Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 16 Today a fairly brief overview of multi unit auctions Everything we ve done so far has assumed unit demand In a couple of cases we ve allowed for the possibility of multiple objects either sold by the same seller or by distinct sellers but we ve continued to assume that each buyer only wants a single object Once we allow for bidders wanting multiple auctions things immediately get much more complicated The Milgrom book divides auctions in this setting into two cases depending on whether they follow the law of one price that is whether identical objects sell for the same price Uniform Price Sealed Bid Auctions The simplest auction for multiple identical units is a uniform price sealed bid auction Each bid is a price quantity pair for example a bid to buy 5 units at a price of 20 each Bidders can submit multiple bids at different prices say 5 units at 20 and another 5 units at 15 The auctioneer aggregates all these bids into a single aggregate demand curve and sets a single market clearing unit price which all winners pay Bids can be seen in one of two ways either as a series of prices bid on individual objects 20 20 20 20 20 15 15 15 15 15 in our example or as a demand function demand is 0 if the price is above 20 5 if the price is between 15 and 20 and 10 if the price is below 15 As with double auctions if there are m objects for sale and the prices bid on the mth and m 1st objects are different there is a range of prices that clear the market We typically assume that price is set at the highest rejected bid as in the second price auction for a single good We generally assume that bidders have weakly decreasing valuations for each successive good that is they value the second good less than the first the third less than the second and so on It s not really clear how well this auction works when this assumption is violated As we ll see complementarities among objects lead to problems in many auction formats With uniform price auctions the main problem to be aware of is demand reduction that bidders who want more than one object have an incentive to shade their demand downwards This can be thought of as bidding less than their valuation for incremental units or by demanding fewer units than they want at a given price It s easiest to see this in an example First note that when a bidder is only interested in a single unit his value for incremental units beyond the first is 0 it is a dominant strategy to bid his true valuation for one unit Same as in the second price auction 1 Now suppose there are two bidders and two objects for sale The first bidder wants only one object and his valuation for it is U 0 1 You re the second bidder and you value the first object at v1 and the second at v2 with 1 v1 v2 0 Suppose you bid b1 on the first object and b2 on the second Since b1 b2 b1 can never set the price since it can never be the highest rejected bid Let t1 be the first bidder s valuation which is also his bid If b2 t1 you win both objects and pay a price of t1 per unit If b2 t1 you win one object and pay b2 So expected revenue can be written as Z 0 Z b2 v1 v2 2t1 dt1 1 b2 v1 b2 dt1 v1 v2 b2 b22 v1 1 b2 b2 1 b2 v1 v2 b2 b2 v1 b2 1 v2 which is strictly decreasing in b2 you re best off bidding 0 for the second unit Obviously this example is extreme in many settings you still want to bid for multiple units However when bidding for any unit after the first your bid has an impact both on the probability of winning that marginal unit and on the expected price you pay for the units you ve already bid on the inframarginal units So it s always optimal to bid less than your true valuation on each unit but the first This is analogous to why monopolists price above marginal cost With a single unit for sale in a second price auction you never set the price when you win so your bid only determines whether or not you win not the price you pay When you re bidding on multiple units your ith bid could set the price while you still win i 1 objects so there is an incentive to shade your bid or to reduce your reported demand In his book Milgrom formalizes this into a theorem in a setting with bidders who want two items each The theorem basically says that it s a dominant strategy to bid your valuation for the first object but there s never an equilibrium where all bidders bid their full value on both and therefore as long as the distribution of bidder valuations has full support there s no equilibrium in which the allocation is always efficient Another problem with uniform price auctions is the existence of low revenue equilibria that is equilibria in which the seller s revenue is very low A simple example suppose there are N objects for sale and N bidders each of whom want up to k 1 item and value each incremental item up to this capacity at 1 At any price below 1 there is demand for N k items so you would expect competition to push the price to 1 per unit However one equilibrium of the uniform price auction is for each bidder to bid 1 for one unit and 0 for all additional units leading to total revenue of 0 When the number of objects doesn t match the number of sellers there are similar low or no revenue equilibria involving some bidders not bidding on any objects Milgrom also constructs an example of similar low revenue equilibria in a setting with divisible goods so the problem has nothing to do with the discrete nature of the items for sale 2 Simultaneous Ascending Auctions Another type of auction sometimes used in FCC spectrum auctions among other settings is the simultaneous ascending auction The items up for sale are treated distinctly that is bidders bid on a particular object not a generic one A simultaneous ascending auction takes place in a series of rounds At the end of each round the auctioneer announces the standing high bid for each object and a minimum bid for each item in the next round usually a fixed percentage above the standing high bid In the next round bidders submit new bids In early simultaneous ascending auctions bidders could submit any bid above the minimum in later versions they were limited to a discrete set of choices to avoid communication via bids If multiple bidders submit the same new high bid …