vpFREE2 Forums

ER/ROR (Expected Return/Risk of Ruin) puzzle

#1

Suppose you have a new single coin $1 VP game that eliminates the
Royal and makes other changes, such that 3 Std Devs (99%+) of players
achieve long term results after about 1 million hands. It is a
positive game returning 101%, and a group of 10,000 rocket scientists
who always play perfect set about playing it with $10,000 bankrolls,
which are deemed adequate for this game using a 10% ROR assumption.
Well, in the course of play 1000 players will go bust and 9000 will
make 1% of $1 million or $10,000. So at the end we have:

9000 players x $10,000 profit = $90,000,000
1000 players x $10,000 loss = -$10,000,000
Net for group = $80,000,000
Total hands played were 9,000,000,000 for the successful players
We don't know exactly for the ruined players, but we know it was less
than 1,000,000,000 and probably was front-weighted, so lets call it
300,000,000.

Thus the group had $80 million gain on 9300 million hands/dollars of
coin in, or 0.86%.
Yet the 101% machines had to have paid out $93 million more than they
took in – after 9.3 billion hands they had to have normalized.

So where did the other $13 billion go?

#2

This should, of course, have read $13 million, not billion.

···

--- In vpFREE@yahoogroups.com, "blaw57" <blaw57@y...> wrote:

So where did the other $13 billion go?

#3

in the real world someone else would step in and play those machines
that became open because the player lost his bankroll
from the casino perspective, the machines would achieve long term
results
from the player perspective, some would quit early because of
bankroll risk of ruin, this short term result would be balanced by
some players who were lucky (got more than expected return) in the
short term
long term players tend toward long term results (er)
short term players are a distribution, some win and some lose, but
their total average tends toward the er (for a sufficiently large
sample)

Suppose you have a new single coin $1 VP game that eliminates the
Royal and makes other changes, such that 3 Std Devs (99%+) of

players

achieve long term results after about 1 million hands. It is a
positive game returning 101%, and a group of 10,000 rocket

scientists

who always play perfect set about playing it with $10,000

bankrolls,

which are deemed adequate for this game using a 10% ROR assumption.
Well, in the course of play 1000 players will go bust and 9000 will
make 1% of $1 million or $10,000. So at the end we have:

9000 players x $10,000 profit = $90,000,000
1000 players x $10,000 loss = -$10,000,000
Net for group = $80,000,000
Total hands played were 9,000,000,000 for the successful players
We don't know exactly for the ruined players, but we know it was

less

than 1,000,000,000 and probably was front-weighted, so lets call it
300,000,000.

Thus the group had $80 million gain on 9300 million hands/dollars

of

coin in, or 0.86%.
Yet the 101% machines had to have paid out $93 million more than

they

···

--- In vpFREE@yahoogroups.com, "blaw57" <blaw57@y...> wrote:

took in – after 9.3 billion hands they had to have normalized.

So where did the other $13 billion go?

#4

blaw57 wrote:

... It is a positive game returning 101%, and a group of 10,000
rocket scientists who always play perfect set about playing it with
$10,000 bankrolls, which are deemed adequate for this game using a
10% ROR assumption. Well, in the course of play 1000 players will go
bust and 9000 will make 1% of $1 million or $10,000.

Under these general assumptions, it's the group of 10,000 players as a
whole who have a 1% return expection. Of the subset of players who
don't bust, they're ER will be greater than 1%.

- Harry

#5

Thanks, Harry and Stuart, that would *have* to be the answer,
wouldn't it? (Jazbo answered on the other board with the same
answer.) The machines have to pay 101%, the ruined players didn't get
it, so there's no one left. Still, that leaves the anamoly of a large
group of players earning more than the ER over a large number of
hands.

Now, I realize in the real world, the gain over ER that we active
players are experiencing due to ruined gamblers is probably too small
to detect and separate from the other factors. But it seems like you
could set up a closed simulation that would present quite a quandary
in its results, unless it's mathematically impossible to ruin more
players than would be swamped by the normalization cycle of the
survivors. That might be an interesting simulation exercise to do.

#6

An idea hit me while I was posting my reply on the other board -
modify my reply above to the following (also John Zaroff just joined
the group who answered the puzzle on the other board.)

...

Now, I realize in the real world, the gain over ER that we active
players are experiencing due to ruined gamblers is probably too small
to detect and separate from the other factors. But it seems like you
could set up a closed simulation that would present quite a quandary
in its results, unless ...

...unless, it is mathematically impossible to ruin players with more
hands than would be swamped by the normalization cycle of the
survivors. My example was invented to fit the puzzle, but I actually
don't know if you could come up with a game that would ruin 1000
players with 300 mil hands while letting another 9000 normalize in
only one million hands each. Maybe those who ruin have to ruin in
1000 hands on average, while the survivors have to play 10 million
hands each to normalize. Then the extra yield the survivors get would
be miniscule.

I bet that's the answer. But it would still be an interesting
simulation exercise to do.

#7

Originally posted:
"Suppose you have a new single coin $1 VP game that eliminates the
Royal and makes other changes, such that 3 Std Devs (99%+) of players
achieve long term results after about 1 million hands. It is a
positive game returning 101%, and a group of 10,000 rocket scientists
who always play perfect set about playing it with $10,000 bankrolls,
which are deemed adequate for this game using a 10% ROR assumption."

And later posted:
"My example was invented to fit the puzzle, but I actually don't
know if you could come up with a game that would ruin 1000 players
with 300 mil hands while letting another 9000 normalize in only one
million hands each. Maybe those who ruin have to ruin in 1000 hands
on average, while the survivors have to play 10 million hands each
to normalize. Then the extra yield the survivors get would be
miniscule."

I didn't want to answer the puzzle originally because the puzzle
had, IMHO, little practical value. Common sense(1) should help in
this situation. Say 10,000 players playing $5 a hand on a game that
returns 101% -- this means the casino is facing $500 in expected
loss per hand. If you assume 600 hph, that translates into a loss
of $300,000 an hour.

Here's the fallacy part of the puzzle that blaw57 overlooked.

If my understanding of math is correct, the overall results of the
players **on each hand** has very low standard deviation from
expected result. At 10,000 hands per trial, the so called "bell-
shape curve" ought to be liken to a very tall isosceles triangle
with a miniscule base. You might want to calculate the std dev
yourself to verify this. Put it this way, within 4 hands, one of the
players probably got a RF. If you recall why a casino makes money,
it's because it has a small edge benefited from massive wagers at
once; blaw57's puzzle simply reverses the role of the casino and
players.

My last thought to blaw57 would be use a system rather than 1 or
10,000 or 100,000 players. If the game has a return of 101% with
perfect play, then the "expected" ending bankroll at ending point in
time is [101% * the dollar action], notwithstanding how the ending
bankroll is ultimately divided.

Cheers.

(1) For example, which casino would allow this situation to go on
indefinitely, i.e. losing an expected $300,000 an hour (less going
forward as some players tap out and can't borrow from the winners)?
Assuming a casino can accomodate 10,000 vp of this game, where can
we find 10,000 hard-core anal-retentive vp players?

···

--- In vpFREE@yahoogroups.com, "blaw57" <blaw57@y...> wrote:

#8

the extra money goes to the casino
gambler's risk of ruin favors the house
if a lot of people are busting out due to insufficient bankrolls, the
house edge is increased
the players who don't bust out, taken as a subgroup, have a higher
average than the group expected return, however as they keep playing
their long term trend is still the expected return
in the long term, the total expected return dominates the variance
and any short term risk of ruin culling

#9

nightoftheiguana2000 wrote:

the extra money goes to the casino
gambler's risk of ruin favors the house
if a lot of people are busting out due to insufficient bankrolls, the
house edge is increased

So a casino should look to put the highest volatilty machines
available on their floor to increase their expected return over the
mathematical ER of the game?

You lost me ...

- H.

#10

blaw57 wrote:

Now, I realize in the real world, the gain over ER that we active
players are experiencing due to ruined gamblers is probably too small
to detect and separate from the other factors ...

Somehow this strikes me as contemplating how fortunate it is that with
each passing year, the total life expectancy of those living who were
born in the same year as me increases, mostly as a consequence of
those who were unfortunate enough to kick off.

I'm definitely behind in getting "thank you" cards in the mail.

- H.

#11

--- In vpFREE@yahoogroups.com, "Harry Porter" <harry.porter@v...>
wrote:

I'm definitely behind in getting "thank you" cards in the mail.

- H.

To Elmer Johnson, deceased
C/O Estate of Elmer Johnson
(wreath enclosed)

Dear Mr. Johnson,

Thank you for dying last month at the age of 45. Together with
another 0.77% of my age group who did me this service during the past
year, you have increased my actuarial lfe expectancy to 79.3 years
from only 78.7, where it stood at this time last year.

Yours sincerely,

blaw57

Harry, that same thing has been gradually dawning on me as I pursue
this line of thought, and I've been trying to think of a good
analogy. That one is perfect.

I think what it says is that there's really no puzzle here, just that
those run the risk but who avoid bombing out early will, in fact,
earn a higher return than the ER, because the ER includes the chance
of bombing out early.

The reason I thought there might be something significant here is the
following: risk of ruin covers all people who drop out of a given VP
segment due to excessive losses. The losses don't have to physically
wipe out their savings and put them in bankruptcy, just convince them
not to play any more. Pyschological ruin is probably many times more
frequent than actually running a bankroll to zero. Now, think of high
limit games ($25, $100) and the experience of people you know of who
have tried them. Many lose thousands and quit, never to return. So,
does that mean that there is an inherent built-in advantage over the
ER for those who approach the games with an adequate bankroll (and a
well-prepared psyche)?

In part, this depends on who benefits from a gamblers ruin - the
casino or other players. I used to think the casino did, along the
lines of what iguana is saying, but then I came up against the
incontrovertible fact that the machines as a group experience the
longest run of all, and thus they *have* to achieve the ER, no more
no less, first of all. So, the casino does *not* benefit from gambler
ruin. Thus, it must be the players who benefit.

Now, does this mean that the adequately bankrolled player can expect
to achieve more than the ER if he plays machines where a significant
part of the play is by under-bankrolled players?

Well, I've run some simulations, using a made-up game with a variance
of only 2.6. As it turns out, it still takes many more trials than I
thought before things start to normalize so my computer wheels are
still spinning, but the answer is starting to become clear, I think.

And the answer is.....

No, the adequately bankrolled player does not benefit from the ruin
either. Over the long run, he achieves the ER and never ruins, and
thus others of his class have no extra yield to divide and so they
achieve the ER as a group as well as individually.

The beneficiary of gambler ruin is other gamblers who ran a risk of
ruin but did not ruin. And you cannot construct a model, even
hypothetical, that includes only ruined gamblers and adequately
bankrolled players - there must always be a portion of the
speculative who succeed and reap the rewards of the ruin going on all
around them.

To go back to the original puzzle, the 9000 survivors could not have
normalized in a time period so short that the losers losses were a
significant percentage of coin-in. The survivors will intially be
ahead by the amount of the losers' losses, and the above average
return would be the beneficial effect of playing on a 90% ruin-free
bankroll instead of a 100% ruin-free one. In the many millions of
more hands required for normalization, the average returns achieved
will gradually bring down this premium to the point of insignificance.

Thus it all comes back to common sense in the end, and no doubt there
will be many who wonder if there was any value in pursuing that
circular path at all. Well, all I can say is I come from a background
where the rule is to question everything, even common sense, as there
are occasionally flaws even in common sense. If a common sense math
propositon is really true, then it should be possible to prove it
mathematically as well as common-sensically.

#12

Risk of ruin is the probability that you will _eventually_ go broke.
If the risk of ruin is 90%, then you have a 10% probability of playing
indefinitly, never having your bankroll drop to zero. Players who
play indefinitely with a 1% edge tend to amass an ever growing
bankroll.

Some ruin quickly, others play for a very long time (millions or
even billions of hands) before being ruined, and the lucky 10%
never lose their entire stake. There is no "normalize" about it.
On average, the amount won by the survivors is the edge
multiplied by the total combined wagers of the entire group, but
ultimately the 10% of survivors each wager an ever-growing
amount until it exceeds the amount wagered by all the losers
combined.

···

On Friday 23 July 2004 10:52 am, blaw57 wrote:

...unless, it is mathematically impossible to ruin players with more
hands than would be swamped by the normalization cycle of the
survivors. My example was invented to fit the puzzle, but I actually
don't know if you could come up with a game that would ruin 1000
players with 300 mil hands while letting another 9000 normalize in
only one million hands each. Maybe those who ruin have to ruin in
1000 hands on average, while the survivors have to play 10 million
hands each to normalize. Then the extra yield the survivors get would
be miniscule.

I bet that's the answer. But it would still be an interesting
simulation exercise to do.

#13

the extra money goes to the casino
gambler's risk of ruin favors the house

Not true. If the game favors the house, then risk of ruin is 100%.
RoR can only drop below 100% if the game favors the player.
There are no exceptions to this rule, and all games that favor
the player will have RoR less than 100%.

if a lot of people are busting out due to insufficient bankrolls, the
house edge is increased

Not so. EV and RoR are independent concepts, so house edge is
not in any sense "due to" insufficient bankroll. If you take a million
players and start them with one unit each, they do not as a group
experience a different house edge than a group of 10 players who
each start with 100,000 units. But, the players who start with only one
unit have a much higher risk of ruin (by a factor of (1/R)^100,000).

···

On Friday 23 July 2004 04:47 pm, nightoftheiguana2000 wrote:

#14

it's a short term effect, as you say only for positive games
a classic example would be a small casino that has always offered
full pay deuces in quarters and nothing playable in dollars, they
decide to put in dollar deuces, they do quite well at first because
they bust out many of the quarter players who are insufficiently
bankrolled - for example a quarter player might quit when he looses
$4,000 and go back to quarters - if players quit when they are on a
losing streak the casino wins even if the game is positive and the
players are playing perfect strategy - but, this is a short term
effect, soon the only players left are the ones who have sufficient
bankroll to ride out the negative streaks and thus the casino begins
to lose money near the expected rate

another way to think of it:
suppose the rule is play dollar deuces until you lose $4,000. for
that the risk of ruin in 60%. even if the other 40% of surviving
players are up an average of $4,000, the casino is still ahead
(it's a short term effect, but risk of ruin is a short term effect,
for positive games)

> the extra money goes to the casino
> gambler's risk of ruin favors the house

Not true. If the game favors the house, then risk of ruin is 100%.
RoR can only drop below 100% if the game favors the player.
There are no exceptions to this rule, and all games that favor
the player will have RoR less than 100%.

i didn't say the game favors the house

> if a lot of people are busting out due to insufficient bankrolls,

the

> house edge is increased

Not so. EV and RoR are independent concepts, so house edge is
not in any sense "due to" insufficient bankroll. If you take a

million

players and start them with one unit each, they do not as a group
experience a different house edge than a group of 10 players who
each start with 100,000 units. But, the players who start with

only one

unit have a much higher risk of ruin (by a factor of (1/R)^100,000).

using http://www.lotspiech.com/GamblersRuin.html i can run a sim
example, quarter deuces, $50 stake, $2300 retire, 2000 hands, results:
prob win product
81% -$50 -$40.50
10% $67.50 $6.75
5.4% $302.50 $16.34
1.3% $537.50 $6.99
0.59% $772.50 $4.56
0.8% $1007.50 $8.06
0.39% $1242.50 $4.85
0.1% $1477.50 $1.48
0.02% $1712.50 $0.34
0.02% $1947.50 $0.39
0.01% $2182.50 $0.22

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:

On Friday 23 July 2004 04:47 pm, nightoftheiguana2000 wrote:

---
net=$9.48 (expected win = 2000 x $1.25 x .0076 = $19)

granted this is not an idea example, ideally you would run out
several royal cycles but such a sim would take a while, but hopefully
i've demonstrated that having a low cutoff decreases the average
return

#15

on second thought, i concede
the short term casino effect is due to less than optimum players
my sim example is wrong in that the *average* number of hands isn't
2000, but something more like 1,000 because those who bust out stop
playing
the money from bankroll busters goes to those who don't bust
those who don't bust are experiencing better than average luck in the
short term, in the long term they tend towards the average
short term is dominated by luck, long term is dominated by skill

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:

it's a short term effect, as you say only for positive games
a classic example would be a small casino that has always offered
full pay deuces in quarters and nothing playable in dollars, they
decide to put in dollar deuces, they do quite well at first because
they bust out many of the quarter players who are insufficiently
bankrolled - for example a quarter player might quit when he looses
$4,000 and go back to quarters - if players quit when they are on a
losing streak the casino wins even if the game is positive and the
players are playing perfect strategy - but, this is a short term
effect, soon the only players left are the ones who have sufficient
bankroll to ride out the negative streaks and thus the casino

begins

to lose money near the expected rate

another way to think of it:
suppose the rule is play dollar deuces until you lose $4,000. for
that the risk of ruin in 60%. even if the other 40% of surviving
players are up an average of $4,000, the casino is still ahead
(it's a short term effect, but risk of ruin is a short term effect,
for positive games)

> > the extra money goes to the casino
> > gambler's risk of ruin favors the house
>
> Not true. If the game favors the house, then risk of ruin is

100%.

> RoR can only drop below 100% if the game favors the player.
> There are no exceptions to this rule, and all games that favor
> the player will have RoR less than 100%.

i didn't say the game favors the house

> > if a lot of people are busting out due to insufficient

bankrolls,

the
> > house edge is increased
>
> Not so. EV and RoR are independent concepts, so house edge is
> not in any sense "due to" insufficient bankroll. If you take a
million
> players and start them with one unit each, they do not as a group
> experience a different house edge than a group of 10 players who
> each start with 100,000 units. But, the players who start with
only one
> unit have a much higher risk of ruin (by a factor of (1/R)

^100,000).

using http://www.lotspiech.com/GamblersRuin.html i can run a sim
example, quarter deuces, $50 stake, $2300 retire, 2000 hands,

results:

prob win product
81% -$50 -$40.50
10% $67.50 $6.75
5.4% $302.50 $16.34
1.3% $537.50 $6.99
0.59% $772.50 $4.56
0.8% $1007.50 $8.06
0.39% $1242.50 $4.85
0.1% $1477.50 $1.48
0.02% $1712.50 $0.34
0.02% $1947.50 $0.39
0.01% $2182.50 $0.22
---
net=$9.48 (expected win = 2000 x $1.25 x .0076 = $19)

granted this is not an idea example, ideally you would run out
several royal cycles but such a sim would take a while, but

hopefully

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:
> On Friday 23 July 2004 04:47 pm, nightoftheiguana2000 wrote:
i've demonstrated that having a low cutoff decreases the average
return

#16

There's an old thought problem that's somewhat related. Assume that
the chance of a woman giving birth to a boy is exactly the same as
giving birth to a girl. Imagine a culture in which every family has
children until the first boy, & then stops. (So one family might
have one son & then stop, another may have 10 daughters & then
finally have a son & stop.)

The question is, what will the ratio of men to women ultimately be
in this society? Even mathematically intelligent people sometimes
have trouble intuiting this, just as intelligent people sometimes
think that Risk of Ruin benefits the casino.

But of course if the birth ratio of boys to girls is 1:1, the ratio
of men to women in the society will be 1:1, & when any individual
couple decides to start or stop having kids won't affect this.
Likewise, the return of any casino game will be the same, regardless
of when any individual starts or stops playing.

Stuart (RandomStu)
http://home.comcast.net/~sresnick2/mypage.htm

#17

Hmmm... I have a couple of suggestions;-)

···

At 01:21 PM 7/23/2004, you wrote:

Assuming a casino can accomodate 10,000 vp of this game, where can
we find 10,000 hard-core anal-retentive vp players?

#18

Have to admit I've only been reading bits and pieces of this thread, but the above statement sounds counter intuitive. Why would a casino benefit from multiple people with insufficient bankrolls? Would not multiple people with insufficient bankrolls at some point be equal to one person with sufficient bankroll? IOW, wouldn't the casino loss or win be determined by the total number of games played (assuming proper strategy) without regard to who and which person's money played the hand?

Please use little words because I'm slow;-)

Chandler

···

At 05:47 PM 7/23/2004, you wrote:

the extra money goes to the casino
gambler's risk of ruin favors the house
if a lot of people are busting out due to insufficient bankrolls, the
house edge is increased

#19

Stuart wrote:

There's an old thought problem that's somewhat related. Assume that
the chance of a woman giving birth to a boy is exactly the same as
giving birth to a girl. Imagine a culture in which every family has
children until the first boy, & then stops. (So one family might
have one son & then stop, another may have 10 daughters & then
finally have a son & stop.)

The question is, what will the ratio of men to women ultimately be
in this society? Even mathematically intelligent people sometimes
have trouble intuiting this, just as intelligent people sometimes
think that Risk of Ruin benefits the casino.

But of course if the birth ratio of boys to girls is 1:1, the ratio
of men to women in the society will be 1:1, & when any individual
couple decides to start or stop having kids won't affect this.
Likewise, the return of any casino game will be the same, regardless
of when any individual starts or stops playing.

Heh, heh. Intuition throwing mathematically intelligent people?? True enough that that happens; it sounds like the Monty Hall dilemma which had math professors quibbling. For those of you who are unfamiliar with that, the question is: in Let's Make a Deal, after a person chooses one door out of three, and Monty Hall doesn't tell them yet whether they've won or not, but instead asks them if they would like to change their minds, are they mathematically better off to change their minds or stay with their original choice? _Intuitively_, since they don't know what is behind any door yet, most folks think that there is still a one in three chance, but mathematically speaking the odds are better if they always change their minds.

I must admit that I did need to think twice about your statement, but you are correct.

Cheers.

Bill Velek

#20

Chandler wrote:

···

At 05:47 PM 7/23/2004, you wrote:
>the extra money goes to the casino
>gambler's risk of ruin favors the house
>if a lot of people are busting out due to insufficient bankrolls, the
>house edge is increased

I don't know who said that, but I have to disagree. The house edge is determined by the odds of the game, regardless of whether or how many people are busting out due to insufficient bankrolls.

Bill Velek