%0 Book Section %1 statphys23_0691 %A Takayasu, H. %A Takayasu, M. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K distribution fair gamble income law statphys23 topic-11 zipf %T St. Petersburg Paradox and Zipf’s law %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=691 %X St.Petersburg paradox is a famous classical problem posed by Bernoulli more than 250 years ago and even now it is a topic of controversy in economics especially in view of behavioral finance. Assume that there is a lottery: You can toss a coin successively until a tail appears. If the number of heads is N, you can get 2 to the power of Nth dollars. While the mathematical expectation of reward is infinite, no one will pay infinite fee for this lottery because the probability of getting finite reward is 1, so the problem makes a paradox. Since Bernoulli's first paper this paradox has been discussed from the view point of rational human behaviors, however, we will consider this paradox from the standpoint of a bookmaker, so that the results are more rigorous. Firstly, I will show that the St. Petersburg lottery is a composite of a fair gamble and a kind of option. The basic gamble is making a bet twice if you have a head by a coin toss otherwise you will lose. You can repeat this gamble free as the expectation value is always the same. In addition to this fair gamble an option can be added, that is, a right to back one coin toss. Namely, when you have a tail at the N-th coin toss, you can stop at the N-1-th coin toss so that you can get 2 to the power of (N-1)th dollars. It is easy to calculate the fair price of this option for each coin toss. The original paradox contains infinite number of options that made the problem being very complicated. By considering a situation that many people are repeating this fair gamble paying additional one dollar each time step, it is shown that the fluctuation of the total rewards can be described by a random multiplicative process with additive term. Applying Takayasu-Sato-Takayasu’s formula for this problem 1, I will prove that there exists a statistically steady state in this stochastic process and so-called Zipf's law realizes in the long time limit. The interesting point is that Zipf’s law can be widely observed in the income distribution of companies in the real world implying that the St. Petersburg paradox is not merely a mathematical quiz but it has a profound meaning in the real world economy. Reference\\ 1) Takayasu, Sato, Takayasu, Phys. Rev. Lett., 79(1997), 966-969.

@incollection{statphys23_0691, abstract = {St.Petersburg paradox is a famous classical problem posed by Bernoulli more than 250 years ago and even now it is a topic of controversy in economics especially in view of behavioral finance. Assume that there is a lottery: You can toss a coin successively until a tail appears. If the number of heads is N, you can get 2 to the power of Nth dollars. While the mathematical expectation of reward is infinite, no one will pay infinite fee for this lottery because the probability of getting finite reward is 1, so the problem makes a paradox. Since Bernoulli's first paper this paradox has been discussed from the view point of rational human behaviors, however, we will consider this paradox from the standpoint of a bookmaker, so that the results are more rigorous. Firstly, I will show that the St. Petersburg lottery is a composite of a fair gamble and a kind of option. The basic gamble is making a bet twice if you have a head by a coin toss otherwise you will lose. You can repeat this gamble free as the expectation value is always the same. In addition to this fair gamble an option can be added, that is, a right to back one coin toss. Namely, when you have a tail at the N-th coin toss, you can stop at the N-1-th coin toss so that you can get 2 to the power of (N-1)th dollars. It is easy to calculate the fair price of this option for each coin toss. The original paradox contains infinite number of options that made the problem being very complicated. By considering a situation that many people are repeating this fair gamble paying additional one dollar each time step, it is shown that the fluctuation of the total rewards can be described by a random multiplicative process with additive term. Applying Takayasu-Sato-Takayasu’s formula for this problem [1], I will prove that there exists a statistically steady state in this stochastic process and so-called Zipf's law realizes in the long time limit. The interesting point is that Zipf’s law can be widely observed in the income distribution of companies in the real world implying that the St. Petersburg paradox is not merely a mathematical quiz but it has a profound meaning in the real world economy. Reference\\ 1) Takayasu, Sato, Takayasu, Phys. Rev. Lett., 79(1997), 966-969.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Takayasu, H. and Takayasu, M.}, biburl = {https://www.bibsonomy.org/bibtex/218d3b091c76957725ad1a06f8eb50333/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {ea41d1a150ddda9484746417e5309903}, intrahash = {18d3b091c76957725ad1a06f8eb50333}, keywords = {distribution fair gamble income law statphys23 topic-11 zipf}, month = {9-13 July}, timestamp = {2007-06-20T10:16:27.000+0200}, title = {St. Petersburg Paradox and Zipf’s law}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=691}, year = 2007 }