Замечание о Санкт-Петербургском парадоксе

Abstract

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.