Simulating Daniel Bernoulli’s St. Petersburg game: Theoretical and empirical consistency

Abstract

Russon and Chang simulated St. Petersburg games and found that their results were inconsistent with their theoretical predictions. In this article, the theoretical outcomes are derived this time using the methodology suggested by Daniel Bernoulli, and games are then simulated. When this is done, it is found that the theoretical and empirical results are consistent.

Considering the Pasadena “Paradox”

“…Generally E{X} is a function of the number of games played N and is E{X} = k/2 + l, when the game is played N=2 k times and ë chosen to give the desired degree of confidence. Vivian (2004) validated empirically the correctness of the equation. 2 The statement was made by Bernoulli, however it is of course not possible, as Menger (infra) pointed out, for the series to extend to infinity, since in this event N = 8 in which case N/2 or N/4 etc becomes meaningless.…”

“…The methodology suggested by Vivian (2003) and demonstrated by simulation Vivian (2004) can be applied to the Pasadena game. The methodology is essentially the same as suggested by Bernoulli (1954/1738) himself after correcting for some errors in his methodology.…”

A Resolution of St. Petersburg Paradox

“…If the optimal lottery, with the largest expected utility, corresponds to n → ∞ and has the infinite value, then any rational player should feel it profitable to spend all available money to buy a ticket allowing for the maximal possible number of tosses. But numerous empirical data drastically contradict this conclusion, since the majority of real players prefer the lotteries with quite a modest number of tosses (Bottom et al (1989), Rivero et al (1990), Vivian (2004), Hayden and Platt (2009), Cox et al (2009), Neugebauer (2010), Cox et al (2018), Nobandegani and Shultz (2020)). This contradiction is the essence of the paradox.…”