Problem Set 3

N-th price Auctions - Submit

Assume there are 12 bidders and that each has a valuation 0 ≤ v_i ≤ $100 for tickets to a Bulls game. Recall that each player submits a bid b_i and that whoever gets the good has utility v_i -t_i, where t_i is what they pay. Everybody who does not get the good pays nothing and has utility of zero. You are player 1 and have a valuation of v₁ = $60, and need to decide how to bid, though you do not know b_-1 = (b₂,b₃,...,b₁₂)

First-Price Auction

Consider a first-price sealed-bid auction (t_i = b_i for the winning bidder i). Describe a situation (ie a set of other bids b_-1) in which it is better for you to bid b₁ = $55 rather than b₁ = $60. You have thus shown that bidding your valuation is not a dominant strategy since in at least one case (the one you have given) it is better to bid something else.

Second-Price Auction (Vickrey Auction)

Consider the second-price auction, in which the person with the highest bid pays the second highest bid.

Consider bidding lower than your valuation (say b₁ = 55). Give a concrete example (again some b_-1) in which you are better o bidding b₁ = 60 rather than b₁ = 55. Also, describe in words (do your best, though this is not easy) why there does not exist b_-1 in which your utility is actually higher by bidding b₁ = 55 rather than your valuation.

Now consider b₁ = 65. Again, give a concrete case of b_-1 in which bidding your valuation is better than this bid. And again, describe why there does not exist an example in which bidding your valuation is worse. You have thus informally shown that it is a dominant strategy to bid your valuation.

Third-Price Auction

Consider a third-price auction: The individual with the highest bid receives the good and pays the third highest bid. Give a concrete case when some bid is better than b₁ = 60. To do so you just need to imagine (and write down) a particular b_-1 in which this turns out to be true. You have thus shown that it is not a dominant strategy to bid your true valuation in a third-price auction.

Public Goods ProblemSuppose you and your roommate are deciding whether or not to buy an espresso machine. The machine costs $50. You each have a value v_i ∈ [0,60]. You also each need to agree to pay some amount t_i such that t₁ + t₂ = 50. If the espresso machine is not bought you each get utility of zero. If you do buy the espresso machine then each roommate gets utility v_i -t_i. Efficiency

First, what is the utilitarian (or efficient) outcome and how does it depend on each roommates' valuation? To answer this question you also need to consider the utility of the seller. Her utility is u_seller = -50+t₁ +t₂ if a sale takes place and zero otherwise.

Split the Cost

Suppose your roommate suggests that you both vote yes or no. If you both vote yes than you both buy the machine and pay $25 each. If either of you votes no then the machine is not bought.

Write down the 2x2 game matrix. Payouts should be in terms of v₁ and v₂.

In this game, what is a (weakly) dominant strategy for a roommate with valuation v_i = 20?

Is there a (weakly) dominant strategy if v_i = 30?

Will the split the cost game always generate the efficient outcome regardless of each roommates' valuation?If yes, explain. If no, give an example (ie a pair of valuations) in which the Nash equilibrium of this game produces a non-utilitarian outcome.

VCG (Vickrey-Clarke-Groves) Mechanism

In this particular situation each roommate states how much they value the machine. Again call this b_i. According to the VCG mechanism, each pays t_i = 50 - b_-i if b₁ + b₂ ≥ 50. In other words, the payment is only made if the machine is bought.

We want to show that bidding your true valuation is a dominant strategy. To make this concrete, supposeyour valuation is v₁ = 40. Explain why, regardless of what your roommate bids, that bidding your valuation (weakly) dominates all bids less than b₁ = 40.

Explain why bidding greater than 40 is weakly dominated by bidding 40.

This is great! Looks like it is a dominant strategy to bid your true valuation and thus the espresso machinewill only be bought when it is efficient to buy it. But whoops, there is just one little (or big) problem with this scheme. What is it?