Bernoulli

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

Pooling strategies for St Petersburg gamblers

Sandor Csörgö and Gordon Simons

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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

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


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References

  • [1] Bernoulli, D. (1738) Specimen theoriae novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5, 175-192. (Reprinted in Die Werke von Daniel Bernoulli, Vol. 2, pp. 223-234, Basel: Birkhäuser, 1982. English translation: Exposition of a new theory of the measurement of risk. Econometrica, 22, 23-36, 1954, reprinted in W.J. Baumol and S.M.
  • [2] Goldfeld (eds.), Precursors in Mathematical Economics: An Anthology. pp. 15-26, London: London School of Economics and Political Science, 1968; in G.A. Miller (ed.), Mathematics and Psychology, pp. 36-52, New York: Wiley, 1964; and in A.N. Page (ed.), Utility Theory: A Book of Readings, pp. 199-214, New York: Wiley, 1968.)
  • [3] Bernoulli, N. (1713) Letter in French to P R de Montmort, 9 September. In P.R. de Montmort Essay d´analyse sur les jeux de hazard, (2nd edn), pp. 401-402. Paris: Jacque Quillau. (Reprinted in New York: Chelsea, 1980.)
  • [4] Csörgö, S. (2002) Rates of merge in generalized St. Petersburg games. Acta Sci. Math. (Szeged), 68, 815-847.
  • [5] Csörgö, S. and Dodunekova, R. (1991) Limit theorems for the Petersburg game. In M.G. Hahn, D.M.
  • [6] Mason and D.C. Weiner (eds.), Sums, Trimmed Sums and Extremes, Progr. Probab. 23, pp. 285- 315. Boston: Birkhäuser.
  • [7] Csörgö, S. and Simons, G. (1996) A strong law of large numbers for trimmed sums, with applications to generalized St Petersburg games. Statist. Probab. Lett., 26, 65-73.
  • [8] Csörgö, S. and Simons, G. (2002) The two-Paul paradox and the comparison of infinite expectations. In I. Berkes, E. Csáki and M. Csörgo? (eds), Limit Theorems in Probability and Statistics, Vol. I, pp. 427-455. Budapest: János Bolyai Mathematical Society.
  • [9] Csörgö, S. and Simons, G. (2008) Resolution of the St. Petersburg Paradox. Book manuscript in preparation.
  • [10] Dutka, J. (1988) On the St. Petersburg paradox. Arch. Hist. Exact Sci., 39, 13-39.
  • [11] Feller, W. (1945) Note on the law of large numbers and 'fair´ games. Ann. Math. Statist., 16, 301-304.
  • [12] Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I (3rd edn). New York: Wiley.
  • [13] Gnedenko, B.V. and Kolmogorov, A.N. (1954) Limit Distributions for Sums of Independent Random Variables. Cambridge, MA: Addison-Wesley.
  • [14] Jorland, G. (1987) The Saint Petersburg paradox 1713-1937. In L. Krüger, L.D. Daston and M. Heidelberger (eds), The Probabilistic Revolution, Vol. I: Ideas in History, pp. 157-190. Cambridge, MA: MIT Press.
  • [15] Martin-Lö f, A. (1985) A limit theorem which clarifies the 'Petersburg paradox´. J. Appl. Probab., 22, 634-643.
  • [16] Samuelson, P.A. (1977) St. Petersburg paradoxes: defanged, dissected, and historically described. J. Economic Lit., 15, 24-55. (Republished as Chapter 298 in K. Crowley (ed), The Collected Scientific Papers of Paul A. Samuelson, Vol. 5, pp. 133-164. Cambridge, MA: MIT Press, 1986.)
  • [17] Zolotarev, V.M. (1978) Pseudomoments. Teor. Verojatnost. i Primenen., 23, 284-294 (in Russian).