Bounded utility functions resolve the St. Petersburg paradox in decision theory.
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While the sources mention that utility functions were defined by Bernoulli and Cramer to help resolve the St. Petersburg paradox, they do not establish that bounded utility functions universally resolve it.
Because the outcomes of repeated investments or gambles involve products of variables, authorities have repeatedly been tempted to the belief that, in a long sequence, maximization of the expected value of terminal utility can be achieved or well-approximated by a strategy of maximizing at each stage the geometric mean of outcome (or its equivalent, the expected value of the logarithm of principal plus return). The law of large numbers or of the central limit theorem as applied to the logs can validate the conclusion that a maximum-geometric-mean strategy does indeed make it "virtually certain" that, in a "long" sequence, one will end with a higher terminal wealth and utility. However, this does not imply the false corollary that the geometric-mean strategy is optimal for any finite number of periods, however long, or that it becomes asymptotically a good approximation. As a trivial counter-example, it is shown that for utility proportional to x(gamma)/gamma, whenever gamma not equal 0, the geometric strategy is suboptimal for all T and never a good approximation. For utility bounded above, as when gamma < 0, the same conclusion holds. If utility is bounded above and finite at zero wealth, no uniform strategy can be optimal, even though it can be that the best uniform strategy will be that of the maximum geometric mean. However, asymptotically the same level of utility can be reached by an infinity of nearby uniform strategies. The true optimum in the bounded case involves nonuniform strategies, usually being more risky than the geometric-mean maximizer's strategy at low wealths and less risky at high wealths. The novel criterion of maximizing the expected average compound return, which asymptotically leads to maximizing of geometric mean, is shown to be arbitrary.
We identify restrictions on a decision maker’s utility function that are both necessary and sufficient to preserve dominance reasoning in each of two versions of the Two-Envelope Paradox (TEP). For the classical TEP, the utility function must satisfy a certain recurrence inequality. For the St. Petersburg TEP, the utility function must be bounded above asymptotically by a power function, which can be tightened to a constant. By determining the weakest conditions for dominance reasoning to hold, the article settles an open question in the research literature. Remarkably, neither constant-bounded utility nor finite expected utility is necessary for resolving the classical TEP; instead, finite expected utility is both necessary and sufficient for resolving the St. Petersburg TEP.
St. Petersburg paradox, formulated by N. Bernoulli in the early 18th century, led to defining the utility function (D. Bernoulli, G. Cramer) as a way to resolve the paradox and played an important role in the development of decision making theory. In the 20th century, the paradox attracted the attention of many researchers, including Nobel Prize winners P. Samuelson, R. Aumann, L. Shapley. N. Bernoulli assumed that payments grow exponentially with the coin toss number. The growth rate of payments is higher than the exponential one in the generalized St. Petersburg paradox. The utility functions of Bernoulli and Cramer don't lead to the resolution of the paradox in this case. In 1934, K. Menger showed the necessity and sufficiency of the boundedness of the utility function for resolving of the generalized St. Petersburg paradox. A brief overview of the subject matter is given, as well as the autors' approach to resolving the classical paradox, based on discounting cash flows, in which the time intervals between consecutive coin tossings play a special role. The adaptation of the proposed approach to the generalized St. Petersburg paradox is also described. The proposed approach is an alternative to the traditional based utility function. It allows to solve, in particular, the inverse problem: to find (ambiguous solution) the moments of possible payments according to the set sizes of payments, the force of interest and the price of the game.
The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the lottery game is
The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the lottery game is infinite but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naïve decision criterion that takes only the expected value into account predicts a course of action that p
the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.
He demonstrated in a letter to Nicolas Bernoulli that a square root function describing the diminishing marginal benefit of gains can resolve the problem. However, unlike Daniel Bernoulli, he did not consider the total wealth of a person, but only the gain by the lottery.
This solution by Cramer and Bernoulli, however, is not completely satisfying, as the lottery can easily be changed in a way such that the paradox reappears. To this aim, we just need to change the game so that it gives even more rapidly increasing payoffs. For any unbounded utility function, one can find a lottery that allows for a variant of the St. Petersburg paradox, as was first pointed out by Menger.
Recently, expected utility theory has been extended to arrive at more behavioral decision models. In some of these new theories, as in cumulative prospect theory, the St. Petersburg paradox again appears in certain cases, even when the utility function is concave, but not if it is bounded.
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