現代数学の系譜工学物理雑談古典ガロア理論も読む83

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586: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2020/02/16(日) 00:07:02 ID:3HCL8TTE

>>580
＞だいたい有理数の循環節が独立同分布のはずがないでしょ

有理数を使うgame2 について解説しておくと
http://www.ma.huji.ac.il/hart/puzzle/choice.pdf?
Choice Games November 4, 2013 Written by Sergiu Hart
(抜粋)
A similar result, but now without using the Axiom of Choice.
Consider the following two-person game game2:
・ Player 1 chooses a rational number in the interval [0, 1] and writes down
　its infinite decimal expansion 0.x1x2...xn..., with all xn ∈ {0, 1,..., 9}.
・ Player 2 asks (in some order) what are the digits xn except one, say xi ;
　then he writes down a digit ξ ∈ {0, 1,..., 9}.
・ If xi = ξ then Player 2 wins, and if xi ≠ ξ then Player 1 wins.
By choosing i arbitrarily and ξ uniformly in {0, 1,..., 9}, Player 2 can guarantee a win with probability 1/10.
However, we have:
Theorem 2 For every ε > 0 Player 2 has a mixed strategy in game2 guaranteeing him a win with probability at least 1 − ε.
Proof. The proof is the same as for Theorem 1, except that here we do not use the Axiom of Choice.
Because there are only countably many sequences
x ∈ {0,..., 9}^N that Player 1 may choose (namely, those x that become eventually
periodic), we can order them-say x
(1), x(2),..., x(m)
,...-and then choose in each equivalence class the element with minimal index (thus F(x) = x(m) iff m
is the minimal natural number such that x 〜 x (m)).
(引用終り)

つまり、game2は
１．Player 1は、区間  [0, 1]の10進の有理数を1つ選ぶ
２．Player 2は、10進の有理数各桁の数字1つ（xi）を除いて、見て良い
３．xi は、普通は的中率 probability 1/10だが、Game1と同じ戦略で、”1 − ε”にできるという

つづく

http://rio2016.5ch.net/test/read.cgi/math/1581243504/586

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