这个作业是根据场景制定最优的拍卖策略，关于资产定价和金融计量经济学
ECON 641 Homework #2
November 2020
1拍卖
1.假设有三个竞标者和两个商品A和B。每个竞标者的
每种商品及其组合{A，B}的评估为
下表给出：
捆绑v1 v2 v3
∅0 0 0
{A} 2 1 1
{B} 2 1 1
{A，B} 2 1 4
（a）什么是福利最大化分配？
（b）假设根据Vickrey-Clark-Groves的规定出售捆绑包
（VCG）拍卖，我们在课堂上进行了讨论。识别收货人
每个投标人及其支付的金额。
2.考虑拍卖单个物品。有两个竞标者。每个投标人
根据[0,100]上的均匀分布得出的值具有一个私人已知的值。
假设要求投标人提交密封投标书。高出价者
在拍卖中胜出，并需要支付获胜者的平均值，
失败的竞标。
（a）解决贝叶斯纳什均衡。
（b）将预期拍卖收入与第一价格进行比较，然后
第二价格密封的竞标拍卖（如课堂讨论）。
3.卖方希望出售一件物品，并且有两个投标人，A和B。
知道该项目对于A来说是理想的，对于B来说是理想的，或者都不是理想的。
前两个事件中每一个的概率为z，并且存在1 − 2z
该对象对两者都不理想的可能性。如果该项目是“理想”的
出价者，出价者将以100来表示获胜。否则，出价者将
价值为50的中奖金额。两个竞标者可以识别出某个项目是否为
对他们来说很理想，因此知道他们的个人价值观。
1个
（a）如果卖方进行第二次价格拍卖，最佳策略是什么
每个竞标者，纳什均衡？
（b）第二次价格拍卖的预期收益是多少？
（c）发布者可以通过设置底价（最低）来增加收入吗？
减多少（提示：可能取决于z）
（d）假设卖方宣布它将首先报价
以pH值卖出，然后如果没有人愿意以该价格购买，则提供
价格pL，并且如果有多个竞标者想要以给定价格购买
价格，它将在决定赢家之间随机分配。找
pH和pL的选择可以最大化卖方的收入，
投标人充分了解机制并做出战略响应。
（e）比较（b），（c）和（d）中卖方的收入。哪个是卖方的收入最大化选项？
2 Matching
1. Consider a matching market with four heterosexual men and three heterosexual women who have the following preferences:
m1 : w1 w2 w3 w1 : m2 m3 m1 m4
m2 : w3 w2 w1 w2 : m4 m2 m3 m1
m3 : w3 w1 w2 w3 : m4 m2 m1 m3
m4 : w1 w3 w2
(a) Use the deferred acceptance (DA) algorithm to find the womanoptimal stable matching.
(b) Are there other stable matchings? If so, what are they?
(c) Is the woman-optimal stable matching Pareto optimal for women?
For men?
(d) What happens if we use the top trading cycles (TTC) algorithm?
Try it first to give women what they want, and then for men. How
does the result related to (a) and (b)? Is it stable?
2. Suppose there are two schools in an area: A and B, and eight students in
an area labeled s1, …, s8. The students are numbered by their test score
rankings, so that s1 has the highest test scores, s2 the second highest, etc.
Each school can accommodate four students. All students would prefer
school A, and both schools rank the students by their test scores.
(a) What is the unique stable matching of students to schools?
(b) Now suppose school A decides to implement a preference program
that favors students who live near the school. Students s2, s6, and
2
s8 are ’local.’ Specifically, suppose slots are allocated using a DA
algorithm, and A makes sure to allocate half of its slots to local
students. What is the resulting match?
(c) Suppose instead A divides itself into two schools A0 and A00, each
with two seats. A0 prioritizes by test score and A00 ranks by local/
not-local and within those categories by test score. What happens
in the DA algorithm if students prefer A00 to A0
?
(d) What if students prefer A0
to A00?
(e) Among (c) and (d), which is more effective in your opinion? Would
you have a different view if s8 preferred school B?