and 
. If Γ1 ⊃ Γ ⊃ Γ2 or Γ2 ⊃ Γ ⊃ Γ1, then Γ is between Γ1 and Γ2.This chapter is about some general principles that apply to gambling houses on any set F of fortunes, and the next is about some general principles that apply only when F is a set of real numbers.
As a preliminary, certain facts and technical terms so natural that they might almost go unmentioned are catalogued. Some of them have been encountered in earlier sections.
If Γ1 and Γ2 have the same space of fortunes F and Γ1(f) ⊃ Γ2 (f) for all f in F (or Γ1 ⊃ Γ2, for short), then Γ1 is a superhouse of Γ2, and Γ2 is a subhouse of Γ1 If Γ1 ⊃ Γ2, the same relation subsists between the leavable closures and . If Γ1 ⊃ Γ ⊃ Γ2 or Γ2 ⊃ Γ ⊃ Γ1, then Γ is between Γ1 and Γ2.
The union 
of a nonvacuous class of gambling houses on a fixed F is defined by
for all f in F. It is a superhouse of each Γα, and 
= 
. Typically, many strategies available in a union are not available in any one term of the union.
“Intersection” and “∩” can almost be substituted for “union” and “∪” in the preceding paragraph, but some differences must be mentioned. Since we have chosen to require, for a gambling house Γ, that Γ(f) be nonvacuous, the intersection of a class of gambling houses can fail to be a gambling house. It always is a gambling house if the elements of the class are leavable, and the intersection of a class is leavable if and only if the elements of the class are. A strategy is available in ∩α Γα at f if and only if it is available in each Γα at f.
Though the gambling house Γ permits the gambler with fortune f to use immediately only the gambles in Γ(f), the gambler can, in effect, generally build up or compose other gambles for himself in the course of time. Since time here is at no premium, these composite gambles are, to all intents and purposes, as available to the gambler as are the elements of Γ(f) themselves. If, for example, γ ∈ Γ(f) and γg ∈ Γ(g), then the new gamble γ′ for which each bounded real-valued υ has the expectation
is effectively available at f. (This γ′ is the distribution of the terminal fortune of an available two-move policy with initial fortune f. The expectation of υ under γ′ is the expectation under γ of the conditional expectation of υ under the conditional distribution γg.)
More generally, if π = (σ, t) is a policy available to the gambler at f, he can compose for himself γ(π), the distribution of ft under σ. Recapitulating (2.9.4),
for all bounded functions υ.
Possibly the operation (2), not to mention its special case (1), does not actually augment Γ(f) for any f. That is, perhaps γ(π) ∈ Γ(f). In this case, Γ is closed under composition. Even the γ′ constructed according to (1) provide an adequate test of closure under composition, as can be shown easily with or without the help of (2.9.7):
THEOREM 1. If, for each f, every γ′, in the sense of (1), is included in Γ(f), then Γ is closed under composition.
A simple example of a gambling house Γυ closed under composition is constructed from any bounded, real-valued υ with F as its domain by letting
this is the full house under υ. Reversal of the inequality in (3) and replacement of the inequality by equality also lead to houses closed under composition—the full house under −υ and the υ-conserving house. The important special case of a house conserved by an exponential function was introduced and exploited in (de Finetti 1939) and is studied in Section 8.7.
An intersection of any number of houses closed under composition is itself closed under composition. In particular, the intersection of any class of full houses is closed under composition and is leavable. A conserving house, for instance, can be viewed as the intersection of two full houses. Full houses will be discussed frequently again.
Any house Γ determines its composition closure Γc, the intersection of all superhouses of Γ that are closed under composition. The composition closure of Γ can be defined much more constructively. Let Γ′′(f) = γ(π): π is available in Γ at f}, where γ (σ, t) is the distribution under σ of ft as in (2).
Proof: Plainly, Γc ⊃ Γ′′. The reverse inclusion holds if Γ′′ is closed under composition. In view of Theorem 1, it is therefore necessary only to show that γ′″ ∈ Γ′′(f), where 
, whenever γ′′ ∈ Γ′(f), and 
for all g. This is done with the help of (2.9.7). 
If Γ is closed under composition, then Γ = Γc, and whatever can be achieved by a policy available at f can also be achieved by a single γ available at f, which sometimes (as in Section 9.2) simplifies the evaluation of U.
THEOREM 3. If Γ is leavable and closed under composition, then U(f) = υ(f) = sup γu over γ in Γ(f).
A function υ from fortunes to real numbers determines a set υ° of gambles,
the gambles permitted by υ. Such permitted sets of gambles are introduced here because of the importance of some permitted houses, that is, houses in which each Γ(f).is the set of gambles permitted by some υf.
A full house Γυ in the sense of (2.3) is, of course, the permitted house with υf = υ − υ(f), and these are the most important examples of permitted houses. Later sections will have much to say about full houses and a little about other permitted houses. The present section concerns mainly the relation between two functions υ and w when the permitted set υ° is a subset of w°, or w is (at least) as permissive as v.
Since unbounded functions υ are often of interest, a clear and appropriate convention is needed for the meaning of the inequality γυ ≤ 0. For this and later purposes, γυ is extended to certain unbounded υ by the following familiar sequence of definitions:
If υ is nonnegative, γυ = lim (γ min (υ, c)) as the number c approaches ∞; with ∞ admitted as a possible value, γυ always exists for nonnegative υ. If υ is nonpositive, γυ = −γ (− υ). For an arbitrary υ, γυ = γυ+ + γυ−, provided at least one of the two terms is finite, where, as usual, υ+ = max (υ, 0) and υ− = min (υ, 0).
In particular, γυ ≤ 0 if and only if γυ+ < ∞ and γυ+ ≤ |γυ−| where |γυ−|may be ∞|.
(a) γυ ≤ 0 for all γ (that is, every gamble belongs to υ°) if and only if υ ≤ 0.
(b) γυ ≤ 0 for no γ (that is, υ° is vacuous) if and only if υ is uniformly positive (that is, there is an 
for which 
).
(c) δ(g) ∈ υ° if and only if υ(g) ≤ 0.
(d) If f+ and f− are fortunes for which υ(f+) > 0 and υ(f−) < 0, then the two-point gamble γ for which
belongs to υ°, and γυ = 0.
(e) If υ is unbounded from below, there is, for every f, a two-point gamble γ in υ° with γ{f} arbitrarily close to 1.
(f) If υ is bounded from below, z = inf υ ≤ 0, and υ(g) > 0, then
and, for all γ ∈ υ° and 
,
Of course, each part of Theorem 1 has practical implications for a gambler constrained to choose a gamble from υ°. For example, part (e) shows that, if υ is unbounded from below and the gambler’s utility u is bounded, then the gambler can practically have his heart’s desire with a single two-point gamble in υ°. Part (f) bounds what he can achieve in υ° when υ is bounded from below.
If υ ≤ aw for some nonnegative constant a, then γw ≤ 0 plainly implies that γυ ≤ 0; in short, υ° ⊃ w°, or υ is as permissive as w. Similarly, if υ = aw for some positive a, then υ° = w°; υ and w permit the same gambles. Several paragraphs lead to the next two theorems asserting almost the converses of these facts.
Evidently, from part (a) of Theorem 1, if w ≤ 0, then υ° ⊃ w° if and only if υ ≤ 0. The condition that υ ≤ 0 can, though artificially, be written υ ≤ 0w.
Suppose, for completeness, that w is nowhere negative. Then γw ≤ 0 if and only if γ{f: w(f) > δ} = 0 for each positive δ. If for each positive there is a positive δ such that 
implies w(f) > δ, then γw ≤ 0 implies γυ ≤ 0, or υ° ⊃ w°. Conversely, if, for some f, υ(f) > 0 and w(f) = 0, δ(f) is in w° but not in υ°; similarly, if there is a sequence of distinct fortunes fi with w(fi) ≤ i−1 but 
, a diffuse gamble on the sequence {fi} belongs to w° but not to υ°. In summary, if w is nowhere negative, then υ is as permissive as w if and only if, for every positive , there is a positive δ such that 
implies w(f) > δ. (When discussion is restricted to countably additive γ on some sigma-field with respect to which υ and w are measurable, it is necessary and sufficient that υ(f) > 0 imply w(f) > 0.)
Example 1. On [0, 1], let u(f) = f(1 − f), υ(f) = f2, and w(f) = 0 or 1 according as f is 0 or positive. Then u° ⊃ υ° ⊃ w°, and neither inclusion can be reversed (though, in a countably additive setting, the second one could be). In this example there is no nonnegative a for which u ≤ aυ.
Suppose, finally, that the two sets of fortunes, F+ where w(f) is positive and F− where it is negative, are both nonvacuous. For any fortunes f+ and f− in F+ and F−, consider that two-point gamble γ carried by those two fortunes for which γw = 0. A necessary condition for υ° ⊃ w° is that γυ ≤ 0. Equivalently, −w(f−)υ(f+) + w(f+)υ(f-) < 0, or
If these necessary conditions obtain for all f+ ∈F+ and F−∈F−, then the range of neither side of (5) is vacuous, that of the right side is nonnegative, and there is some nonnegative number a between the supremum of the range of the left side and the infimum of the range of the right side. For such an a, υ(f) ≤ aw(f) if w(f) ≠ 0. If w(f) = 0, δ (f)w = 0, and δ(f)υ ≤ 0 implies υ(f) ≤ 0. So υ ≤ aw. Consequently, the condition that υ ≤ aw, which is always sufficient for υ° ⊃ w°, is, in this case, also necessary.
Condensing the three possibilities:
THEOREM 2. γw ≤ 0 implies γυ ≤ 0, or υ is as permissive as w, if and only if one of the following two conditions obtains:
(a) w is nowhere negative, and for every positive there is a positive δ such that 
implies w(f) > δ.
(b) w is somewhere negative, and there is a nonnegative a such that υ(f) ≤ aw(f) for all f.
(The hypothesis of Theorem 2 says that no linear functional separates υ from the cone generated by w and all nonpositive functions, so that υ is in the “natural closure” of the cone, and part (b) says that then υ must be in the cone itself. However, a rigorous formulation and proof along this line would not fall under the most elementary general theories; the need for part (a) seems symptomatic of a need for special handling.)
Except for two extreme and not very important cases, υ° = w° if and only if υ and w are positive multiples of each other:
THEOREM 3. γυ ≤ 0 is equivalent to γw ≤ 0, or υ° = w°, if and only if one of the following three conditions obtains:
(a) υ and w are nowhere positive.
(b) Condition (a) of Theorem 2 applies in both directions.
(c) For some positive number a, υ = aw.
Proof: If υ or w is nowhere positive, part (a) of Theorem 1 establishes condition (a) of the present theorem.
If υ and w are nowhere negative, Theorem 2 implies condition (b) of the present theorem.
If w, say, is somewhere positive and somewhere negative and υ° = w°, then υ ≤ aw for some nonnegative a, according to Theorem 2. The possibility that υ is nowhere positive is excluded, and so a is positive. The possibility that υ is nowhere negative is now also excluded; so, according to Theorem 2, w ≤ bυ for some nonnegative b. Since υ ≤ aw ≤ abυ and since υ(f) can be positive and can be negative, ab = 1, and υ = aw.
It is of some interest for gambling problems to determine, for a bounded function u, the supremum of γu over the γ permitted by a function w. This is plainly the least number b (or − ∞ if and only if w is uniformly positive) for which γu ≤ b for all γ in w°, that is, the least number b for which (u − b) is as permissive as w. According to Theorem 2, there are two cases to consider.
First, suppose w ≥ 0. For which numbers b′ does (u − b′) satisfy condition (a) of Theorem 2? For those such that for each positive there is a positive δ such that wherever 
, w(f)) > δ; that is, numbers as large as
The number (6) is therefore b in this case. If competition is restricted, in a countably additive spirit, to gambles for which γw ≤ 0 implies that γ{f: w(f) > 0} = 0 (as when γ is discrete), then (6) is replaced by
And this value can obviously be approached by means of one-point gambles.
Second, if w is somewhere negative, then b is the least number for which there is a nonnegative number a such that u ≤ aw + b. The discussion on which Theorem 2 is based shows that the value of b in this case is not affected by restricting competition to one-point and two-point gambles.
The description of b in the second case actually applies in the first case as well. The b thus described is obviously large enough. To see that it is not too large, suppose that w ≥ 0 and b′ exceeds the number (6). There is a δ such that, wherever u(f) > b′, w(f) > δ. Suppose a ≥ sup (u/δ). Then u ≤ aw + b′.
THEOREM 4. For a bounded u, sup γu over γ in w° is the least b such that, for some nonnegative number a, u ≤ aw + b. If w is nonnegative, (6) is an alternative evaluation. In this special case, (6) must be replaced by (7) if the competition is restricted to gambles for which γ{f: w(f) > 0} =0; and this latter bound remains valid even if the competition is restricted still further to one-point gambles. Except in the special case, neither the sup γu nor b is decreased if competition is restricted to one-point and two-point gambles.
The following modifications of Theorems 2, 3, and 4 are left to the interested reader.
THEOREM 5. γw = 0 implies γυ < 0 if and only if one of the following two conditions obtains:
(a) w is nowhere negative or nowhere positive, and for every positive there is a positive δ such that implies |w(f)| > δ.
(b) w is somewhere positive and somewhere negative, and there is a real number a for which υ ≤ aw.
If υ is nonnegative (positive) at some f where w is positive, then a in condition (b) must be nonnegative (positive); if υ is nonpositive wherever w is negative, then a can be nonnegative. Therefore, if υ (f+) ≥ υ(f−) wherever w(f+) > 0 > w(f−), then a can be nonnegative; so if υ is of the form υ′(w) with υ′ nondecreasing on the range of w, then a can be nonnegative.
COROLLARY 1. γw = 0 implies γυ = 0 if and only if one of the following two conditions obtains:
(a) w is nowhere negative or nowhere positive, and for every positive there is a positive δ such that 
implies |w(f)| > δ.
(b) w is somewhere positive and somewhere negative, and there is a real a for which υ = aw.
COROLLARY 2. γw = 0 is equivalent to γυ = 0 if and only if condition (a) of Corollary 1 applies in both directions or condition (b) of Corollary 1 applies with a unequal to 0.
Every part of Theorem 4 can be easily adapted to the set of gambles for which γw = 0, but only the main counterparts are explicitly expressed in the next theorem.
THEOREM 6. For a bounded u sup γu over {γ: γw = 0} is the least b such that, for some real number a, u ≤ aw + b. If u is a nondecreasing function of w, then it is enough to admit nonnegative values of a. If w is nonnegative or nonpositive, (6) applied to |w| in place of w is an alternative evaluation of sup γu (whether u is a nondecreasing function of w or not).
According to (2.3), Γυ, the full house under υ, is the house for which Γυ(f) = (υ − υ(f))°. The assumption in Section 2 that υ is bounded was made for momentary simplicity only and is now dropped. Clearly, Γυ is the largest house for which υ is excessive. If both υ and − υ are excessive for a house Γ so that γυ = υ(f) for all f and all γ Γ(f), then Γ conserves υ. The largest house that conserves υ, the intersection of Γυ and Γ−υ, is the υ-conserving house defined in Section 2.
If υ is excessive for Γ, then all aυ + b, with b real and a positive, also are. Therefore, υ and aυ + b define the same full and conserving houses. The converse is also true.
THEOREM 1. The full house under υ is the full house under w if and only if
for some real b and some positive real a. The υ-conserving house is the w-conserving house if and only if (1) holds for some real a and b with a ≠ 0.
Proof: As already mentioned, the “if” part of the first assertion is obvious. To prove the “only if” part, consider first the situation in which υ or w takes on at least three values. There is no further loss of generality in assuming that, for some f, υ − υ(f) is both somewhere positive and somewhere negative. As follows immediately from Theorem 3.3, if Γυ = Γw, then
for some positive a. This is equivalent to (1) and disposes of the situation in which υ or w takes on at least three values. The remaining possibilities for the first assertion are trivial. Proof of the assertion about conserving houses is left to the reader.
The assertion of Theorem 1 about conserving houses is close to (Hardy, Littlewood, and Polya 1934, Theorem 83, p. 66).
Two simple principles:
THEOREM 2. If u ≤ aυ + b, then U ≤ aυ + b for any subhouse Γ of the υ-conserving house and, provided a > 0, for any subhouse of the full house under υ as well.
Proof: Immediate from the basic Theorem 2.12.1.
THEOREM 3. If u coincides with aυ + b at two fortunes, say f and h, and if Γ is a superhouse of the υ-conserving house, then U(g) ≥ aυ(g) + b at every g for which aυ(g) + b is between u(f) and u(h).
Proof: The assertion is trivial unless the numbers u(f), u(h), and aυ(g) + b are all different, in which case there is no loss of generality in assuming u(f) < aυ(g) + b < u(h). Then, according to part (d) of Theorem 3.1, there is a υ-conserving two-point gamble γ in Γ(g) carried on f and h such that γu = aγυ + b = aυ(g) + b, which proves the assertion. Explicitly,
Example 1. Let υ be any function on [0, 1] that attains its minimum at 0 and its maximum at 1, and let u(1) = 1 and u(f) = 0 for f in [0, 1). According to Theorem 1, no generality is lost in assuming that υ(0) = 0 and υ(1) = 1. According to the first principle, U is at most υ for any subhouse of the full house Γ; according to the second, this bound is achievable, even in the υ-conserving house, by one-move, two-point policies.
Example 2. Let υ be defined on (−∞, 0], with 0 ≤ υ ≤ 1, υ(0) = 1, and with inf υ = 0. This contrasts with Example 1 only in that a partially diffuse gamble may be needed where a two-point gamble could be used before. If partially diffuse gambles are excluded, there remain arbitrarily good one-move, two-point policies; there are still optimal strategies but no optimal policies.
Example 3. On the interval [0, 10], let υ be convex and increasing, and let u(f) be the largest integer that is at most f. For any Γ between the υ-conserving house and the full house under υ and for f in [n, n + 1] (n = 0, · · · , 9),
as the two principles show.
This example, unusual in this book in that it solves a problem with a many-valued utility u, admits a generalization that, in a sense, solves all gambling problems for conserving and full houses. The rest of this section is about this generalization (not used in later sections). Theorems 2.3, 3.4, and 3.6 are the tools for the generalization.
THEOREM 4. If Γ is the full house under υ, then
over those numbers a and b for which a ≥ 0 and aυ + b ≥ u. At any f at which υ attains its minimum,
THEOREM 5. If Γ is the υ-conserving house, then
over those numbers a and b for which aυ + b ≥ u. If u is a non-decreasing function of υ, then it is enough to admit nonnegative values of a. At any f at which υ attains its maximum or minimum,
It is easy to see how (6) and (7) have to be modified if Γ is replaced by its subhouse of one-point and two-point gambles.
Some interesting gambling houses are constructed from a family of functions υf from the fortunes to real numbers, one function for each f, by letting Γ(f) = {γ: γυf ≤ 0}. In the terminology of Section 3, Γ is a permitted house, and 
. Full houses are interesting examples of permitted houses, as are income-tax casinos (defined in Section 9.2), but later sections rely so little on the present section that some may prefer to skip or defer it.
Practically any family υf permits some gambling house Γ. What is needed is simply that no υf be uniformly positive so that every 
is nonempty. However, υf(f) must be nonpositive if Γ is to be leavable.
Immediately from Theorem 3.3:
THEOREM 1. If Γ is permitted by υf and υf assumes both positive and negative values for each f, then Γ is also permitted by 
if and only if 
for some positive a(f).
The solution of gambling problems often hinges on finding appropriate excessive functions.
THEOREM 2. Suppose that Γ is permitted by υf, a(f) ≥ 0, and b(f) is real. If, for all f and g, b(g) ≤ a(f)υf(g) + b(f), then b is excessive for Γ.
Theorem 2 will be illustrated in Section 9.2. We are largely in the dark as to when nontrivial functions a and b exist.
A permitted house can be closed under composition, as has already been illustrated by the case of full houses.
THEOREM 3. If υf − υf(g) is at least as permissive as υg for all f and g, then the house permitted by the family υf is closed under composition.
Proof: Suppose γυf ≤ 0 and γgυg ≤ 0 for all g. Under these assumptions and the hypothesis of the theorem,
Now apply Theorem 2.1.
COROLLARY 1. If υf(f) ≤ 0 and υf − υf(g) is at least as permissive as υg for all f and g,
Under a mild hypothesis, Theorem 3 has a converse.
THEOREM 4. If each υf assumes both positive and negative values and if υf(f) ≤ 0 for all f, then the house Γ permitted by υf is closed under composition if and only if υf − υf(g) is at least as permissive as υg for all f and g.
Proof: The “if” clause is more than covered by Theorem 3.
As for the “only if” clause, let γ be a two-point gamble carried by, say, h and g, such that γυf = 0 and γ{g} > 0; such a gamble exists for an arbitrary f and g under the preliminary hypotheses of the theorem. Let γe = δ(e) if e ≠ g, and let γg be any gamble for which γgυg ≤ 0. Thus γ ∈ Γ(f) and γe ∈ Γ(e) for all e. If Γ is closed under composition, then γ′ = ∫ γe dγ(e) is an element of Γ(f), and
Since γ{g} > 0, γg(υf − υf(g)) = − υf(g) + γgυf ≤ 0.
Theorem 3.4 implies the final theorem and evident refinements of it.
THEOREM 5. If the house permitted by υf is leavable and closed under composition, U(f) is the infimum of those numbers b for which u ≤ aυf + b for some nonnegative a.
