Answer the problems below as best as you can – you can answer formally, or informally; however, when possible, formal answers might be more precise, and to that effect convincing.

problem: As in class there is some set N of bidders in an auction of one object, which the seller of the object can deliver at 0 cost. Suppose each bidder i’s valuation of the object is given by some parameter v_i ≥ 0 which is only known to bidder i.

1) Suppose the auction format is the first-price auctions, where bidders simultaneously prepare their bids, submit them, and the winning bidder (the one with the highest bid) pays her bid. Would bidders want to bid their valuations, below their valuations, or above their valuations?

2) Suppose now that the auction proceeds as follows: initially some low price is announced, say p₀ = 0, and it keeps being increased continuously through time, e.g., at time t, p(t) = t. At each moment, all biders who still want to remain in the auction keep their hands raised; at the moment a bidder wants to drop out she lowers her hand (and she can’t raise it again). The object is awarded to the last bidder with the raised hand. At what price would a bidder want to lower her hand?

3)  Suppose that each bidder i’s valuation of the object is now given by mini_∈N v_i, where still only v_i is known to bidder i (so now in most cases she can’t be certain about the value of the object). All bidders again simultaneously submit their bids (e.g., in sealed envelopes), and the object is awarded to the highest bidder at the price p = 1/N ∑_i∈N b_i, where bi is the bid of bidder i. How would the bidders want to bid now?

4) For the case where each bidders’ valuation is given by v_i (only known to i), what would you suggest to the seller if he wanted to devise the auction to raise as much revenue as possible?

problem: There are now a number of sellers from set M, and buyers from set N. Each seller j can deliver one unit of the good at some price c_j ≥ 0which is only known to him, and each buyer i desires to buy at most one unit of the good, which she values at v_i, only known to her.

1) Suppose there is only 1 seller. He runs an auction in which the winning bidder (the highest bidder) pays the price equal to the second-highest bid. If the seller wanted to make sure to not make a loss, what would he want to add to the rules of the auction?

2) Suppose there is only 1 buyer and the sellers now simultaneously propose different auction formats to her. She then decides with which seller she will go. What kind of auction formats would sellers propose to the buyer?

3) To make things simple suppose now that 0 ≤ c₁ ≤ c₂ ≤ ...c_m ≤ 1 and 0 ≤ v_n ≤v_n−1... ≤ v₁ ≤ v0. Think of sellers as the supply and buyers as the demand.

Further assume that k is the number such that c_k < v_k but c_k+1 > v_k+1, and that it so happens that c_k < v_k+1 < c_k+1 < v_k.

i) What does k represent? (Draw a graph)

ii) Suppose sellers and buyers simultaneously prepare their asks (sellers) and bids (buyers) and submit them. The price pa that the sellers obtain is determined as the minimal submitted bid (call it b), such that all sellers who deliver the goods at that bid were actually asking less than b (suppose that the number of sellers with such asks were!), and such that there were exactly l buyers who were bidding more than b. The price pb is similarly determined as the maximal ask (call it ¯ a), such that there are buyers who bid more, and l sellers who ask for less. In the con?guration above (sellers and buyers don’t know that is the con?guration) would sellers want to prepare down their costs as their asks and would buyers want to prepare down their valuations as their bids? Why or why not?

(iii) In the market system considered in part (ii) would the final allocation be Pareto efficient? Why or why not? (Do not forget to take into account important details).