The Mathematics of Gambling by Dr. E. O. Thorp
It is somewhat ridiculous to discuss an optimal money management strategy when the
player has a negative expectancy. As indicated in chapter 8, with an enforced house
maximum and minimum wager, there is no way to convert a negative expectation into
a positive expectation through money manipulation. Any good money management
plan says not to wager in such a situation. Players facing a negative expectancy
should look elsewhere for a gambling game or, at the very least, bet insignificant
amounts and write off in their mind the expected loss as "entertainment."
After the gambler has discovered a favorable wagering situation, he is faced with the
problem of how best to apportion his limited financial resources. A rule or formula
exists, which you can use to decide how much to bet. I will explain the rule and tell
you the benefits that are likely if you follow it.
Let's begin with a simple illustration that I deliberately exaggerated to better get the
ideas across. Suppose you have a very rich adversary who will let you bet any
amount on heads at each toss of a coin that you both know that the chance of heads is
some number "p" greater than 1/2. If your bet pays even money, then you have an
edge. Now suppose "p" = 0.52, so you tend to win 52 percent of your bets and lose
48 percent. This is similar to the situation in Blackjack when the ten-count ratio is
about 1.5 percent. Suppose, too, that our bankroll is only $100. How much should
you bet? You could play safe and just bet one cent each time. That way, you would
have virtually no chance of ever losing your $100 and being put out of the game. But
your expected gain is 0.4 per unit or .04 cents per bet. At 100 one cent bets an hour,
you expect to win four cents per hour. It's hardly worth playing.
Now look at the other extreme where you bet your whole bankroll. Your expected
gain is $4 on the first bet, more than if you bet any lesser amount. If you win, you
now have $200. If you again bet all of it on your second turn, your expected gain is
$8 and is more than if you bet any lesser amount. You make your expected gain the
biggest on each turn by betting everything, but if you lose once, you are broke and out
of the game. After many turns, say 20, you have won 20 straight tosses with
probability, 0.5220 = 0.000002090 and have a fortune of $104,857,600 or you have
lost once with probability 0.999997910 and have nothing. In general, as the number
of tosses increases, the probability that you will be ruined tends to 1 or certainty. This
makes the strategy of betting everything unattractive.
Since the gambling probabilities and payoffs at each bet are the same, it seems
reasonable to expect that the "best" strategy will always involve betting the same
fraction of your bankroll at each turn. But what fraction should this be? The "answer"
is to be "p" - (1-"p") = .052 - 0.48 = 0.04, or four percent of your bankroll each time.
Thus you bet $4 the first time. If you win, you have $104, so you bet 0.04 x $104 =
$4.16 on the second turn. If you lost the first turn, you have $96, so you bet 0.04 x
$96 = $3.84 on the second turn. You continue to bet four percent of your bankroll at
each turn. This strategy of "investing" four percent of your bankroll at each trial and
holding the remainder in cash is known in investment circles as the "optimal
geometric growth portfolio" or OGGP. In the 1962 edition of Beat the Dealer, I
discussed its application to Blackjack at some length. There I called it the Kelly
system, after one of the mathematicians who studies it, and I also referred to it as
(optimal) fixed fraction (of your bankroll) betting.
Why is the Kelly system good? First, the chance of ruin is "small." In fact, if money
were infinitely divisible (which it can be if we use book-keeping instead of actual
coins and bills, or if we use precious metals such as gold or silver), then any system
where you never bet everything will have zero chance of ruin because even if you
always lose, you still have something left after each bet. The Kelly system has this
feature. Of course, in actual practice, coins, bills or chips are generally used, and
there is a minimum size bet. Therefore, with a very unlucky series of bets, one could
eventually have so little left that he has to bet more of his bankroll than the system
calls for. For instance, if the minimum bet were $1, then in our coin example, you
must over bet once your bankroll is below $25. If the minimum bet were one cent,
then you only have to over bet once your bankroll falls below 25 cents. If the bad
luck then continues, you could be wiped out.
The second desirable property of the Kelly system is that if someone with a
significantly different money management system bets on the same game, your total
bankroll will probably grow faster than his. In fact, as the game continues
indefinitely, your bankroll will tend to exceed his by any pre-arranged multiple.
The third desirable property of the Kelly system is that you tend to reach a specified
level of winnings in the least average time. For example, suppose you are a winning
card counter at Blackjack, and you want to run your $400 bankroll up to $40,000.
The number of hands you will have to play on average to do this will, using the Kelly
system, be very close to the minimum possible using any system of money
To summarize, the Kelly system is relatively safe, you tend to have more profit and
you tend to get to your goal in the shortest time.