How to Gamble If You Must: Inequalities for Stochastic Processes (Dover Books on Mathematics) by Dubins Lester E. & Savage Leonard J

Author:Dubins, Lester E. & Savage, Leonard J. [Dubins, Lester E.]
Language: eng
Format: azw3
Publisher: Dover Publications
Published: 2014-08-03T16:00:00+00:00

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.

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