Optimal Betting (Money Management in a Positive Expectation Game)
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 management.

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.
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