SECT. 18.] Insurance and Gambling. 389 



tatioii 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. Ber 

 noulli) 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 1 . 

 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 



1 Bernoulli himself does not seem every psychologist. It is what is 



to have based his conclusions upon commonly called Fechner s Law, 



actual experience. But it is a note- which he has established by aid of 



worthy fact that the assumption with an enormous amount of careful ex- 



which he starts, viz. that the sub- periment in the case of a number of 



jective value of any small increment our simple sensations. But I do not 



(dx) is inversely proportional to the believe that he has made any claim 



sum then possessed (x), and which that such a law holds good in the 



leads at once to the logarithmic law far more intricate dependence of 



above mentioned, is identical with happiness upon wealth, 

 one which is now familiar enough to 



