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Generalized $n$-Paul paradox. (English) Zbl 1138.60026
The present paper has as starting point the results of S. Csörgö and G. Simons (2002--2006) on the classical St. Petersburg(1/2) game, played by two gamblers with an unbiased coin. The author considers the generalized St. Petersburg($p$) game with $p\in (0, 1)$ as the probability of the “heads” at each throw of a possibly biased coin. An interesting result is that, while the stochastic dominance is preserved for the case of two players and an arbitrary parameter $p\in(0, 1)$, for three or more players, the admissibly pooled winning strategies generally fail to stochastically dominate the individual strategies. The main result of the article consists in determining the best admissible pooling strategies for a rational value of $p$, illustrating also the algebraic depth of the problem for an irrational value of the parameter $p$.

##### MSC:
 60E99 Distribution theory in probability theory 60G50 Sums of independent random variables; random walks
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##### References:
 [1] Csörgő, S.; Simons, G.: A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games. Statist. probab. Lett. 26, 65-73 (1996) · Zbl 0859.60030 [2] Csörgő, S.; Simons, G.: The two-paul paradox and the comparison of infinite expectations. Limit theorems in probability and statistics, 427-455 (2002) · Zbl 1027.60030 [3] Csörgő, S.; Simons, G.: Laws of large numbers for cooperative St. Petersburg gamblers. Period. math. Hungar. 50, 99-115 (2005) · Zbl 1113.60026 [4] Csörgő, S.; Simons, G.: Pooling strategies for St. Petersburg gamblers. Bernoulli 12, 971-1002 (2006) · Zbl 1130.91018