有趣而费解的题: 到底该换还是不该换?

sean9991

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.www.ddhw.com 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.  www.ddhw.com 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.    www.ddhw.com

sean9991

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.   www.ddhw.com 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. www.ddhw.com

sean9991

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.

sean9991

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.  www.ddhw.com 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.  www.ddhw.com 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... www.ddhw.com