有趣而费解的题: 到底该换还是不该换?

...... · 发表于 2010-9-18 19:10:43

观察值和期望值的南辕北辙总得有个说法.

别得说多了没用.

HF: · 发表于 2010-9-18 20:31:24

QL 是啥意思？

<冷眼看戏的Lili®>：回复：长学问了

连这种帖子都能贴到脑坛来？可怜的QL！

冷眼看戏的Lili · 发表于 2010-9-18 20:35:34

您不是已接受了学生哥帖子中的“期望值”和“观察值”的论断并予赞扬了吗（见https://www.topchinesenews.com/readpost.aspx?topic_id=9&msg_id=8767&level_string=0z11z02z03z01z01&page=1）？怎么又反悔了？这回是笔误还是真反悔，请予澄清，以免别人误解了您，影响坛上声誉。

冷眼看戏的Lili · 发表于 2010-9-18 20:40:39

谢谢理解！

俺跟学生哥商量商量，尽力而为。

冷眼看戏的Lili · 发表于 2010-9-18 20:58:10

HF哥您好！有一段时间没在坛上见您了，请多指教。
那位朋友理屈词穷，竟拿白马非马的典故来说俺是诡辩，因而无词以对（而退场）。所以俺就回敬他一个成语，让他动动脑子，就光写了个缩写。HF哥您别笑话俺啊。

sean9991 · 发表于 2010-9-18 22:07:00

First of all, this question is not about the convergence of the distribution. It's about human behavior. So we'll put that discussion aside.

To understand this paradox, we have to understand utility function. Money has nomination value, but it's not equal to its utility. For example, $200 may be about twice as valuable as $100, but $20,000,000 might not be twice as valuable as $10,000,000. Mathmatically, we say a normal human being has concave utility function, which means the marginal increment of utility is decreasing as the nomination value increses. The paradox uses a hidden assumption, i.e., the utility of money equals to its face value U(x) = x. And it calculates expection based on this identity utility function. Therefore, it's not normal human behavior, and the paradox is created.

Consider this scenario, you picked an envelop and it has $10,000 inside, you expect to get more money if you switch based on the expection. But your decision is based on your utility function at the moment. Let's assume, you have $9,000 debt and it's due today. So identity utility function is not for you, because, the utility of $1000 is way less than $10,000, the utility of $20,000 is not much better than $10,000 in hand. Now the expectation of the utility of the outcome of swiching is less than the utility of $10,000 at hand. So you'll not switch.

冷眼看戏的Lili · 发表于 2010-9-18 23:56:03

在许多涉及相对地过大金额（或财富、产品等）的实际问题中，引进效函数的概念于决策过程中确会使结论更符合人们的感觉，是一种行之有效的“人性化”方法。
本题所示矛盾，其根源并不在于未引进效函数，而是在于推理过程中的概念偷换。俺希望有人能选一个效函数来试试，逐步计算概率和条件概率，再通过效函数来决定换不换，看看能否解决矛盾。当然，拿常数（譬如，0）当作效函数，结论必然是“不用换”，从而没矛盾了（这样，本题也就不是个脑坛题了）。
事实上，在本题给定数据的基础上，不需要引进效函数，仅用已有的概率论方法，经正确推导，就能得到合理的答案，不会产生那种“换来换去”的“悖论”。待俺跟学生哥商量后，贴出整理清楚的答案（答案基本上都在上面的一些帖子中给出了，但因当时需要针对异议答辨，分析、推理比较分散）供讨论。大家再看看，那答案是否有错。

sean9991 · 发表于 2010-9-19 00:31:32

The fundamental of this paradox IS the utility function. And people will make decision based on the expected utility of the outcome. The paradox is generated when the assumption of identity utility function is not normal human behavior. So we feel something is wrong, but cannot explain.

We don't need a specific utility function to solve this paradox. As long as we know the common property of utility function, i.e., it's concave (when the money amount is high enough) for a normal person (aka risk adverse), then it's enough to solve this paradox. Generally, what will happen is, when the amount is small, people will tend to switch, because you don't care. The explanation in math: the utility function is approximately identity utility function U(x) = x at this stage. But when the amount is big, people will stick to what they got. Especially in the scenario I described, the debt will generate a big curve on the utility function. At this stage, the expect utility is not higher if you switch. So any risk adverse person will stop switching when the amount is large enough. But a real gambler (risk seeking) will always switch no matter what. Why? His utility function is convex.

In all, I don't believe this is another explanation without using utility function or some sort. If you got one, I'll be interested to hear about it.

别的人 · 发表于 2010-9-19 00:37:12

Hu兄问题的具体Answer在哪里？能不能把解决Hu兄问题的utility function给出来？
看完帖子的感觉好像是用utility把Hu兄问题河蟹掉了。也就是说，把问题中已给定的决策准则（比较两个期望值或观察值的大小，取大者）推倒重给，就象楼上所举的特例，取常数函数作为utility function，那就河蟹了，什么矛盾也没有了

。这是Hu兄问题的本意吗？

冷眼看戏的Lili · 发表于 2010-9-19 01:33:08

您还是没有给出具体的解法。按眼下俺对您的主意的理解是：举个例子，在本题中，如果俺仅需要马上交房租$500，但打开的信封中只有$1、 $10、或不少于$1000，俺就选“不换”，因为不是钱已够付房租就是再换还是不够；仅当信封中是$100时才说“换”，而且换了以后就不用再考虑换不换了，这也是因为不是钱已够付房租就是再换还是不够。这就避免了“换来换去”。再举个例子，如果俺现在根本不需要钱，那就抽到多少拿多少，一概说“谢谢，不用换啦！”。这样确实很河蟹（用别的人的话）

。不知俺的理解对不对，请指教。
但俺觉得构题的人的原意不是这样的，而且用不用效函数不是本问题的实质。麻烦您等等，俺跟学生哥沟通后，看看谁来贴个完整解答比较方便，也欢迎别的朋友（包括那位别的人）来总结

。俺有点忙，有一篇paper的deadline是下月初。

sean9991 · 发表于 2010-9-19 08:28:42

The answer is already given. And more detail is provided in my reply to your post. When you are not concern about money, then you are risk nutral or risk seeking. You will always switch. I suggest you read a little more on utility function. BTW, it's translated as 效用函数.

冷眼看戏的Lili · 发表于 2010-9-19 10:12:41

俺还是看不出来您是如何具体地解决Hu兄问题的

，好像只是泛泛地讲了效用函数在风险决策中的运用。能不能在Hu兄问题中举具体的数据，例如，所选信封打开后见到$100，按您的观点和方法，来决定要不要换？如果要换，再回答“那么为什么我们不一开始就选择另外那个信封呢？”这个问题。
多谢指教！

HF: · 发表于 2010-9-19 18:11:46

I don't think utility function plays an essential role in the problem. It is clear from the question that the player seeks to maximize expected outcome -- so his utility function is U(x)=x (or any linear function with postive slope).

Whether you introduce such an unitility function or just use the simple expectation-maximization argument, the conclusion is the same: you should switch no matter what the amount of money in your envelope is. But this is not what the question is about. What really drives people nuts is what you can derive from this: since you always switch not matter what you see in your envelope, why do you bother opening your envelope at all? you should simply switch the envelope without looking, and you will always end up better (better expected utility, if you will) -- but on the other hand, this is strange because the two envelopes are 'symmetrical', there is no reason to believe you gain anything by switching.

原贴：

文章来源: sean9991® 于 2010-9-18 16:31:32 (北京时间: 2010-9-19 4:31:32)
标题：回复：回复：Answer

The fundamental of this paradox IS the utility function. And people will make decision based on the expected utility of the outcome. The paradox is generated when the assumption of identity utility function is not normal human behavior. So we feel something is wrong, but cannot explain.

We don't need a specific utility function to solve this paradox. As long as we know the common property of utility function, i.e., it's concave (when the money amount is high enough) for a normal person (aka risk adverse), then it's enough to solve this paradox. Generally, what will happen is, when the amount is small, people will tend to switch, because you don't care. The explanation in math: the utility function is approximately identity utility function U(x) = x at this stage. But when the amount is big, people will stick to what they got. Especially in the scenario I described, the debt will generate a big curve on the utility function. At this stage, the expect utility is not higher if you switch. So any risk adverse person will stop switching when the amount is large enough. But a real gambler (risk seeking) will always switch no matter what. Why? His utility function is convex.

In all, I don't believe this is another explanation without using utility function or some sort. If you got one, I'll be interested to hear about it.

...... · 发表于 2010-9-19 19:54:14

经过一天的外出歇息, 头脑开始逐渐恢复清醒.

如果说世界上有鬼迷心窍的话, 上周肯定是发生在我的身上了. 怎么自己象变了个人似的. 现在自己回头都看着都丑陋.

现在我正在同心中的魔鬼斗争, 理智开始逐渐占据上风.

在此先向诸位朋友说声对不起. 对所有批评我全部接受.

特别是对Lily, 学生哥, 请接受我深深的歉意. 对楼上我所有的出言不驯无条件收回, 还望海涵.

除歉意外, 还有感谢, 正是两位耐心教诲, 帮助, 才使我能尽快的从迷失中恢复理智.

虽说自己才疏学浅, 不过厚脸皮地说, 脑筋还不笨. 关于楼主的题目, 我正在从另一角度去尝试放弃原来的坚持, 不过目前仍有不少疑问没有理清. 也许以后有结果可以拿来与大家分享, 如果不嫌弃的话.

冷眼看戏的Lili · 发表于 2010-9-19 23:50:26

赞一个！不争不相识，坛上仍是高手朋友。
争论中俺也有过激不适之词，收回作废，也望多多包涵。
讨论中，俺也会犯错，还请今后多指教。

冷眼看戏的Lili · 发表于 2010-9-20 00:20:12

不知Sean兄看出来了没有，即使按俺对您的思路而写的“换了以后就不用再考虑换不换了，这也是因为不是钱已够付房租就是再换还是不够您的”看上去象河蟹了，但其中的推理还存在原来那种“换来换去”时曾有的错误。

冷眼看戏的Lili · 发表于 2010-9-20 12:25:59

Hu大哥：经与学生哥商讨，俺已写好题解(MS Word)。因有数学式子，没法拷到脑坛上。现存成pdf文件，但不知如何上传到脑坛。请告诉俺怎么办。谢谢！
别的朋友若能指点，十分感谢！

...... · 发表于 2010-9-20 20:27:23

不可实现的迷思:

每一个观察值是有限的, 但期望值却是无穷大.

这个命题的确有些不可思议, 直觉上感到这个概率分布有夸大其辞之嫌. 好象在说, 你有理由期望太阳从西边升起似的.

由此引出的"换一个会更好, 何不开始就选择另外那个信封"的问题. 在无法找到原因时, 不免归咎于这个期望值为无穷大的概率分布本来就是"不可实现"的.

现在换一个角度, 就算期望值为无穷大的概率分布真的"不可实现", 何不换成期望值为有限的概率分布? 这一点并不难做到: 将题目稍加改动, 把同一箱子里两个信封的钱数由10:1换成小于2的, 比如1.25:1, 其它条件不变, 即

1元和1.25元有1/2概率

1.25元和1.25^2元有1/4概率,

1.25^2元和1.25^3元有1/8概率,

......

这时候的期望值分别为E(A)=1/2*1+1/4*1.25+1/8*1.25^2...=

E(B)=1.25*E(A)

都是有限值. 都是"可实现的"

这时候如果拿到一个信封有x元的话, 换一个信封, 期望值将变成0.8x*2/3+1.25x*1/3=0.95x

假设信封里的钱代表罚款的话, 仍然会出现

"换一个会更好, 何不开始就选择另外那个信封"的问题.

把问题咎于概率分布"不可实现"就再也占不住脚了.

...... · 发表于 2010-9-20 20:28:58

愧不敢当.

sean9991 · 发表于 2010-9-21 07:25:41

First of all, people based his decision on some (his/her own) utility function. The function is set before we proceed to select envelop. I hope you agree on this. The evil of this paradox is "switching is always good no matter what the amount is in the envelop you picked". Here I'll try to break this. My point is, with a "normal" utility function, i.e., utility function that represents a normal person, then the evil can be eliminated.

Now we assume a normal person (risk adverse) with a concave utility function. Based on this assumption, you cannot come to the conclusion that switching is always good. To the extreme, if $X can get you the whole world, is $(X+1) still better then $X? As in the example I gave earlier, if you, with $9000 debt, see $10,000 in the envelop, then switching will give you lower expected return, then you will stay with the envelop. This will break the chain that switching is always good. So your decision will based on the money that is in your hand. You will no longer blindly making decision.

So now, you still want to ask, what if I have identity utility U(x)=x. I want that X+1 even if X will get me the whole world? I'll say with unresonable assumption, you will get unreasonable conclusion, and that is the paradox.

There are still a lot to be said. One of them is people keep asking for specific utility function. A utility function is like a rule in each person's heart. All we can say is its general property. It's concave. It has a upper-bound. It might even start to go decreasing after certain threshold...

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有趣而费解的题: 到底该换还是不该换?

有所期待.

QL 是啥意思？

回复："南辕北辙"? 怎么又反悔了？

回复：回复：白马是马

回复：QL 是啥意思？

Answer

回复：Answer

回复：回复：Answer

Answer在哪里？

回复：回复：回复：Answer

回复：回复：回复：回复：Answer

回复：回复：回复：回复：回复：Answer

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正在闭门思过... 如有跟贴, 暂不回应

赞一个！[:-Q][:-Q][:-Q]

回复：回复：回复：回复：Answer

求助：如何上传Word或pdf文件到脑坛。

突破"不可实现"的迷思

愧不敢当.[:>]

Try to answer several posts in this one