Bernoulli

• Bernoulli
• Volume 12, Number 6 (2006), 971-1002.

Pooling strategies for St Petersburg gamblers

Abstract

Peter offers to play exactly one St Petersburg game with each of $n ≥2$ players, Paul$1$, $n$, Paul$pp n =(p 1 ,n,...,p n ,n)$, whose conceivable pooling strategies are described by all possible probability distributions $pp n$. Comparing infinite expectations, we characterize among all $V pp n =∑ k =1 np k ,nX k$ those admissible strategies for which the pooled winnings, each distributed as $1$, yield a finite added value for each and every one of Paul$n$, $X 1 ,...,X n$, Paul$S n =X 1+…+X n$ in comparison with their individual winnings $pp n$, even though their total winnings $H (pp n)$ is the same. We show that the added value of an admissible $pp n *$ is just its entropy $n ≥2$, and we determine the best admissible strategy $pp n$. Moreover, for every $S pp n =V pp n -H(pp n)$ and $S pp n$ we construct semistable approximations to $n →∞$. We show in particular that $max {p 1 ,n,…,p n ,n}→0$ has a proper semistable asymptotic distribution as $pp n$ along the entire sequence of natural numbers whenever $S n /n$ for a sequence $S pp n *$ of admissible strategies, which is in sharp contrast to Peter offers to play exactly one St Petersburg game with each of $n ≥2$ players, Paul$1$, ..., Paul$n$, whose conceivable pooling strategies are described by all possible probability distributions $pp n =(p 1 ,n,...,p n ,n)$. Comparing infinite expectations, we characterize among all $pp n$ those admissible strategies for which the pooled winnings, each distributed as $V pp n =∑ k =1 np k ,nX k$, yield a finite added value for each and every one of Paul$1$, ..., Paul$n$ in comparison with their individual winnings $X 1 ,...,X n$, even though their total winnings $S n =X 1+…+X n$ is the same. We show that the added value of an admissible $pp n$ is just its entropy $H (pp n)$, and we determine the best admissible strategy $pp n *$. Moreover, for every $n ≥2$ and $pp n$ we construct semistable approximations to $S pp n =V pp n -H(pp n)$. We show in particular that $S pp n$ has a proper semistable asymptotic distribution as $n →∞$ along the entire sequence of natural numbers whenever $max {p 1 ,n,…,p n ,n}→0$ for a sequence $pp n$ of admissible strategies, which is in sharp contrast to $S n /n$, and the rate of convergence is very fast for $S pp n *$. , and the rate of convergence is very fast for $n ≥2$.

Article information

Source
Bernoulli, Volume 12, Number 6 (2006), 971-1002.

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1165269147

Digital Object Identifier
doi:10.3150/bj/1165269147

Mathematical Reviews number (MathSciNet)
MR2274852

Zentralblatt MATH identifier
1130.91018

Citation

Csörgö, Sandor; Simons, Gordon. Pooling strategies for St Petersburg gamblers. Bernoulli 12 (2006), no. 6, 971--1002. doi:10.3150/bj/1165269147. https://projecteuclid.org/euclid.bj/1165269147