Krigman and ACDOC - the truth

ACDOC has been implying that gambling write Al Krigman, whose articles
are often posted here by Mason, has the same view of "Bankroll
Volatility," or "Stochastic Volatility," as he now calls it, as he
does. Nothing could be further from the truth, of course.

I went back and dug up ACDOC's post of last November 26, when he
quoted Krigman in support of his assertion that DP w/odds had higher
"Bankroll Volatility" than pass w/odds.

First, here is what he quoted from Krigman's article, which, BTW is
called "Why What You Got Ain't What You Expected: Edge, Volatility and
Risk of Ruin in Gambling." (This is still shown, BTW, as a "work in
progress" on the bjmath.com web site.)

[QUOTE]
"Expected value, however, is meaningful only as a long-term
phenomenon. Not only does it usually represent a small fraction of
each transaction, which requires many decisions to accumulates into
money worth mentioning, but it gets buried in the nominal amounts won
or lost on individual decisions. That is, players betting $10 on the
five at craps are subject to an edge of 4%. The house earns a
theoretical $0.40 on the action. But the players either win $14 or
loses $10, and rarely think about the fact that in a "fair" game, the
house had an edge because the bet could win four ways and lose six, so
the $10 at risk should actually have paid $15."

"Volatility, on the other hand, characterizes the up- and down-swings
likely to be encountered during the course of the action. It also
affords a means of quantifying the probability of deviating from the
expected value at any juncture. Statisticians traditionally measure
volatility by the variance -- the second moment of the distribution --
or by its more useful square root, the standard deviation."

*************************************************************************NOTE
OMMISSION BY ACDOC

***(IMPORTANT)
"In the course of the action, the cumulative expectation increases
linearly with the number of decisions while the overall standard
deviation rises with the square root of this quantity. This is the
effect that accounts for volatility dominating performance over the
short haul, while edge becomes overriding in the long term."

"Players understand that they will experience upswings and downswings,
but think of the phenomenon as a manifestation of luck (or mystical
forces) rather than a characteristic of the game that can be described
analytically. However, unlike casinos, players are seriously impacted
by the effects of volatility during normal sessions -- much more so
than by edge, even on the most outrageous "sucker bets." Problems
which accrue to players because of failure to account for volatility
generally involve depleting bankrolls during what should be considered
the normal downswings of a game, unrealistic aspirations for earnings,
and misconceptions about the effects of betting progressions."
[END QUOTE]

*************************
Here is the text that ACDOC cleverly left out where I noted it, above:

[QUOTE directly from Krigman article]
In the course of the action, the cumulative expectation increases
linearly with the number of decisions while the overall standard
deviation rises with the square root of this quantity. This is the
effect that accounts for volatility dominating performance over the
short haul, while edge becomes overriding in the long term.
[END QUOTE]

The reason he left this out is that it states quite clearly that it is
the STANDARD DEVIATION that increases with the number of decisions. In
the calculations in ACDOC's post, he (or rather his idol, DiMauro)
applies the square root of the number of decisions to a maximum amount
that could be lost or won (a range). His calculations do not include a
standard deviation at all.

I continued to read Krigman's article, and here is an example he uses,
from blackjack:

[QUOTE Krigman]
Drawing inferences from expected values and variances of distributions
is standard practice in broad areas of commerce and politics. The
subject is well-covered in the literature of many fields. A simple
example will show the implications for gambling and highlight the
dichotomy between short- and long-term phenomena.

At a six-deck blackjack game with reasonable rules, the edge and
standard deviation for each hand following basic strategy are about
-0.5% and 1.12, respectively.

A) After 100 hands at $10 each, the player's expectation is to lose
0.5% of the $1,000 handle or $5. The standard deviation for the
session is the standard deviation per unit bet, times the size of the
bet, times the square root of 100 hands or 1.12 x $10 x 10 = $112. The
so-called "empirical rule" of statistics says that for wide ranges of
situations like this, approximately 68% of all players will be within
one standard deviation of the mean. In this case, after 100 hands, the
range is therefore from a loss of $117 to a win of $107. The $5
represented by edge is buried in the $112 standard deviation.

B) After 10,000 hands at $10, the player's expectation is to lose 0.5%
of the $100,000 handle or $500. The standard deviation is $1,120.
About 68% of all players can now expect to be from $1,620 behind to
$620 ahead. The $500 biases the range of anticipated results, but
volatility is still dominant.

C) After 1,000,000 hands, the expected loss would be $50,000 and the
standard deviation is $11,200. Roughly 68% of all players can expect
to be from $61,200 to $38,800 in the hole. Edge has surpassed
volatility in effect.
[END QUOTE]

Note that he multiplies the standard deviation of one bet (1.12 * the
bet amount) times the square root of the number of decisions. Note
also, that the information he provides is identical to the information
I provide, albeit about craps, not blackjack. The +1 SD and -1 SD
figures are symmetrical about the ev, just as mine are.

Later in the article, Krigman discusses "Risk of Ruin."
[QUOTE]
Risk of ruin is the probability of depleting a bankroll and being
unable to continue. This quantity, when qualified with bankroll, bet
size, and either a profit target or projected duration of play,
characterizes a gamble in a manner that incorporates the effects of
expectation and volatility while being directly meaningful to players
and casinos alike. The concept has two interpretations, both of which
are relevant to short-term gambling performance."
[END QUOTE]

Another quote about "Risk of Ruin:"
[QUOTE]
Survival-based risk of ruin is a function of bankroll, bet size,
number of decisions, and the probabilities and payoffs associated with
various possible results. The relationship involves straightforward
statistical principles, although the traditional calculations have
recently been shown to be conceptually erroneous. The fallacy lay in
considering only the points at which player could expect to be at the
end of their sessions, rather than the downswings which might put them
out of business before the specified number of decisions could be
completed and from which they might recover if they had additional
capital.
[END QUOTE]

The fallacy he is referring to would be if you applied the theoretical
standard deviation, which has nothing to do with bankroll, and then
plug in a starting bankroll. It's clear that a standard deviation for
a number of decisions assumes all the decisions are reached; if you
run out of money before that, you have to stop playing. My risk of
ruin program actually simulates the sessions and counts up the busts,
times reaching the win goal, etc. Krigman's article is about directly
calculating the "Risk of Ruin." He uses simulations to test his
results.

The formulas are quite complex, but the edge (per decision) and the
variance (per decision) are both factors, as you would expect.

After showing some of the derivation, he gives some examples, one of
which is relevant to craps and, particularly, DiMaurCOC's assertion
that lay bets are much more negatively volatile than buy bets.
[QUOTE]
A "buy" bet on the four at craps is a wager with three ways to win and
six to lose, and a payoff of 2-to-1, except that the house collects a
vigorish equal to 5% of the wager regardless of outcome. Therefore, to
buy the four for $20, players must drop $21 on the layout. If they
win, the house returns the $20 nominal bet along with a $40 payoff.
For all practical purposes, therefore, this is a wager of $21 to win
$39, with a probability of success equal to 0.333%. At this level, on
a per unit basis, the edge on the bet is -0.04761%, the standard
deviation is 1.346, and the skewness is +0.707.

A "no four" bet at craps, also known as a lay bet against the four, is
a wager with six ways to win and three to lose, and a payoff of
1-to-2, except that the house collects a vigorish equal to 5% of the
expected payoff regardless of outcome. Therefore, to bet a nominal $40
against the four, players must drop $41 on the layout. If they win,
the house returns the $40 along with a $20 payoff. In reality,
therefore, this is a wager of $41 to win $19 with a probability of
success equal to 0.667%. At this level, on a per unit basis, the edge
on this bet is -0.02430, the standard deviation is 0.6898, and the
skewness is -0.707.

These bets are essentially the inverses of one another, and exhibit
small skewnesses of equal magnitude but opposite direction. Note that
the differences in edge and standard deviation results because of the
inverse probability-payout relationship between the bets. The
theoretical casino win is the vigorish, equal to $1 for nominal $20
buy or $40 lay bets.

Earnings-based risk of ruin, calculated for these bets using Equations
1, 4, and 5 at various levels of play, all yielded essentially the
same results. Simulations were also performed for these wagers, to
verify the analytical predictions. The criteria for the simulations
differed from those in the analysis in one dimension that could be
expected to impact results. Namely, the end points in the simulation
were levels when players' fortunes equalled or were less than zero, or
equalled or exceeded the established target. The end points in the
analysis are assumed to be exactly zero or the target earnings.

The following tables, A and B, compare risk of ruin determined
analytically and by simulation for the two propositions, with bets and
win goals of various amounts, and bankrolls of $1,000.

Table A

Risk of Ruin for Buy Bets
on the Four at Craps
bet wingoal risk of ruin risk of ruin
simulated analytical
21 100 26% 23%
63 100 17% 13%
105 100 20% 12%
21 500 73% 73%
63 500 48% 48%
105 500 46% 42%
21 1000 93% 92%
63 1000 71% 70%
105 1000 63% 62%

Table B

Risk of Ruin for Buy Bets
against the Four at Craps
bet wingoal risk of ruin risk of ruin
simulated analytical
41 100 26% 23%
123 100 15% 13%
205 100 18% 12%
41 500 74% 73%
123 500 48% 47%
205 500 43% 42%
41 1000 93% 92%
123 1000 70% 69%
205 1000 62% 62%

The tables show the greatest discrepancies between the analytically
predicted and simulated results when bets are large relative to the
win goal. The effect is ascribed to the win levels at which play is
terminated overshooting the nominal value by an amount averaging half
the payout, such that a greater risk of ruin would be predicted were
the actual quitting points used in the equations. The tables also
suggest that the differences in skewness do not affect risk of ruin
calculations. Results are essentially the same for the analytically
derived values on the buy and lay bets, as would be expected because
skewness is ignored. However, the results of the simulations are also
similar for the alternate bets.
[END QUOTE]

Note, particularly the last three sentences.

THE DIFFERENCES IN SKEWNESS DO NOT AFFECT RISK OF RUIN CALCULATIONS.

I think this should set everyone's mind at rest about whether Krigman
would agree with DiMaurCOC.

BTW, guess what author Krigman quotes in the article? You guessed it!
Epstein.

So, I am all through with ACDOC. Enjoy the rest of your life in
ignorance!
Cheers,
Alan Shank






.

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