Steve Jacobs wrote:
The "usual" min-risk goal for positive games is to minimize RoR which
really means to maximize the probability that you will play indefinitely
while your bankroll grows without bound. In both this case and the
max-ER case, the optimal strategy is fixed and doesn't change due
to bankroll.
Changing bet size, or shedding risk by taking a partner, do a better job of
accomplishing the goal as stated. Trivially, so does not playing and
instead depositing the bankroll so as to compound the interest. The trouble
is that "bankroll grows without bound" without any indication of the rate of
growth is meaningless, for reasons similar to your criticism of naiive Kelly
advocacy; as Keynes said, in the long run, we are all dead.
Mixing a min-risk strategy with small bankrolls and a max-ER strategy
with large bankrolls gives a results that yields less ER that using
max-ER alone, and yields greater risk than using min-risk alone.
Risk of Ruin under a particular strategy falls exponentially as current
bankroll grows (doubling the bankroll squares the RoR, turning a 1% RoR into
a .01% RoR). So the "greater risks" incurred by changing to MaxER will, at
some point, be small enough to be less important than the faster growth that
MaxER strategy affords.
There are a number of goals that are equivalent to one another, however,
that call for slightly different strategies depending on the bankroll.
Those are the Kelly criterion goals, and they include minimizing the
expected time to reach a particular level of winnings, or maximizing the
chance that you end up with more winings than any other prestated strategy
over a sufficiently large number of trials.This type of strategy change is somewhat artificial, and imposed by
the constraint that bet size has to match the units accepted by the
machine.
I don't understand that to be artificial; we were discussing optimal
strategy given some constraints, and I took that to be one of them.
I assure you that I have participated in team decisions about what was the
correct strategy to play for a particular opportunity. We used the same
risk-profile "bankroll" as we do for analyzing whether, or which,
opportunity to play. I believe that was the best way to make those
decisions.
Kelly betting seeks to wager the same fixed fraction of bankroll on every
play, and if the machines allowed arbitrary bet sizes so that this would
be possible, then the resulting playing strategy would once again become
unchanged as the bankroll varies, and that strategy would differ from both
min-risk and max-ER.
If the bet size were continuously variable, and there were no other
constraints, I believe the strategy you call min-risk would be optimal.
But I was referring to Kelly in the more general sense, as the strategy best
suited, under all given constraints, to minimize the expected "time" --
number of plays -- to achieve a particular bankroll target (or any of the
equivalent goals). In these cases, I generally assumed there was only one
positive game available, and that it was only positive for a particular bet
size. Our only choices are to play or not, and what cards to hold after the
initial deal.
The Kelly optimal strategy is "don't play". Your probability of
having the bankroll grow unbounded is non-zero if one continues
to play.
That's true. The last desparate fling of the ruined gambler...
Based on your discussion here, you seem to have bought
into the idea that "Kelly is everyting". I don't subscribe to that
theory (and yes, I have Thorp's paper).
I'm sorry I gave that impression. I consider Kelly a useful mathematical
model, which is made economically nonsensical by its isolation of bankroll
from any changes except those due to winning and losing. If one can never
spend one's winnings, what's the point? If one can never replenish a
bankroll, what was its original source, immaculate conception? So I treat
bankroll not as a physical quantity, but as a parameter of the mathematical
model which has some correspondence to the bettor's willingness to accept
risk.
One would try to change unit sizes before that happens, but once
you've reached the largest unit size, and the bankroll grew to a
point where the unit size is a tiny fraction of the proper Kelly wager,
the strategy would change as you describe, for a player who
wishes to maximize log(bankroll). But at that point, you aren't really
doing Kelly betting any more, because you're constrained by unit
size.
Your definition of Kelly betting is too restrictive to be of much use for a
VP player. Given constraints, such as the bet unit size, I prefer the
strategy which best achieves the Kelly goals, one of which, as you state, is
to maximize the expected log of bankroll.
From a Kelly perspective, what you say is true. From the perspective
of a player who seeks only to minimize RoR, rather than minimizing expected
log(bankroll), the min-risk strategy remains optimal for all bankroll sizes.
Optimal in a rather theoretical sense; it doesn't take too much bankroll to
make risk of ruin vanishingly small for either strategy, while MaxER still
shows a clear return advantage over MinRisk.
Well, if the time allowed is finite, these latter goals are
Kelly-equivalents; a bankroll-sensitive strategy will dominate the
bankroll-insensitive min-risk strategy.I disagree. Kelly results in general are asymptotic, based on assumption
that time is unlimited. Kelly results dominate "eventually". In my opinion,
when time is finite, many Kelly results crumble like a house of cards, and
fixed fraction wagering morphs from a beneficial guiding principle to a
set of shackles that prevent one from devising an strategy that is truly
optimal for the finite boundries of realistic problems.
The proofs are asymptotic, but the short version is that Kelly is "faster,"
in expected time to any particular target, than any other strategy. (Note
that Ruin comes into that as infinite time to the goal, so a strategy that
has any risk of Ruin blows up because that term dominates).
Agreed, especially in games like VP. As the bankroll approaches the
goal, payoffs that would overshoot the goal can be treated as if they
are smaller than they really are, and the strategy adjusted accordingly.
I wasn't looking at finite target / overshoot effects, but rather at the
faster growth of the bankroll under MaxER, while the RoR increase is held
low by the grown bankroll.
It is asymptotically the best strategy for a positive game when the
bankroll becomes sufficiently large.I'll now turn your words against you -- "best" in pursuit of what goal?
In maximizing the amount of money available to spend while controlling the
risk of ruin. But you're right, I substituted my own goal for the unknown
goal of the ill-defined problem.
I think you may be using Kelly as a "golden standard" for judging when
other strategies are "appropriate". I personally believe that is flawed
concept, and that too much is being read into the interpretation of
Thorp's Kelly paper. Kelly is a terrific model if you live forever and
are never confined by minimum/maximize bet sizes. It breaks down
for goals that are inherently finite in nature, such as maximizing
the probability of growing a bankroll to a fixed target size.
The Kelly goals are surprisingly robust under real-world constraints. When
a casino announces triple cashback for a particular day, and you are looking
at playing the 5-coin $100 7-10 Double Bonus with .75% cashback plus .15%
cash-equivalent from the comps, you have set up very tight constraints, but
the Kelly framework is still useful in choosing what playing strategy to
use within those constraints.
In fact, the average of *all* the max-ER players, including zeros for those
busted flat, will exceed the average of *all* min-risk players, though the
latter group will indeed include fewer bustouts.I originally started to claim just the opposite, then backed off because I
decided I coun't easily prove it if challenged. If you can show this using
_arithemetic mean_ for the average, I'd appreciate seeing the derivation.
Sure. We'll use the venerable coin-flip game, with a very generous (and
infinitely-bankrolled) opponent who will pay us 2 to 1 (3 for 1) on the .5
probability that we win.
Start with our population of N players, with ten units each of bankroll.
Their MaxER strategy is to bet their entire bankroll at every opportunity.
The MinRisk players play only one unit, no matter how high their bankroll
gets.
After 10 rounds, every MinRisk player has made a bet on every round, so a
total of 10N units have been bet. 5N have been lost, and the other 5N won,
so the expected aggregate bankroll is now 15N, so the expected average
bankroll after 10 plays is 15.
After 10 rounds, an expected 1023N/1024 of the MaxER platers are busted, and
have zero. The other N/1024 players have tripled their bankroll on each
play, and so have 10*3^10 = 590,490 units each. That gives an average
bankroll of 576.65 units. Clearly the average of all the MaxER players far
exceeds the average of the MinRisk players.
A large number of players who overbet their bankrolls by less than 2X
will as a group win more money than an equal number of players who
always bet the perfect Kelly fraction. The winnings will tend to be
more concentrated into the hands of a smaller number of players, but
the group _will_ have more money.
Yes, I agree, though the 2 * Kelly is approximate, not exact (there's a
curve that passes through the origin, rises to a max at the full-Kelly
point, and then falls back through the X-axis; if it were exactly parabolic,
it would fall through at 2*Kelly, but it's only approximately parabolic).
The zero-risk strategy is really boring.
The Caro roulette strategy.
Randy, this discussion isn't about Kelly, and never has been, largely
because VP isn't a very good fit for the Kelly model. At best, Kelly
can only be applied to the game in a very crude way, and a large
fraction of VP players are using a small enough bankroll that the
minimum unit size is a major impediment to subdividing the bankroll.
This isn't about Kelly-optimal betting, but it is about Kelly-optimal
strategy, the strategy which maximizes the expected log of the bankroll.
I didn't mean to get so deeply into this, just to object to your assertion
that the optimal strategy was blind to bankroll size. If the optimal
strategy is to balance risk of ruin against win rate, the larger the
bankroll is, the less important the inherent risk of the strategy is,
because RoR is exponential in bankroll size, while gain (with fixed wager
sizes) is independent of bankroll size.
Translation: For professionals with an adequate bankroll, none of this
matters much.
Actually, for just about anybody, this doesn't matter much. I just didn't
want you to think nobody was paying attention to what you and Harry were
writing.
But declaring the difference "minor to nonesistent" doesn't make it so,
especially for those who aren't playing from your professional perspective.
OK, take a casual VP player. She comes to LV with a $1000 trip stake, and
stays at the Palms. She needs to play 20,000 max-coin hands on the .25*5
FPDW to cover her 3-night room comp (I don't know what Palms actually
expects, that's a f'rinstance).
She has a choice of at least three potential strategies: Reset (MaxER for a
royal of exactly $1000), MinRisk (MaxER for a royal of $565, except hold all
five cards of any dealt quint with three deuces), or Aggressive (MaxER for a
royal of whatever the royal is at that point in her play). You say the
player is generally best off playing the MinRisk strategy. I say it doesn't
much matter, but if she's doing well, late in the trip she has no real
concern about RoR, and probably should prefer at least the Reset strategy,
if not the Aggressive strategy. And, that the point at which she changes
over depends on the bankroll size.
···
--
Randy Hudson