Problem 4

Let agents 1 (Alice) and 2 (Bob). The optimal case is when gets and gets if , and vice versa, thus maximum welfare for efficient allocation is

will demand both initially when but will drop out once

Case . Then

And will not demand anymore. This will lead to two cases:

  • If auction continues since both chooses . will drop out eventually, as will be raised until while still demands as long. Total welfare is .
  • If then will drop out, will get with total welfare . Same can be shown for .

Problem 5