Auctions Supplemental Material

In case you haven't noticed: Auctions are Everywhere! eBay → simple auctions for a single item AdWord Auctions → advertisers bid on keywords for Google searches, pay-per-click determined by next highest bidder Procurement Auctions → government contracts, B2B procurement, open-package inventory clearance Electricity Auctions → long-, medium-, and short-term contracts for wholesale electricity power most of USA Spectrum Auctions → which telecom companies get to use which frequencies in each region

Auction Goals Efficiency: the items being auctioned go to those who value them most. Price Discovery: an iterative process allows competitors to learn about supply-demand pressures to determine market value of items Fairness: No benefit to “gaming,” prices are fair to the seller, ex-post satisfaction for bidders (competitive equilibrium)

Classic auction theory English auction: 1 item, price rises until only one willing buyer remains –Provably optimal strategy: stay in until price reaches your true value –Winner pays one bid increment above second highest bidder’s value Sealed-bid auction: –Each bidder submits value for item –First-price variation: pay-what-you bid –Second-price variation: winner pays second highest bid

Auction Environments Forward Auction - Many Buyers, One Seller (like eBay)Forward Auction - Many Buyers, One Seller (like eBay) Reverse Auction - Many Sellers, One Buyer (More common in B2B commerce)Reverse Auction - Many Sellers, One Buyer (More common in B2B commerce)

Auction Types English Auction – Auctioneer raises prices and bidders indicate they are willing to buy at that price, until only one bidder is leftEnglish Auction – Auctioneer raises prices and bidders indicate they are willing to buy at that price, until only one bidder is left Most Common (Like eBay)

Sealed-bid Auction Types First-Price Auction – All bidders submit a sealed-bid. The bidder with the highest bid wins and pays her bid.First-Price Auction – All bidders submit a sealed-bid. The bidder with the highest bid wins and pays her bid. Second-Price Auction – All bidders submit a sealed-bid. The bidder with the highest bid wins and pays the amount of the second highest bid.Second-Price Auction – All bidders submit a sealed-bid. The bidder with the highest bid wins and pays the amount of the second highest bid.

Sealed-bid Auction Types Generalized-Second-Price (GSP) AuctionGeneralized-Second-Price (GSP) Auction –Used by Google for their AdWords process –All bidders for a keyword bid –kth-highest bidder gets the k-th highest slot –Price is the bid of the k+1 st highest bidder –Only pay if the searcher clicks on the link

Properties of the Second-price Sealed-bid Auction for 1 item Individual Rationality (IR): Bidders each expect a non- negative payoff Efficiency: the highest bid wins Dominant Strategy Incentive Compatibility: Misreporting value never gives an advantage The “Core” property: no coalition can form a mutually beneficial renegotiation among themselves Strategic Correction property: The winner pays as if she had bid optimally in a First-price auction Competitive Equilibrium: bidders get what they want, given the market price(s)

What is a Combinatorial Auction? Any auction for multiple items in which bidders may bid on combinations of items, rather than placing bids on items individually. Advantages: –Complements: don’t have to get stuck with less than what you want –Substitutes: don’t have to get stuck with more than what you want Disadvantages: –Potential computational difficulty for determination of winners, payments, and strategies

Best Packages

Combinatorial Clock Auction Primary Stage –Prices announced and bidders indicate their preferred package of spectrum –Prices rise on excess demand categories until demand can be satisfied Supplementary Bidding –All primary bids become exclusive package bids –Additional Package bids are submitted –Allocation and Payments determined by the techniques presented here Feb UK 800MHz and 2.6GHz combined auction

More details: In the Principal Stage, bids are for generic lots in each category or band, not the specific frequency This makes communication of bids drastically less complicated, and the winner-determination process computationally manageable In a secondary Assignment Stage, bidders winning lots in the same category compete to decide their actual position within the band Principal Stage raised £2.34 billion Assignment Stage raised £27 million Feb UK 800MHz and 2.6GHz combined auction

Results after clock rounds (narrow), and after supplementary bidding/ final allocation

The Beautiful Generalization: The Vickrey-Clarke-Groves (VCG) Mechanism There is a unique mechanism that satisfies: –Individual Rationality –Efficiency –Dominant Strategy Incentive Compatibility for a general set of items with arbitrary preferences. Each winning bidder j gets a discount equal to: wd J – wd J \ j or, the difference between the highest collection of bids with and without bidder j

VCG example (substitutes) b 1 (A) = 4, b 1 (B) = 3, b 1 (AB) = 6 b 2 (A) = 3, b 2 (B) = 4, b 2 (AB) = 5 Efficient solution: bidder 1 gets A, bidder 2 gets B Discount to bidder 1: wd(1,2) = 8, wd(2) = 5, discount = 8 – 5 = 3, payment = 1. Discount to bidder 2: wd(1,2) = 8, wd(1) = 6, discount = 8 – 6 = 2, payment = 2. Interpretation: Each bidder pays the min. amount necessary to take her good away from the other

What’s wrong with VCG? The Quintessential Example (Ausubel and Milgrom) b 1 (A) = 2, b 2 (B) = 2, b 3 (AB) = 2 VCG outcome: Bidder 1 and 2 pay zero The seller would be better off bargaining with 3 for non- zero payment (both would prefer it) Thus, this outcome is not “in the Core” This example can also demonstrate a bidder shill bidding strategy or a vulnerability to collusion

A Practical Generalization: Core-Selecting Mechanisms Prevailing attitude in Mechanism Design literature: Incentive Compatibility must be upheld (a constraint.) Since VCG is not practically viable we must drop DS Incentive Compatibility as a hard constraint –(IR and Efficiency must stay) The perspective of core-selecting mechanisms: Incentive compatibility is an objective –Maintain IR, efficiency, and the core property (with respect to submitted bids) as constraints –minimize the incentives to misreport Note: 2012 Nobel Prize given for Matching Algorithm which sacrifices incentives for the core property, similar to our approach

The Core An Allocation / Payment outcome is blocked if there is some coalition of bidders that can provide more revenue to the seller in an alternative outcome that is weakly preferred to the initial outcome by every member of the coalition. An unblocked outcome is in the core. A Core-Selecting Mechanism computes payments in the core with respect to submitted bids.

5 bidder example with bids on {A,B} b 1 {A} = 28 b 2 {B} = 20 b 3 {AB} = 32 b 4 {A} = 14 b 5 {B} = 12 Winners VCG prices: p 1 = 14 p 2 = 12

The Core b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment The Core Efficient outcome

The Core b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment VCG prices VCG prices: How much can each winner’s bid be reduced holding others fixed? Problem: Bidder 3 can offer seller more (32 > 26)!

The Core b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment VCG prices Bidder-optimal core prices: Jointly reduce winning bids as much as possible Bidder-optimal core Problem: bidder- optimal core prices are not unique!

Unique core prices b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment VCG prices Core point closest to VCG prices Minimize incentive to distort bid!

Bidder Types Private Value – Each bidder knows her own value for the item being auctionedPrivate Value – Each bidder knows her own value for the item being auctioned Common Value – Each bidder has some estimate of the value of the item, but no one knows for sureCommon Value – Each bidder has some estimate of the value of the item, but no one knows for sure

Auction Facts Winner’s Curse – In a common value auction, being the winner often means that your estimated value of the auction item was too highWinner’s Curse – In a common value auction, being the winner often means that your estimated value of the auction item was too high