Actually the percent of hands you lose and push does change some with the count. It is your win percentage that changes little.Originally Posted by Kat
Yes, indeed I was referring to the certainty equivalent. It's been explained before, probably better than I can say it, but I'll give it a shot.
First, I'll explain the concept of certainty equivalent with an example. Assume I am offering you a fair coin flip where you have a 50% chance of winning $1100 and a 50% chance of losing $1000. Your EV is 0.5*(1100)+0.5*(-1000)=$50. If I said you had your choice between that coin flip, or me just giving you $50, anyone that is risk averse will accept the fixed $50. If I offered the coin flip or $1, you'd probably take the coin flip if your bankroll is reasonably large. I can offer different values between $1 and $50, and at some point, you will be indifferent between accepting the fixed payout and accepting the gamble that has a higher EV. That point is the certainty equivalent of that bet.
The more technical explanation is that the certainty equivalent is the fixed dollar amount that, when added to your bankroll, gives you the same utility as the expected utility of your bankroll after making the bet. In addition to the expected value, we can calculate the expected utility of this bet, given by 0.5*U(bankroll+1100)+0.5*U(bankroll-1000), for your given utility function U(x). So the CE for this bet is the value such that U(bankroll+CE)=0.5*U(bankroll+1100)+0.5*U(bankroll-1000). For example, for a full Kelly bettor, U(x)=log(x), and if we assume a bankroll of $20,000, the CE would be calculated as follows:
U(20000+CE)=0.5*U(20000+1100)+0.5*U(20000-1000)
log(20000+CE)=0.5*log(21100)+0.5*log(19000)
log(20000+CE)=0.5*4.3243+0.5*4.2788
log(20000+CE)=4.3015
20000+CE=10^4.3015
20000+CE=20022.49
CE=22.49
So if you are a full Kelly bettor with a $20,000 bankroll, you would be equally happy taking my coin flip bet or accepting $22.49 instead. The formula I gave before should estimate this pretty accurately:
Variance=0.5*(1100-50)^2+0.5*(-1000-50)^2=1102500
CE=EV-Variance/2kB
CE=50-1102500/(2*1*20000)
CE=50-27.56
CE=$22.44
It's not exact, but it's a pretty good approximation. As far as understanding the implication of certainty equivalent and how it applies to the ante situation, this approximate formula makes it easier to see how the ante hurts us. The short explanation is the variance stays the same, but our EV decreases by the amount of the ante, which ultimately means our CE decreases by the amount of the ante.
For example, I'm going to talk about a single bet placed at TC +2 with the rules mentioned previously in this thread. With TC +2, our edge is 0.98% and our standard deviation is 1.151. With full Kelly betting, our optimum bet is approximately bankroll*edge/variance=20000*0.98%/1.151^2=$148. EV is bet*edge=148*0.98%=$1.45. Using our CE estimation formula, CE=EV-Var/2kB=1.45-(148*1.151)^2/(2*1*20000)=1.45-29018/40000=$0.72. So with no ante, our CE for a full Kelly bet at TC +2 and a $20000 bankroll is $0.72. With a $0.50 ante, our CE is $0.22, and I’ve heard that some casinos charge a $1 ante on bets over $100, which makes our CE negative (-$0.28). This means that you would rather just pay the casino $0.25 (or better yet, just not play) than make a $148 bet, even though the EV of the bet is still positive at +$0.45 after paying the $1 ante.
For a full look at the game (not just a single hand), CVCX reports your CE, which gives us our starting point. If this game is dealt to 5/6 penetration, our full Kelly bettor with a $20k bankroll has a CE of $38.41/hr. This is assuming a 1-12 spread, play all, hilo, sweet 16. Since we pay $50 per hundred hands in ante (if the ante is a fixed $0.50), our CE with the ante included is now negative, at -$11.59 per hour. A negative CE means we are better not playing at all. So let’s see what wonging does for us... We've already determined that we can make a bet with a positive CE at TC +2, so let’s play all hands +2 or better and spread from 150-500. I'm tweaking the bet sizes slightly for ease of use, but I get a CE (before antes) of $61.64/hr. We are playing 18.2% of all hands seen, so our actual CE is $61.64-18.2*0.50=$52.54/hr. If the ante is $1 for bets over $100, our CE is $61.64-18.2*1.00=$43.44/hr. Keep in mind this all assumes a $20,000 bankroll and full Kelly betting, as well as fairly good penetration. Just dropping the pen to 4.5/6 reduces the CE to the low $30s with a $0.50 ante or mid $20s with a $1 ante. With this kind of bankroll, in Oklahoma, you are probably better off learning $2/5NL holdem.
Things are worse with a smaller bankroll and less risk tolerance. With $10000 and betting half Kelly, assuming a fixed $0.50 ante and 5/6 pen, you'd have to play only hands at +3 and above, and spreading up (slowly) from $50, which results in a CE of $9.32/hr after factoring in the ante. Drop the pen to 4.5/6 and you are eeking out a mere $5.07/hr in CE. Consider also that you must subtract your expenses from whatever CE you generate with your play.
The end conclusion of all of this is that newer players with small bankrolls can backcount profitably against an ante game, barely, but generally would be better off either learning poker or working extra hours or getting a second job to save money for a larger bankroll that can sustain travel expenses to play better games. Players with larger bankrolls can make more money, but still the drag of $50/hr on their results, coupled with relatively low table max bets makes the games unattractive compared to just about any other state.
TL;DR – Don’t play ante games.
As Tthree said, the effect on house edge depends on how much you are betting, and house edge isn't really the correct way to think about the ante. Instead, focus on the cost per hour for the ante and how that compares to your certainty equivalent. If you are betting optimally, your certainty equivalent will be roughly half of your EV. If CE/hr - ante/hr is still enough money to be worth playing for, then play. But if you were playing all and flat betting, the increase in house edge is simply ante/bet, so if you are betting $5, the ante adds 10% to the house edge. If you are betting $500, it only adds 0.1%. The 0.4% increase you calculated is only correct if your average bet is $125. Since we change our bets depending on the count, it's much easier to just look at the fixed cost of the ante and ignore how it affects the house edge of the game. Hopefully my last post explains this a little bit without being overly confusing.
Fantastic! Not exactly what i was shooting for but a perfect explaination and breakdown. I was figuring the difference of .4% at around the $100 bet mark, so nice to see my math isnt as bad as my bankbook might reflect! The GP is to gather a sufficient BR to begin the travelling caravan of surf, sand and natural hands, only a blue moon away. In the interim, may standard deviation be a lady tonight...I've seen the way she treated other guys she's been with...
a'int that the truth Tthree!!!
but in my case, unlike you, i know just enough maths to be dangerous, lol.
so but yeah! having a guy like Nyne around, even better actually knowing him, priceless man, priceless.
my only problem is i owe him a pizza, lol.
any of you math wizards ever thought of starting a business? After milling through many of these posts you all seem like people persons and extremely intelligent. Just saying...
Dharmaprija- I agree with one of your previous posts...since the OK casinos are the only close casino to me and apparently you, and limited BR, also a newbie...I just want some experience. I guess its like we are paying someone to train us for $50 an hour in live casino experience and CCing.
I will venture to vouch for the fr0g...perhaps Nyne is a new user ID from the old world (BlackjackINFO).
Sagefr0g is well respected, a generally accepted nice amphibian and has a "not to loud" RIBBIT, and an analytical and humorous disposition.
One would do well to listen to "the night sounds".........
That is my take.... on this fr0g, mileage doesn't matter
thank you very much metronome, ditto right back at you.I will venture to vouch for the fr0g...perhaps Nyne is a new user ID from the old world (BlackjackINFO).
Sagefr0g is well respected, a generally accepted nice amphibian and has a "not to loud" RIBBIT, and an analytical and humorous disposition.
One would do well to listen to "the night sounds".........
That is my take.... on this fr0g, mileage doesn't matter
but believe me i'm a lost puppy compared to most on this site and compared to most whom i've met.
i just yack a lot and most of the time only get it half right, but one thing i am is a good judge of character, and i do slug it out in the trenches, lol.
Nyne? trust me he's a giant of an intellect.
I have to interject that there are quite a few here who strike me as candidates for Mensa. I rethink my abilities daily and wonder if I have a chance with that kinda competition out there. With that said, it benefits those of us lower on the mental faculties totem to have them as a resource. Thanks again to all for their sound advise.
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