A Mathematical Odyssey From Las Vegas to Wall Street
UCI Talk May 3, 2012
I would like to thank the UCI mathematics department
for inviting me and to Professor Sarah Eichhorn for
making all the arrangements. It’s a pleasure to see so
many students and faculty from UCI and from Sage Hill
School, including three of my grandchildren.
Today I’ll tell you how I used mathematics in roulette,
casino blackjack and the stock market.
V1
For each, I’ll talk about the idea, the math behind the
idea, the implementation, and what happened afterwards.
So that you don’t have to take any notes, I have posted an
outline and the images for this talk on my website.
It all started with roulette, back in 1947 when I was a
15 year old high school junior interested in anything to do
with science and math. While studying physics on my own
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I thought about predicting the motion of an orbiting
roulette ball.
V2 Here’s a picture of a casino roulette wheel.
Describe a play of the game: Rotor spins, bettors place
chips, ball is launched, spins, slows, croupier calls “no
more bets,” ball falls into rotor, stops in one of 38 pockets
in the U.S. or 37 in Europe.
In the U.S. if you bet a dollar on any number you get $36
if that number comes up and nothing if does not. If the
ball falls at random, meaning all numbers are equally
likely, then the casino edge is 2/38 or 5.3%.
In this case mathematicians proved there is no winning
system.
My idea was to measure motion of the ball and the rotor
to predict which pocket the ball would fall into. You don’t
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have to predict exactly – you only need to move win
probability from 1/38 to better than 1/36 to get an edge.
A few years went by, and then in the summer of 1955,
just after I got my MA in Physics, I was sitting in the
dining room of the UCLA co-op where I lived drinking tea
on a Sunday afternoon. As usual, a debate started – this
one on whether you could beat the casinos.
I argued that physical prediction of roulette would
probably succeed. Most thought not: If you could do it,
someone would have done it by now and they would have
changed the game.
Or, there are too many randomizing factors, making
prediction impossible. This reignited my interest.
(a) First, I tried to show proof of principle through
experiments on a half-sized wheel.
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Here’s a frame from a film of one of these experiments.
What you see is: ball, stopwatch, playing card, labeling
each spin of the ball.
V3 website, film
My plan for going ahead:
(b) the set-up: two people, one with prediction device
who transmits results to second one, who bets
(c) the math behind it: differential equations for motion;
probability and statistics for errors and distributions of
outcomes
(d) About this time, my wife and I were invited to a 1956
summer party given by Richard Feynman’s girlfriend.
RF was sitting in an alcove playing the bongo drums;
asked RF if any way to beat roulette; answer “no.” Thus
encouraged (!), I continued my experiments.
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During one of them, my in-laws arrive for dinner. Where
am I?
Launching balls down inclines and across the floor in the
kitchen.
Math Ph.D. in the spring of 1958, then joined the faculty.
December 1958
Vacation trip to Las Vegas – it was inexpensive since I
didn’t gamble, and I wanted to observe roulette wheels.
An unexpected interruption to pursuing roulette: Hearing
that I was going, Prof. Robert Sorgenfrey told me about a
JASA article; played blackjack for the first time, tried their
blackjack strategy in a casino; did comparatively well –
lost $8.50 of my $10 in 40 minutes while all others lost
heavily. Realized game likely beatable.
Rules of blackjack
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Players gets two cards, dealer gets one, up, one down.
Goal is to get total of 21 or less, player ins if total beats
dealer, loses if total is less, ties if total is the same.
Cards count A=1, 11; 2-10 face value, J,Q,K=10.
More cards may be drawn to improve total, other detailed
rules.
V4 You can see part of a blackjack layout in this Life picture
of me playing at Tropicana, 1964
The idea for beating the casino: several rounds of play are
dealt from the pack of one or more decks before
reshuffling. For rounds after the first, the missing cards
change the odds. I saw that this would often be
substantial, sometimes favoring the casino and
sometimes favoring the player. This meant I could
overcome the average casino edge by: betting big when
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player has edge, betting 0 or small when casino does. I
win the majority of the big bets, the casino wins the
majority of the small bets.
Math: 𝑬(Player) is very complex rational function of the
quantities of each type of card, the ten 𝑸(𝑰). 10
dimensional lattice model for 𝑬(Player) function. 𝑬(Player)
turns out to depend mainly on the fraction 𝒇(𝑰) = 𝑸(𝑰)/𝑸 of
each of the ten card values.
My reasoning showed that there must be frequent large
favorable fluctuations so my plan had to work,
theoretically.
M.I.T. Summer 1959: I attempt to solve with desk
calculators. Complete solution needs 33 million subsets
of one deck calculations of 10 or more man years each;
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330 million years; 33 million page catalog. Use math to
reduce and simplify.
Fall 1959 – Summer 1960: IBM 704; I code the solution in
Fortran.
V5 computer program
Result: Card counting as a simplified practical
approximation. Example: point count, complete
(normalized) point count.
My results give all possible simplified card-counting
systems, their strategies, their edges.
Math: gradient, function of 10 variables (multi-variable
calculus)
November 1960
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Shannon meeting; describe; Proc. NAS paper on
blackjack; embark on roulette project. Shannon’s
basement, full size roulette wheel on a pool table,
strobes, timers from M.I.T. labs.
November 1960 – Summer 1961
Now working on roulette, blackjack simultaneously.
First, what happened with roulette:
Shannon and I build the world’s first wearable computer.
“What makes you tick?” Our lab shows giant expected
return of 44%. Shannons and Thorps go to Las Vegas for
a week in August 1961.
V6, Shannon
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Computer works perfectly; casino play confirms 44%
expected return but we have one hardware issue:
continual breaking of wires to the speaker in my ear. A
“bug” emerges from my ear canal. Future problem for the
project is a killer potential countermeasure: requiring all
bets be placed before the ball is launched.
V8, Here’s what we built: The world’s first wearable
9,
computer: pictures from the Nixdorf Computer Museum
exhibit.
V10 Supposed to be featured on Travel Channel: Mysteries at
the Museum (May 2012 or later is possible first run date).
Back to Blackjack
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National publicity when my results were announced;
Scoffers: casino, press
Washington Post editorial: send $1 for a sure fire weed
killer.
1961 April, M.I.T. spring break: bankrolled by X & Y.
Trip: 10K → 21K in 20 hours; in today’s dollars, multiply
by 8.
Sports Illustrated “Bye, bye Blackjack”
Life 1964 eight page spread “The Professor Who Breaks
the Bank”
New York Times Best-seller Beat the Dealer
V11 (cover of Beat the Dealer)
V12 Training for casino play.
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Whenever I needed $, could get some from the casinos.
But it was never about the money – rather, the challenge,
the math, and the ideas.
1962-1964
Found ways to win at other games, then got interested in
the stock market. No mafia knee breakers.
Summer 1964 Taught self about the market.
Summer 1965 Common stock purchase warrants: a safe?
way to win!
Fall 1965
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UCI opens, founding math department faculty member;
Sheen T. Hassouf and I meet and collaborate.
V13 Beat the Market 1967
I manage accounts for self, others, including UCI admin
and faculty. I discover and use what later will be known
as the Black-Scholes options formula.
V14 Selecting warrant hedges.
1968
Ralph Waldo Gerard, Dean of the Graduate School,
introduces me to Warren Buffett.
Buffett is closing his hedge fund; he OKs me; the Gerards
move $ from Buffett to me.
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1969
PNP – one of the earliest hedge funds (explain hedge
funds as defined then and now); then there were less
than 200, most of which closed in the 1969-1971
downturn.
First market neutral fund designed to do equally well
whether the market goes up or down; first derivatives
fund; first to be 100% hedged.
Math of BS formula:
my version “integral model,” risk neutral world, intuitive
derivation, kept secret for benefit of my investors and
myself.
Academia:
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they used stochastic differential equations;
PNP stayed ahead of the marching legions of academia
and investment rivals for the next 20 years.
Nobel Prize to Scholes & Merton 1997 for Black-Scholes
formula.
V15 1969-1988 record: no losing years or even losing
quarters; only 3 negative months (declines were <1%)
out of 230.
The big test: Monday, October 19, 1987 market down
23%, PNP flat to -1%; next day was up for PNP.
1992-2002 Encore: statistical arbitrage.
Most up, most down; factor analysis; principal
components.
V16 Unleveraged stat arb performance
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Status Today
Investing:
Quants “rule”; Many descendants of Beat the Market,
Beat the Dealer, e.g. locally, the bond king, Bill Gross;
other billionaires like Citadel’s Ken Griffin, hedge fund
stars Brian Stark, Paul Singer, and more.
Read about it in:
V17 The Quants by Scott Patterson
There also was an impact from my financial ventures on
the UCI community. Several of the students, staff and
faculty I hired became multimillionaires and one peaked
at around a billion dollars.
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UCI also has received gifts of at least $15 million as a
result of Beat the Market.
The Kelly criterion – how to exploit favorable gambling or
investment situations – which I first used in blackjack bet
sizing and then helped popularize in gambling and
investing is now of widespread academic interest, e.g.
V18 Fortune’s Formula – story by Poundstone.
V19 The math (cover of The Kelly Capital Growth Investment
Criterion).
Blackjack and Roulette:
The Blackjack Ball (Newsweek, Feb. 20, 2012)
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The 50 year blackjack war between advantage players
and casinos continues; the original Beat the Dealer
methods still are working.
Blackjack and roulette teams with wearable computers
prowl the world. Computer in a tooth?
V20 To see an outline of this talk, with visuals, and the math I
used for this and other things, visit my website:
I understand we have a few minutes for questions.
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