Pooling strategies for St Petersburg gamblers

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 $n\ge 2$.