This was a thread from quite awhile back, but I am
looking at this subject again. I would like Steve
Jacobs and others answers to this question:
I've been lurking on this list for a little while now,
and this seems as good a time as any to emerge and
yammer about topics near and dear to my heart. 
I wasn't here for the previous discussion, so I'll
clarify my assumptions explicitly when possible.
I know that a minimum risk strategy will vary from
the Max ER one for the same game.
I take "Max ER" strategy to be the unique strategy
which maximizes the expectation from a single "play."
There are a very large number of other strategies that
can be employed; each of these other strategies
deviate from the Max ER strategy is some way. Some of
these reasons are:
-Simplicity
For example, the strategies that most players normally
play, which are slightly simplified versions of the
Max ER strategy.
-No reason at all
Other strategies deviate from the Max ER strategy for
no reason at all -- as for example occurs with errors
or not knowing the correct play in a given situation.
-Variance reduction
It is possible for some games to construct strategies
that reduce volatility for the sake of small amounts
of EV. For example, imagine a game where a full house
paid just a tiny amount less than the expectation of a
draw to say, three aces (because of jackpots or the
like). Then risk-averse players might simply choose to
sacrifice the tiny amount of expectation to reduce
volatility by holding the full house. I'll call these
VR strategies for short.
However, would a
minimum risk player (who believed in the math)
sometimes choose a lower EV game even when a higher
one is available?
The thing is that in one sense playing a VR strategy
is the same as choosing a different game -- one with
lower volatility and lower expectation.
It's worth noting, however, that volatility and risk
are not synonymous, although they're related.
For example, increasing the royal payout on JoB to
5000 instead of 4000 would increase the volatility or
variance of the game substantially -- but it would
have a positive or zero effect on any risk metric you
can imagine.
The thing is that there isn't a single measurement of
"risk" that we can just slap a number on and work
with. There are a lot of different reasonable ways to
define risk -- some of which you bring up in a moment.
First, there's "risk of ruin." This is a
mathematically derived quantity that uses the
following assumptions:
You start with some bankroll amount.
You play the game with your strategy forever.
What's the chance that you ever go broke?
Now the answer to this question can be revealing in a
number of ways. First of all, if the game is negative
expectation, then the answer is always 100%. But if
the game is positive, the answer isn't 0% - we might
lose all our money before we get ahead enough to be
safe.
So risk of ruin is essentially a *long-term* risk
measurement, for considering questions like "I want to
play video poker for a living and have a 5% chance of
losing my starting stash. How much bankroll do I
need?"
The usual way that RoR is calculated in VP (I've
gathered) is from simulations, although it can be
directly calculated algebraically as well.
For example, choose JoB instead
of FPDW because of its lower volatility - they would
be more likely to last longer on a smaller session
bankroll although if they did this every session, in
the long term (infinity) they would be net losers
because JoB is a negative game. Would they be
better to stick to FPDW and vary their strategy to a
lower-risk one?
So here we get into an idea of "session bankroll,"
which is a measurement of "how often and how quickly
will I go broke from a given bankroll?" This would be
primarily of use to the occasional player. This is an
altogether different measurement than risk of ruin,
because it is primarily concerned with the outcomes
that occur in a specified timeframe.
Consider two games. In the first game, the player
flips a coin, and if the coin comes up heads, the
player wins $.99. If the coin comes up tails, the
player loses $1.
In the second game, we spin a wheel numbered 1 to 10.
If the wheel comes up 1, the player wins $10. If the
wheel comes up any other number, the player loses $1.
Now the risk of ruin of the first game is 100%,
because it is a negative expectation game. So if you
play every day, you will end up a net loser. The risk
of ruin of the second game, by contrast, is something
entirely different. The game is positive expectation,
and it turns out that the RoR of the game is
e^(.0194398*b).
Now, suppose we have a player with $50 who is trying
to decide what to play. If they are going to play one
game or the other for a long time, and often, they
should choose the second game. But let's say that they
really just want to play SOME game for a few hours and
not go broke. Then choosing the second game is likely
wrong -- because they will go broke playing forever
about 37.8% of the time, and almost all of that will
occur during the first few hours.
How does this apply to video poker? Well, the chance
of losing a fixed bankroll stake in a fixed, modestly
chosen number of hands (a session) is dependent almost
exclusively on the volatility of the game, and not the
expectation.
Remember before I characterized playing VR strategies
as choosing a different game, with lower expectation
and lower volatility. One problem here is that you
can't convert these two quantities very freely; that
is, the cost of reducing volatility in expectation
terms grows as you do it more and more. A strategy
will have at most a few places where you can easily
reduce variance by giving up expectation. After that,
the cost becomes even greater.
Say our hypothetical player will play sessions of
approximately 10,000 coin-in.
If our hypothetical risk-averse player chooses to play
JoB (99.5% return, but a 10% chance of going broke on
a session bankroll) instead of FPDW (100.7% return,
but a 30% chance of going broke on a session
bankroll), then he/she is effectively paying 1.2% of
10,000, or 120 coins, for the pleasure of not going
broke 20% of the time.
This type of comparison holds true no matter whether
the two games are JoB and FPDW, or FPDW and FPDW
played with a VR strategy.
Another example, choosing between two negative
games. Would the wise minimum-risk player choose
8/5 Bonus over 9/6 JoB because he has a chance for
several mini-jackpots. In this case, he is choosing
more volatility as well as lower EV. But is this a
wise choice for his goal?
Almost certainly not. Because all video poker payouts
are within some narrow band, the more mini-jackpots
there are, the *lower the payouts for run-of-the-mill
hands will be*, and the more likely a player is to go
broke from any bankroll level.
I can see between two positive-expectation games
that choosing the less volatile one might be a wise
choice for someone whose goal is to last longer in
one session because he has a small bankroll.
Although one's required long-term bankroll gets
smaller the higher the game EV.
If you have a small bankroll, and your goal is simply
to make that bankroll last, expectation is a lot less
critical than variance. For example, let's say I have
a $500 bankroll to play $1 VP. It doesn't matter
whether I play FPDW or 8/5 JoB, I'm probably going
broke in short order anyway. Now, if I have a $50,000
bankroll for $1 VP, it makes a ton of difference. But
let's say I have a $1,500 bankroll and I want it to
last through 50k coin-in. I likely have a much better
shot at achieving this goal playing 9/6 JoB than I do
playing 10/7 DB, expectation be damned.
But I pay for this privilege - the EV difference
between my EV at JoB and at DB. In the same way, if I
compel the VP strategy gurus to provide a risk-averse
strategy, I pay for the privilege of playing that game
as well - the difference between Max ER EV and the VR
strategy's EV.
I still think we are on a slippery slope when we
discuss this "in front of the children," but I need
to understand this thoroughly so I can give newbies
and people with different goals the most useful
advice. It seems to me that most people who deviate
from Max EV strategy or choose games like negative
DDB instead of positive games are actually
increasing their risk as well as losing EV. Is this
not true?
Choosing a lower expectation game doesn't change your
session risk that much. It changes your long-term risk
a lot, of course. But if you derive a significant
enough chunk of utility from the ability to keep
playing for longer, then it may be rational to choose
a lower EV and lower variance game. However, choosing
a lower EV but *higher* volatility game, such as DDB,
is almost certainly worse in all cases.
Jerrod Ankenman
···
--- Jean Scott <QueenofComps@frugalgambler.biz> wrote:
____________________________________________________
Start your day with Yahoo! - make it your home page
http://www.yahoo.com/r/hs
