bankroll via utility theory

[BillC]

You should have about

100/k big bets

where k is your Kelly fraction in blackjack (perhaps as implied by your acceptable risk of ruin). This is equivalent to a utility function u(x)=-x^(1-1/k) for k>1 and log(x) if k=1. The utility specifies risk tolernace and can be expressed in terms of risk of ruin and other drawdowns. So if you play .5 Kelly, k=.5 (and you will eventually be reduced to half your original BR about half the time). Since most teams use around .3 Kelly. That puts the required bank at around 300 big bets, pretty close to the standard Malmuthian 300 BB, arrived at totally differently in GTOT. Virtually all bj teams have k between .25 and .35.

I am assuming one big bet per hour and a variance of 100 big bets per hour. The 100 can be replaced by var./EV in general.

The 100/k expression means that a full Kelly bettor (i.e. one with a log utility function) should be playing a game where the big bet is exactly 1% of the BR. This then is optimal Kelly betting for poker.

the value of k is a question of risk tolerance, which in turn a question of utility function. A convenient way to get a utility function for poker is to use "implied utility" from BJ. There is also a complicated estimator for the value of k based on historical betting and returns (I don't know if it is known in the lit.)

I jsut did this on the back of an envelope, so it might be wrong. Details to follow if there is any interest.

[daryn]

ahhhh.. thank you very much. i got it now..

[Nottom]

Since I am have no more than a passing knowledge in Card Counting and beating blackjack, I really have no clue what a Kelly number is.

I do wonder if what you are saying basically boils down to the same calculation as Mason's but just goes about it in a different way.

[BillC]

My post is an attempt to put the bankroll question on a firmer foundation,
compared with the seat-of-the-pants sort of thing (using a simple 2 or 3 sigma
above the mean computation) I've often seen.

I can't give a full treatment here, so I will have to refer you to bjmath.com for some
articles on Kelly, utility, etc. My treatment is probably not equivalent to the one in
GTOT. It's virtue is that it give an empirical way of pegging a utility function
from the behavior of BJ players, rather than asking the intangible
"what risk of ruin do you find acceptable?". One's BR
depends on risk tolerance which is best described using utility theory.
It relates to risk of ruin and risk of losing a specified portion of one' BR, etc.
It also relates to optimal betting (e.g. in BJ) where one bets an an amount
propotional to one's advantage (divided by variance). Proportional betting is
the the same but the bet is scaled back by the factor "k". If you know what
your k is, then you can determine your BR. Blackjack teams almost all have k
between .25 and .33. Some futures quants recommend k=1/6 (yeah this stuff
is well-established in finance and econ.)
Thus most people will want to have at least 300 to 400 BBs, up to a floor of 600 BB, in order
to be able to sleep at night.
It all boils down to a subjective factor (or medical factors,
such as hypertension, congenital neurosis). In fact, the futures value of k is lowered because, first,
they don't want their clients to freak, and secondly, they really aren't sure what their EV and
variance are. The beauty of BJ is that you can get a relatively accurate bead on EV and variance.
Given the futures vs. BJ analogy, I would be tempted to recommend a BR on the higher end (at least 400 BBs).

Ideally you want to go to a smaller game as your BR decreases
and ramp up as your BR increases (assuming constant variance/EV).

Here is the optimal strategy (k=1, Full Kelly) for var=100 BB, EV=1 BB: Play in a game
with BB=1% of your BR, and always round down. If you have 1400$, do not
play 8-16, and round down 5-10. This strategy results in optimal BR growth rate
but is too volatile for humans; even k=.5 is too wild. (Keep in mind that your BR is all your
money minus expenses)

I do realize that many of you are more than sufficiently bankrolled for your game of choice
so this stuff is irrelevant (or is it?)