vpFREE2 Forums

Interesting MultiStrike Statistics

#1

As I've posted in various groups, I've been of the impression that the
overall variance of MultiStrike is sizable. An analysis of the game
suggests my suspicions are off.

(And if you want to skip the detailed discussion of MS game variance
totally, you can skip to the "Summary" section, i.e. what this means
with respect to play, for my conclusions of these findings on the game.
Of course, you're going to miss the underlying explanation of those
implications.)

···

------

There are two components to the game that quite sizably add to the game
volatility.

The first of these is the conditional advancement from one Level to the
next (i.e., you must have a win or a Free Ride to advance). This means
that in most cases you'll lose a portion of your wager without even a
shot at a win on that respective Level. This is a risk that isn't posed
by almost all other vp games. Of course, this potential loss is offset
by the fact that when you do advance your payoffs on a win are
multiplied. Nonetheless the added risk is present.

The second is this progressive payoff. Achieving the expected return on
the game is dependent upon attaining the expected number of wins/Free
Rides on each level. If results are disproportionately skewed to lower
Level wins (with under performance on higher levels, i.e. losses) then
actual return will be subpar. Conversely, if wins are skewed to higher
Level wins actual returns will be superior. Again, this is a source of
volatility not presented in other vp games and increases variance,
likely substantially.

But in absence of hard data surfacing, I was ultimately driven to write
a program to analyze the 20,000+ possible outcomes of a play. (I'm a
little obsessive, as if that wasn't already apparent.) The results are
fascinating.

------

As it turns out, MultiStrike play has only a nominally greater variance
than single line play with the same total wager. (This means, for
example, a comparison of the volatility of $.25 MS with $1 single line,
where the total wager risked per play is $5).

In the case of 9/6 Jacks or Better, whereas the variance of single line
play is 19.5 (expressed in units of bets-squared), MultiStrike variance
is 21.0. (My methodology and result has been confirmed by LED Gaming.)

The explanation I've arrived at is that in the underlying play all lines
are played independently of each other. The independency of MS Levels
means that $.25 MS base variance is equivalent to single line $.25 play
(i.e. 1/4 that of $1 play). (This differs from standard multiplay where
the results of each individual line are related because they all are
generated from the same initial deal. This "covariance" serves to
increase volatility, say of $.25 multiplay, relative to single line $.25
play.) Consequently while the two factors that increase MS variance do
indeed act as substantial contributors (as noted above), that's opposite
a sizably reduced base variance and the combined result is this moderate
increase vs. single line play.

This finding provides me considerably greater comfort at taking an
extended stab at this game (despite my initial intimidating $.25 MS JB
loss of $800 in under an hour :).

------

There's another outcome of this analysis that I find even more
fascinating. (Mind you, these findings represent an extension of the
analysis that hasn't been fully vetted by LED yet).

When only 2 or 3 Levels are played, the variance is actually smaller
than a single line game played at the identical total wager (e.g.
comparing 2 Level $.25 play with single line $.50 play). The only
contribution to variance that differs from that of 4-Level play is in
the conditional advancement. With fewer Levels played, the contribution
of this factor is reduced. The other variance factors remain the same.

In the case of 3 Level play, the variance is 16.6. For 2 Level play,
the variance is 15.2. Mind you, the variance of 1 Level play is still
the 19.5 of single line play. The total wager isn't divided over
multiple Levels but, of course, the factors that increase variance
(conditional advancement and progressive payoffs) are absent.

When the variance associated with a total wager in one game is lower
than another game, in this case say 2-Level $.25 play vs. $.50 single
line play, and yet the ER of both games is largely identical, the
2-Level game becomes advantageous in presenting smaller risk. (And
2-Level play ER is actually slightly higher than single line JB, 99.60 -
mind you, this assumes an optimized strategy for 2-Level play is used
that's different from 4-Level. The 2-Level strategy can be determined
by adjusting the "2-4-6" strategy, using the method discussed in the
Dancer article))

------

Summary:

As noted in the above detailed discussion, MS variance when played with
4 Levels is only modestly higher than single line play (comparing a
comparable total wager per play, say $5 - $.25 MS vs. $1 single line).
This makes the game very approachable by the player of the single line
player (again, assuming the same total wager and the same game, e.g. JB,
is played).

When it comes to playing lesser Levels, the variance (and therefore
risk) of play is lower than single line play of the same total wager.
This suggests that one might prefer to play 2 $.25 MS levels vs. $.50
single line (and it's the case that the ER is slightly higher, with
strategy optimized for 2-Level MS play). Of course, MS speed per play
would be slower given the mechanics and lower total coin-in would
result.

Bottom line, I find MS much more fascinating and look to give it an
extended trial.

- Harry

Note: I seldom distribute a post to multiple vp groups. Each group has
it's own unique focus and generally it's not appropriate to "broadcast"
a post. In this case the MultiStrike topic has drawn considerable
interest in each group to which this is distributed and I've made an
exception. I apologize that some will receive multiple emails of this
post.

#2

--- In vpFREE@yahoogroups.com, "Harry D. Porter" <harry.porter@v...>
wrote:
(snip)
<this assumes an optimized strategy for 2-Level play is used
<that's different from 4-Level. The 2-Level strategy can be
<determined by adjusting the "2-4-6" strategy, using the method
<discussed in the Dancer article))

···

********************************************************************

THANK YOU so much for this analysis!
Can you clarify the point you made above? That is,
1)can one practice the 2-level optimal strategy by adding 2 to the
Winpoker pay table?
2)Since free ride frequencies are slightly different in level 1 and
level 3, would this have an effect?

Thanks again,
L.Wluiki

#3

Interesting post Harry! A couple of comments below...

When the variance associated with a total wager in one game is lower
than another game, in this case say 2-Level $.25 play vs. $.50 single
line play, and yet the ER of both games is largely identical, the
2-Level game becomes advantageous in presenting smaller risk.

Careful. Lower variance does not automatically imply smaller risk,
even when the ER is identical. This only holds for games that are
favorable to the player. For unfavorable games, lower variance
usually implies greater risk.

[snip]

When it comes to playing lesser Levels, the variance (and therefore
risk) of play is lower than single line play of the same total wager.

Same comment. Although it is tempting (and fairly common) to think
of risk and variance as if they were "similar", this isn't mathematically
correct. Whenever variance is used to compute risk, the computed
value for risk should be considered an approximiation.

···

On Friday 23 January 2004 02:34 am, Harry D. Porter wrote:

#4

Thanks for the feedback, Steve. I'd be interested in further comment on your
statement: "For unfavorable games, lower variance usually implies greater
risk." This isn't immediately clear to me. Let me paint my grasp of the
topic and I'd appreciate it if you would provide further feedback (and you'll
forgive me if I'm a little long-winded here in the interest of clarity).

I'll first own up to "risk" and "variance" as not being fully equivalent.
But from a practical standpoint variance isn't a bad starting point in
assessing the potential damage a game might do to your bankroll. Of course
the return of a game is another key factor. I'll also admit that some of the
statements I might make here are approximations at best, but I think they're
sufficiently on target for purposes of getting a handle on this subject.

Now, I'm interpreting variance as expressing, among other things, the
relative downside to actual return vs. expected. If two games have identical
returns, then the game with the higher variance has the larger downside (and,
in my book, "risk"). However, if the returns aren't equal things become
murkier.

If variances of the two games are identical but one game has an inferior
return, then while both games pose essentially similar probability of falling
short of EV by a given number of bets (or "downside" relative to ER).
However, clearly the game with the lesser return presents the greater
probability of a loss or, in other words, potential bankroll damage. So, if
a game possesses a greater variance and a greater return the potential
impairment to bankroll relative to the other game becomes uncertain without
further analysis of the numbers. But it's still the case that this game
poses the greater risk of falling short of ER.

But, in the case of 2 or 3 Level MS play, variance is lower and return is
nominally higher than the equivalent single line game. I see both of these
serving to reduce MS risk of bankroll impairment vs. single line play -
whether the games have positive or negative returns.

If your comments are correct, then there's obviously something missing from
this conceptual picture. In particular, I see these statements as holding
true irrespective of a positive or negative return. Care to steer me on a
stronger course, Steve?

- Harry

Steve Jacobs wrote:

···

Careful. Lower variance does not automatically imply smaller risk,
even when the ER is identical. This only holds for games that are
favorable to the player. For unfavorable games, lower variance
usually implies greater risk.

[snip]

Same comment. Although it is tempting (and fairly common) to think
of risk and variance as if they were "similar", this isn't mathematically
correct. Whenever variance is used to compute risk, the computed
value for risk should be considered an approximiation.

#5

lwluiki wrote:

Can you clarify the point you made above? That is,
1)can one practice the 2-level optimal strategy by adding 2 to the
Winpoker pay table?
2)Since free ride frequencies are slightly different in level 1 and
level 3, would this have an effect?

The answer to the first question is Yes.

In 4-Level play, the optimal strategy for Level 1 can be determined by
adding 6 to the paytable for each coin played. The equates to adding
2 for each potential Level of advancement. When playing just 2
Levels, you'd only add 2.

Note, though, that the strategy for Level 2 (when only 2 Levels are
played) involves no further advancement and therefore standard game
strategy is appropriate.

As far as Free Ride frequencies, they do have an impact on the
analysis. But from a strategy standpoint addition of +6/4/2 to pays
for 4-Level play and +2 for 2-Level play will both get you very close
to optimal ER. (Note: In both cases these don't achieve the
absolutely best ER's possible, but they both get you very close.)

- Harry

#6

Steve Jacobs wrote:

Careful. Lower variance does not automatically imply smaller risk,
even when the ER is identical. This only holds for games that are
favorable to the player. For unfavorable games, lower variance
usually implies greater risk.

Right. As a confirmed MS junkie, I can attest that playing all four lines of $.25 MS definitely requires a $1-sized bankroll. It is not unusual to win or lose $1000 in an hour or two.

John

#7

John Kellywrote:

Right. As a confirmed MS junkie, I can attest that playing all four
lines of $.25 MS definitely requires a $1-sized bankroll. It is not
unusual to win or lose $1000 in an hour or two.

And I hope it didn't come across that I was suggesting anything but
that. My statement is that the MS variance is only modestly higher
(but definitely higher) than single line play of the same total wager.

That means that the comparison of $.25 MS play is to $1 single line
play (a $5 total wager per play and assumes the same game and paytable
in each case). I'm suggesting that $.25 MS is approachable by the
player comfortable with $1 play, although they should expect a modest
degree of additional play volatility.

As I note, this is a MS aspect that wasn't intuitively obvious to me
at the outset.

- Harry

#8

Oh, OK. Sorry about that Harry, I didn't read carefully enough. So actually my experiences are in agreement with your analysis.

John

Harry Porter wrote:

···

John Kellywrote:

Right. As a confirmed MS junkie, I can attest that playing all four lines of $.25 MS definitely requires a $1-sized bankroll. It is not unusual to win or lose $1000 in an hour or two.

And I hope it didn't come across that I was suggesting anything but
that. My statement is that the MS variance is only modestly higher
(but definitely higher) than single line play of the same total wager.

That means that the comparison of $.25 MS play is to $1 single line
play (a $5 total wager per play and assumes the same game and paytable
in each case). I'm suggesting that $.25 MS is approachable by the
player comfortable with $1 play, although they should expect a modest
degree of additional play volatility.

As I note, this is a MS aspect that wasn't intuitively obvious to me
at the outset.

#9

Thanks for the feedback, Steve. I'd be interested in further comment on
your statement: "For unfavorable games, lower variance usually implies
greater risk." This isn't immediately clear to me. Let me paint my grasp
of the topic and I'd appreciate it if you would provide further feedback
(and you'll forgive me if I'm a little long-winded here in the interest of
clarity).

Part of the difficulty with talking about risk is that gambling authors
generally treat risk and variance as different expressions of the same
thing. It turns out that this is wrong, and I've argued "against the
current" on this previously with limited success, but now I believe I
can demonstrate this mathematically.

In the past I've argued that when playing a neg-EV game, variance
is the player's only friend, because reducing variance to zero would
force the player to simply pay the casino a fixed amount of EV each
time the game is played. Variance is what gives the player a chance
to come out ahead. Although this is true, this argument isn't perfect.
It isn't correct for a neg-EV player to seek variance just for the sake
of variance. But, in a game like Roulette, where all the plays have
the same EV, plays with high payoff (like 35:1 for a wager on a
single number) are lower risk than plays with lower payoff (like 1:1
for red/black/even/odd wagers).

I'll first own up to "risk" and "variance" as not being fully equivalent.
But from a practical standpoint variance isn't a bad starting point in
assessing the potential damage a game might do to your bankroll.

It isn't a bad starting point for pos-EV games. It is a terrible starting
point for neg-EV games, as the Roulette example illustrates.

Of course the return of a game is another key factor.

Seems that way, but one can actually ignore EV and variance completely
when studying risk. In addition, if your objective is risk based (example:
turn initial bankroll B units into goal G units, or go bust trying), then EV
and variance don't matter at all, only the risk matters. The probability
of success for this objective is given by p(success) = (1 - R^B) / (1 - R^G)
where R is the risk of ruin (or, for negative games, the inverse of the
casino's risk of ruin).

Roulette also illustrates this point -- the EVs of the plays are all the
same, but the risk varies from a high of R=1.11111 for even money
bets, to a low of R=1.00306536 for bets on a single number. If your goal
happens to be to turn $1 into $36, then betting it on a single number
on the roulette wheel gives you a 1/38 chance (double zero assumed).
If you try to turn $1 into $36 by betting on red/black/even/odd, your
probability of success drops to 1 in 390.5. Using the low variance
plays reduces your probability of reaching the goal by more than
a factor of 10 in this example.

I'll also admit that some of
the statements I might make here are approximations at best, but I think
they're sufficiently on target for purposes of getting a handle on this
subject.

Now, I'm interpreting variance as expressing, among other things, the
relative downside to actual return vs. expected. If two games have
identical returns, then the game with the higher variance has the larger
downside (and, in my book, "risk").

True for positive EV games, false for negative EV games.

However, if the returns aren't equal things become murkier.

Right. So, I would claim that if you really want to talk about risk, it
is better to just forget about EV and variance and compute risk
directly. This gets rid of the "murk" from using variance to approximate
risk, and it also gets rid of the "murkier" situations when returns aren't
equal.

If variances of the two games are identical but one game has an inferior
return, then while both games pose essentially similar probability of
falling short of EV by a given number of bets (or "downside" relative to
ER). However, clearly the game with the lesser return presents the greater
probability of a loss or, in other words, potential bankroll damage.

Although this may seem clear, it isn't necessarily true, due to the fact
that EV and variance cannot give a perfect indication of risk.

So,
if a game possesses a greater variance and a greater return the potential
impairment to bankroll relative to the other game becomes uncertain without
further analysis of the numbers. But it's still the case that this game
poses the greater risk of falling short of ER.

I think your reasoning on this falls well in line with conventional wisdom
on the subject. I also believe the conventional wisdom is flawed.

But, in the case of 2 or 3 Level MS play, variance is lower and return is
nominally higher than the equivalent single line game. I see both of these
serving to reduce MS risk of bankroll impairment vs. single line play -
whether the games have positive or negative returns.

Much of my point is that when it comes to risk, things that hold for positive
games do not hold, and often get turned around completely, for negative
games.

In positive games, minimizing risk means that you make the smallest bets
you can to maximize the number of units in your bankroll. The more units,
the higher your probability of reaching any pre-defined goal, before going
broke. Here you want to spread your wagers out as much as possible.

In negative games, minimizing risk means that you want to make as few
bets as possible, playing with "maximum boldness" to give you the best
shot at reaching your goal quickly and going home a winner.

When thinking about risk, it may be helpful to view the game from the
perspective of the casino. If the game is negative for you, then it is
positive for the casino. Any action you take that results in reducing
the risk for the casino will increase your own risk, and vice-versa.
If the casino has the edge, then they minimize risk if bets are small.
So, if you bet small, they will grind you down with minimum risk to
their bankroll. If variance increases the casino downside, then it
gives the player a better shot at coming out ahead. So, on a
Roulette table where the house has the edge, the high variance
bets are riskier for the casino and less risky for the player.

If your comments are correct, then there's obviously something missing from
this conceptual picture. In particular, I see these statements as holding
true irrespective of a positive or negative return. Care to steer me on a
stronger course, Steve?

I've been working on developing a new framework for thinking about
probability, based on a concept that I call "virtual payoffs." I'm going
to walk through this (too quickly, I'm afraid) and then try to show how
this allows us to view risk.

A while back I posted an article titled "Equivalent Games". That article
included the following table:

         <---- Coin Flip Subgame ----> <------ Overall Outcome ----->
Return % Tries Sub-cycle % Hit Cycle

···

On Friday 23 January 2004 08:37 am, Harry D. Porter wrote:
--------------------------------------------------------------------------
1000 4.762161445 1735.498092607 0.002743974 36443.495514000
   50 0.558915312 51.268241981 0.010901784 9172.810426000
   25 5.962619841 25.307949781 0.235602642 424.443457000
    8 9.224490842 8.028576209 1.148957250 87.035440178
    5 5.522918550 5.010195394 1.102335960 90.716445466
    4 4.523411750 4.006115160 1.129126740 88.564017180
    3 22.284452726 3.003056542 7.420590460 13.476016570
    2 25.809295114 2.001018502 12.898079200 7.753092414
    1 21.351734420 1.000000000 21.351734420 4.683460277
--------------------------------------------------------------------------

This game is 8/5 JoB, played with a max-EV strategy. The column labeled
"Sub-cycle" can be interpreted in a lot of interesting ways. One way is to
treat these numbers as "virtual payoffs" that would make the game fair, so
that it paid back exactly 100.000000%. In other words, a breakeven game.
Given any set of final probabilities for a game+strategy, as shown in the
last two columns, there are an infinite number of ways to assign virtual
payoffs to give a breakeven game. This particular set of virtual payoffs
gives a "risk view" of the strategy. The virtual payoffs for this view are
related to the actual payoffs in such a way that each case has the same
risk. For a risk view, the actual payoff AP and virtual payoff VP are related
by the equation:

VP = (1 - R^AP) / (1 - R).

For this strategy, the risk parameter is very close to R=1.0010185.
So, for example, an actual payoff of 1000 gives:

VP = (1 - 1.0010185^1000) / (1 - 1.0010185) = (1 - 2.7676) / (1 - 1.0010185)
  = 1735.49

This non-linear relationship between the actual and virtual payoffs
defines what "risk" truly means. Favorable games have R values
less than one, while unfavorable games have R values greater
than one.

Now for some magic. As I said above, these virtual payoffs would
give a return of 100% if they were applied to the actual game and
the (normal) max-EV strategy for 8/5 JoB was played. If we pretend
that this game has payoffs that match the virtual payoffs, and compute
a strategy that maximizes the "virtual EV", we will raise the virtual EV
above 100% and get a new playing strategy which has a lower risk!
In fact, this new strategy reduces the risk from 1.0010185 to 1.000962459.

This new "better risk" strategy has numbers that look like this:

pay % Try T Cycle prob Cycle
----------------------------------------------------------------------------
1000 5.15745432686 1679.98348585437 0.00306994347 32573.88974064100
   50 0.54971783713 51.19737578622 0.01073722683 9313.39176725000
   25 5.94586199097 25.29087965863 0.23509905829 425.35261829200
    8 9.21072403444 8.02700079635 1.14746768664 87.14842358029
    5 5.59918933828 5.00963386043 1.11768434466 89.47069937749
    4 4.46467662610 4.00577846182 1.11455904730 89.72158114211
    3 22.23687507540 3.00288830414 7.40516223823 13.50409306142
    2 25.78246346011 2.00096245927 12.88503107126 7.76094364437
    1 21.05303731071 1.00000000000 21.05303731071 4.74990845854
----------------------------------------------------------------------------
      100.00000000000 0.40546996070 44.97184792739 2.22361331830

Notice that the royal cycle has shrunk from 36443 to 32574. This
is to be expected, since we were playing as if the royal had a
payoff of 1735 -- effectively trying for royals more often than we
did in the max-EV strategy. The column with virtual payoffs is
now labeled "T Cycle" because these numbers come from a
program that I wrote to compute risk and cost "views" from
probability numbers (or cycle numbers) for a VP game. These
represent new, improved "virtual payoffs" that represent the
risk view of this lower-risk strategy.

Now let's do the same thing again -- use the "T Cycle" column as
virtual payoffs, and find a new strategy which maximizes "virtual EV".
This gives the following table:

pay % Try T Cycle prob Cycle
----------------------------------------------------------------------------
1000 5.15280360155 1679.95880300094 0.00306722022 32602.81068143500
   50 0.55010832794 51.19734366634 0.01074486074 9306.77487799900
   25 5.94577440676 25.29087191802 0.23509566717 425.35875376100
    8 9.21046326880 8.02700008174 1.14743530273 87.15088315836
    5 5.60010808033 5.00963360570 1.11786779655 89.45601645259
    4 4.47000911963 4.00577830908 1.11589029016 89.61454444205
    3 22.23567891425 3.00288822781 7.40476408955 13.50481916650
    2 25.78052093421 2.00096243385 12.88406043919 7.76152832191
    1 21.05453334653 1.00000000000 21.05453334653 4.74957095245
----------------------------------------------------------------------------
      100.00000000000 0.40546994776 44.97345901283 2.22353366174

The risk parameter for this strategy has a value of R=1.000962433846, which
is only a tiny improvement compared to the previous iteration. Now if you use
these virtual payoffs and try to iterate again, you get the same strategy
back! This means that we can't take any more "steps" to reduce risk. In
addition, the "virtual EV" for these payoffs is 100.00000% and this
is also maximum EV for these (virtual) payoffs. The strategy that produced
these numbers is the min-risk strategy for this game.

The royal cycle for this min-risk strategy is 32602.8, compared to a royal
cycle of 36443.5 for the max-EV strategy. The min-risk strategy is found
by pretending that the payoffs are all "stretched" according to the formula
which represents an equal risk parameter for all payoffs. If the actual
payoffs were set to the values of the virtual payoffs, we would get a
breakeven game. What this boils down to is that minimizing risk is just
like pretending that the payoffs were "risk adjusted" and the strategy
optimized for the virtual payoffs, resulting in a breakeven game. For
neg-EV games, this process results in virtual payoffs that are larger
than the real payoffs, causing the min-risk strategy to "try harder" to
get big payoffs. In contrast, pos-EV games result in virtual payoffs
that are smaller than the real payoffs, causing the min-risk strategy
to "avoid" big payoffs. So, to reduce risk in pos-EV games, we play
a less agressive strategy as if the royal was worth significantly less
than its true value. In neg-EV games, minimizing risk requires us to
do just the opposite -- pretend the royal payoff is larger than it really
is, and play more agressively.

The concept of virtual payoffs can be applied to a wide variety of playing
objectives. The min_cost_royal strategy keeps most payoffs "real" while
using a virtual payoff for the royal. If we play the max-EV strategy for
8/5 JoB, the virtual payoff for a royal would need to be 1798.6 units
to give a breakeven game. This is the cost of a royal for this strategy.
This min_cost_royal strategy reduces cost(royal) to 1733.142 units, and
this is the virtual payoff for the royal when using that strategy. This is
not very far removed from the 1679.95 virtual payoff given by the
min-risk strategy. In fact, the risk for min_cost_royal strategy is
1.00096246595, and the "risk view" of the min_cost_royal strategy
gives a virtual payoff of 1679.9899 for the royal flush, very close to
the value used to minimize overall risk.

I've gone through this rather quickly, so I've probably totally lost
anyone trying to read this. I'd appreciate any feedback in case
anyone is still with me. I need to learn how to describe this in
a way that is easy to understand, because I believe this is an
extremely powerful way to approach a broad spectrum of
problems involving optimal strategies.

Once one becomes used to thinking in terms of virtual payoffs, it
becomes obvious that minimizing risk means "play an overly
cautious strategy if you have the advantage, play agressively
if the house has the advantage."

#10

Hi!

I really did try to follow this through, but, alas with little, or
no, success. I do think, however, that I sense what you are saying,
with one needing to be "prudent" for positive games and "agressive"
for negative games.

Now accepting this, and letting the more mathematically inclined to
joust with you as far as truth or falsity goes, how does the
strategy vary, from what we get when using "vpstrat", when playing
FPDW (a positive game), and NSUD (a negative game)?

Thanks!

.....bl

#11

Steve Jacobs wrote:

Part of the difficulty with talking about risk is that gambling
authors generally treat risk and variance as different expressions
of the same thing. It turns out that this is wrong ...

Steve, I appreciate the detailed discussion. It merits further
discussion on a number of points, although perhaps under a subject
heading more general and pertinent than "MultiStrike". I hope to take
some time to digest it more fully and give you some feedback.

Concerning my MultiStrike post, it's clear to me that what you
stringently reacted to was my inclusion of the term "risk".
Admittedly, my use was cavalier and applied in a way that ran afoul of
its more specific denotation.

It would have best if I'd steered clear of the word since it wasn't
necessary to make my point -- namely the relative attractiveness of MS
now that I find it to have variance in the ballpark of the underlying
game (comparing a single line game at 4x the base denomination of the
MS play, a comparable total wager per play in each case).

I've assumed from a very basic practical perspective that you don't
have any difficulty with the discussion along this line in my post,
without getting into a discussion of "risk", per se.

- H.

#12

Steve Jacobs wrote:

<snip>

In positive games, minimizing risk means that you make the smallest bets
you can to maximize the number of units in your bankroll. The more units,
the higher your probability of reaching any pre-defined goal, before going
broke. Here you want to spread your wagers out as much as possible.

In negative games, minimizing risk means that you want to make as few
bets as possible, playing with "maximum boldness" to give you the best
shot at reaching your goal quickly and going home a winner.

I can think of very unusual situations in which this would be
applicable, such as if one needed $100 for a bus ticket that one
valued far more than $100, if one had $99, and if one thought the best
way to get another $1 was to play roulette. But I can't think of an
ordinary situation in which playing a game which had negative EV would
be better than not playing. Correct me with facts if I'm wrong, since
I'm only speaking from intuition, but in the context of playing almost
exclusively positive EV games, playing, say, a few hands of a negative
EV game, for, say, social purposes, it's still better to minimize
variance. I don't care to do the math, but which of the following
would be higher?

1) The chance of winning 100,000 units before losing 100,000 units by
playing full pay deuces wild and, once, playing 10 hands of bad pay
deuces wild.

2) The chance of winning 100,000 units before losing 100,000 units by
playing full pay deuces wild and, once, paying the casino what the EV
would be of playing 10 hands of bad pay deuces wild.

I believe the chance of success would be slightly higher in number 2.

A card counter who makes "cover" bets in the context of playing at an
overall advantage is in a similar situation. Wouldn't s/he rather
just pay the casino the EV of those bets?

<snip more detailed analysis that I'll read later>

#13

Steve Jacobs wrote:
> Part of the difficulty with talking about risk is that gambling
> authors generally treat risk and variance as different expressions
> of the same thing. It turns out that this is wrong ...

Steve, I appreciate the detailed discussion. It merits further
discussion on a number of points, although perhaps under a subject
heading more general and pertinent than "MultiStrike". I hope to take
some time to digest it more fully and give you some feedback.

Concerning my MultiStrike post, it's clear to me that what you
stringently reacted to was my inclusion of the term "risk".
Admittedly, my use was cavalier and applied in a way that ran afoul of
its more specific denotation.

I don't think that I was particularly keyed on the term "risk" itself, more
on the (mistaken) idea that it was always good to reduce variance, even
if the game is negative EV.

I suppose it is possible that there are situations in negative EV games
where reducing variance helps in some way. But, hopefully the risk
examples make it clear that reducing variance isn't always in the
player's best interest.

It would have best if I'd steered clear of the word since it wasn't
necessary to make my point -- namely the relative attractiveness of MS
now that I find it to have variance in the ballpark of the underlying
game (comparing a single line game at 4x the base denomination of the
MS play, a comparable total wager per play in each case).

I've assumed from a very basic practical perspective that you don't
have any difficulty with the discussion along this line in my post,
without getting into a discussion of "risk", per se.

I don't think eliminating risk from the discussion would get rid of my
objection. In fact, off the top of my head, I can't think of any
statistical parameter for which it would be correct to say "if two
games have the same EV, it is always better to choose the game
with [higher/lower] X." It certainly isn't true for X=variance or
(equivalently) X=standard_deviation, as shown (hopefully) by
my examples with risk.

···

On Sunday 25 January 2004 03:31 pm, Harry Porter wrote:

#14

Steve Jacobs wrote:

<snip>

>In positive games, minimizing risk means that you make the smallest bets
>you can to maximize the number of units in your bankroll. The more units,
>the higher your probability of reaching any pre-defined goal, before going
>broke. Here you want to spread your wagers out as much as possible.
>
>In negative games, minimizing risk means that you want to make as few
>bets as possible, playing with "maximum boldness" to give you the best
>shot at reaching your goal quickly and going home a winner.

I can think of very unusual situations in which this would be
applicable, such as if one needed $100 for a bus ticket that one
valued far more than $100, if one had $99, and if one thought the best
way to get another $1 was to play roulette. But I can't think of an
ordinary situation in which playing a game which had negative EV would
be better than not playing.

I'm not claiming that anyone should play negative games unless compelled
to do so. But, if one chooses to play negative games (for whatever reason)
and also wishes to minimize risk, then bold play gives the highest probability
of reaching a fixed target bankroll, regardless of how close it is to the
starting bankroll.

Correct me with facts if I'm wrong, since
I'm only speaking from intuition, but in the context of playing almost
exclusively positive EV games, playing, say, a few hands of a negative
EV game, for, say, social purposes, it's still better to minimize
variance. I don't care to do the math, but which of the following
would be higher?

1) The chance of winning 100,000 units before losing 100,000 units by
playing full pay deuces wild and, once, playing 10 hands of bad pay
deuces wild.

2) The chance of winning 100,000 units before losing 100,000 units by
playing full pay deuces wild and, once, paying the casino what the EV
would be of playing 10 hands of bad pay deuces wild.

I believe the chance of success would be slightly higher in number 2.

I'm not sure either way, but I'll give this more thought and try to reach
a solid conclusion. I think the best thing, if it were allowed, would be
for the player to play a single hand with a 10 unit wager to cover the
negative bets, then play the favorable bets one unit at a time.

A card counter who makes "cover" bets in the context of playing at an
overall advantage is in a similar situation. Wouldn't s/he rather
just pay the casino the EV of those bets?

I don't think so, but I'm not positive. I'll give it more thought.

···

On Sunday 25 January 2004 03:54 pm, Tom Robertson wrote:

#15

Steve Jacobs wrote:

<snip>

Correct me with facts if I'm wrong, since
I'm only speaking from intuition, but in the context of playing almost
exclusively positive EV games, playing, say, a few hands of a negative
EV game, for, say, social purposes, it's still better to minimize
variance. I don't care to do the math, but which of the following
would be higher?

1) The chance of winning 100,000 units before losing 100,000 units by
playing full pay deuces wild and, once, playing 10 hands of bad pay
deuces wild.

2) The chance of winning 100,000 units before losing 100,000 units by
playing full pay deuces wild and, once, paying the casino what the EV
would be of playing 10 hands of bad pay deuces wild.

I believe the chance of success would be slightly higher in number 2.

I'm not sure either way, but I'll give this more thought and try to reach
a solid conclusion. I think the best thing, if it were allowed, would be
for the player to play a single hand with a 10 unit wager to cover the
negative bets, then play the favorable bets one unit at a time.

It could be simplified. Which of the following is greater?

1) The chance of winning 100,000 units before losing 100,000 units by
playing full pay deuces wild, starting out 1 unit behind.

2) The weighted chance of winning 100,000 units before losing 100,000
units by playing full pay deuces wild, starting out behind by 101
units 50% of the time and starting out 99 units ahead 50% of the time.

<snip>

···

On Sunday 25 January 2004 03:54 pm, Tom Robertson wrote:

#16

I'm not claiming that anyone should play negative games unless

compelled

to do so. But, if one chooses to play negative games (for whatever

reason)

and also wishes to minimize risk, then bold play gives the highest

probability

of reaching a fixed target bankroll, regardless of how close it is

to the

starting bankroll.

I think you're defining "risk" as risk of not attaining bankroll
growth. Risk can be defined in many ways, but the more common concept
is risk of ruin, i.e. the risk of losing a bankroll. Higher variance
always increases the risk of ruin. (risk of ruin is defined for
negative games if you limit the plays, for infinite plays it
approaches 100%)

···

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

#17

I think you're defining "risk" as risk of not attaining bankroll
growth. Risk can be defined in many ways, but the more common concept
is risk of ruin, i.e. the risk of losing a bankroll.

The risk that I refer to is mathematically equivalent to the concept
of "risk of ruin" and is based on the exact equation that some
call the "Sorokin Equation" (I prefer not to call it that, since it
was discovered by ancient mathematicians who died long before
Soroking was ever born). I prefer to not call it "risk or ruin" for
several reasons, mostly because the same parameter that can be
used to compute the probability of ruin can also be used to compute
the probability of turning a bankroll of B units into a bankroll
of G units, which I find more useful since none of us live long
enough to play endlessly while our bankrolls grow without bound.

I also choose to avoid the phrase "risk of ruin" because a lot
of gambling literature virtually defines RoR in terms of EV and
variance, using a formula which is actually just an approximation,
while the so-called Sorokin Equation is in fact exact.

Higher variance
always increases the risk of ruin. (risk of ruin is defined for
negative games if you limit the plays, for infinite plays it
approaches 100%)

Right, RoR isn't very meaningful for unfavorable games, since
it is always 100%, but risk of failing to turn B units into G units
(by going bust) is meaningful whether the game is favorable or
unfavorable.

In the specific case of unfavorable VP games, the strategy
which minimizes the probability of going broke before turning
B units into G units does in fact have higher variance than
the max-EV strategy for the same game. So, changing from
the max-EV strategy to the min-risk strategy causes an
increase in variance. This implies that higher variance
doesn't always increase risk of ruin.

In addition, in favorable VP games, if you deviate from the
min-risk strategy in any way you will increase your risk
of ruin. Although I don't have a specific example with
numbers, I believe it would be possible to deviate from
the min-risk strategy in such a way that variance would
decrease. For example, if you deviate from the min-risk
strategy by playing slightly less aggressively, I believe
your variance will decrease. In fact, if you start with the
min-risk strategy and play less aggressively while trying to
decrease risk, your EV and variance will both decrease
while moving toward the min-risk strategy. Once the
min-risk strategy is reached, continuing to decrease the
level of aggression will continue to decrease both EV
and variance, but I believe RoR will now increase. So,
decreasing variance doesn't always imply a decrease
in RoR.

The strategy that minimizes variance will also make
EV very small. In Blackjack, it is possible to reduce
variance completely to zero, but doing so requires
the player to intentionally lose every hand. A VP
player can't reduce variance to zero, because
some hands are impossible to play in a way that
guarantees a loss (example: any suited hand will
always have some non-zero probability for a flush,
no matter how you play the hand).

···

On Monday 26 January 2004 12:45 am, pal16r8 wrote:

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

#18

Actually, this is exactly my plan. I don't have to live very long, I
play about 1,000 hours per year at about 1,000 hands per hour, that's
a million hands per year, that's enough to have a high chance of
seeing average returns. It's not that complicated, I only play
positive games and I only play within my bankroll. (In case anyone
wants a very rough rule of thumb, you need about $5,000 to play
quarters, $20,000 to play dollars, $100,000 to play five dollars ...)

On the contrary, your plan assumes someone attains a bankroll, and
then hits the casino, and if they happen to say double their bankroll
by sheer luck, they immediately stop gambling and never gamble again
in their life. Most likely that person will be right back at the
casino the next day, seeing if they can repeat their previous
success. In other words, they are not playing to a positive goal,
they are playing until their bankroll runs out.

···

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

I prefer to not call it "risk or ruin" for
several reasons, mostly because the same parameter that can be
used to compute the probability of ruin can also be used to compute
the probability of turning a bankroll of B units into a bankroll
of G units, which I find more useful since none of us live long
enough to play endlessly while our bankrolls grow without bound.

#19

Steve,

Let me attempt to paraphrase part of your claim. You seem to be
saying that one should consider playing a non-max ER strategy to
reduce risk. If the game has a negative ER, this reduced risk
strategy has increased variance compared to the max ER strategy.
Since we can't define risk as ROR in a negative ER game, you have
proposed using the probability of not achieving a bankroll goal
before busting as risk. Assuming I got the jest of your argument,
perhaps the following makes some sense.

I think many people are having a hard time accepting your alternative
definition of risk. I would think that often, a person's goal is to
play for a fixed number of hands with minimum damage to their trip
stake. In this case I believe that variance is not your friend.

Seems to me that there is an inherent problem in analyzing the "best"
way to play a negative game, when the obvious solution is to not play
at all. The increased variance seems to help only if it can motivate
you to stop participating in a self-destructive behavior more
quickly. I agree with Tom's argument that if you are forced to make
(a fixed number of) negative cover bets, you should prefer lower
variance, not higher variance.

AJ

#20

Steve,

Let me attempt to paraphrase part of your claim. You seem to be
saying that one should consider playing a non-max ER strategy to
reduce risk. If the game has a negative ER, this reduced risk
strategy has increased variance compared to the max ER strategy.

Correct.

Since we can't define risk as ROR in a negative ER game, you have
proposed using the probability of not achieving a bankroll goal
before busting as risk. Assuming I got the jest of your argument,
perhaps the following makes some sense.

I don't have an alternate _definition_ of risk, I'm simply pointing out
that the mathematical framework for ROR permits one to draw
meaningful conclusions not only for favorable games, where ruin
can be (sometimes) avoided, but also for unfavorable games where
ruin is inevitable if the game continues without end.

I think many people are having a hard time accepting your alternative
definition of risk.

No doubt. People have a hard tim accepting much of what I say about
any kind of alternative strategy.

I would think that often, a person's goal is to
play for a fixed number of hands with minimum damage to their trip
stake. In this case I believe that variance is not your friend.

If you want to have minimum damage from an unfavorable game, then
the best thing to do is not play. If you want to play a fixed number of
hands, then playing max-EV minimizes the average number of dollars
lost per hand played. Minimum risk is a more appropriate strategy when
the player is trying to reach a particular bankroll level before quitting,
without regard to how many hands are played.

This raises a potentially interesting point. I believe it is probably true
that max-EV is best for cases where "time" (or the number of hands played)
is used to measure progress. Risk based strategies tend to work best
when time isn't relavent, and the player measures progress solely on
the basis of current bankroll. Cost based strategies are different than
either of these, and measure progress according to dollars lost in
exchange for dollars won. These are all fundamentally different ways
of measuring the outcome of some series of wagers, and the inherently
lead to different optimal strategies. This is tightly related to the fact
that different objectives call for different methods to achieve optimal
results.

I suspect that people will always be uncomfortable with the very idea
that different situations call for different strategies, and different ways
of playing "perfectly." Human nature seems to want things to work
in such a way that there is only one "best" way to play, but that isn't
what mathematics reveals.

Seems to me that there is an inherent problem in analyzing the "best"
way to play a negative game, when the obvious solution is to not play
at all.

There is no inherent problem. The fact that one might be compelled to
play a negative game, for whatever reason, does not prevent the
player from seeking to make the most of a bad situation by minimizing
losses or minimizing the probability of an undesireable outcome. For
the most part, the same math still applies. Things just work out a bit
differently than they do for favorable games, and the "usual" rules
don't always apply.

The increased variance seems to help only if it can motivate
you to stop participating in a self-destructive behavior more
quickly.

It isn't about motivation, it is about squeezing the most you can
from the situation. If you need desperately to turn $1 into $36
and the only game available is Roulette, then the mathematical
reality is that placing a single wager on one number gives you
the maximum probability of reaching that specific goal. Any
other play (barring "don't play") will lead to a lower probability
of reaching the goal, and therefore a lower overall expected
value for the player's final bankroll. The fact that higher
variance turns out to be "better" is a side effect.

Please think about that for a moment. If your specific objective
is to reach some target bankroll or go bust trying, then the
min-risk approach not only maximizes your probability of
success, but also maximizes the average value of your outcome
based on the specified constraint of "hit the goal or go bust".
When situations call for a min-risk strategy, that strategy
maximizes the expected value of the _permitted_ outcomes.
In effect, minimizing risk maximize _constrained_ ER.

I agree with Tom's argument that if you are forced to make
(a fixed number of) negative cover bets, you should prefer lower
variance, not higher variance.

Tom may be correct, I'm still thinking that through. However,
if that particular situation calls for lower variance in order to
minimize overall risk, I have absolutely no doubt that other
situations exist for which the opposite is true.

···

On Monday 26 January 2004 03:51 pm, AJ wrote: