二人猜数

fzy · 发表于 2005-9-10 01:36:31

难度：++

二人猜数。甲在1到9中选一个奇数，乙在2到8中选一个偶数。如果两数差1，乙胜，否则甲胜。

问两人的最佳策略各是什么？如果两人都采取最佳策略，获胜的概率各是什么？

QL · 发表于 2005-9-11 06:14:44

This can be formulated as a max-min problem..., is there an simpler approach?

husonghu · 发表于 2005-9-11 06:39:29

Could you give some details about your max-min approach?

寒潭清 · 发表于 2005-9-11 09:20:45

欢迎QL回来,问好..

QL · 发表于 2005-9-11 20:41:16

Thanks.

QL · 发表于 2005-9-11 20:49:49

Assume 甲's strategy is to randomly pick an odder number with probablity p1,p3,...,p9

and 乙's stragetgy is to randomly pick an even number with probability q2,q4,...q8

Then 乙 pick q's so as to maximize

p1*q2+p3*(q2+q4)+....p7*(q6+q8)+p9*q8

and 甲 minimize the resulting formular.

fzy · 发表于 2005-9-14 19:51:01

The best strategy for A is to choose 1, 5, and 9 with probablity 1/3 each. It can be calculated that A can win at least 2/3 of the time no matter what B does.

The best strategy for B is to choose 2 and 8 with probability 1/3 each, and 4/6 with a combined probability 1/3. It can be calculated that B wins at least 1/3 of the time no matter what A does.

husonghu · 发表于 2005-9-15 03:14:31

.....A can win at least 2/3 of the time .....
.....B wins at least 1/3 of the time .....

Are these two statements a little contradicting?

fzy · 发表于 2005-9-15 19:29:56

These two statements are not contradicting to each other, since we have >= and <=. Together they guarantee that both strategies are optimal: If applied together, A wins 2/3 of the time, B 1/3, which answers part 2.

原贴：

文章来源: husonghu® 于 2005-9-14 19:14:31
标题：Good But a small question for you ....

.....A can win at least 2/3 of the time .....
.....B wins at least 1/3 of the time .....

Are these two statements a little contradicting?

野菜花 · 发表于 2005-9-22 21:07:06

Thanks a lot for your expensive flower!!

[@}

fzy · 发表于 2005-9-22 22:07:05

我的花和别人一样钱呀。（莫非别人有discount？

）

野菜花 · 发表于 2005-9-22 23:11:35

really worth.

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二人猜数

回复：二人猜数

[@};-][@};-][>:D<]Nice to you again, QL. .....

欢迎QL回来,问好..[@};-][@};-][>:D<]

回复：欢迎QL回来,问好..[@};-][@};-][>:D<]

回复：[@};-][@};-][>:D<]Nice to you again, QL. .....

Answer

Good[@};-][@};-] But a small question for you ....

回复：Good[@};-][@};-] But a small question for you .

Thanks a lot for your expensive flower!! [@};-][@}

我的花和别人一样钱呀。（莫非别人有discount？[:((]）

You sold your posts for less money than they