Steve Jacobs wrote:
> I don't believe it is generally correct to change playing strategy based
> on the size of your bankroll. One playes max-ER because the objective
> calls for it, not because the bankroll size "calls for" or "allows"
> max-ER strategy.
"Correct" in pursuit of what goal? If you are playing a positive game, and
your goal is to maximize how much you expect to gain over any fixed number
of plays, then a max-ER strategy will achieve that (unless the bankroll is
very low relative to the advantage and variance). If your goal is to
minimize the chance you will eventually go broke, the best strategy is to
not play.
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.
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.
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. 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.
Other types of boundary conditions can also call for changes in
strategy. That is why I qualified my statement by saying "generally
correct" rather than "always correct".
For these strategies, as the bankroll falls, the strategy becomes
increasingly risk-averse, as long as it retains positive expected
log-bankroll change. If the bankroll falls enough, that optimal strategy
will be not to play.
Agreed -- this is another type of boundary condition, based on
a minimum allowed wager.
If the bankroll falls enough, that optimal strategy
will be not to play.
The Kelly optimal strategy is "don't play". Your probability of
having the bankroll grow unbounded is non-zero if one continues
to play. 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).
If the bankroll grows enough, that optimal strategy will be the
max-ER strategy.
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.
In between, the strategy you refer to as
min-risk will be optimal for some range of bankroll, but except in a few
degenerate cases, it won't be the "best" strategy independent 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.
> One plays min-risk because they want the highest probability of having
> their bankroll last until a goal is reached. Min-risk does this best
> whether the bankroll is a single unit or a million units. This remains
> true whether the goal is "play indefinitely" as per risk of ruin, or
> the goal is "multiply initial bankroll by N' or just "build bankroll up
> to G units."
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.
> 3) If the min-risk strategy says that the required bankroll is N units,
> no other strategy can give a smaller required bankroll for the same goal.
No other fixed strategy can, but a strategy that adapts to the current
bankroll level may.
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.
> max-ER doesn't belong on an altar (or other special pedestal). It isn't
> _the_ best strategy, but only one of a number of strategies which are
> each best in their own way.
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?
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.
> If you were to take a million max-ER players and start them each with
> $1000, and take a million min-risk players and start them each with
> $1000, and let all the players play for a very long time, you would
> find that the average _surviving_ max-ER player would have more
> money than the average surviving min-risk player,
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.
If you're making this claim based on Kelly results, then I would respond
by saying "that only holds if you force the use of geometric mean for
the averaging".
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.
> Your max-ER bias is showing, and calling this "magic bullet strategies"
> seems like an emotional appeal. The fact is, the max-ER promise of "more
> money" has a clause hidden in the fine print -- you do get more money,
> but only if your bankroll survives. The max-ER strategy does _not_ give
> you the best chance for survival.
Not playing gives the best chance of survival. What Steve is calling
min-risk is a strategy optimal for minimizing the chance of bustout,
conditioned on a required bet size.
Correct. That's why I call it min-risk instead of zero-risk.
The zero-risk strategy is really boring.
> Q: Which is more important to you personally, having a higher
> "wage" rate in $$$/hour, or having a higher probability of being
> allowed to play as long as you wish, having your bankroll grow
> without bound?
Your bankroll grows without bound playing any positive-ER strategy.
True, but the _probability_ of having that happen always has bounds.
The
question is, what is to be the tradeoff between risk of ruin and rate of
growth?
> Those who choose $$$/hour should use max-ER strategy. Those
> who chose better probability of getting rich should use min-risk
> strategy.
No; for any fixed, finite monetary level defined as "getting rich", a
Kelly-optimal strategy minimizes the expected time to reach that goal.
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.
Those bemused by this wrangling can be reassured that in most cases, for
most players, the differences between strategies is minor to nonexistent,
and the real effects on the chance of going busted, or the expected time to
reach a goal, are even less noticeable.
Translation: For professionals with an adequate bankroll, none of this
matters much.
But declaring the difference "minor to nonesistent" doesn't make it so,
especially for those who aren't playing from your professional perspective.
···
On Monday 23 February 2004 06:34 am, ime@panix.com wrote: