COS 444 Internet Auctions: Theory and Practice

COS 444Internet Auctions:Theory and Practice Spring 2009 Ken Steiglitz ken@cs.princeton.edu

Multi-unit demand auctions(Ausubel & Cramton 98, Morgan 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

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, price=$8@ If buyers pay this, surplus (1) = $4 revenue = $16

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 2nd unit  first rej. bid If buyers pay this, surplus (1) = $10 (larger) surplus (2) = $ 8 (larger) revenue = $ 0! … and auction is inefficient

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!

Multi-unit demand auctions • Ausubel & Cramton 98 prove, in a simplified model, that this example is not pathological: 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

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?

The VCG auction for multi-unit demand • 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 kirejected bids (if bidder i gets ki items) --- the “displaced” bids = her “externality” Notice: this reduces to Vickrey for single item

The VCG auction for multi-unit demand 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

The VCG auction for multi-unit demand 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

The VCG auction for multi-unit demand 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 $6 @$10 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)

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”

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-unitand multi-itemauctions. • 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.

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

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, 2006. Setup:

Combinatorial auctions • To illustrate notation: Suppose there are 4 bidders. The supply vector and a feasible assignment are: = 1 3 4 2 (supply vector) x1 = 0 2 1 0 bidder 1 gets two of item 2, etc. x2 = 0 0 1 1 x3 = 0 0 0 1 x4 = 1 0 1 0

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

Combinatorial auctions • So in VCG (truthful bidding) the bidder pays the difference between the max. possible welfare of others with n absent(αn), 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, thesocial cost of n’s presence.

Combinatorial auctions • Theorem: In a VCG mechanism truthful bidding is dominant, and truthful bidding is efficient(maximizes total welfare). • 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 .

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

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

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!

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

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.

NP-complete in a nutshell (2) • See the list of NP-complete problems in wikipedia for > 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 Αand a positive integer B, is there a subset of Α whose sum is precisely B ?

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

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 Αand set the target value to B .

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.)

Single-seller auctions

COS 444 Internet Auctions: Theory and Practice