Steve,
You are correct. I am NOT calculating risk of ruin (RoR) and nor am
I trying to! (BTW, your desription of ROR is right on and I didn't
mean to imply that I was calculating RoR)
I am computing something different: The Risk of loosing a certain
amount of money for a fixed number of hands for a particular game and
then I am comparing this quantity to one computed for another game.
But I am doing so with certain CRITICAL assumptions
I am assuming that the player of both games already has sufficient
bankroll to avoid ruin in the long-term (that is, a large enough bank
roll to play "forever" or until someother event with a certain
probabilily of ruin). This RoR bankrool is a large number (a very
large number!). [Aside: If the player doesn't have this large
bankroll, for both games, thenperhaps they shouldn't be considering
playing the game at all?]
BTW, When I play, I don't ever carry that large bankroll into a
casino. Instead, I take a few sessions worth of bank roll with me.
My hope is that for any single (short) session, I won't use up all
the cash I have before I have to go. In other words, I am assuming
that the cash I have is enough to withstand the swings of a typical
finite-length session so that I will get to play for a somewhat
predictable amount of time and therefore a predictable number of
hands. Hence, for me at least, I can generally expect to reach a
predetermined number of hand in a session. [Aside: I
wrote "generally expect", yet I've found that about 3/4 session are
successful in the casino, though my calculations show that it should
lower, depending upon game. I just don't feel comfortable carrying
that much cash around to increase the percentage that much]
So when I compare two different games that I might choose to play,
each for a certain amount of time (and therefore a predictable number
of hands, but may differ for different games), I am interested in the
PDF (or CDF) of the games, which as you correctly pointed out hides
the excursions (times series) that might occur during the session.
Indeed, the reason why I have as much backroll as I do, and the
reason why one should even concern him/herself with RoR is to make
sure that the overall bankroll is so large that one need not be
worried that the excursions will wipe then out. I just want to know
that after a lot of say 3 hr sessions, what my typical session will
look like for game 1 or game 2. I want to be able to take the results
of my sessions and compare them to something. So I look at my max
excursions (both + and -) and final session outcomes and compare them
(over many sessions) to the RoR-like computations and the PDFs/CDFs
(for the appropriate number of hands). Then I have some (only a
little) insight into the questions: did I actuallu play well enough?
Is something odd going on? What caused the poor/good outcome? BTW,
these are statistical questions... I never evey play enough to really
know the answers. Truth be told, when other people see the swings
(both up and down) I experience during one session, they are often
shocked. But so far, I haven't experienced anything that really
seems to be unexpected.
[Aside: Now, I admit, one my infer a bit Rob Singer-ism in how I
play. One must be careful of a hidden fallacy here. Surely I could
say to myself, why not always stop my seesion when I have a postive
winning excursion of a certain magnitude. Then I would "always" make
money! Not true. There is a certain probability that during that
finite length session that I run out of monery before reaching that
win goal. Sure, I could increase the amount of cash I had in
pocket. That would reduce that probability of a ruined session, but
not to zero. As you know (and I am writing this for other people
than you), you would need the "classical" infinite run RoR bankroll
just to guarentee the ability to play for ever.. and that wouldn't
guarentee reaching the win goal (more likely we would die from
exhaustion first!) I have done plenty of time-series simulations (an
ensemble of which would produce esitmates of RoR numbers for a finite
length session) and I admit that one can be easily seduced by looking
at small set of such series that such a stop-and-always win stratgey
is possible. Caveat Emptor: always, always use large enough data
sets]
gotta run now
Message: 1
Date: Mon, 22 Aug 2005 09:50:04 -0600
From: Steve Jacobs <jacobs@xmission.com>
Subject: Re: Re: Max ER VS "other" strategies... long> BTW I
> apologize in advance-- I don't mean to be picking on anyone in
> particular. In this email, I will focus on just one, oft-misused
> term, "lower risk"I don't think you need to apologize for challenging something
that you feel is incorrect.> "Lower Risk"
> The term "lower risk" has been thrown around a bit lately,
> and most
> often it has been used in a misleading way. Each of us probably
has a
> different personal definition of "Risk" and, without
> qualification,
> the comparative phrase "lower risk" has little meaning. To
> me, risk
> is the probability of a negative event occurring. So if I ask
what
> is MY risk of loosing my entire bank roll (for a particular
session
> and game, etc), I am looking for the probability of the loss
> occurring.I agree with the statement above. I would generalize it somewhat
by saying that risk is the probability of going broke before
reaching
some specific goal. Any risk calculation can also be framed in
terms
of computing the probability of success rather than the probability
of
failure. Classical "risk of ruin" uses the goal "play
indefinitely" but
the same mathematical formulation works when the goal is a specific
target bankroll. Another example would be the risk of losing your
bankroll before hitting a royal flush -- I prefer to frame this in
terms
of "probability of success" and call this "best shot at royal," but
it is
the same as doing a certain type of risk calculation. I think it
is fair
to claim that any optimal strategy that is based on optimizing a
probability falls under the broad category of "risk."> Mathematically, the probability is equivalent to the
> integral (or sum, also known as "the area under the curve")
> of the
> probability distribution function (PDF) for the game I am actually
> playing from minus infinity to size of my back roll (in other
words
> it is 1-the probability that from my bankroll to positive
infinity).
I'm sorry, but that is not how risk of ruin is calculated. If you
take
an initial bankroll of N units and compute a PDF from a large number
of trials (say 10*N) then the PDF will have a "tail" on the left
which
represents a net loss larger than the initial bankroll. In
addition, some
results in the positive side of the PDF will represent cases where
the
bankroll went negative but eventually returned to the positive side.
A proper RoR calculation will not include any cases where the
bankroll was negative at any time. Including such cases means
that play continued after the player has gone broke. Clearly, any
calculation which included such bankroll "paths" is flawed.[text deleted]
> Ok, so practically speaking, what does this all mean? First,
> statements like,
>
> >> Whenever a player compares a negative game to a positive game,
the
> >> positive game will _always_ be lower risk than the negative
game.
>
> are wrong. "Will _always_ be" is just too strong of a
> phrase.My statement is true. A negative game ALWAYS yields 100% RoR (based
on the "classical" definition) while a positive game ALWAYS yields a
RoR less than 100%. This is not merely my opinion, it is a
mathematical
fact.
It is simply impossible for a negative EV game to go below the 100%
RoR
"barrier" and it is impossible for a positive EV to have 100% (or
higher)
RoR. Of course, if the positive game has a very tiny advantage,
the RoR
will be almost 100%, but it will still be below 100% by some small
fraction.> To
> understand this, we turn to the trusty ol' CDF or PDF and
> consider
> two fictitious vp games. Game 1 (vp1) has PDF that looks like a
> (truncated) Gaussian (otherwise known the "bell curve" or
> "normal
> distribution") at least for a very large number of hands.
> Gaussians
> have the property that the mean (EV), mode (peak or most likely
> outcome) and median (50% CDF point) all occur at the same value
(how
> nice!). [ BTW The CDF of this curve is called the error function
or
> ERF]. Assume for now that the vp1 game has a 0 EV. Hence the
most
> likely outcome for this game is 0 as is the EV.
If the EV is 0, then half of the curve is negative, so clearly play
has
been
allowed to continue after players have gone broke. That isn't
allowed for
a proper RoR computation.
I don't have time right now to provide a tutorial on the correct
method
for
computing RoR, but by way of a hint it involves the "characteristic
equation"
which is based on the exact probability distribution for a single
play.
I'm
sure I've described this somewhere in a previous post, but I don't
have
a reference. In general, if p(N) is the probability that the game
returns
N units to the player from a single unit wagered, and R is the risk
of
ruin,
then R is given (recursively) by:R = p(0) + p(1)*R + p(2)*(R^2) + ... p(1000)*(R^1000) + ...
This equation is exact.
For a simple coin flip, where W = p(win) and L = p(lose), the
possible
outcomes are zero or two units, and this reduces to:
R = L + W*(R^2)
or
R*(L + W) = L + W*(R^2)
or
W*(R - R^2) = L*(1 - R)
or
W*R*(1 - R) = L*(1 - R)
or
R = L / W
So a coin flip that favors the player by 10% has W=0.55 and L=0.45
to give R = 0.45/0.55 = 9/11. The probability of ruin is thus
81.8181%
···
On Monday 22 August 2005 02:12 pm, cdfsrule wrote:
and the probability of playing forever is 9.0909%.The RoR equation for VP games is a bit more complex, but works the
same way.