A surprising result from a low-cost strategy with reduced variance; can anyone explain this?

[Bill_Velek]

I've always been interested in learning some of the statistical aspects of VP and striving to understand and appreciate some of the nuances in that regard. After a fairly long absence from these groups except for some fairly frequent lurking, I was once again intrigued by some of the recent threads about 'Risk of Ruin' and minimal-cost strategies by some members, including Harry Porter's analysis. I'd like to pursue this further, and hope some of you are interested.

Despite criticism from some members for playing short-coin, that's what I usually do. That's because I play for recreation in Tunica where there are no positive JOB machines, and I don't mind losing a limited bankroll which I've budgeted for the trip. Therefore, my interest is pretty much in making my bankroll last as long as reasonably possible, although I haven't really considered my strategy to be strictly 'minimal-cost', per se, because it still yields maximum EV for short-coin play rather than making any additional concessions beyond that in order to reduce variance.

Well, I decided to experiment a bit to see how much I could reduce variance without dramatically reducing EV, and with just a very simply change on the pay-table in WinPoker, I was very surprised with the results and want to double check this with the math wizards on these groups.

Most of you probably have either WinPoker or FrugalVP, so you can double check my figures for yourselves.

I started with JOB 9/6 single-coin which WinPoker indicates has an EV of 98.3735% and a variance of 4.92681. I decided to reduce variance by shifting strategy away from Royals toward the more common low-paying hands, and I hoped to do this by changing the pay table to 50 coins for a Royal on a single-coin bet -- the same as a regular Straight-Flush. I then ran an analysis with WinPoker, and it yielded an EV of 98.0270% with the variance reduced to 3.76258; however, since the machines actually do pay 250 coins rather than only 50 for a short-coin Royal, I went back and corrected the winnings to arrive at the true EV for a 9/6 machine when playing perfectly with the new strategy; I did this by multiplying the frequency of the hand by the true payout of 250 and also multiplying the frequency for all other hands times their respective values, totaling all expected winnings, and dividing the total by the approximately 2.6 million hand total of hands played. The readjusted EV is 98.3435, which is only .03% lower than normal short-coin strategy.

Therefore, with this new 'lower cost' strategy, a player sacrifices only .03% of EV for a seemingly dramatic drop in variance of more than 20%. This suggests to me that for a cost of only thirty-cents per thousand dollars of coin-in (4,000 short-coin games on a quarter machine), I could dramatically reduce my risk-of-ruin, which seems to be a bargain if developing and mastering the new strategy is not too complicated.

However, I also realize that by falsifying the value of a Royal, the variance computed by WinPoker is probably not correct, although I don't know how variance is actually computed. But here were my thoughts and questions on that, which no doubt reveal my complete lack of understanding about variance:

Could the use of an artificially LOW payout for a Royal affect the resulting variance and standard deviation such that the resulting figures and 'risk of loss' calculations would be too optimistic? It is hard for me to understand how that could be the case when missing the Royal, or any winning hand for that matter, only reduces bankroll by the single-coin that was bet -- regardless of whether the payout for a Royal is deliberately undervalued or not, whereas hitting the Royal actually generates five times the payout that the variance and standard deviation calculations were based upon. In other words, wouldn't this sort of be like just pretending that any hit Royals really only paid out the 50 coin value that had been used to compute the variance, and just ignoring the other 200 coins -- as if you just found them on the floor. As I see it, if I don't hit Royals as often as are statistically predicted, then my losses aren't mounting any faster than what was expected anyway, but if I do hit the Royals as often as expected (or more), than I'm just that much further up on the game than what the variance and standard deviation would predict, and even safer from risk of ruin.

The other observation that I had with this modified strategy is that, although WinPoker indicates a dramatic drop in variance (which I had always equated with 'volatility'), the frequency of the various hands surprisingly changed to what I would think would lend to an even MORE volatile game. Specifically, by changing from optimum short-coin strategy (Royals having a value of 250) to the modified strategy (Royals having a value of only 50 and a substantial reduction in variance), I expected to see a trade off of fewer Royals in favor of more frequent low-paying hands; however, that is not really the case according to WinPoker. Comparing both strategies, playing a total of about 2.6 million hands each (the totals differing less than half a hand), there is a net REDUCTION in the total number of WINNING hands, with an actual shift toward the middle of the pay table with most of the losses being made up with Flushes and Straights. Here are the following approximate differences in hand distributions:

Using the modified strategy, per approx. 1.6 million hands, results in:

    9.38 fewer Royals (-- I expected this)
    5.66 fewer Straight Flushes
    6.88 fewer Full Houses
   64.68 fewer Three of a Kinds
  343.66 fewer Two Pair
2364.81 fewer Jacks or Better (-- this surprised me)

and

    2.27 more Four of a Kinds
  521.32 more Flushes
  492.02 more Straights
1779.40 more 'Nothing' Hands (-- this _really_ surprised me)

This means that you have more frequent one-coin loses, fewer draws or gains of just one or two coins, but more of the relatively less frequent gains of 3, 5, or 24 coins. I can't see why this wouldn't cause wider swings in your bankroll, which to me means greater volatility despite the lower variance.

I hope someone can explain this to me, especially if I'm in error with my methods here somewhere.

Thanks.

Bill Velek

[Harry_Porter]

Bill Velek wrote:

Well, I decided to experiment a bit to see how much I could reduce
variance without dramatically reducing EV, and with just a very simply

change on the pay-table in WinPoker, I was very surprised with the
results and want to double check this with the math wizards on these
groups.

Bill,

Bob Dancer published an article on this very subject. I'll follow up
with the date and publication.

In that article, he reviewed the fact for certain games it was indeed
possible to alter strategy in the manner that you suggest to reduce
variance to a significant extent with an almost insignificant reduction
in return (relative to the variance reduction). This was done, again,
similarly to what you suggest -- using a strategy that de-emphasized the
value of the RF.

I believe that in the case of certain "super pay" games at the Strat
(10/6 JB and 9/7 JB - both of which I believe have since bit the dust),
such a result was achieved.

I reviewed this article with some fascination and was curious how such
an approach would affect other games. I reviewed this with a full
analysis and found that for other games the resulting reduction in
return was far more significant. It escapes me at this time why the
super pay games produce more optimal results (aside from the fact that a
high proportion of the return lies in the FH/F hands).

Bill, in the next day or two I'll revisit that analysis and summarize
the key facts.

Now, what I've indicated here makes me suspect your results. On the
other hand, in playing short coin the contribution of the RF to game
variance is substantially reduced, as the reduced variance of short coin
obviously attests. But if my head is on straight about this, again not
having looked at this in some time, the consequence should be the
opposite of what you're reporting as a result.

As noted, the reduced value of a RF in playing short coin means that it
has a reduced contribution to total variance. Reducing the amount of
return contributed by a RF, by altering strategy as just discussed,
would result in the variance reduction being appreciably smaller than in
the full-coin situation.

However, the overall situation is complex. In case of the reduced RF
value, the full coin case entailed a reduction per coin from 800 to 250,
if I recall -- or about 2/3. Your single coin case is a reduction from
250 to 50, or a 4/5 reduction. This means that the related reduction in
variance contribution isn't immediately apparent, at least to me.

I would expect the extent to which an altered strategy shifts return
away from the RF to a FH/F is much smaller in the one coin case would be
smaller due to the smaller absolute payoffs. However, at this moment
this is not an obvious fact to me and I need to review both situations
to make a definitive statement.

Bottom line, you may be correct in your conclusions. It is the case
that you're talking about a much larger relative reduction in RF
emphasis. But my gut feeling is that the reduction in variance and
return would both be relatively insignificant to warrant an altered
strategy. One potential problem with your results is that I'm pretty
sure that you're methodology for determining the reduction in variance
is flawed. My mind can't really make the leap from the correct detailed
calculation and the shortcut method you employed.

Bill, give me until the weekend to take time out for a full analysis of
this.

To be honest, I have little interest in the practical application in
this case. I understand your rationale for playing short coin and don't
question the fact that you do it. But I consider this recreational play
at it's fullest. Application of a fairly complex approach to play such
as this does not seem warranted. I suppose I should reserve judgment on
that until I see the result of my analysis.

However, I think most realize by now that I can't help but be interested
in this problem sheerly out of curiosity and a certain satisfaction I'll
derive in evaluating this. I guess I'm a real academic in that respect.

- Harry

[Harry_Porter]

Bill,

Bob Dancer published an article on this very subject.

... with a red face, I find that it was Skip Hughes's article ...

- H.

[AJ1]

Bill,

Seems like Harry is on the case big time, but here are a few
thoughts.

Variance is the sum of the probability of each result times the
square of the result. The variance reported in winpoker can be
adjusted to the real variance of the adjusted strategy by adding the
probability of the royal reported times 60,000.
(250*250-50*50) = 60,000. I don't currently have access to software,
but I think you will find that with this correction, the variance is
very close to original 4.9.

The increase in flushes might be due to keeping suited "kickers"
along with two suited honors. This would partially explain the loss
of both royals and high pairs. It seems plausible to me that
the "royal avoidance" approach could actually (very slightly) hurt
your chances of trip bankroll survival over some reasonable playing
intervals.

For almost all players, financially there isn't much to get excited
about here. That loss of 2365 sets of high pairs per 1.6 million
hands is about one hand (25 cents) per hour. Every few hours you get
an extra straight or flush in compensation. The "feel" of the game
will be virtually unchanged. Bringing an extra $5 bill will have
much greater impact on probability of busting out over a long day
than any of these strategy changes.

Strategically avoiding royals, lowering variance with minimal loss of
ER, and playing to conserve bankroll are different goals and can lead
to significantly different sets of playing strategy variations.

In some circumstances, particularly short trials, variance is very
poorly correlated to smoothness of the ride or probability of trip
bankroll surviving.

AJ

I've always been interested in learning some of the statistical

aspects

of VP and striving to understand and appreciate some of the nuances

in

that regard. After a fairly long absence from these groups except

for

some fairly frequent lurking, I was once again intrigued by some of

the

recent threads about 'Risk of Ruin' and minimal-cost strategies by

some

members, including Harry Porter's analysis. I'd like to pursue

this

further, and hope some of you are interested.

Despite criticism from some members for playing short-coin, that's

what

I usually do. That's because I play for recreation in Tunica where
there are no positive JOB machines, and I don't mind losing a

limited

bankroll which I've budgeted for the trip. Therefore, my interest

is

pretty much in making my bankroll last as long as reasonably

possible,

although I haven't really considered my strategy to be strictly
'minimal-cost', per se, because it still yields maximum EV for
short-coin play rather than making any additional concessions

beyond

that in order to reduce variance.

Well, I decided to experiment a bit to see how much I could reduce
variance without dramatically reducing EV, and with just a very

simply

change on the pay-table in WinPoker, I was very surprised with the
results and want to double check this with the math wizards on

these groups.

Most of you probably have either WinPoker or FrugalVP, so you can

double

check my figures for yourselves.

I started with JOB 9/6 single-coin which WinPoker indicates has an

EV of

98.3735% and a variance of 4.92681. I decided to reduce variance

by

shifting strategy away from Royals toward the more common low-

paying

hands, and I hoped to do this by changing the pay table to 50 coins

for

a Royal on a single-coin bet -- the same as a regular Straight-

Flush. I

then ran an analysis with WinPoker, and it yielded an EV of

98.0270%

with the variance reduced to 3.76258; however, since the machines
actually do pay 250 coins rather than only 50 for a short-coin

Royal, I

went back and corrected the winnings to arrive at the true EV for a

9/6

machine when playing perfectly with the new strategy; I did this by
multiplying the frequency of the hand by the true payout of 250 and

also

multiplying the frequency for all other hands times their

respective

values, totaling all expected winnings, and dividing the total by

the

approximately 2.6 million hand total of hands played. The

readjusted EV

is 98.3435, which is only .03% lower than normal short-coin

strategy.

Therefore, with this new 'lower cost' strategy, a player sacrifices

only

.03% of EV for a seemingly dramatic drop in variance of more than

20%.

This suggests to me that for a cost of only thirty-cents per

thousand

dollars of coin-in (4,000 short-coin games on a quarter machine), I
could dramatically reduce my risk-of-ruin, which seems to be a

bargain

if developing and mastering the new strategy is not too complicated.

However, I also realize that by falsifying the value of a Royal,

the

variance computed by WinPoker is probably not correct, although I

don't

know how variance is actually computed. But here were my thoughts

and

questions on that, which no doubt reveal my complete lack of
understanding about variance:

Could the use of an artificially LOW payout for a Royal affect the
resulting variance and standard deviation such that the resulting
figures and 'risk of loss' calculations would be too optimistic?

It is

hard for me to understand how that could be the case when missing

the

Royal, or any winning hand for that matter, only reduces bankroll

by the

single-coin that was bet -- regardless of whether the payout for a

Royal

is deliberately undervalued or not, whereas hitting the Royal

actually

generates five times the payout that the variance and standard

deviation

calculations were based upon. In other words, wouldn't this sort

of be

like just pretending that any hit Royals really only paid out the

50

coin value that had been used to compute the variance, and just

ignoring

the other 200 coins -- as if you just found them on the floor. As

I see

it, if I don't hit Royals as often as are statistically predicted,

then

my losses aren't mounting any faster than what was expected anyway,

but

if I do hit the Royals as often as expected (or more), than I'm

just

that much further up on the game than what the variance and

standard

deviation would predict, and even safer from risk of ruin.

The other observation that I had with this modified strategy is

that,

although WinPoker indicates a dramatic drop in variance (which I

had

always equated with 'volatility'), the frequency of the various

hands

surprisingly changed to what I would think would lend to an even

MORE

volatile game. Specifically, by changing from optimum short-coin
strategy (Royals having a value of 250) to the modified strategy

(Royals

having a value of only 50 and a substantial reduction in variance),

I

expected to see a trade off of fewer Royals in favor of more

frequent

low-paying hands; however, that is not really the case according to
WinPoker. Comparing both strategies, playing a total of about 2.6
million hands each (the totals differing less than half a hand),

there

is a net REDUCTION in the total number of WINNING hands, with an

actual

shift toward the middle of the pay table with most of the losses

being

made up with Flushes and Straights. Here are the following

approximate

differences in hand distributions:

Using the modified strategy, per approx. 1.6 million hands, results

in:

    9.38 fewer Royals (-- I expected this)
    5.66 fewer Straight Flushes
    6.88 fewer Full Houses
   64.68 fewer Three of a Kinds
  343.66 fewer Two Pair
2364.81 fewer Jacks or Better (-- this surprised me)

and

    2.27 more Four of a Kinds
  521.32 more Flushes
  492.02 more Straights
1779.40 more 'Nothing' Hands (-- this _really_ surprised me)

This means that you have more frequent one-coin loses, fewer draws

or

gains of just one or two coins, but more of the relatively less

frequent

gains of 3, 5, or 24 coins. I can't see why this wouldn't cause

wider

swings in your bankroll, which to me means greater volatility

despite

the lower variance.

I hope someone can explain this to me, especially if I'm in error

with

···

--- In vpFREE@yahoogroups.com, Bill Velek <billvelek@a...> wrote:

my methods here somewhere.

Thanks.

Bill Velek

[Harry_Porter]

AJ wrote:

Seems like Harry is on the case big time, but here are a few
thoughts.

Skip Hughes saved me the trouble with an update of his analysis to
review this scenario. He's detailed it over on the winpoker group.

Bringing an extra $5 bill will have much greater impact on
probability of busting out over a long day than any of these
strategy changes.

You got it, AJ. Skip's numbers work out to be a 5% reduction in
variance. If Bill's single coin play tends to risk no more than $100
in a day, that $5 bill will add as much comfort to play as adjusting
strategy.

- Harry