Guess the Larger Number
Digital Object Identifier (DOI): 10.14708/ma.v44i1.1205
Abstract
B. I myself have invented a game. Well, think of a number.A. I got a number.B. Me too. Now, tell me yours.A. Seven.B. Seven. Mine is eight – I won.Sergey Solovyov, Assa (conversation of Bananan and Alika)
We discuss variations of the zero-sum game where Bob selects two distinct numbers, and Alice learns one of them to make a guess which of the numbers is the larger.
We discuss variations of the zero-sum game where Bob selects two distinct numbers, and Alice learns one of them to make a guess which of the numbers is the larger.
Keywords: Game of Googol; best choice problem; secretary problem; full-information;
Subject classification: 91A60 (91A05)
References
1] B. Berezovsky and A. Gnedin, The best choice problem, Moscow, Nauka, 1984. MR 0768372
[2] D. Blackwell (1951), On the translation parameter problem for discrete variables, Annals of Mathematical Statistics 22: 391–399. doi: 10.1214/aoms/1177729585; MR 0043418
[3] D. Blackwell and M.A. Girshick, Theory of games and statistical decisions, Dover, N.Y., 1979.
[4] F.T. Bruss (1996), The fallacy of the two-envelopes problem, Mathematical Scientist 21: 112–119.
[5] F.T. Bruss (2003), Playing a trick on uncertainty, Newsletter of the European Mathematical Society, issue 50, pp. 5 and 7-8 (online).
[6] G. Campbell and S.M. Samuels (1981), Choosing the best of the current crop, Advances in Applied Probability 13: 510–532. MR 0615949
[7] Y.S. Chow and H. Teicher, Probability theory: Independence, interchangeability, martingales, Springer Texts in Statistics, Springer, New York, 1997. doi: 10.1007/978-1-4612-1950-7; MR 1476912
[8] T. Cover (1987), Pick the largest number, In: Open Problems in Communication and Computation, T. Cover and B. Gopinath eds, Springer, p. 152.doi: 10. 10071/ 978-1-4612-4808-8
[9] L. Devroye, Non-uniform random variate generation, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8643-8; MR 836973
[10] T. Ferguson (1989), Who solved the secretary problem? Statistical Science 4: 282–289. MR1015277
[11] A. Gnedin (1994), A solution to the game of googol, Annals of Probability 22:1588–1595. MR1303655
[12] A. Gnedin (1996), Das Spiel Googol, Nachr. Akad. Wiss. G¨ottingen Math.-Phys. Kl. II. MR1422387
[13] A. Gnedin (1995), On a class of exchangeable sequences, Statistics and Probability Letters 25: 351-356. MR 1363235
[14] A. Gnedin and U. Krengel (1995), A stochastic game of optimal stopping and order selection, Annals of Applied Probability 5: 310–321. MR 1325055
[15] A. Gnedin and U. Krengel (1996), Optimal selection problems based on exchangeable trials, Annals of Applied Probability 6: 862–882. MR 1410118
[16] A. Gnedin and G. Olshanski (2012), The two-sided infinite extension of the Mallows model for random permutations, Advances in Applied Mathematics 48: 615-639. MR 2920835
[17] B.M. Hill (1968), Posterior distribution of percentiles: Bayes’ theorem for sampling from a population, Journal of the American Statistical Association 63: 677–691. MR 0238430
[18] T.P. Hill (2009), Knowing when to stop, American Scientist 97, No.2 (online).
[19] T.P. Hill and U. Krengel (1992), On the game of googol, International Journal of Game Theory 21: 151–160. MR 1203008
[20] H.P. Kirschner (1976), On the risk-equivalence of two methods of randomization in statistics, Journal of Multivariate Analysis 6: 159-166.
[21] D. A. Lane and W.D. Sudderth (1978), Diffuse models for sampling and predictive inference, Annals of Statistics 6: 1318–1336. MR 0523766
[22] T.M. Liggett, Interacting particle systems, Springer, 2005.
[23] I. Molchanov, Theory of random sets, Springer, 2005.
[24] J. Petruccelli (1980), On a best choice problem with partial information, Annals of Statistics 8: 1171–1174. MR 585717
[25] D. Samet, I. Samet and D. Schmeidler (2004), One observation behind two-envelope problems, American Mathematical Monthly 111: 347–351. MR 2057189
[26] S.M. Samuels (1989), Who will solve the secretary problem? [Comment on Ferguson] Statistical Science 4: 289–291.
[27] S.M. Samuels, Secretary problems, In: Handbook of Sequential Analysis, B.K. Ghosh and P.K. Sen eds, Marcel Dekker, NY, 1991. ISBN: 0-8247-8408-1; MR 1174296
[28] S. Sorin, A first course on zero-sum repeated games, Springer, 2002.
[29] T.J. Stewart (1978), Optimal selection from a random sequence with learning of the underlying distribution, Journal of the American Statistical Association, 73: 775–780. MR 0521326
[30] K. Szajowski (1984), Optimal stopping of a sequence of maxima over an unobservable sequence of maxima, Applicationes Mathematicae 18: 359–374. MR 0757886
[31] P. Winkler (2001), Games people don’t play, In: Puzzlers Tribute: A Feast for the Mind, D. Wolfe and T. Rodgers eds., AK Peters. ISBN10 : 1-56881121-7
[2] D. Blackwell (1951), On the translation parameter problem for discrete variables, Annals of Mathematical Statistics 22: 391–399. doi: 10.1214/aoms/1177729585; MR 0043418
[3] D. Blackwell and M.A. Girshick, Theory of games and statistical decisions, Dover, N.Y., 1979.
[4] F.T. Bruss (1996), The fallacy of the two-envelopes problem, Mathematical Scientist 21: 112–119.
[5] F.T. Bruss (2003), Playing a trick on uncertainty, Newsletter of the European Mathematical Society, issue 50, pp. 5 and 7-8 (online).
[6] G. Campbell and S.M. Samuels (1981), Choosing the best of the current crop, Advances in Applied Probability 13: 510–532. MR 0615949
[7] Y.S. Chow and H. Teicher, Probability theory: Independence, interchangeability, martingales, Springer Texts in Statistics, Springer, New York, 1997. doi: 10.1007/978-1-4612-1950-7; MR 1476912
[8] T. Cover (1987), Pick the largest number, In: Open Problems in Communication and Computation, T. Cover and B. Gopinath eds, Springer, p. 152.doi: 10. 10071/ 978-1-4612-4808-8
[9] L. Devroye, Non-uniform random variate generation, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8643-8; MR 836973
[10] T. Ferguson (1989), Who solved the secretary problem? Statistical Science 4: 282–289. MR1015277
[11] A. Gnedin (1994), A solution to the game of googol, Annals of Probability 22:1588–1595. MR1303655
[12] A. Gnedin (1996), Das Spiel Googol, Nachr. Akad. Wiss. G¨ottingen Math.-Phys. Kl. II. MR1422387
[13] A. Gnedin (1995), On a class of exchangeable sequences, Statistics and Probability Letters 25: 351-356. MR 1363235
[14] A. Gnedin and U. Krengel (1995), A stochastic game of optimal stopping and order selection, Annals of Applied Probability 5: 310–321. MR 1325055
[15] A. Gnedin and U. Krengel (1996), Optimal selection problems based on exchangeable trials, Annals of Applied Probability 6: 862–882. MR 1410118
[16] A. Gnedin and G. Olshanski (2012), The two-sided infinite extension of the Mallows model for random permutations, Advances in Applied Mathematics 48: 615-639. MR 2920835
[17] B.M. Hill (1968), Posterior distribution of percentiles: Bayes’ theorem for sampling from a population, Journal of the American Statistical Association 63: 677–691. MR 0238430
[18] T.P. Hill (2009), Knowing when to stop, American Scientist 97, No.2 (online).
[19] T.P. Hill and U. Krengel (1992), On the game of googol, International Journal of Game Theory 21: 151–160. MR 1203008
[20] H.P. Kirschner (1976), On the risk-equivalence of two methods of randomization in statistics, Journal of Multivariate Analysis 6: 159-166.
[21] D. A. Lane and W.D. Sudderth (1978), Diffuse models for sampling and predictive inference, Annals of Statistics 6: 1318–1336. MR 0523766
[22] T.M. Liggett, Interacting particle systems, Springer, 2005.
[23] I. Molchanov, Theory of random sets, Springer, 2005.
[24] J. Petruccelli (1980), On a best choice problem with partial information, Annals of Statistics 8: 1171–1174. MR 585717
[25] D. Samet, I. Samet and D. Schmeidler (2004), One observation behind two-envelope problems, American Mathematical Monthly 111: 347–351. MR 2057189
[26] S.M. Samuels (1989), Who will solve the secretary problem? [Comment on Ferguson] Statistical Science 4: 289–291.
[27] S.M. Samuels, Secretary problems, In: Handbook of Sequential Analysis, B.K. Ghosh and P.K. Sen eds, Marcel Dekker, NY, 1991. ISBN: 0-8247-8408-1; MR 1174296
[28] S. Sorin, A first course on zero-sum repeated games, Springer, 2002.
[29] T.J. Stewart (1978), Optimal selection from a random sequence with learning of the underlying distribution, Journal of the American Statistical Association, 73: 775–780. MR 0521326
[30] K. Szajowski (1984), Optimal stopping of a sequence of maxima over an unobservable sequence of maxima, Applicationes Mathematicae 18: 359–374. MR 0757886
[31] P. Winkler (2001), Games people don’t play, In: Puzzlers Tribute: A Feast for the Mind, D. Wolfe and T. Rodgers eds., AK Peters. ISBN10 : 1-56881121-7
Pages: 183-207
Full Text: PDF
