investigate the effect of collusion in the second price sealed bid auction. Throughout, you may assume that the bidders' are drawn independently from a uniform distribution on [0, 1] and that bids outside the interval [0, 1] are inadmissible.
Assume that impediments to collusion are minimal, in the following sense. All bidders involved in a bidding ring can costlessly determine each other's valuations. Furthermore, the ring has no problem making (and enforcing) agreements on side any payments necessary to guarantee voluntary participation of all ring members. Thus the objective of the ring is to maximize the joint surplus of its members.
(a) Consider first the case of two bidders. If they form a ring, what two bids should they optimally submit? Explain your answer.
(b) Suppose now that this ring faces an enforcement problem because ring members cannot be prevented from breaching the agreement. If the ring has to guard against this possibility, will it require different bids from its members?
(c) Suppose instead that there is a third bidder, who does not participate in the ring. Does the presence of this extra bidder change the bidding behavior of the ring members?