there are fair games that will not permit you to reach 1 from f with probability greater than e, as was discovered by Donald Ornstein (Section 11.2).Imagine yourself at a casino with $1, 000. For some reason, you desperately need $10, 000 by morning; anything less is worth nothing for your purpose. What ought you do? The only thing possible is to gamble away your last cent, if need be, in an attempt to reach the target sum of $10, 000. There may be a moment of moral confusion and discouragement. For who has not been taught how wrong and futile it is to gamble, especially when short of funds? Yet, gamble you must or forgo all chance of the great purpose that can be achieved only at the price of $10, 000 payable at dawn. The question is how to play, not whether.
As is well known, any policy of compounding bets that are subfair to you must decrease your expected wealth. Consequently, no matter how you play, your chance of converting $1, 000 into $10, 000 will be less than 1/10. How close to 1/10 can you make it and by what strategy? That is the sort of problem this book attacks. The particular problems treated are highly simplified and idealized, especially when we get down to cases, and we do not leave our armchairs far enough to discover whether many of them would be applicable in such complicated situations as obtain for any real gambler.
The fantasy with which we have introduced the general problem of optimal gambling systems has no immediate practical importance. But the problem, once proposed, cries out for attention as pure mathematics. As such, we have found it stimulating and hope you will.
A rough idea of some of the more colorful problems and conclusions of this book can be imparted quickly.
Suppose the only kind of gamble available to you in your effort to convert a sum of money into a larger target sum is to bet, at fixed, subfair odds, on independent repetitions of some fixed kind of event, such as drawing a red card or a spade. You are free to choose for yourself the amount to stake on each bet, subject only to the restriction that you can never stake what you do not possess. Under these circumstances, one—not usually the only—optimal strategy for you is to play boldly; that is, always to stake on each bet either all the money in your possession or just enough to arrive immediately at the target sum in case you win the bet, whichever of these two stakes is the smaller (Sections 5.3 and 6.3).
If your object is, as before, to reach a target sum and if you are free at each play to stake any amount within your means, dividing the stake equally among as many numbers on a roulette table as you wish, then it is ill-advised to bet on more than one number at a time (Theorem 6.9.2). This may be true even if you are free to stake different amounts on different numbers simultaneously, but that is an open and promising question.
In games that have several possible outcomes, the gambler does not merely lose his stake or win back a predetermined multiple of it, as when he simply bets on an event. Instead, he wins back some random multiple (quite possibly 0) of the stake that he has deposited, so his bet is described as a random variable—his winnings, possibly negative—with an essential minimum not less than the negative of the stake. A popular index c of unfairness of a bet is the ratio of its expected loss to its stake. What can be said about the probability of converting a fortune f into the larger target sum 1 if the house permits only bets of index at least c? The probability is at most 1 − (1 − f)1−c, which can almost be attained by, for example, repeatedly betting very small stakes on very improbable events at such odds that winning one of the bets will bring the gambler at once to the target sum 1 (Section 9.3).
A second index α of unfairness is the ratio of the expected value of the loss to the variance. The gambler’s probability of converting f into 1 with only bets of index at least a is at most f/(1 + α − αf). This bound also is nearly attained by successively placing small stakes on the chance of leaping to the goal (Section 9.4).
A third index of unfairness ß is the ratio of the expected loss to the first absolute moment. This index was introduced by Donald Ornstein, who showed that the least upper bound is (1 − ß)f/(l + ß − 2ßf) and that it is attained by staking the whole of f on a single chance of attaining the goal (Section 9.2).
When a gambling situation permits you to make fair bets, you may be able to reach 1 from f with probability f. In particular, if there is available a fair game that you can repeatedly play, choosing the stake each time at your own discretion (within your means), you “ought” to be able to reach 1 with probability arbitrarily close to f. Surprisingly, though, for each positive there are fair games that will not permit you to reach 1 from f with probability greater than e, as was discovered by Donald Ornstein (Section 11.2).
This book does not consist entirely of the study of such concrete and colorful problems, primarily because to formulate and solve such problems satisfactorily requires considerable abstraction and generalization. We came gradually to adopt a certain concept of finitely additive, time-discrete stochastic processes as the model of a gambling strategy. Since finitely additive measures have not been popular, the mathematical fundamentals of such processes had to be explored, and this produced some analytic novelties (Chapter 2). The abstract formulation, once achieved, suggested abstract questions (Chapters 2, 3, and 7), and it had to be related to other abstract structures already familiar to probability theory (Chapter 12).
Many concrete problems—in particular, those reported on in this section—refer to a moderately general kind of gambling house, here called a casino, associated with nonnegative stochastic processes, and these problems suggested a theory of casinos of interest in itself (Chapter 4).
Many facts about gambling can be translated with almost no loss into the language of stochastic processes. For example, the limitation on a gambler constrained to use bets for which the expected loss is at least α times the variance says scarcely more than this: If X0, X1, ··· is an expectation-decreasing semimartingale for which Xn ≥ 0 for all n, X0 < 1, and the conditional expectation of Xn − Xn+1 given the past is at least α times its conditional variance given the past, then X0/(1 + α − αX0) is an upper bound for the probability that supn Xn ≥ 1. And this bound is sharp.
In short, one thing has led to another, as happens in mathematics, and it is not possible to say satisfactorily in advance what all the parts of the book are about. But the introduction of each chapter offers additional summary information.
Probabilists have been interested in gambling since mathematical probability began; so it is surprising that problems like the first one mentioned in the preceding section had not been solved, or even much discussed. Perhaps moral disapproval has focused so much light on those respects in which gambling is futile as to leave other aspects of gambling in darkness.
The idea that large bets are sometimes advantageous for attaining a fixed goal is in the gambling air. The earliest mathematical reference to it that we know is by Coolidge (1908–1909). He studies the first problem of the preceding section, generalized to include the possibility that there is a legal upper limit to your stake, and purports to prove that your only optimal strategy is always to stake the legal limit, or all that you possess, or enough to reach the target, whichever is least. The assertion of uniqueness is incorrect. Whether that of optimality is correct when there is a legal limit, we do not know. We have not seen how to rescue much from Coolidge’s faulty demonstration.
The only other reference we know in this immediate area is a passage in which Feller (1950, p. 285, first edition only) comes to the not quite accurate conclusion that bold play is uniquely optimal when there is no legal limit.
Knowledge that small bets are contraindicated in certain situations is widespread. As De Moivre (1711, Problem IX) and James Bernoulli (1713) were quite capable of showing, if you engage in only the smallest allowable bets on red-and-black, say $1 or less, you are all but certain to lose your entire $1, 000 before reaching $10, 000. An interesting paper by Blackwell (1954) and a sequel to it by Weingarten (1956) can be interpreted as greatly generalizing the method discovered by De Moivre and as treating certain problems about how to gamble if you must relatively completely.
An extension of the recommendation of large bets is the recommendation of bets for which the variance is large compared with the expected loss. This idea goes back at least to de Finetti (1939, 1940), whose exposition of it we paraphrase in Section 8.7. Incidentally, our work on gambling might never have begun had one of us (Savage) not been intrigued by hearing of this rule indirectly through Jesse Marcum. De Finetti’s papers (1939, 1940) are also descendants of De Moivre’s Problem IX.
One very important idea about gambling strategies permeates the whole history of probability. It might be called the “conservation of fairness”, and it says that any combination or concatenation of bets that are at most fair, when viewed as a single, composite bet, must be at most fair. For example, De Moivre takes the conservation of fairness for granted in his Problem IX, and we find it our most important tool.
Bachelier (1900, 1912, 1914, 1938) deliberately makes the conservation of fairness central to his theory of speculation in stocks and commodities. This work is apparently the oldest sustained study of continuous-time stochastic processes. As we have heard it put, Brownian motion, a gem of modern physics, is a botanical discovery the mathematical theory of which originated in the social sciences.
Some formulations of conservation of fairness in the measure-theoretic framework date from (Halmos 1939). These are carried forward in Doob’s (1953) treatise on stochastic processes.
While we were working on this book, which is largely motivated by unfavorable games, Leo Breiman was independently working on how to get rich quick at favorable games. His interesting paper (Breiman 1961) and this book complement each other.
When the theory of gambling systems is viewed very abstractly, it tends to merge, in principle, with the theory of sequential decisions and with the general idea of dynamic programming, a broad concept of strategy optimization developed by Bellman (1952, 1957). The relation of the theory of gambling strategies to dynamic programming and to the establishing of inequalities for stochastic processes is discussed in Chapter 12.