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 .