, 0 < w < 1. Formally, if ρ is the measure that attaches probability w to 1 and to − 1, so that [sρ + f] attaches probability w to f + s and to f − s, then Γ(f) = {[sρ + f]: 0 ≤ s ≤ f}; equivalents, Θ(f) = {[sρ]: 0 ≤ s ≤ f}. This chapter is mainly about a very special kind of casino, where the only game offered is that of red-and-black. The gambler can stake any amount s in his possession. He wins back his stake and as much more again with probability w, and he loses his stake with probability , 0 < w < 1. Formally, if ρ is the measure that attaches probability w to 1 and to − 1, so that [sρ + f] attaches probability w to f + s and to f − s, then Γ(f) = {[sρ + f]: 0 ≤ s ≤ f}; equivalents, Θ(f) = {[sρ]: 0 ≤ s ≤ f}.
The possibilities w = 0 and w = 1 could be included, but they are trivial and uninteresting. In the terminology of Section 4.4, the casino is easily seen to be superfair, fair, or subfair (but not trivial) according as w exceeds, equals, or is exceeded by 1/2; correspondingly, in (0, 1), U(f) is 1, f, or positive but less than f. Attention will be focused on the subfair possibility w < 1/2.
It is natural to generalize ρ to a distribution confined to −1 and some arbitrary positive value instead of to − 1 and +1, and that will be done in the next chapter, but the relative simplicity of the theory of red-and-black justifies this separate chapter. A parallel special case with w = 1/2 but with the possible gain different from the possible loss is also briefly treated in the present chapter.
If w < 1/2, the strong law of large numbers suggests that a consistent policy of small bets is a bad strategy. This may stimulate an interest in large bets. How large? It is illegal for the gambler to bet more than he possesses and plainly unwise for him to bet more than just enough to reach his goal if he wins the bet. The gambler may, then, be said to be playing boldly if he always bets his entire fortune f or enough to reach the goal, whichever is least. That is, if f ≤ 1/2, it is bold to stake f, but if 1/2 ≤ f ≤ 1, it is bold to stake 
. In short, it is bold to stake the minimum of f and , or, equivalently, to use the bold gamble
The gamble-valued function γ determines for each f a stationary strategy, called the bold strategy at f and designated by σ(f). Since γ(f) ∈ γ(f) for all f, σ(f) is available in the red-and-black casino Γ at f.
The main objective of this chapter is to prove that the bold strategy at f is optimal for w < 1/2. Incidental objectives are to establish the identity of U and a function well known in analysis and probability theory, to say something about what happens when the gambler is required to terminate at the nth move, and to treat briefly a kind of casino that might be called dual to red-and-black.
As a bonus, the study leads to new facts about a prominent family of singular distribution functions and about Bernoulli, or binomial, processes.
This section is about the utility u(σ(f)) of the family of bold strategies as a function of f for 0 ≤ f ≤ 1. This function will be called Qw, when necessary, but usually simply Q. By definition then, Qw(f) = Q(f) = u(σ(f)).
The bold strategy is almost stagnant. For both 1 − wn and 
converge to 1 as n goes to infinity, and the probability of stagnation by the nth gamble is at least as large as one of these two numbers. On this account, it is easy to verify that Q(f) is the (Eudoxus) probability of stagnating at 1 under σ(f).
If f ≤ 1/2, the gambler’s first move will leave him at 2f with probability w and at 0 with probability . In the former case, the rest of his play is described by σ(2f); in the latter case, play stagnates with u = 0. This fact and a dual one for f ≥ 1/2 lead, through Theorem 3.2.1, to the conclusion that
There is one and only one bounded function Q on [0, 1] that satisfies (1); Q is continuous and strictly increasing; Q(0) = 0, and Q(1) = 1. These facts are easy to prove and are explicitly proved by de Rham in an interesting paper (1956–1957). Since the probability Q(f) is bounded between 0 and 1, Q is characterized by (1) and has the properties cited. If w = 1/2, the function f obviously solves (1); so Q1/2(f) = f for f in [0, 1].
An interesting interpretation of (1) was pointed out to us by W. Forrest Stinespring. Let 
be the probability that the sequence of 0’s and 1’s generated by independent tosses with a w-coin (that is, one which produces 0 with probability w and 1 with probability ) represents a number in the binary notation that does not exceed f. If f is at most 1/2, the sequence of tosses can generate a number as small as f only if the first binary digit is a 0 and the remaining digits represent a number as small as 2f (aside from one possibility of probability 0); if f is at least 1/2, an initial 0 ensures success, and success can also be had after an initial 1 if and only if the remaining digits represent a number as small as 2f − 1. This all means that 
satisfies (1) and is therefore the same as Q. This interpretation shows strikingly that if w ≠ 1/2 then Q considered as a monotonie function—more specifically, the distribution function of a probability measure—on [0, 1] is singular with respect to Lebesgue measure. For, according to the strong law of large numbers, the probability measure associated with Qw attaches probability 1 to the set of numbers in whose binary representation the relative frequency of 0’s is w. Thus, as is well known, every Qw is singular with respect to every other. In particular, if w ≠ 1/2, then Qw is singular with respect to Q1/2, which corresponds to Lebesgue measure. Once it is proved that Q is U for w < 1/2, Q will be seen as a casino function which is as irregular, in terms of everyday intuition, as a monotonie function can be.
Interpretation of Q as a distribution function brings out a computational formula (which can of course be derived directly from (1)).
where k is a nonnegative integer less than 2n, a is the number of l’s in the binary representation of k2−n, and b is n − a. The probability argument for (2) is this: To toss a sequence representing a number between k2−n and (k + 1)2−n, it is sufficient and (almost) necessary to begin with the n binary digits that represent k 2−n.
The duality
can easily be established probabilistically from the definition of Q or algebraically by verifying that the right side of (3) also satisfies (1).
Several points about Q are illustrated by Table 1 for f = k/8, k = 0, · · · , 8, and for a few other values of f, listed in binary as well as in more conventional notation. First, Q is computed by application of (1), beginning with k = 0 and 8 and proceeding through k = 4, k = 2 and 6, and k = 1, 3, 5, and 7. Next, Q is rewritten to illustrate the duality (3). Finally, the column of differences illustrates (2).
The entry for f = 1/3, for example, can be computed from the infinite series delivered by repeated application of (2), namely, 
. But the calculation 
is also of interest. Both types of calculation apply to express Qw(f) as a rational function of w for each rational f.
Table 1 Selected values of Qw(f)
Figure 1 Values of Qw(f) for w = .2 and .4.
THEOREM 1. The bold strategy at f is optimal for red-and-black casinos with w ≤ 1/2.
Proof: In view of Theorem 2.12.1, it suffices to show that
for 0 ≤ f − s ≤ f ≤ 1 and w ≤ 1/2, where it is to be understood that Qw(f + s) = 1 for f + s> 1.
The possibility that f + s > 1 can be set aside. For in view of the monotony of Qw, if (1) were to fail for some 
, it would also fail for 
. Therefore, throughout the rest of the proof let it be assumed that 
.
Since Qw is continuous, it suffices to establish (1) for binary-rational values of f and s, that is, for numbers of the form k2−n with k and n nonnegative integers and k ≤ 2n. A number of this form will be said to be of order at most n. It will frequently be convenient to consider that all other numbers in the unit interval have infinite order.
At least one of the transformations f → 2f or f → 2f − 1 carries f back into [0, 1]; both do only if f = 1/2. These transformations carry numbers of order n into numbers of order n − 1, except that the numbers of order 0, that is, 0 and 1, are left invariant.
It is trivial to verify (1) if f and s are of order 0 or of order 1. Next, suppose that (1) has been established for all binary rationals f and s of order at most n, and study an f and an s of order at most n + 1. There are four cases to be considered, and it is helpful to rewrite (1) as
Case 1, f + s ≤ 1/2. The first functional equation of (2.1) and the inductive hypothesis justify the calculation: 
.
Case 2, f − s ≥ 1/2. The proof is just as in Case 1, except that the second functional equation of (2.1) is used instead of the first.
Case 3, f − s < f ≤ 1/2 < f + s. Calculate thus:
Now f is necessarily greater than 1/4, for otherwise s ≤ 1/4 and f + s ≤ 1/2. Therefore 2f > 1/2, and the sequence of equalities (3) can be continued thus:
Since 
, this last expression is greater than or equal to both
and
It is strictly greater than both unless w = 1/2 or f − s = 0. By the inductive assumption, the first or the second of these expressions is nonnegative, according as 2f + 2s − 1 ≥ 2f − 1/2 or 2f − 2s ≥ 2f − 1/2, that is, s ≥ 1/4 or s ≤ 1/4.
Case 4, f − s ≤ 1/2 < f < f + s. The proof for this case is similar to that of the preceding one. 
Though (1) is but a very special case of the casino inequality (4.2.1), by implying that Q is the U of red-and-black casinos for w ≤ 1/2, it implies the whole casino inequality. This has certain consequences for the binomial process corresponding to independent tosses with a w-coin. For example, the special casino inequality Q(fg) ≥ Q(f)Q(g) implies that the conditional probability of generating a binary number less than fg, given that the number is less than g, is at least the unconditional probability that such a number is less than f.
In general, facts of this chapter and the next can be interpreted as new facts about the Bernoulli process and about the functions Qw, which have an independent analytic interest as singular distribution functions. For literature on the Qw see (de Rham 1956-1957).
The following instance of strict inequality in (2) will soon be used.
THEOREM 2. 
if 0 < f − s < 1/2 < f + s < 1 and w < 1/2.
Proof: There are two cases corresponding to Cases 3 and 4 in the proof of Theorem 1. Assume first that f ≤ 1/2. The calculation for Case 3 of Theorem 1 shows that 
is strictly greater than a quantity which, in view of Theorem 1, is now known to be nonnegative. A dual argument applies for f ≥ 1/2.
A stake s for the fortune f is conserving if the lottery θ that wins s with probability w and loses s with probability 
is such that [f + θ]conserves U at f, that is, if [f + θ]Q = Q(f). Thus the stake s is conserving at f if and only if
which clearly requires that f + s ≤ 1 or f − s ≥ 1.
What are the conserving stakes available at f in (0, 1), that is, the s ≤ min 
which satisfy (1)? The functional equations (2.1) imply that, for f ≤ 1/2, s = f is conserving, whereas, for 1/2 ≤ f ≤ 1, 
is conserving. Let S1(f) = f for f ≤ 1/2 and 
for 1/2 ≤ f ≤ 1, that is, S1(f) = min 
. Then S1(f) is a conserving stake for the fortune f. The following observation leads to the discovery of a sequence of conserving stakes. Suppose that a gambler with a fortune f < 1/2 adopts a strategy σ that first maximizes the probability of reaching 1/2 before bankrupting the gambler, and then, if he has had the good fortune to reach 1/2, goes on to maximize the probability of reaching 1. The probability of reaching 1/2 from f in any casino with a casino function U is at least U(2f). Thus the probability of reaching 1 under σ is at least
Therefore, σ is itself an optimal strategy.
Since red-and-black is an inclusive casino, contraction by the factor 1/2 of an optimal strategy at 2f is optimal for reaching the fortune 1/2 from f. Therefore, 
is a conserving stake for f when the space of fortunes is [0, 1/2] and the goal is 1/2. But as the preceding discussion shows, it is therefore also a conserving stake for the original red-and-black casino. Moreover, if 2f > 1/2, S2(f) is unequal to S1(f). Thus bold strategies are not the only optimal ones.
Suppose next that a gambler with a fortune f, 1/2 ≤ f ≤ 1, adopts a strategy σ that first maximizes the probability of attaining 1 before the fortune becomes as low as 1/2, and then, if the fortune drops to 1/2, goes on to maximize the probability of reaching 1 by adopting the bold strategy. The probability of reaching 1 under such a strategy is 
. Therefore, this strategy is also optimal. Again, since S1 is a conserving-stake-valued function for red-and-black, 
is conserving for [1/2, 1] as the space of fortunes with 1 as the goal. Therefore, for each f in the unit interval, S2(f) is a conserving stake for the original red-and-black casino.
Define Sn+1 by induction: 
or 
according as f ≤ 1/2 or f ≥ 1/2. The argument already given shows inductively that, for each n and f, Sn(f) is a conserving stake.
It is easy to verify that Sn+1(f) is the distance between f and the closest nth-order binary not necessarily different from f. Therefore Sn(f) is a monotone decreasing sequence of nonnegative functions that converges to 0. For any f other than a binary rational, Sn(f) > 0 for all n. Therefore, for any such f there are arbitrarily small, but positive, conserving stakes, though naïve intuition about the law of large numbers might erroneously suggest that small stakes are to be avoided. It is convenient to let S∞(f) = 0; for 0 is a trivial conserving stake.
There are in fact no conserving stakes other than Sn(f) and 0. Suppose otherwise. Then equality holds in (1) for some f and some positive s other than any Sn(f). Choose such a pair f, s for which s is more than half as large as the supremum of all such s. Then f − s and f + s are strictly separated by 1/2. Why? If, for example, f + s ≤ 1/2, an application of the first of the functional equations (2.1) implies that equality holds in (1) for 2f and 2s. Furthermore, 2s is not Sn(2f) for any n, for otherwise 
, contrary to the assumption that s ≠ Sn+1(f). A similar argument is applicable if 1/2 ≤ f − s; so f − s < 1/2 < f + s. Furthermore, f − s is not 0, nor is f + s equal to 1, for otherwise s = S1(f). But for such f and s, as Theorem 3.2 states, equality does not hold in (1). Therefore s is a conserving stake for f if and only if s = Sn(f) for some n. This is recorded in the next theorem.
THEOREM 1. For 0 ≤f − s ≤f ≤ 1 and w ≤ 1/2,
Moreover, for 0 < w < 1/2, equality holds in (2) if and only if s = Sn(f) for some n (including n = ∞).
Now that all conserving stakes and hence all conserving gambles have been determined, it is easy to characterize all optimal strategies.
THEOREM 2. A strategy σ, available at f in (0, 1) in a subfair red-and-black casino, is optimal if and only if, immediately following each of the at most denumerably many partial histories of positive σ-probability (including the vacuous one), none but conserving stakes are used, and stagnation does not occur after any such partial history except at 0 and 1.
Proof: Theorems 3.4.1 and 3.5.3 apply. The main point to check is that, if σ satisfies the hypotheses, then the probability that fm is a binary rational converges to 1 as m approaches ∞; once fm is a binary rational, each succeeding gamble reduces the order with certainty, until order 0 is attained.
(The situation for fair red-and-black casinos is clearly quite different from that described in Theorem 2 for subfair ones.)
A stationary family of strategies 
is tantamount to a stake-valued function s, with s(f) ≤ f. The characterization of optimal strategies in Theorem 2 implies that s corresponds to a stationary family of optimal strategies if and only if, for every f in (0, 1), s(f) is a conserving stake for f other than the trivial stake 0, and for every f ≥ 1, s(f) ≤ f − 1.
COROLLARY 1. A stake-valued function s determines a stationary family of optimal strategies if and only if, for every f in (0, 1), there is a nonnegative integer n = n(f) less than the order of f such that s(f) is the distance between f and the closest nth-order binary rational (and s(f) ≤ f − 1 for f ≥ 1).
There clearly exist wildly irregular (for example, non-Lebesgue-measurable), conserving-stake-valued functions as well as many that are only mildly irregular. However, only the bold one (corresponding to n(f) = 0) is continuous, as is easily seen.
As Aryeh Dvoretzky pointed out to us, even under the requirement that play be limited to m moves, bold play is still optimal. He pointed out also that this is not true of the somewhat more general casinos to be studied in the next chapter. Here is his argument for the red-and-black casino.
Let Qm(f) be the probability of winning in m moves with the bold strategy beginning at f. The Qm satisfy functional equations like (2.1):
for 0 ≤ f ≤ 1.
The proof given for the inequality (3.1) also shows that
for binary rationals f and s if 0 ≤ f − s ≤ f ≤ 1.
That (2) holds also when f and s are not binaries may be seen as follows. Let fm denote for the moment the largest binary rational of order m that does not exceed f. Then
This fact and (1) imply by induction that
Finally, compute thus,
The first inequality depends on the arithmetic inequality,
which is true simply because the right side of (6) is a binary of order m + 1 not larger than f. Thus (2) holds without the restriction to binaries. But equality holds in (2) for the bold value of s; so
Let Um(f) be the optimal probability of winning in m moves beginning at f. Then the sequence Um satisfies (7) in the role of Qm. Since U1 = Q1, it follows by induction that Qm = Um for all m, and bold strategies are seen to be optimal for limited playing time. Of course this immediately implies that Qw = Uω This and Theorem 4.5.1 provide a variant of the proof of Theorem 3.1 given earlier that Q = U.
Consider a casino in which the only game available is to bet on the outcome of a fair coin. A fixed fraction of the stake is collected as tax (or entrance fee); the remainder is paid back twice over if and only if the coin falls heads. It is convenient to denote the ratio of the possible gain to the stake by 
, where 
. Thus Γ(f) = {[sρ′ + f]: s ≤ f}, where 
.
Once more the bold strategy—that for which s(f) is the minimum of f and 
for f in (0, 1), and s(f) = 0 elsewhere, so that every bet has a chance to be decisive—is interesting. The utility of the bold strategy satisfies an equation closely analogous to (2.1).
which is tantamount to the pair of equations,
Equation (1) has at most one bounded solution, because the discrepancy between any pair of solutions is doubled by passing from f to f/r or to 
, whichever lies in [0, 1]. But if Qw is the function defined by (2.1), then clearly 
satisfies (1) so 
.
The fundamental inequality for Qr with r > 1/2 is obtained by applying the duality (2.3) to the inequality (3.1).
for 0 ≤ f − s ≤ f + s ≤ 1 and 1 ≥ r ≥ 1/2.
This is equivalent to
for 0 ≤ g − tr ≤ g ≤ 1.
Therefore the bold strategy is optimal for this casino, and R is its casino function. It is also easy to describe all optimal strategies and to show that bold strategies are optimal even for limited playing time.
To do the latter, prove that
where Rm(f) is the probability of success in m moves with the bold strategy beginning at f, and R(f)m is the largest binary rational of order m that does not exceed R(f). Since any further details would bore you and us, we move on to the next chapter and a more general type of casino.
