pot odds in terms of Certainty Equivalence

[BillC]

I calculated the break even point for CE for utility function -1/x (which corresponds to 1/2 Kelly, or equivalently a 2% risk of ruin (with a continuous model)), for various drawing odds. Most blackjack teams are more conservative than this, often choosing a Kelly fraction of 1/3 to 1/4). I chose this function b/c it is not too far from a
realistic utility and is easy to work with algebraically.

I then calculated the excess (over the pure EV based required pot size n) you would need for 1:n shot, assuming a one unit call to see the next card. The formula is
Bn/(B-n-1) for the pot size needed to break even on a 1:n shot at cost of 1 unit, with bank B. Subtracting n gives the "CE = offset":

n(n+1)/(B-n-1)

The chart below shows the excess pot required (to break even in CE) for 1:n shots, n=1,...,30 and for Bankrolls 50, 100, 200, 300 units.

E.g. suppose you are drawing to a 1:22 shot, paying 1 unit to see the one card, and your bankroll is 300 units;
then you need 22 units in the pot to break even in EV. In CE
terms, you need an extra 1.8 units to break even, so you'd need pot odds of 23.8:1.

The moral of the first column with a 50 unit bank is that
one shouldn't play 'undercapitalized'. Keep in mind that a unit is not a big bet or a small be; it's how much you must pay to draw.

This is a first attempt to quantify the idea that you should
avoid borderline hi-variance draws if your bankroll is not relatively huge.

-BillC


n/B 50.0 100.0 200.0 300.0 600.0
1.0 0.0 0.0 0.0 0.0 0.0
2.0 0.1 0.1 0.0 0.0 0.0
3.0 0.3 0.1 0.1 0.0 0.0
4.0 0.4 0.2 0.1 0.1 0.0
5.0 0.7 0.3 0.2 0.1 0.1
6.0 1.0 0.5 0.2 0.1 0.1
7.0 1.3 0.6 0.3 0.2 0.1
8.0 1.8 0.8 0.4 0.2 0.1
9.0 2.3 1.0 0.5 0.3 0.2
10.0 2.8 1.2 0.6 0.4 0.2
11.0 3.5 1.5 0.7 0.5 0.2
12.0 4.2 1.8 0.8 0.5 0.3
13.0 5.1 2.1 1.0 0.6 0.3
14.0 6.0 2.5 1.1 0.7 0.4
15.0 7.1 2.9 1.3 0.8 0.4
16.0 8.2 3.3 1.5 1.0 0.5
17.0 9.6 3.7 1.7 1.1 0.5
18.0 11.0 4.2 1.9 1.2 0.6
19.0 12.7 4.8 2.1 1.4 0.7
20.0 14.5 5.3 2.3 1.5 0.7
21.0 16.5 5.9 2.6 1.7 0.8
22.0 18.7 6.6 2.9 1.8 0.9
23.0 21.2 7.3 3.1 2.0 1.0
24.0 24.0 8.0 3.4 2.2 1.0
25.0 27.1 8.8 3.7 2.4 1.1
26.0 30.5 9.6 4.1 2.6 1.2
27.0 34.4 10.5 4.4 2.8 1.3
28.0 38.7 11.4 4.7 3.0 1.4
29.0 43.5 12.4 5.1 3.2 1.5
30.0 48.9 13.5 5.5 3.5 1.6

[Louie Landale]

[1] You make the assumption, the same poor assumption in several of Malmuth's Bankroll requirements calculations, that hero will continue to make or not make EV-near-zero draws as his bankroll diminishes or expands.

No. You CAN make these calls (in your example, getting between 22:1 and 23.8:1) without risk-of-ruin with a non-huge bankroll, so long as you are willing to stop making them as your bankroll diminishes. You can also make them if you can make other serious adjustments, such as playing in a lower limit game, or passing those starting hands that tend to "improve" to close EV call situations.

[2] Put a little more effort into saying these tricky things in English. Few people, including myself, can follow these advanced math concepts. Without your last paragraph this post would be useless.

[3] Never-the-less, this looks like a great start for an advanced concept worthy of insertion into future Poker theory books.

- Louie

[BillC]

[1]. I don't agree here. Maybe the time has come already (given your BR) to "stop making some plays". The CE analysis trys to answer this type of question. It is simply a way of accounting for the negative effect of variance.

Incidently, the usual req'd BR calculation is flawed since it ignores the possibility that you will bust out before your EV overwhelms your SD. You have to look at your BR as a random walk, not just as normal random variable. This changes things by a factor of about 2!-- But that's for another thread (so this a preview).

[2] Sorry for being so technical. The topic of CE is standard in Economics and pro BJ, and has been written about a lot. You can get an intro and references suitable for card players at

http://www.bjmath.com/bjmath/kelly/kellyfaq.htm



[ QUOTE ]
[1] You make the assumption, the same poor assumption in several of Malmuth's Bankroll requirements calculations, that hero will continue to make or not make EV-near-zero draws as his bankroll diminishes or expands.

No. You CAN make these calls (in your example, getting between 22:1 and 23.8:1) without risk-of-ruin with a non-huge bankroll, so long as you are willing to stop making them as your bankroll diminishes. You can also make them if you can make other serious adjustments, such as playing in a lower limit game, or passing those starting hands that tend to "improve" to close EV call situations.

[2] Put a little more effort into saying these tricky things in English. Few people, including myself, can follow these advanced math concepts. Without your last paragraph this post would be useless.

[3] Never-the-less, this looks like a great start for an advanced concept worthy of insertion into future Poker theory books.

- Louie

[/ QUOTE ]

[Bozeman]

I think the general problem with your approach is that edge in poker is constantly varying over a wide range, unlike BJ, for example. This can have dramatic influence on this calculation.

Imagine a good player, you state that this player should forgo 23:1 odds on a 22:1 draw. However, over the course of this game, this player will have many opportunities to make decisions, and very few that are this close. Thus his risk of ruin for these plays will be very tiny, since his BR should have increased significantly by the time he approaches enough of these marginal decisions to even possibly bankrupt him.

In the extreme limit, if you know what your opponent has, you will always make decisions that give you a 0-100% edge. Avoiding the 0-1% edge ones will not significantly change your RoR, but will measurably decrease your win rate.

Craig

[muck_nutz]

The normal BR calculations are wrong by far more then then "bust before you fly problem" (schlesinger's "grand canyon" illustration). Other then a model posted by Bill Chen to RGP I havn't seen one which deals with the drain of capital from your BR. For people where the BR problem is real (those living out of their BR) that is a far bigger issue.

As far as applying CE to making poker decisions I think that the number of factors which contain error that go into a poker decision make assigning yet another error source undesirable. Instead I think that poker players should stay away from games they are not well capitalized to play in.

Your work might be applicable to computer poker play (online bot) where the other factors are explicitly quantified.

[BillC]

The risk of ruin figure is in the model to fix the utility
function, and that is a way of encoding risk tolerance. It has nothing to do with actual ruin. CE is (again) just an adjustment for variance. I think my table shows signifigance for some decisions, even for an "adequate" BR.
(think about pot and no-limit poker too!)

I apologize for my attack on "bankroll requirements"--I have
certainly seen computations based soley on the normal distribution.

As for other posts saying that there is too much other uncertainty, I say that this is more information, just like
other fine parameters like counting exact no. of outs, tainted outs, player profiles, estimating implied odds, etc. Expert poker is a game of close decisions...When you split a lot of hairs, well, you've got a lot of split hairs.

[Louie Landale]

[1a] I didn't mean to suggest this stuff should be ignored, I was suggesting your conclusive thresholds were off since they presume a CONSISTENT strategy until your BR is either healthy or zero.

[1b] Are you saying that Malmuth's BR conclusions (IIRC, you are safe (within 3 SDs) of never going broke) are way too small and he made a serious mathematical snafoo? Prove it. I look forward to the fireworks on THAT great thread.

- Louie

[BillC]

[1] I might help to notice that the numbers depend on BR.
So if your BR grows, the offset decreases and are essentially zero for a big bankroll. Otoh if you have a bad run, the offset increases. The CE formulation depends on a subjective utility and doesn't really assume anything about your future strategy. How one gets the utilty function is the tricky part. It's not really about ruin; rather the choice from a class of utility function corresponds to ruin tolerance. I don't want to write a primer on CE, since this has been done by others.

I have a 50K BR but sometimes play 3-6 online and in fact none of these CE calculations are relevant to me. But I know there are a lot of people who play slightly underbankrolled, say playing with a 200 BB bankroll. Suppose you do that and then have a really bad run...

[2] All I am saying is that the way the BR calculation is often done is flawed. I reserve comment about haw MM did it.