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Lenny Frome Article

#1

Here's another example where two different players view the
same game differently: In Craps, a $6 place bet on number 8
is paid $7 if an 8 is rolled before the next 7. These bets can
be placed or removed at any time. Jim defines the end of a
game as a roll of 8 or 7, where the 8 results in a $7 win and
a 7 results in a $6 loss. On average a seven is rolled 6 times
for every 5 times that the roll is eight. Jim computes his EV
as 100 * [($7 * 5) - ($6 * 6)] / ($6 * 11) = -1.515%.

Coming from a craps background, I like the example from Steve. The crux of
the problem is whether or not double-up is a dependent or separate,
independent event. No matter how one chooses to parse it, they are both.
Without a win, there can be no double-up option. But the win is re-bet on
double-up at the discretion of the player, and whether or not it's a 50-50
prop BEFORE the first card is chosen doesn't matter. They are two separate
bets, with two separate results, pickiness over what the exact EV
notwithstanding.

On the other hand, if you choose to think that every parlay is part of the
same bet, then it's no use arguing otherwise.

Exactly why would it be viewed as not independent? Is it because the
double-up option pops up automatically after a win if it's set to pop up?
In that case, what about the bet button that automatically reappears after
each decision--the one you push to get your next hand? Is that also a
dependent event?

It serves no good purpose, at least in my opinion, to lead novice and
recreational vp players to believe that this is a "good" bet. The funniest
argument in favor of double-up that I've seen so far is that it eats up
some time while they're preparing your W2G. What???!! My advice is to
avoid the double-up.

Here's how much sense it makes:

A student takes a true-false exam and doesn't know the answers, so he
decides to flip a coin for each question. Heads will be true, tails will
be false. He completes the test. Then he starts flipping the coin again
and someone asks why he's doing it. He says, "I want to recheck my
answers!"

So maybe it doesn't prove my point, but it's funny.

lb
"variance giveth, variance taketh away"

···

Here's another example where two different players view the
same game differently: In Craps, a $6 place bet on number 8
is paid $7 if an 8 is rolled before the next 7. These bets can
be placed or removed at any time. Jim defines the end of a
game as a roll of 8 or 7, where the 8 results in a $7 win and
a 7 results in a $6 loss. On average a seven is rolled 6 times
for every 5 times that the roll is eight. Jim computes his EV
as 100 * [($7 * 5) - ($6 * 6)] / ($6 * 11) = -1.515%.

#2

Ok ... suggestion - skip this post if you couldn't give a damn if
doubling in vp is dependent, independent, or on the verge of a full
scale revolt for it's own game selection.

···

------

The key to independence of two events is that there's no way to link
them as a single conjoined event mathematically. The occurence of one
has no bearing on the other.

The simple fact that in order to double in vp it's necessary to first
have a win is sufficient for the events to be dependent.

One consequence of this dependency is that it's possible to calculate
the resulting play variance if you always double a win once in your
play, repeating a double when the first is a push (for all wins, or a
selected subset ... say any win of less than 100 credits).

To do this you alter the table of payouts and frequencies so that you
split each hand into two outcomes, a win of twice the wager and no win
- dividing the standard hand frequency in half for each. Variance is
then calculated manually in the same manner that you would for the
standard paytable/hand frequencies.

The point to this exercise is simply to illustrate the dependency
between the two wagers in vp -- an exercise that couldn't be completed
in one pass otherwise.

It also demonstrates that any vp double increases game variance.

- Harry

#3

Harry Porter wrote:

The point to this exercise is simply to illustrate the dependency
between the two wagers in vp -- an exercise that couldn't be
completed in one pass otherwise.

One additional note: The same analysis demonstrates that doubling
doesn't change play ER.

The point is that when you choose to double, you're extending the same
play -- you haven't collected the hand win and then rebet it. The ER
of the ultimate pays, whether you double or not, are unchanged.

I'm sure there will be those that are unwilling or unprepared to
accept this statement. However, the math stands for itself even if
the reasoning is unclear.

Now, Steve made a valid point earlier -- there is more than one
perspective from which to approach a problem. However, the critical
point is that if the problem is treated within a consistent system of
relationships, the outcome of each approach will be identical.

- Harry

(and any "Bosco" comments from the acvpp crowd are unnecessary :slight_smile:

#4

The point is that when you choose to double, you're extending the

same
play -- you haven't collected the hand win and then rebet it. The ER
of the ultimate pays, whether you double or not, are unchanged.<<

That's the problem. You have WON the hand, regardless of whether or
not you decide to collect it or RE-BET it on double-up. By your
logic, any play is dependent on the prior play if losing would
prevent further play, rather than viewing each bet as independent.
Not only is it a different bet, it's a different GAME. Oranges ain't
apples, even if they're in the same basket.

lb
"variance giveth, variance taketh away"

#5

lawrenceboxer wrote:

By your logic, any play is dependent on the prior play if losing
would prevent further play, rather than viewing each bet as
independent.
Not only is it a different bet, it's a different GAME. Oranges
ain't apples, even if they're in the same basket.

That's exactly what I'm asserting and I'll stand by my logic. A
relationship creates dependency. Lack of relationship defines
independency.

I want to emphasize the fact that analysis of a subdivided paytable
that reflects doubling of a win, under any assumptions you care to
make, will result in the same play ER. If you question this and wish
me to construct an example, I'll forward it to you for your critique.

If you can produce an alternative analysis/approach that produces a
different ER as a consequence of doubling I'd like to see it so that I
can reconsider my position. Keep in mind the fact that a double is a
100% ER proposition when it stands alone doesn't address the inherent
dependency in this case. Dependent events when paired represent a
subset of paired events were they independent.

Key point: If the double decision is effectively made independent of
a win, for example, if you're presented the opportunity to make a
50-50 wager of a given amount after each play, win or lose, ER is
affected.

But even in this case that opportunity can be treated as a dependent
wager, divided into the case of a win and the case of a loss. (Which
is an alternate approach for this scenario, but will produce the same
result.) It's the potential for a net win between the paired wagers
that presumably results in the influence on ER in that case.

Bottom line, a calculation that fully models vp hand play and any
subsequent doubling is necessary to evaluate EV properly.

- Harry

#6

Key point: If the double decision is effectively made independent of

a win, for example, if you're presented the opportunity to make a
50-50 wager of a given amount after each play, win or lose, ER is
affected.<<

Your construct in this case doesn't make any sense. How do you
double up a loss? A 50-50 chance of winning what, if you've already
lost your bet? Conversely, you cannot opt to double-up unless you've
already won your bet. In simplistic terms, you've got a 50-50 chance
of losing what you've already won by re-betting the win. Smart? I'll
leave it to others to decide how they want to approach it.

I'm not going to continue arguing this, but rather concede in good
humor that, as someone else noted here, we are not going to agree.

lb
"variance giveth, variance taketh away"

#7

lawrenceboxer replied to my statement:

>>Key point: If the double decision is effectively made independent
of a win, for example, if you're presented the opportunity to make a
50-50 wager of a given amount after each play, win or lose, ER is
affected.<<

with:

Your construct in this case doesn't make any sense. How do you
double up a loss? A 50-50 chance of winning what, if you've already
lost your bet?

The example was one under which ER might be effected. It substitutes
a double with a general 50-50 wager, as stated - necessarily provided
as an option after every bet at the wagerers option. If you want,
make the wager after the win a true "double". Make the wager after a
loss whatever you wish.

My point was I'm not arguing that interspersing a 50-50 wager between
each bet wouldn't affect ER. However, a 50-50 wager that's dependent
on an initial win won't affect ER - whether in the form of a double,
or in any other form.

I'm not going to continue arguing this, but rather concede in good
humor that, as someone else noted here, we are not going to agree.

I've no problem with that, but if you want to examine the spreadsheet
to make a solid challenge of the ER calc with doubling, I'm always
game for someone to prove me wrong and give me a chance to learn :).

So far, though, we've tossed words at each other without concrete
backup and little chance to benefit from each other.

- H.

#8

The fact that "the math stands for itself" proves only that your view
is _one_ valid way to look at the situation. Demonstration that one
view is mathematically correct is not evidence that other mathematically
correct views do not exist. In fact, the alternate strategies that I talk
about are all derived from viewing the same game in a way which is
different than the "one true way" (cough) that people have grown
accustomed to thinking about.

Bob and Jim disagree about how to compute EV for video poker.
In particular, they both play 9/6 JoB. Bob uses one of the popular
VP programs to compute EV in the "standard" way. Jim feels that
when a high pair is returned as a "push" that this doesn't constitute
the end of a game, because anyone who played and only got their
money back would play again and again until they either lost or
received a higher payoff.

A huge argument ensues. Bob says "what's the difference, it shows
up as a credit so you were paid." Jim counters with "you haven't
been paid until the coins are in the tray." Bob says "fine, I'll just
cash out every payoff so that you'll see my coins dropping into
the tray and know that I had a payoff. Clearly, the frequency of
payoffs will approach their mathematically expected values, and
the EV will be as predicted."

Jim respondes "Fine, but that isn't the only way to view the game.
I can choose to replay the "push" payoffs until they either lose
or result in a higher payoff, then cash out so that you will see
the coins dropping into my tray and know that I had a payoff."

The situation I've described above is no different than what comes
up in craps or baccarat or any number of other games. Craps
players who play "don't" bets seem to prefer to treat a push as
a non-event which hasn't resolved the wager, rather than an
event which marks the end of a game. Both views are mathematically
valid, and they lead to different numerical values for game EV. Still,
they are mathematically equivalent views.

The VP situation above is similar. If you choose to view a one unit
payback as the end of a game, then you will compute a value X
for EV. A player who replays pushes until resolved will compute
a value Y for EV. Same games viewed in two different ways and
resulting in two different values for EV. Both views are mathematically
correct. In fact, if Bob and Jim carefully determine optimal playing
strategies from their different views, they will arrive at identical playing
strategies. The overall probability distributions will look different,
but they are really two different forms of the same thing. Jim's
probability distribution can be described in terms of Bob's probability
distribution in the following way: Jim's probabilities are equal to
the conditional probabilities of Bob view, given that no push events
have taken place.

It boils down to this -- if P is the overall probability of getting a high
pair, then in Bob's view this value appears as the probability of
"1 unit returned". Jim's probability distribution can be derived from
Bob's by doing the following:

1) Remove the "High Pair" payoff from the distribution, but remember
the value of P which represents the probability of a high pair.

2) For all other payoffs, divide the probability by (1 - P).

3) Divide the overall EV by (1 - P) to get Y = X / (1 - P).

This procedure "transforms" Bob's view into Jim's view, and the
same procedure can be applied "in reverse" to transform Jim's
view into Bob's view. This is exactly analogous to using
different coordinate systems in Geometry to describe locations.
One view may be Cartesian coordinates while another uses
Polar coordinates. The numbers come out different, but the
real mechanics of what is underneath are the same.

Now, if you've followed that and it makes sense, then here
is something to think about. In Jim's view, there _is_ no
payback of one unit. Jim has chosen to regard all games
that follow a high pair as if they are "dependent" on winning
that high pair. In fact, we could describe this more carefully
by painting the coins that come out according to which kind
of hand was associated with the payoff. Gold colored coins
for royals, silver colored coins for straight flushes, and so
on. The coins paid back for high pairs could be a dull gray
color. So, when Jim gets a dull gray coin, he immediately
feeds it back in and keeps playing until the hand does NOT
result in a dull gray coin. Jim considers those "extra" plays
as intermediate steps in a single "game". Mathematically
there is nothing wrong with viewing things this way.

The type of dependency that Jim sees above is no different
than the double up debate. The two views lead to different
values of EV and different probability distributions, but they
remain mathematically equivalent in the sense that they
ultimately yield the same playing strategy when each
player strives to maximize their own definition of EV.

In short, Dan and Harry are both flat out wrong when they
claim that there is only one way to correctly view the
double up situation.

I'm out of time for now, but I plan to write up a detailed
description of how to compute EV from both views. I
don't know if I'll be able to come up with a simple way
to explain how they remain mathematically equivalent
in spite of the fact that they have different EV and
different variance, but it really is no different than the
corresponding situation in Craps or Baccarat. You
can view a "push" as ending a game or as a non-event,
and the view you choose alters EV and variance and
the overall probability distribution, yet underneath the
views are still mathematically equivalent.

There is more than one view, and the different views
do not necessarily yield the same numbers.

···

On Thursday 01 April 2004 11:00 pm, Harry Porter wrote:

The point is that when you choose to double, you're extending the same
play -- you haven't collected the hand win and then rebet it. The ER
of the ultimate pays, whether you double or not, are unchanged.

I'm sure there will be those that are unwilling or unprepared to
accept this statement. However, the math stands for itself even if
the reasoning is unclear.

#9

Steve Jacobs wrote:

In short, Dan and Harry are both flat out wrong when they
claim that there is only one way to correctly view the
double up situation.

Without further discussion, I'm quite willing to concede this may be true.

However, backtracking a little, it's my sense that there was a larger
question in this discussion than the ER impact of doubling. That
would be whether doubling is a sensible activity for the negative game
player.

The argument in favor was that it increases ER and that drove the
ensuing ER discussion. Setting aside the ER question, the more
immediate question is whether it reduces that player's expected loss.

To the extent that doubling reduces the number of negative vp plays
and therefore expected loss, there's clearly potential merit to it.

But determination if it's truly beneficial needs to take into account
whether it exposes the player to a greater potential loss. I beleive
it does (as, I sense, D. Paymar does).

I don't feel that the smaller expected loss compensates sufficiently
for the greater loss potential. I'm sure others will view this
differently. But I'd suggest that a negative game player simply ease
up on their play, consider more frequent breaks, and spend more time
enjoying casino amenities as a means of reducing their expected loss
rather than playing the double option.

- Harry