For a 6-max full table FullTilt would pay 36c per dollar raked in bonus. For the full-ring, 0.5 Personal Rake Factor, and 8.5 player tables you can see that you are getting 100% of your actual rake back in bonus. For the loose 6-max case you get about 26% of your actual rake back in bonus. So us tighties get a pretty good deal on reload bonuses, but not so good a deal on rakeback (100% vs. say 32%). For loose players it does not matter as much, but they are getting less credit than they deserve (26% vs. 32%).

Why does any of this matter? When you are trying to clear a reload bonus the most cost effective way is by tight/aggressive full ring play. You can get close to all of your personal rake paid back in bonus this way. You may be able to clear the bonus faster playing 6-max, but you will get much less of your personal rake back. Playing styles and full-ring vs. 6-max can have a huge impact on the amount of rake that you actually pay and what you get credit for. The rake is much/much larger for 6-max tables vs. full ring. It is easier to beat full-ring vs. 6-max tables with everything being equal due to the huge difference in rake.

Your personal rake factor is difficult to derive, but if you have PokerTraker, you can figure out what it is when completing a bonus. During my $300 bonus challenge, it took $411 in personal rake to clear the $300 bonus while averaging 8.5 players/table. My personal rake credit per hand (based on $20 pot) is below.

PRC/hand = ($20 * 5%) / 8.5 = 11.8 cents / hand
Bonus earned = 6 cents / hand
I earned 73% in bonus of what I paid in rake. I actually paid
($411/$300)(6c)= 8.2c / hand

My Personal Rake Factor = 8.2c / 11.8c = 0.69

This means, I pay about 69% of average person's rake. If I am paying less than somebody (a loose player) is paying more than 100%. Because I was not running as well as normal during the challenge this factor was higher than normal. If you play a few big pots and lose the factor can be thrown off a bit, so I will still say my long-term PRF is in the 0.5 to 0.6 range.

If you are paying less than the average person's rake, this is extra profit in your pocket. This may be a place where some of us are leaving money on the table.

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At 8:32 PM, Blogger Mr Subliminal said...

An interesting and well thought out analysis.

One question - what were your assumptions in getting a 0.008 factor for a player that always folded pre-flop? I get a figure at least 10 times greater.

At 7:57 AM, Blogger Blinders said...

I assumed that you fold all hands so your only contrribution to the rake is the rake on your blinds. I think I got it right, but I will check the work.

At 10:39 AM, Blogger Blinders said...

yep, I was off by about an order of magnitude. 0.075 is about right for folding all hands. 4.25 is the factor for playing every pot heads up at a 8.5 player table. I updated the post in blue.


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