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%I #30 May 01 2024 16:11:26
%S 1,3,9,2,20,12,23,27,31,35,187,1461,485,105,64,69,67,18,11,41,87,23,
%T 97,828,251175,497650,1582733,480083,3070955,139927,1253,1301,160,83,
%U 172,89,184,181,187,193,199,205,211,217,223,229,235,241,247,253,259,265
%N Numerators of the Nash equilibrium of guesses for the number guessing game for n numbers.
%C Consider two players: one player picks a number between 1 and n, and another player guesses numbers, receiving feedback "too high" or "too low". The number picker is trying to maximize the expected number of guesses, whereas the number guesser is trying to minimize the expected number of guesses. While a binary search would in expectation be the optimal strategy if the number were chosen randomly, it is not the case if the number is chosen adversarially.
%H R. Fokkink and M. Stassen, <a href="https://doi.org/10.1007/978-3-642-25280-8_10">An Asymptotic Solution of Dresher's Guessing Game</a>, Decision and Game Theory for Security, 2011, 104-116.
%H Robert Fokkink and Misha Stassen, <a href="https://www.gamesec-conf.org/2011/files/10_STASSEN_GameSec2011_11142011.pdf">Dresher's Guessing Game</a>, conference presentation, 2011.
%H Michal Forisek, <a href="https://ipsc.ksp.sk/2011/real/solutions/booklet.pdf">Candy for each guess</a>, p. 15-19, IPSC 2011 booklet.
%H Michal Forisek, <a href="https://ipsc.ksp.sk/2011/real/problems/c.html">Candy for each guess</a>.
%e a(n)/A286677(n): 1, 3/2, 9/5, 2, 20/9, 12/5, 23/9, 27/10, 31/11, 35/12, 187/62, 1461/470, 485/152, 105/32, 64/19, 69/20, 67/19, 18/5, 11/3, 41/11, 87/23, ...
%e For n=3, the Nash equilibrium of guesses is 9/5. This is attained when the number picker chooses 1 with 2/5 probability, 2 with 1/5 probability, and 3 with 2/5 probability. The number guesser guesses the numbers 0,2,1 in order with 1/5 probability, 2,0,1 in order with 1/5 probability, and 1,0,2 (i.e., binary search) with 3/5 probability.
%Y For denominators see A286677.
%K nonn,frac,changed
%O 1,2
%A _Lewis Chen_, May 12 2017
%E More terms from _Lewis Chen_, Oct 29 2019
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