## Bernoulli

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

### Pooling strategies for St Petersburg gamblers

Sandor Csörgö and Gordon Simons

#### Abstract

Peter offers to play exactly one St Petersburg game with each of $n\ge 2$ 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 ${V}_{{pp}_{n}}={\sum}_{k=1}^{n}{p}_{k,n}{X}_{k}$ 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${S}_{n}={X}_{1}+\dots +{X}_{n}$ in comparison with their individual winnings ${pp}_{n}$, even though their total winnings $H\left({pp}_{n}\right)$ is the same. We show that the added value of an admissible ${pp}_{n}^{*}$ is just its entropy $n\ge 2$, and we determine the best admissible strategy ${pp}_{n}$. Moreover, for every ${S}_{{pp}_{n}}={V}_{{pp}_{n}}-H\left({pp}_{n}\right)$ and ${S}_{{pp}_{n}}$ we construct semistable approximations to $n\to \mathrm{\infty}$. We show in particular that $max\{{p}_{1,n},\dots ,{p}_{n,n}\}\to 0$ 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 $n\ge 2$ 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 ${V}_{{pp}_{n}}={\sum}_{k=1}^{n}{p}_{k,n}{X}_{k}$, 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 ${S}_{n}={X}_{1}+\dots +{X}_{n}$ is the same. We show that the added value of an admissible ${pp}_{n}$ is just its entropy $H\left({pp}_{n}\right)$, and we determine the best admissible strategy ${pp}_{n}^{*}$. Moreover, for every $n\ge 2$ and ${pp}_{n}$ we construct semistable approximations to ${S}_{{pp}_{n}}={V}_{{pp}_{n}}-H\left({pp}_{n}\right)$. We show in particular that ${S}_{{pp}_{n}}$ has a proper semistable asymptotic distribution as $n\to \mathrm{\infty}$ along the entire sequence of natural numbers whenever $max\{{p}_{1,n},\dots ,{p}_{n,n}\}\to 0$ 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 $n\ge 2$.

#### 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

**Keywords**

added value asymptotic distributions best admissible pooling strategies comparison of infinite expectations several players St~Petersburg games

#### 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