Split the cost

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Problem Set 3

N-th price Auctions - Submit

Assume there are 12 bidders and that each has a valuation 0 ≤ vi ≤ $100 for tickets to a Bulls game. Recall that each player submits a bid bi and that whoever gets the good has utility vi -ti, where ti 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 (ti = bi 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 vi ∈ [0,60]. You also each need to agree to pay some amount ti 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 vi -ti. 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 useller = -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 vi = 20?

Is there a (weakly) dominant strategy if vi = 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 bi. According to the VCG mechanism, each pays ti = 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?

Reference no: EM13715497

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