File:Bernoulli utility.png

English: $7. Therefore, let AB represent the quantity of goods initially possessed. Then after extending AB, a curve BGLS must be constructed, whose ordinates CG, DH, EL, FM, etc., designate utilities corresponding to the abscissas BC, BD, BE, BF, etc., designating gains in wealth. Further, let m,n, p, q, etc., be the numbers which indicate the number of ways in which gains in wealth BC, BD, BE, BF [misprinted in the original as CF], etc., can occur. Then (in accord with $4) the moral expectation of the risky proposition referred to is given by: Now, if we erect AQ perpendicular to AR, and on it measure off AN = PO, the straight line NO -AB represents the gain which may properly be expected, or the value of the risky proposition in question. If we wish, further, to know how large a stake the individual should be willing to venture on this risky proposition, our curve must be extended in the opposite direction in such a way that the abscissa Bp now represents a loss and the ordinate po represents the corresponding decline in utility. Since in a fair game the disutility to be suffered by losing must be equal to the utility to be derived by winning, we must assume that An = AN, or po = PO. Thus Bp will indicate the stake more than which persons who consider their own pecuniary status should not venture.