We now have a fuller understanding of the tradeoff between risk and reward for fractional Kelly versus full Kelly and for Kelly subject to minimizing the underperformance of a benchmark or specified desired wealth path. (LocationÂ 222)

Kelly (1956) is given credit for the idea of using log utility in gambling and repeated investment problems, as such it is known as the Kelly criterion. Kellyâ€™s analyses use Bernoulli trials. Not only does he show that log is the utility function which maximizes the long run growth rate, but that this utility function is myopic in the sense that period by period maximization based only on current capital is optimal. (LocationÂ 424)

Once one has an edge, with positive expectation, Thorp outlines a general theory for optimal wagering on such games. (LocationÂ 459)

Optimal consumption is linear and increasing in current wealth and in the present value of the noncapital income stream. (LocationÂ 481)

The optimal asset mix depends only on the probability distribution of returns, the interest rate and the investorâ€™s one-period utility function of consumption. (LocationÂ 482)

so as to maximize the expected growth rate of capital plus the present value of the noncapital income stream. (LocationÂ 485)

general agreement on the following proposition: Expected values are computed by multiplying each possible gain by the number of ways in which it can occur, and then dividing the sum of these products by the total number of possible cases wherey in this theory, the consideration of cases which are all of the same probability is insisted upon (LocationÂ 545)

utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed (LocationÂ 593)

Since on our hypothesis we must consider infinitesimally small gains, we shall take gains BC and BD to be nearly equal, so that their difference CD becomes infinitesimally small. (LocationÂ 629)

Any gain must be added to the fortune previously possessed, then this sum must be raised to the power given by the number of possible ways in which the gain may be obtained; these terms should then be multiplied together. Then of this product a root must be extracted the degree of which is given by the number of all possible cases, and finally the value of the initial possessions must be subtracted therefrom; what then remains indicates the value of the risky proposition in question (LocationÂ 643)

Whether or not the goal is reached depends on future occurrences, but, in any event, the subgoal of maximization of the expected value of the payouts expressed in utiles is logically related to the goal of maximization of the forthcoming payout also expressed in utiles. (LocationÂ 1089)

Wealth-holders have the option of holding their wealth in many different combinations of stocks, bonds, and cash. (LocationÂ 1103)

then choosing the bet with the highest mathematical expectation of winning. (LocationÂ 1143)

In terms of subgoals as defined in this study, Bernoulli showed that use of the expected-value subgoal did not always lead to choices that seemed rational to him and proposed instead the use of the expected-utility subgoal. (LocationÂ 1149)

Possible payouts range from 2.22 per ducat risked to 0. Payouts with ranges such as thisâ€”indeed, much greater rangesâ€”are ordinary economic occurrences. (LocationÂ 1159)

Bernoulli used the lottery-ticket example to show that the expected values of the payouts are not good guides in making choices involving large risks. (LocationÂ 1187)

The decision-maker who is interested in maximizing his wealth at the end of a long series of choices should ask himself how he would come out in the long run if he made the same choice on the same terms over and over again. (LocationÂ 1211)

The first group contains those to whom each risk is a unique event either because they do not expect it to recur or because they keep its effects entirely separate from the results of other risks. (LocationÂ 1230)

The second class of wealth-holders includes those who expect to be faced repeatedly with risks of the same general type and magnitude. (LocationÂ 1236)

Bernoulli states that the wealth-holder should ask himself whether the added satisfaction associated with the expected gain justifies undertaking the risky venture. (LocationÂ 1244)

According to maximum-chance analysis, the wealth-holder or portfolio manager should ask himself how he can maximize his chances of getting as good or better return than can be obtained with any other specified plan, assuming that he risks the same proportion of his portfolio (LocationÂ 1247)