Page:EB1911 - Volume 22.djvu/400

man must possess some fortune, or its equivalent, in order to live. To estimate now the value of a moral expectation. Suppose a person whose fortune is a to have the chance p of obtaining a sum α, q of obtaining β, r of obtaining γ, &c., and let

p + q + r + . . . = 1,

only one of the events being possible. Now his moral expectation from the first chance—that is, the increment of his moral fortune multiplied by the chance—is

$pk\left\{\log {\frac {a+\alpha }{h}}-\log {\frac {a}{h}}\right\}=pk\log {(a+\alpha )}-pk\log {a}$.

Hence his whole moral expectation is^[1]

E = kp log (a + α) + kq log (a + β) + kr log (a + γ) + . . . - k log a;

and, if Y stands for his moral fortune including this expectation, that is, k log (a/h) + E, we have

Y = kp log (a + α) + kq log (a + β) + . . . - k log h.

To find X, the physical fortune corresponding to this moral one, we have

Y = k log X − k log h.

Hence

X = (a + α)^p(a + β)^q(a + γ)^r,

and X − a will be the actual or physical increase of fortune which is of the same value to him as his expectation, and which he may reasonably accept in lieu of it. The mathematical value of the same expectation is^[2]

pα + qβ + rγ + . . .

70. Gambling and Insurance.—These formulae are employed, often with the aid of refined mathematical theorems, to demonstrate received propositions of great practical importance: that in general gambling is disadvantageous, insurance beneficial, and that in speculative operations it is better to subdivide risks—not to “have all your eggs in one basket.”

71. These propositions may be deduced by the use of a formula which perhaps keeps closer to the facts: viz. that utility or satisfaction is a function of material goods not definitely ascertainable, defined only by the conditions that the function continually increases with the increase of the variable, but at a continually decreasing rate (and some additional postulate as to the lower limit of the variable), say y = ψ(x) (if x as before denotes physical fortune, and y the corresponding utility or satisfaction); where all that is known in general of ψ is that ψ′(x) is positive, ψ″(x) is negative; and ψ(x) is never less, x is always greater than zero. Suppose a gambler whose (physical) fortune is a, to have the chance p of obtaining a sum α and the chance q( = 1 − p) of losing the sum β. If the game is fair in the usual sense of the term pα = qβ. Accordingly the prospective psychical advantage of the party is pψ(a + α) + qψ(a − β) = pψ(a + α) + qψ{a − (p/q)α}, say y_α. When α is zero the expression reduces to the first state of the man, ψ(a), say y₀. To compare this state with what it becomes by the gambling transaction, let α receive continually small increments of Δα. When α is zero the first differential coefficient of (y_α − y₀), viz. pψ′(a) − pψ′(a), = 0. Also the second differential coefficient, viz. pψ″(a) + p²/qψ″(a), is negative, since by hypothesis ψ″ is continually negative. And as α continues to increase from zero, the second differential coefficient of (y_α − y₀), viz. pψ″(a + α) + p²/qψ″(a + p/qα), continues to be negative. Therefore the increments received by the first differential coefficient of (y_α − y₀) are continually negative; and therefore (y_α − y₀) is continually negative; y_α < y₀^[3] for finite values of α (not exceeding qa/p).^[4]

72. To show the advantage of insurance, let us suppose with Morgan Crofton^[5] that a merchant, whose fortune is represented by 1, will realize a sum ε if a certain vessel arrives safely. Let the probability of this be p. To make up exactly for the risk run by the insurance company, he should pay them a sum (1 − p)ε. If he does, his moral fortune becomes, according to the formula now proposed ψ(1 + p_ε), since his physical fortune is increased by the secured sum ε, minus the payment (1 − p)ε; while if he does not insure it will be pψ(1 + ε) + (1 − p)ψ(1). We have then to compare ψ(1 + p_ε), say y₁, with pψ(1 + ε) + (1 − p)ψ(1), say y₂. By reasoning analogous to that of the preceding paragraph it appears that (y₂ − y₁) is zero when ε = 0 and continually diminishes as ε increases up to any assigned finite (admissible) value. Similarly it may be shown that it is better to expose one's fortune in a number of separate sums to risks independent of each other than to expose the whole to the same danger. Suppose a merchant, having a fortune, has besides a sum ε which he must receive if a ship arrives in safety. Then, if the chance of the ship arriving = p, and q = 1 − p, his prospective advantage is pψ(1 + ε) + qψ(1). Now instead of exposing the lump sum ε to a single risk, let him subdivide ε into n equal parts, each exposed to an independent equal risk (q) of being lost. As n is made larger^[6] it becomes more and more nearly a certainty that he will realize p_ε out of the total ε exposed to risk. Therefore his condition (in respect of the sort of advantage which is under consideration) will be approximately ψ(1 + p_ε). Then we have to compare ψ(1 + p_ε), say y₁, with pψ(1 + ε) + qψ(1), say y₂. By reasoning analogous to that which has been above employed—observing that (p − p²)ψ″(1) is negative for all possible values of p—we conclude that y₂ < y₁.

73. The Petersburg Problem.—The doctrine of “moral fortune” was first formulated by Daniel Bernoulli^[7] with reference to their celebrated “Petersburg Problem,” which is thus stated by Todhunter^[8]: “A throws a coin in the air: if head appears at the first throw he is to receive a shilling from B, if head does not appear until the second throw he is to receive 2s., if head does not appear until the third throw he is to receive 4s., and so on, required the expectation of A.” So many lessons are presented by this problem that there has been room for disputing what is the lesson. Laplace and other high authorities follow Daniel Bernoulli. Poisson finds the explanation in the fact that B could not be expected to pay up so large a sum. Whitworth, who regards the disadvantage of gambling as consisting mainly in the danger of becoming “cleaned out,”^[9] finds this moral in the Petersburg problem. All have not noticed what some regard as the principal lesson to be obtained from the paradox: viz. that a transaction which cannot be regarded as one of a series—at least a “cross-series”^[10]—is not subject to the general rule for expectations of advantage whether material or moral.^[11]

Section IV.—Geometrical Applications.

74. Under this head occur some interesting illustrations of principles employed in the preceding sections; in particular of a priori probabilities and of the relation between probability and expectation.

75. Illustrations of a priori Probabilities.—The assumption which has been made under preceding heads that the probability of certain alternatives is approximately equal appears to rest on evidence of much the same character as the assumption which is made under this head that one point in a line, plane or volume is as likely to occur as another, under certain circumstances. Thus consider the proposition: if a given area S is included within a given area A, the chance of a point P, taken at random on A, falling on S is S/A. In a great variety of circumstances such a size can be assigned to the spaces, and “taking at random” can be so defined that the proposition is more or less directly based on experience. The fact that the points of incidence are equally distributed in space is observed, or connected by inference with observation, in many cases, e.g. raindrops and molecules. There is a solid substratum of evidence for the premiss employed in the solution of problems like the following: On a chess-board, on which the side of every square is a, there is thrown a coin of diameter b (b < a) so as to be entirely on the board, which may be supposed to have no border. What is the probability that the coin is entirely on one square?^[12] The area on which the coin can fall is (8a − b)². The portion of the area which is favourable to the event is 64(a − b)². Therefore the required probability is (a − b)²/(a − 1/8b)².

76. Random Lines.—Speculative difficulties recur when we have to define a straight line taken at random in a plane; for instance, in the following problem proposed by Buffon.^[13]

A floor is ruled with equidistant parallel lines; a rod, shorter than the distance between each pair, being thrown at random on the floor, to find the chance of its falling on one of the lines. The problem is usually solved as follows:—

Let x be the distance of the centre of the rod from the nearest line, θ the inclination of the rod to a perpendicular to the parallels, 2a the common distance of the parallels, 2c the length of rod; then, as all values of x and θ between their extreme limits are equally probable, the whole number of cases will be represented by

$\int _{0}^{a}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\,dxd\theta =\pi a.$

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1. It is important to remark that we should be wrong in thus adding the expectations if the events were not mutually exclusive. For the mathematical expectations it is not so.
2. This paragraph is taken from Morgan Crofton's article on “Probability,” in the 9th edition of the Ency. Brit.
3. Cf. Marshall, Principles of Economics, Mathematical Appendix, note ix.
4. Or should we rather say, not exceeding the limit at which ψ(a − pα/q) becomes 0? (The value of ψ(0) may be regarded as −∞.) Neither of the proposed limitations materially affects the validity of the theorem.
5. Loc. cit. par. 25.
6. See above, par. 25 (James Bernoulli's theorem).
7. Specimen theoriae novae de mensura sortis (16), translated (into German) with notes by Pringsheim (1906).
8. Op. cit. art. 389.
9. Choice and Chance, pp. 211, 232. The danger of a party to a game of chance being “ruined” (by losing more than his whole fortune), which forms a separate chapter in some treatises, is readily deducible from the theory of deviations from an average which will be stated in pt. ii.
10. Above, par. 5.
11. Above, par. 20.
12. Whitworth, Exercises, No. 500.
13. Cf. Morgan Crofton, loc. cit.