主题：两个女生和一台二手电视机引来的game theory 和 auction 问题 -- Ready-Go
这题目够晕吧，我起的 
，为什么起这么个名字，让朋友逼的，误交匪类呀，大家一定要吸取我的惨痛教训。
这要从昨天说起，未名空间贴出一个贴子，说的是一件小事情。
原文如下
两个女生A和B，在一起住了一年。开始的时候凑钱160$买了一个新电视机。
（每人平摊80$）
现在B要搬走了，不想要电视机，希望能够返还一部分钱，由此产生了纠纷：
B认为：假设使用一年电视磨损后还有80%价值，希望A给他40%的原价。
A认为：二手电视机这里不太好卖，根本卖不到80%，叫B自己去花80%的价格
买一个用过一年的电视机估计她自己也不愿意。B用了一年电视才给16$，
A感到不划算。A只赞成返还30%。
虽然就是十多元的差异，但是搞得两个人心中很不爽。
我给的建议就是：现在把让B去电视卖掉平分钱。
不卖的话，大家又何建议？
我朋友，对，就是那匪类，看了这贴子特兴奋，完全是纳什当年挖掘纳什平衡的架式，昨天和我为这个 case 讨论一天，最后提出他的 ultimate solution，原文如下
I have a very good solution for this problem from business game theory
perspecitve. No 3rd party will be involved.
I propose a Dutch auction for the TV between the 2 roommates A and B.
The start bidding price will be $160 and each time the bidding price will be
dropped, say $10, until someone take the TV.
Let's suppose A eventually won the bid for $100.
Then A need to pay $100 for the TV, and (160-100)/2=30 for the depreciation
cost. A paid $80 dollars already when they purchased the TV. Therefore, A only
need to pay 100+30-80=50 to B.
B originally needs to pay (160-100)/2=30 for the depreciation cost.But since B
has paid $80 when the TV was purchased. So she got back 80-30=50 from A.
I believe this is the fairest solution. And any challenge will be welcome!
我的这个朋友今天和我吵了一路，说他的解法是最公平的，我说不是，但我还没有经过计算，论据不足。他让我贴上，让大家帮他找出这个理论公平的论据，RG是请大家帮着找出推翻他的方法的理由。他要是有一天因为这个得了诺贝尔奖，分给找出理由的网友二分之一奖金，要不你们两个也来个 auction，dutch auction 还是 english auction 随便。
许多事情以小见大，这位估计要从这个小 case 里建个模型发展出套管理学原理。昨天晚上我 min max 了半天也没给他弄出来，最后只好委曲求全，丧权辱国地承认他的办法最公平。否则我就没办法看书了。
但是，本姑娘没理狡三分，为什么 dutch auction 就比 english auction 和 Vickrey Auction 合理，为什么 open bid 在这个 case 里就比 seal bid 强，推给他让他想破脑袋去
最有趣的小品文。我认为只要双方接受，规则没有更公平、最公平之分（其实是自己笨，回答不了姑娘的问题，等着高人啦。）
从你朋友的分析来看，A显然是花了130大洋买了一个二手电视。
如果能以80大洋从别处买到一个类似质量的二手电视，A显然不会bid for $100。
我觉得这个问题的关键不在于采用那种拍卖方式，而在于二手市场价格。BTW，我那台二十五?嫉亩?手大彩电，才20大洋，呵呵。想想如果A能以20大洋买到另一个二手电视，他会bid多少？在加上他明知道B不想要，他完全可以尽量压低价格。
所以，你朋友那个suppose没有一点根据，自然结果也就没有什么意义。
难道是张三，李四，王五，赵六，我是孙二麻子
I agree with Jlanu.
From the Auction, at price 100, if A buy the TV, she got net value: 100-(80-30)=50;
While B got the net value: 80-30=50
But if u check the original argument:
if discounted at 80%, A got net value:160*0.80-80+(160*0.2)/2=64; B got the net value:80-(160*0.2)/2=64. It is fair enough,right?
If discounted at 60%,similarly, both of them got the net value:48.
Thence, if they can sell the TV higher than 100, the Aucion(selling at 100) is not fair to B;
if sell lower than 100, then the Auction is unfair to A.
What matters here is the possible selling price.
So, ur method might be the most fair one---Just sell it.(I guess.)
不过RG的方式倒一定是错的.题目开头就说明了二手不好卖.(所以说明好买吗?)
偶的看法倒很简单,因为这个Game一点也不公平.B不想要电视机,均衡点在于A的良心,也就是他肯给B多少钱.
要谈Game Theory,最近的知识经济是个很好的题材呢.
先说一句好不,如果说这方法错了呢,就说RG朋友错了,如果说这方法对了,就说RG对了,公开卖友求荣也是经济学一个CASE哈。
关于这个方法对B不公平的问题，他的解释如下：
I already have the answer for your question.
Let us compare what A and B want.
B: 1.Want to get as much money back as possible.
2.Do not want to keep the TV.(It is called OPTION in Business terms)
A: 1. Only want to give as little money back to B as possible, and don't care
about the option of whether keeping the TV or not.
B should pay more because B not only wants the money back but also
wants the OPTION of not keeping the TV(B愿意不怕麻烦卖掉这个电视). The extra
money paid is called PREMIUM in business term.
Therefore, it is not unfair for B even if the TV is sold below market value
Firstly, we assume the auction price is a fair maket price.
If B thought she suffered from a loss due to so called Option, she could ask to sell the TV, if the selling price is fair.---Now, A got the same Option to sell or to keep the TV,right? Should this Option be considered?
I think we can not say the loss which B would like to accept is due to OPtion.
If u insist there be a loss, it might be from the buying and selling difference (Repurchase cost). If A and B do sell the TV, A will need to buy another one, that might cost A money and the trading fees. (But the result is the same as the Auction, that is, A keeps the TV,B gets the money.) Such a loss, A wanna share with B by underestimating the market price(Auction price), then it is unfair.
在市场上寻找同种型号或类似型号二手货的价格,作为该电视机的估价,然后付一半的价钱给B.
最后只好委曲求全，丧权辱国地承认他的办法最公平。否则我就没办法看书了。
但是，本姑娘没理狡三分，为什么 dutch auction 就比 english auction 和 Vickrey Auction 合理，为什么 open bid 在这个 case 里就比 seal bid 强，推给他让他想破脑袋去
――――――――――――――――――――
这个世界，什么是能惹得，什么是不能惹得。
没啥说的，以后在MM面前，俺会坚定不移的低眉顺眼，老实厚道来着。
那位匪类同志，您的牺牲是有价值DI，西西河的男同胞们，会永远地的记住您DI
……（光辉形象？丑态，前车之覆……
不敢再往下想了）
What is fair and what is unfair is a
philosophical question, which depends
on people's ideology and cannot be
settled by game theory.
What they are disputing her is that the
joint utility derived from the TV by A
and B decreases because B is leaving.
Because the second-hand market is
illiquid and transaction cost, they
cannot return to their original state.
Someone must bear this loss of value,
the central question is how to divide
this loss fairly.
I can think of at least 2 ways to
define fairness
1. The loss of value is caused by B, so
B should bear the whole cost.
2. Everyone should bear equal share of
the loss.
Your second friend's argument that the
Dutch Auction format is unfair is based
on the second definition. Because B has
to pay the transaction cost to sell the
TV, and A has to pay transaction cost
to buy the TV, the TV worths more to B
to A. Therefore whatever auction format
you choose, B will be better off than
A, so it is unfair.
Now on auction formats.
1. Dutch auction is equevalant to first-
price sealed bid.
2. This auction game is highly
irregular, if you want to model it
realistically.
(1). The bidder not only need to pay
their bid, but also (160-x)/2, which
depends on their bids. The Revenue
Equvalence theorem breaks down. Assume
it is a symmetric private value
auction, (more on that later), and
uniform distribution of types, denote V
as valuation, you can prove the optimal
strategy in Dutch auction is 3V/4
(higher than the standard case which is
V/2), while the optimal strategy in
English Auction (and Vickey Auction) is
2V-160 (lower than the standard case
which is V). Dutch auction generates
more revenue than English auction. B is
better off, and A is worse off in Dutch
Auction. (Because A is more likely to
win the auction.)
Furthermore, if V is less than 80, they
want to bid a negative number, the
Vickey auction is not viable.
(2). However, because of the second
hand market, there is a large part of
common value in this auction. Milgrom
and Weber (1982) shows that in this
case, English auction generates more
revenue than VIckey Auction, and Vickey
Auction generates more revenue than
first-price auction. So B is worse off
and A is better off in Dutch auction.
(3). Because we know B probably values
the TV less than B, the valuation is
asymmetric. A will bid more agressively
in first-price auctin than in English
auction. However, I don't know how this
will affect A and B's utility.
In conclusion, as to the choice of
auction format, the answer is I don't
know. It needs further assumptions and
calibration about how A and B's
valuation is distributed and correlated.
I can do the research for them, if they
give me a fair payment.
应该说不仅仅b有option，a也有，就是不愿意麻烦去再买个新的（也可能是盼望着把这台dump掉好换个更新款的，视实际情况而定）。两人还有个共同的愿望，就是不想真的把电视去买了。由于信息不平等的问题（卖的人知道这是好电视，确实是一年前买的，买的人却不能肯定），所以售价一定比其实际价值低。
所以一方面一人想多拿到钱，一人想少付钱，两人是竞争的关系；另一方面，又尽量不能让谈判破裂，以至于真的把电视卖了，那对双方都是损失。在既合作又竞争的情况下应该用什么模型来解决？我不知道。但经济学家们一定研究过的。而且，我有个感觉，顺着这条路想下去，纳什同学真的要来啦。似乎和他的那套东东是相关的，只不过我一下子想不清楚。请高手帮我理顺吧。
还有一件事要考虑，事情是B引起的。如果她不搬，不就什么事也没有？所以她应该多付出一点。越来越复杂了。