MJ: CVCX Question: Goals vs Actual Results calculators

[MJ]

I have been studying the Goals and Actual Results widgets and am trying to explain a discrepancy.

Let me first provide the necessary fields:

BR: 20k
Trip BR: 1k
Hours for trip: 4
Goal for trip: $500
Win Rate: $14/Hr and $56/Trip

Ok, here is my question. According to the Goals widget, there is a 52% chance of winning $500(goal) after 4 hours of play. This projection seemed a bit optimistic to myself, so I decided to input a win of $500 into the Actual Results calculator to see what it comes up with. Well, the AR calculator spits out 27%.

So, why a difference of 25% (52%-27%) between the widgets?
The answer no doubt lies in the assumptions behind the respective widgets.

The Goals widget: "The results is the probability of reaching that goal before going bankrupt or hitting the specified number of hours". Basically, there are 3 stop points at work (triple barrier) here and they are as follows:

1) Loss of trip br - (in this case $1k)
2) Time - Playing for 4 hours
3) Reaching goal - $500

The first 2 stop points only serve to reduce the chances of reaching our goal of $500 should they occur. If either one occurs before reaching our goal, the experiment stops prematurely. With regard to #2, if we are ahead $400 ($100 short of our goal) but have played 4 hours, then we must stop! Thus, the time constraint can potentially hinder us in trying to reach our goal. The 3rd stop point is extremely useful, because once we win $500, we stop playing and lock up the win!

Now let us examine the Actual results widget. This has a couple of assumptions:

1) Playing with entire Br, not some limited Trip Br
2) Time constraint - 4 hours
3) No Goal constraint - so there is no stop loss limit here

With the AR widget, we have the advantage of virtually never tapping out because we are playing with the entire BR (20k). In fact, the Trip Ruin calc gives the chances of losing $1k to be about 15% (Double Barrier). That means that with the Goals widget, 15 out of 100 counters will tap out by the time 4 hours have expired with no chance of coming back to reach the goal ($500). Again, the AR widget has the advantage in the sense of playing with the entire Br and never tapping out.

Looking at assumption #2, this Time constraint is imposed on both widgets so it is neither advantageous or disadvantageous.

I suppose constraint #3 is the big difference between the widgets in the sense that if a counter is up $500 or more, the experiment does not cease. Some counters will continue to win more, and some will drop down below $500. When you average the performance of many counters, I guess only 27 out of 100 will be ahead $500 or more.

The question here is whether constraint #3 is that much more advantageous than constraint #1 so as to justify 25 more counters being ahead $500 with the Goals widget than with the AR widget at the end of a 4 hour trip.

Your comments please...
**************************************

MJ

[Don Schlesinger]

> I have been studying the Goals and Actual Results
> widgets and am trying to explain a discrepancy.

There is none! :-)

> Let me first provide the necessary fields:

You haven't provided all of them. Where's the standard deviation?

> BR: 20k
> Trip BR: 1k
> Hours for trip: 4
> Goal for trip: $500
> Win Rate: $14/Hr and $56/Trip

> Ok, here is my question. According to the Goals
> widget, there is a 52% chance of winning $500(goal)
> after 4 hours of play. This projection seemed a bit
> optimistic to myself, so I decided to input a win of
> $500 into the Actual Results calculator to see what it
> comes up with. Well, the AR calculator spits out 27%.

> So, why a difference of 25% (52%-27%) between the
> widgets?
> The answer no doubt lies in the assumptions behind the
> respective widgets.

> The Goals widget: "The results is the probability
> of reaching that goal before going bankrupt or hitting
> the specified number of hours". Basically, there
> are 3 stop points at work (triple barrier) here and
> they are as follows:

> 1) Loss of trip br - (in this case $1k)
> 2) Time - Playing for 4 hours
> 3) Reaching goal - $500

> The first 2 stop points only serve to reduce the
> chances of reaching our goal of $500 should they
> occur. If either one occurs before reaching our goal,
> the experiment stops prematurely. With regard to #2,
> if we are ahead $400 ($100 short of our goal) but have
> played 4 hours, then we must stop! Thus, the time
> constraint can potentially hinder us in trying to
> reach our goal. The 3rd stop point is extremely
> useful, because once we win $500, we stop playing and
> lock up the win!

> Now let us examine the Actual results widget. This has
> a couple of assumptions:

> 1) Playing with entire Br, not some limited Trip Br
> 2) Time constraint - 4 hours
> 3) No Goal constraint - so there is no stop loss limit
> here

> With the AR widget, we have the advantage of virtually
> never tapping out because we are playing with the
> entire BR (20k). In fact, the Trip Ruin calc gives the
> chances of losing $1k to be about 15% (Double
> Barrier). That means that with the Goals widget, 15
> out of 100 counters will tap out by the time 4 hours
> have expired with no chance of coming back to reach
> the goal ($500). Again, the AR widget has the
> advantage in the sense of playing with the entire Br
> and never tapping out.

> Looking at assumption #2, this Time constraint is
> imposed on both widgets so it is neither advantageous
> or disadvantageous.

> I suppose constraint #3 is the big difference between
> the widgets in the sense that if a counter is up $500
> or more, the experiment does not cease. Some counters
> will continue to win more, and some will drop down
> below $500. When you average the performance of many
> counters, I guess only 27 out of 100 will be ahead
> $500 or more.

> The question here is whether constraint #3 is that
> much more advantageous than constraint #1 so as to
> justify 25 more counters being ahead $500 with the
> Goals widget than with the AR widget at the end of a 4
> hour trip.

> Your comments please...

There is virtually nothing to be said. You've said it all. And the numbers are what they are, and they are correct; that much I can assure you.

Instead of wondering whether it can be so, you should be commenting: "It's very interesting that this is the case." But, I discussed this phenomenon at length in BJA3 (see below).

The short version of your recital is that, in neither case is tapping out a big problem, so it simply comes down to the fact that, in one case, you're looking at the (smaller) end-point probability of being at a certain goal, but only at the conclusion of your play. In the other case, it counts as reaching your goal if you hit it at any time at all along the way.

As I stated in BJA3 (see the "Premature bumping into the barrier syndrome" discussions) the end-point probability is about half of the achieve-the-goal-at-any-time probability.

Don

[MJ]

Thanks for responding!

> You haven't provided all of them. Where's the standard
> deviation?

The SD really has no bearing on either of the widgets being discussed. But since you asked, SD for 1 Hr = $367.52. Hence, the SD for 4 Hrs = sq rt (4) x $367.52 = $745.04.

> There is virtually nothing to be said. You've said it
> all. And the numbers are what they are, and they are
> correct; that much I can assure you.

I don't doubt it, I just want to make sure I understand what is going on at a conceptual level (I like to know why things work). Are there any inaccuracies in what I stated? Do you consider the Goals widget to be triple barrier as it imposes 3 constraints - time, br, and goal?

> Instead of wondering whether it can be so, you should
> be commenting: "It's very interesting that this
> is the case."

It is very interesting that this is the case! :-) Good stuff.

>But, I discussed this phenomenon at length in BJA3 (see below).

I do have BJA3. Just didn't get very far into it yet.

> The short version of your recital is that, in neither
> case is tapping out a big problem,

Whoa, wait a minute. I'm not sure what you mean by "big problem", but in the case of the Goals widget, there is a 15% chance of tapping out with a trip bank of $1k. 15% is a big problem, at least IMHO. :-) As I wrote above, this means that 15 out of 100 counters will die a premature death and will not be eligible to reach the Goal!!

Tapping out will not affect the AR calculator (in this case), so it does have a bit of an edge in trying to reach the goal, at least looking at it purely from a BR perspective.

> so it simply comes down to the fact that, in one case, > you're looking at
> the (smaller) end-point probability of being at a
> certain goal, but only at the conclusion of your play.
> In the other case, it counts as reaching your goal if
> you hit it at any time at all along the way.

Yea, I get it.

> As I stated in BJA3 (see the "Premature bumping
> into the barrier syndrome" discussions) the
> end-point probability is about half of the
> achieve-the-goal-at-any-time probability.

Can you kindly provide the p.# for this article? I can't seem to find it.

Thanks for your comments.

MJ

[Don Schlesinger]

> The SD really has no bearing on either of the widgets
> being discussed. But since you asked, SD for 1 Hr =
> $367.52. Hence, the SD for 4 Hrs = sq rt (4) x $367.52
> = $745.04.

I'd have to take a look, but I don't understand that at all. How can you possibly discuss ROR or reaching a goal without providing a standard deviation? It isn't possible, unless you're assuming some sort of flat betting, and the s.d. is already factored in (c. 1.14 units per hand, depending on rules).

> I don't doubt it, I just want to make sure I
> understand what is going on at a conceptual level (I
> like to know why things work). Are there any
> inaccuracies in what I stated?

Not that I recall.

> Do you consider the
> Goals widget to be triple barrier as it imposes 3
> constraints - time, br, and goal?

As I write in the book (please do read pp. 135-146, upon which all of Norm's goal widgets are based), these are "double-barrier" formulas. The barriers are considered the goal and the zero (ROR). Time is a "constraint," but not a "barrier."

> It is very interesting that this is the case! :-) Good
> stuff.

> I do have BJA3. Just didn't get very far into it yet.

READ the book!

> Whoa, wait a minute. I'm not sure what you mean by
> "big problem", but in the case of the Goals
> widget, there is a 15% chance of tapping out with a
> trip bank of $1k. 15% is a big problem, at least IMHO.
> :-) As I wrote above, this means that 15 out of 100
> counters will die a premature death and will not be
> eligible to reach the Goal!!

But the other 85 will have a chance to reach the goal early, and then fall back, thereby seemingly not achieving the goal at the end, but having touched it, nonetheless, during the trip. That's what you're seeing with the higher number.

> Tapping out will not affect the AR calculator (in this
> case), so it does have a bit of an edge in trying to
> reach the goal, at least looking at it purely from a
> BR perspective.

See above. Not as big as being able to touch the goal at any time, rather than insisting on being there just at the end.

> Yea, I get it.

Good. :-)

> Can you kindly provide the p.# for this article? I
> can't seem to find it.

That's what they make indexes for!! :-) Try looking up "premature bumping into the barrier syndrome," on p. 532. :-)

> Thanks for your comments.

You're welcome. Now, do your homework! :-)

Don

[MJ]

Ok, I read the suggested passages. Interesting stuff.

On page 124, you state that the probability of being behind by at least a certain amount at some point during a trip is a little more than double the probability of being behind by that same amount at exactly the end of a specified time.

On page 125, 3 out of 4 of your estimations seem to agree quite well with Wong's simulations. There is one estimate of probability for 4,000 hands played that is a wee bit off, however.

By your estimate, the p of being down $1,100 or more should be around 32%; actually slightly greater. But it turns out to be 42.6% according to the simulation. This difference seems to be rather significant. How do you explain it while the other estimates were so close?

MJ

[Don Schlesinger]

> Ok, I read the suggested passages. Interesting stuff.

Thanks.

> On page 124, you state that the probability of being
> behind by at least a certain amount at some point
> during a trip is a little more than double the
> probability of being behind by that same amount at
> exactly the end of a specified time.

It varies. Later, if you keep reading, you'll see that it could be considerably more than double. Stop reading just sections; read the whole damn chapter! Don't be lazy ;-)

> On page 125, 3 out of 4 of your estimations seem to
> agree quite well with Wong's simulations. There is one
> estimate of probability for 4,000 hands played that is
> a wee bit off, however.

Sue me! :-) They're estimates. The more precise stuff comes later, with all the new formulas.

> By your estimate, the p of being down $1,100 or more
> should be around 32%; actually slightly greater. But
> it turns out to be 42.6% according to the simulation.

And, I did write, "Not too suprisingly," no?

> This difference seems to be rather significant. How do
> you explain it while the other estimates were so
> close?

Just keep reading.

Don

[MJ]

On a side note, I also read pp 135-146, and was quite pleased with how the Goal formulas performed when juxtaposed with CVData in the charts. Now I understand why Norm says if you want precise figures, use CVData. However, the CVCX Goal calculators seem to have more than enough accuracy for a weekend warrior.