week 111 COS 444 Internet Auctions: Theory and Practice Spring 2009 Ken Steiglitz

week 112 Multi-unit demand auctions (Ausubel & Cramton 98, Morgan 01)Ausubel & Cramton 98Morgan 01 Examples: FCC spectrum, Treasury debt securities, Eurosystem: multiple, identical units Important questions: Efficiency (do items go to buyers who value them the most?); Pay-your- bid (discriminatory) prices vs. uniform-price; optimality of revenue The problem: conventional, uniform-price auctions provide incentives for demand- reduction

week 113 Multi-unit demand auctions Example 1: (Morgan) 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Suppose bidders bid truthfully; rank bids: $10 bidder 1 10 bidder 1 8 bidder 2  first rejected bid, If buyers pay this, surplus (1) = $4 revenue = $16

week 114 Multi-unit demand auctions Example 1: But bidder 1 can do better! Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Suppose bidder 1 reduces her demand: $10 bidder 1 for her first unit 8 bidder 2 for first unit 0 bidder 1 for her 2 nd unit  first rej. bid If buyers pay this, surplus (1) = $10 (larger) surplus (2) = $ 8 (larger) revenue = $ 0! … and auction is inefficient

week 115 Multi-unit demand auctions Thus, uniform price  demand reduction  inefficiency The most obvious generalization of the Vickrey auction (winners pay first rejected bid) is not incentive compatible and not efficient Lots of economists got this wrong!

week 116 Multi-unit demand auctions Ausubel & Cramton 98 prove, in a simplified model, that this example is not pathological: Proposition: Proposition: There is no efficient equilibrium strategy in a uniform-price, multi-unit demand auction. The appropriate generalization of the Vickrey auction is the Vickrey-Clark-Groves (VCG) mechanism… it turns out to be incentive- compatible

week 117 The VCG auction for multi-unit demand Return to example 1: 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Suppose bidders bid truthfully, and order bids: $10 bidder 1 10 bidder 1 8 bidder 2 Award supply to the highest bidders … How much should each bidder pay?

week 118 The VCG auction for multi-unit demand Define:Define: social welfare = W (v) = maximum total value received by agents, where v is the vector of values Then the VCG payment of i is defined to be W( v -i ) − W -i (v) = welfare to others when bidder i drops out (bids 0), minus welfare to others when i bids truthfully = sum of highest k i rejected bids (if bidder i gets k i items) --- the “displaced” bids = her “externality” Notice: this reduces to Vickrey for single item

week 119 The VCG auction for multi-unit demand Example 1: Example 1: 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 If bidder 1 bids 0, welfare to others = $8, and is $0 when 1 bids truthfully…  1 pays $8 for the 2 items

week 1110 The VCG auction for multi-unit demand Example 2 Example 2: 3 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Bidder 3: capacity 1, value $6 $10 bidder 1 10 bidder 1  bidder 1 gets 2 items 8 bidder 2  bidder 2 gets 1 item 6 bidder 3 Welfare to others when 1 bids 0 = $14 Welfare to others when 1 bids truthfully = $8  1 pays $6 for the 2 items

week 1111 The VCG auction for multi-unit demand Example 2 Example 2, con’t 3 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Bidder 3: capacity 1, value bidder 1  bidder 1 gets 2 items 8 bidder 2  bidder 2 gets 1 item 6 bidder 3 Welfare to others when 2 bids 0 = $26 Welfare to others when 2 bids truthfully = $20  2 pays $6 for the 1 item (notice that revenue = $12 < $18 =3x$6 in uniform-price case, so not revenue optimal)

week 1112 Summary VCG mechanisms are efficient incentive-compatible (truthful is weakly dominant) …as we’ll see individually rational (nonnegative E[surplus]) max-revenue among all efficient mechanisms … but not optimal revenue in general, and prices are discriminatory, “murky”

week 1113 Combinatorial auctions In the most general kind of single-seller auction, there are multiple copies of multiple items for sale… we thus distinguish between multi-unit and multi-item auctions. Typically these come up in important situations like spectrum auctions, where buyers are interested in bundles of items that interact synergistically. The textbook example: a seller offers “left shoe”, “right shoe”, “pair”. Clearly, the pair is worth much more than the sum of values of each.

week 1114 Combinatorial auctions Here’s an example of common practice in real- estate auctions [Cramton, Shoham, Steinberg 06, p.1]: Individual lots are auctioned off Then packages are auctioned off If price(package) > Σprice(lots in package), then the package is sold as a unit; else the constituents are sold at their individual prices

week 1115 Combinatorial auctions The VCG mechanism generalizes to combinatorial auctions very nicely, which we now show. We follow L.M. Ausubel & P. Milgrom, “The lovely but lonely Vickrey auction,” in Combinatorial Auctions, P. Cramton, Y. Shoham, R.Steinberg (eds.), MIT Press, Cambridge, MA, Setup:

week 1116 Combinatorial auctions To illustrate notation: Suppose there are 4 bidders. The supply vector and a feasible assignment are: = (supply vector) x 1 = bidder 1 gets two of item 2, etc. x 2 = x 3 = x 4 =

week 1117 Combinatorial auctions Define the maximizing assignment x* by Each buyer n pays: α n is the max. possible welfare of others with bidder n absent, doesn’t depend on what bidder n reports in general relative to the reporting function of others …thus efficient if truthful

week 1118 Combinatorial auctions So in VCG (truthful bidding) the bidder pays the difference between the max. possible welfare of others with n absent (α n ), minus the welfare of others that results from the maximization with n present. You can think of this as the externality of bidder n, or, perhaps, the social cost of n’s presence.

week 1119 Combinatorial auctions Theorem:Theorem: In a VCG mechanism truthful bidding is dominant, and truthful bidding is efficient (maximizes total welfare). Proof:Proof: Fix everyone else’s report, (not necessarily truthful). Denote by x * and p * the allocation and payment when bidder n reports truthfully; and by and the allocation and payment when bidder n reports.

week 1120 Combinatorial auctions Bidder n’s surplus when she reports is which is n’s surplus when reporting truthfully. □

week 1121 Combinatorial auctions Bidder n’s surplus when she reports is which is n’s surplus when reporting truthfully. □ true value by def. of x * by def. of p * by def. of x ^

week 1122 Combinatorial auctions Ausubel & Milgrom 2006 discuss the virtues of the VCG mechanism (for general combinatorial auctions) : Very general (constraints easily incorporated) Truthful is a dominant strategy Efficient Maximum revenue among efficient mechanisms …Sounds good, but as we’ve seen, revenue can be disastrously low!

week 1123 Combinatorial auctions Weaknesses of VCG (for general combinatorial auctions): Low (or even zero) revenues Non-monotonicity of revenues as functions of no. of bidders and amounts bid Vulnerability to collusion of losing bidders Vulnerability to use of multiple bidding identities by a single bidder Prices are discriminatory Loses dominant-strategy property when values not private In general case bid expression, winner determination, payoff calculations become computationally intractable

week 1124 NP-complete in a nutshell (1) We think of problems as language-acceptance questions: a problem is a set of strings that describe YES-instances, and if there is a Turing machine that accepts that string in polynomial time, the problem is “easy”, that is, in P. Example: is node a connected to node b in a given graph? The class of problems NP are those problems whose YES- instances can be checked in polynomial time. Example: Does a given graph have a Hamilton circuit? Easy to check, hard to find! Cook’s theorem: All problems in NP reduce in poly. time to Boolean Satisfiability: does a given logical expression have a satisfying truth assignment? That is, if there is a fast algorithm for recognizing satisfiable expressions, then P=NP. Any problem in NP with this property is called NP- complete.

week 1125 NP-complete in a nutshell (2) See the list of NP-complete problems in wikipedia forthe list of NP-complete problems > 3000 such problems. It is widely believed that NP- complete problems are intractable in the sense of having no poly.-time algorithms. The usual way to show that a problem is NP-complete is to reduce a known NP-complete problem to it. By transitivity of reduction such a problem is as hard as any in NP. Example: SUBSET SUM: given a finite set of integers Α and a positive integer B, is there a subset of Α whose sum is precisely B ?

week 1126

week 1127 complexity classes P and NP may all be one class! (but everyone thinks P≠NP)

week 1128 Payment calculation in VCG is NP-complete Consider the step in VCG where the maximizing assignment is calculated: Formulate this as the recognition problem: Is there an assignment that achieves a given total value B ? SUBSET SUM clearly reduced to this; just make the values equal to the set members of Α and set the target value to B.

week 1129 An example of a (real) intractable auction Here’s a real example of an auction in which winner determination is intractable. It’s suggested by a comment on Frank Robinson’s mail-bid sale: “You can bid with a budget limit, or with alternate choices.” A coin dealer conducts an auction by mail as follows. She sends out an illustrated catalog describing n items k customers. Each customer then returns a list of integer bids for the items; that is, the maximum amount she is willing to spend for each item. (Bids may be 0.) In addition, the customer sends an integer limit, which is a limit on the total amount of money she is willing to spend on all the items she purchases. She wants to award items in such a way as to respect the bids, limits, and at the same time maximize the total revenue realized. (Proof of NP-completeness of winner determination is left as an exercise.)

week 1130 Single-seller auctions