§ 17. The most interesting class of attempts to prove the disadvantages of gambling appeal to what is technically called ‘moral expectation’ as distinguished from ‘mathematical expectation.’ The latter may be defined simply as the average money value of the venture in question; that is, it is the product of the amount to be gained (or lost) and the chance of gaining (or losing) it. For instance, if I bet four to one in sovereigns against the occurrence of ace with a single die there would be, on the average of many throws, a loss of four pounds against a gain of five pounds on each set of six occurrences; i.e.[ TN: space] there would be an average gain of three shillings and fourpence on each throw. This is called the true or mathematical expectation. The so-called ‘moral expectation’, on the other hand, is the subjective value of this mathematical expectation. That is, instead of reckoning a money fortune in the ordinary way, as what it is, the attempt is made to reckon it at what it is felt to be. The elements of computation
therefore become, not pounds and shillings, but sums of pleasure enjoyed actually or in prospect. Accordingly when reckoning the present value of a future gain, we must now multiply, not the objective but the subjective value, by the chance we have of securing that gain.
With regard to the exact relation of this moral fortune to the physical various more or less arbitrary assumptions have been made. One writer (Buffon) considers that the moral value of any given sum varies inversely with the total wealth of the person who gains it. Another (D. Bernoulli) starting from a different assumption, which we shall presently have to notice more particularly, makes the moral value of a fortune vary as the logarithm of its actual amount.[9] A third (Cramer) makes it vary with the square root of the amount.
§ 18. Historically, these proposals have sprung from the wish to reconcile the conclusions of the Petersburg problem with the dictates of practical common sense; for, by substituting the moral for the physical estimate the total value of the expectation could be reduced to a finite sum. On this ground therefore such proposals have no great interest, for, as we have seen, there is no serious difficulty in the problem when rightly understood.
These same proposals however have been employed in
order to prove that gambling is necessarily disadvantageous, and this to both parties. Take, for instance, Bernoulli's supposition. It can be readily shown that if two persons each with a sum of £50 to start with choose to risk, say, £10 upon an even wager there will be a loss of happiness as a result; for the pleasure gained by the possessor of £60 will not be equal to that which is lost by the man who leaves off with £40.[10]
§ 19. This is the form of argument commonly adopted; but, as it stands, it does not seem conclusive. It may surely be replied that all which is thus proved is that inequality is bad, on the ground that two fortunes of £50 are better than one of £60 and one of £40. Conceive for instance that the original fortunes had been £60 and £40 respectively, the event may result in an increase of happiness; for this will certainly be the case if the richer man loses and the fortunes are thus equalized. This is quite true; and we are therefore obliged to show,—what can be very easily shown,—that if the other alternative had taken place and the two fortunes had been made still more unequal (viz.[** TN: space] £65 and £35 respectively) the happiness thus lost would more than balance what would have been gained by the equalization. And since these two suppositions are equally likely there will be a loss in the long run.
The consideration just adduced seems however to show