I study the use of the D'Alembert gambling system for whitewashing money at a Casino.

1. “R”goestothecasino
The m
a
gic of the d’Alembert (*)
Rich
a
rd Gill, 3 M
a
y 2021
(*) subtitle borrowed from Crane and Shafer (2020)

2. Jean-Baptiste le Rond d’Alembert
(1717 – 1783).
French mathematician, mechanician,
physicist, philosopher, and music
theorist.
Together with Denis Diderot, a co-
editor of the Encyclopédie.
D'Alembert's formula for obtaining
solutions to the wave equation is
named after him.
The wave equation is sometimes
referred to as d'Alembert's equation.
The Fundamental theorem of
algebra is named after d'Alembert in
French.
d’Alembert’s principle was due to
Lagrange.
d’Alembert wrote the articles on
probability in the Encyclopédie. And is
famous for giving the wrong answer to
a simple problem … (!?)

3. Diderot and d’Alembert (1751)
Encyclopédie
NB: this is after Christiaan Huygens, Jacob Bernoulli,
Abraham de Moivre; before Laplace.
Denis Diderot

4. Encyclopédie:Croixoupile
Annexe I
Extrait commenté de l’Article “Croix ou Pile”
écrit pour l’Encyclopédie par D’ALEMBERT.
CROIX OU PILE, (analyse des hasards). Ce jeu, qui est très connu, & qui n’a
pas besoin de définition, nous fournira les réflexions suivantes. On demande
combien il y a à parier qu’on amènera croix en jouant deux coups consécutifs.
La réponse qu’on trouvera dans les auteurs, & suivant les principes
ordinaires, est celle-ci. Il y a quatre combinaisons.
Premier coup. Deuxième coup.
Croix. Croix.
Pile. Croix.
Croix. Pile.
Pile. Pile.
De ces quatre combinaisons, une seule fait perdre & trois font gagner,
il y a donc 3 contre 1 à parier en faveur du joueur qui jette la pièce. S’il pariait
en trois coups, on trouveroit huit combinaisons, dont une seule fait perdre, &
sept font gagner ; ainsi, il y aurait 7 contre 1 à parier. Voyez COMBINAISON &
AVANTAGE. Cependant cela est-il bien exact ?43
Car, pour ne prendre ici que
le cas de deux coups, ne faut-il pas réduire à une les deux combinaisons qui
donnent croix au premier coup ? Car, dès qu’une fois croix est venu, le jeu est
fini, & le second coup est compté pour rien. Ainsi, il n’y a proprement que
trois combinaisons de possibles :
Croix, premier coup
Pile, Croix, premier & second coup.
Pile, pile, premier & second coup.
Donc il n’y a que 2 contre 1 à parier. De même, dans le cas de trois
coups, on trouvera :
Croix
Pile, croix.
pile, pile, croix.
Pile, pile, pile.
Donc il n’y a que 3 contre 1 à parier. Ceci est digne, ce me semble, de
l’attention des calculateurs, & irait à réformer bien des règles
unanimement reçues sur les jeux de hasard43
.
Toss a coin a maximum of three times,
stop at the
f
irst “heads”.
What is the probability of seeing heads?
d’Alembert’s answer:
there are four possible outcomes,
three ending in heads, one not.
So the chance is 3/4
I think d’Alembert was not stupid.
I think he deliberately shows
that blindly counting
# favourable and # unfavourable
outcomes can be dumb.
You *must* also argue why
those outcomes are equally likely.

5. •One of the oldest casino betting systems is the one that calls for the player to begin with a unit
bet, increase the size of the bet by one unit after every loss, and decrease it by one unit, except
when it is already only one unit, after every win. Since the late 19th century, this system has been
called the d’Alembert, on the erroneous theory that it was invented by the mathematician Jean Le
Rond d’Alembert. Whatever its origin, the d’Alembert was already one of the most popular systems by
the 1790s.
•As Bertrand reported in 1798, admirers of the d’Alembert thought it was bound to succeed because the
numbers of wins and losses will eventually equalize. Consider what happens when you play the
d’Alembert after having bet one unit and lost. Whenever the number of subsequent wins and
losses, including this
f
irst loss, are equal, your net gain will be equal to the number of wins (or,
equivalently, equal to one-half the number of rounds played).
•The flaw in this venerable argument for the d’Alembert is that you may run out of money before your
wins and losses equalize. But as Jacques-Joseph Boreux explained in 1820, the system often appears to
work in practice. If you set a modest goal and play the d’Alembert repeatedly, playing each time until
the goal is reached or you are forced to stop (because you run out of money, the séance is over, or your
proposed bet exceeds the house limit), you can expect a string of relatively quick successes before you
ever fail, and the successes will not usually require taking too much money out of your pocket.
Bettingsystems
Risk is r
a
ndom: The m
a
gic of the d’Alembert
H
a
rry Cr
a
ne
a
nd Glenn Sh
a
fer (2020)
https://rese
a
rchers.one/
a
rticles/20.08.00007

6. • Lawyer: lawyer’s client is accused of laundering money
• Client claims in each of 5 successive years to have won close to
€10.000,– playing red/black and low/high at roulette at Holland Casino
• Client pays no income tax on his winnings, since Holland Casino pays
the income tax on behalf of all players
• No capital-gains tax had to be paid because below threshold
(client submitted proof of his gains to the tax inspector)
• The prosecution says this is impossible, and anyway would take much
too much time
• Client says he used the d’Alembert system
Animaginaryconsultationproject
Im
a
gine, my client is
a
crimin
a
l defence l
a
wyer

7. • Two simultaneous games are played: red/black and low/high
• Initial bet in each game = 1 unit (€50.–)
• Initial capital = 25 units for each game
• Maximum number of rounds = 21
• Quit either game if game capital falls below 16 units
• Roulette at Holland Casino has 36 numbered slots and one
“zero”; if zero comes up, then stakes on even odds bets are
shared 50/50 between player and house
• To avoid working with half units and dependence between the
two games I’ll pretend two independent games each with actual
odds 36:37 in favour of the house
Client’sstrategy

8. startKapitaal <- 25
eersteInzet <- 1
noodstopKapitaal <- 16
aantalBeurten <- 21 # Aantal beurten in een spel
set.seed(12345)
boxes <- rep(0, 51)
J <- 100000
Capitals <- 0:50
for (j in (1:J)) {
huidigeKapitaal <- startKapitaal
huidigeInzet <- eersteInzet
spelVerloop <- rep(0, aantalBeurten)
for(i in 1:aantalBeurten) {
uitkomst <- sample(x = c(-1, +1), prob = c(37, 36), size = 1)
if(huidigeInzet > 0){
stap <- uitkomst * huidigeInzet
huidigeKapitaal <- huidigeKapitaal + stap
huidigeInzet <- max(1, huidigeInzet - uitkomst)
if(huidigeKapitaal < noodstopKapitaal) huidigeInzet <- 0
}
spelVerloop[i] <- huidigeKapitaal
}
boxes[huidigeKapitaal + 1] <- boxes[huidigeKapitaal + 1] + 1
}
Rsimulations
Core code

9. 0 3 6 9 12 16 20 24 28 32 36 40 44 48
Percent chance of final capital
Simulation with N = 100,000
Capital at end of game
Probability
(percent)
0
2
4
6
8
10
Data: 100,000 simulated games

10. Rmovie(100games)
library(animation)
ani.options(interval = 0.2, nmax = 101)
options(warn=-1)
startKapitaal <- 25
eersteInzet <- 1
noodstopKapitaal <- 16
aantalBeurten <- 21
dalembert <- function () {
nmax = ani.options("nmax")
for (J in 0:(nmax - 1)) {
plot(x = -2, y = -1, ylim = c(5, 45), xlim = c(0, 23),
xlab = "Round", ylab = "Capital (in game units)”}
if (J > 0) {
set.seed(12345)
boxes <- rep(0, 51)
for (j in (1:J)) {
####
####
#### Earlier code to simulate one game…
####
####
}
}
ani.pause()
}
}
saveGIF({
ani.options(nmax = 101)
dalembert()},
interval = 0.2, movie.name = 'dalembert.gif', ani.width = 800, ani.height = 600)
saveVideo({
ani.options(interval = 0.2, nmax = 101)
dalembert()}, video.name = "dalembert.mp4", ani.width = 800, ani.height = 600,
other.opts = "-pix_fmt yuv420p -b 300k")

12. • Markov process with states: (Capital, Stake)
• State space = {0, 1, …, 49} × {0, 1, …, 9}
• Initial state = (25, 1) ; 21 time steps.
• Possible moves: for s > 0
“Up”: (c, s) → ( (c + s) ∧ 49) , (s – 1) ∨ 1 )
“Down”: (c, s) → ( (c – s) ∨ 0), (s + 1) ∧ 9)) ,
unless c – s < 16, in which case new state is ((c – s) ∨ 0, 0)
“Stopped”: (c, 0) → (c, 0)
• “Up” and “Down” probabilities are 36 / 73 and 37 / 73 respectively
Whysimulate?

13. Capitals <- 0:49
Stakes <- 0:9
Stop <- 16
nCaps <- length(Capitals)
nStakes <- length(Stakes)
nStates <- nCaps*nStakes
P <- array(0, dim = c(nCaps, nStakes, nCaps, nStakes))
for (i in 1:nCaps) {
for (j in 2:nStakes) {
P[i, j, min(i + j - 1, nCaps), max(2, j-1)] <- +36
newCapIndex <- max(i - Stakes[j], 1)
if (Capitals[newCapIndex] < Stop) {newStakeIndex <- 1} else
{newStakeIndex <- min(j+1, nStakes)}
P[i, j, newCapIndex, newStakeIndex] <- +37}
P[i, 1, i, 1] <- 73}
Pmat <- array(P, dim = c(nStates, nStates))
Pmat <- Pmat/73
stateMat <- array(0, dim = c(nCaps, nStakes))
stateMat[(1:nCaps)[Capitals==25], (1:nStakes)[Stakes==1]] <- 1
state <- array(stateMat, dim = c(1, nStates))
for (i in (1:21)) {state <- state%*%Pmat}
stateMat <- array(state, dim = c(nCaps, nStakes))
finalCapState <- apply(stateMat, 1, sum)
R trick:
same vector of numbers can be seen as a
50 x 10 x 50 x 10 array and as a
500 x 500 matrix
Computing
f
inalstate
probabilities

14. 0 3 6 9 12 16 20 24 28 32 36 40 44 48
Percent chance of final capital
Capital at end of game
Probability
(percent)
0
2
4
6
8
10
Theory:
Compute
𝜋
P
21
P is a 500 × 500 matrix,
𝜋
is a row vector of length 500.
Group states according to capital

16. • Chances of loss/gain about 50-50
• Mean gain is 9.5 if there’s a gain
• Mean loss is 10.5 if there’s a loss
• One loses on average (10.5 – 9.5)/2 = 0.5 per game (that’s 2% of initial capital)
• [mean net gain = –0.53, standard deviation 10.68]
• You may as well pay half a unit to the bank and bet 10 units at even odds, i.e., win
10 or lose 10 with probabilities 50-50
• Play 40 games, win about 20, winnings = about 200 units, at €50,– Euro per unit
that’s about €10.000,–
• Playing two games simultaneously, twice an evening only needs 10 visits to the
casino
Thenumbers
In round numbers

17. • The lawyer’s client’s assertions would, at face value, have been
plausible
• He would most likely have lost more money than he gained
• The chance of a net gain of 200 units (or more) in 40 games
(that’s €10.000,– in €50,– units) is about 0.0005
• The chance of doing this year after year for
f
ive years is, for
practical purposes, nihil
• It is a bit cheaper and takes less time, but would be more
noticeable, to bet 10 units just once in each game [Why?]
Conclusions

18. •The lawyer could cherry-pick from my
f
indings
•I would charge the lawyer (and hence his client) for my work
•In forensic statistics the current dogma is “we should report
a likelihood ratio” (or a Bayes factor?); p-values are out!
•Is this a good way to launder money?
•We want a simple and fast game-plan with low costs
and a big chance of netto gain per game
Questions
Ethic
a
l, principle, m
a
them
a
tic
a
l

19. Sources
www.researchers.one/article/2020-08-32
Risk is random:
The magic of the d’Alembert
Harry Crane and Glenn Shafer
Rutgers University
August 23, 2020
Abstract
The most common bets in 19th-century casinos were even-money bets on red or
black in Roulette or Trente et Quarante. Many casino gamblers allowed themselves
to be persuaded that they could make money for sure in these games by following
betting systems such as the d’Alembert. What made these systems so seductive?
Part of the answer is that some of the systems, including the d’Alembert, can give
bettors a very high probability of winning a small or moderate amount. But there
is also a more subtle aspect of the seduction. When the systems do win, their return
on investment — the gain relative to the amount of money the bettor has to take
out of their pocket and put on the table to cover their bets — can be astonishingly
high. Systems such as le tiers et le tout, which offer a large gain when they do
win rather than a high probability of winning, also typically have a high upside
return on investment. In order to understand these high returns on investment, we
need to recognize that the denominator — the amount invested — is random, as it
depends on how successive bets come out.
In this article, we compare some systems on their return on investment and
their success in hiding their pitfalls. Systems that provide a moderate gain with a
very high probability seem to accomplish this by stopping when they are ahead and
more generally by betting less when they are ahead or at least have just won, while
betting more when they are behind or have just lost. For historical reasons, we
call this martingaling. Among martingales, the d’Alembert seems especially good
at making an impressive return on investment quickly, encouraging gamblers’ hope
that they can use it so gingerly as to avoid the possible large losses, and this may
explain why its popularity was so durable.
We also discuss the lessons that this aspect of gambling can have for evaluating
success in business and finance and for evaluating the results of statistical testing.
https://www.gsimulator.net/dalembert