Auctions -1 Debasis Mishra QIP Short-Term Course on

Auctions -1 Debasis Mishra QIP Short-Term Course on Electronic Commerce Indian Institute of Science, Bangalore February 15, 2006 1

Outline n Single-item auctions n Models of bidder behavior n Multi-item auctions n References February 15, 2006 QIP Course on E-Commerce : Auctions- 1 2

Auctions - Introduction n Auction - comes from Latin word auctus to mean increase. – Not every auction has increasing prices. n Among one of the first engaging tales - Sale of Roman empire to the highest bidder in 1764. n A market institution that works on the concept of competition. n Natural discovery of price and buyers. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 3

Auctions - Why and Why Not? n Why auctions? – Seller unsure about how much the price should be. – It can be used to sell almost anything -universal. – Buyers learn, in some auctions, about the information of other buyers - leads to more efficient and revenuegenerating markets. n Why not auctions? – Overhead of time and infrastructure. – Fixed price methods are simple. – Values of bidders are almost known. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 4

Auction Settings Forward auctions: a seller selling items to buyers (bidders). n Reverse auctions: a buyer buying items from sellers/suppliers (bidders). n Both the settings are natural transpose of each other: n – Bidders compete in both settings. – At low (high) price many buyers (sellers) demand (supply) items in forward (reverse) auctions. – Highest (lowest) price buyer (seller) wins in forward (reverse) auctions. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 5

Auctions in Practice Selling of flowers (Holland), tobacco, fish, tea, art objects and antique pieces (Sotheby's). n Transfer of assets from public to private: Sale of industrial enterprises in Eastern Europe, transportation system in Britain, timber rights all over the world, and off-shore oil leases. n Auction of spectrum rights worldwide - US, Europe, and even India. n Internet auctions of consumer goods (amazon. com, ebay. com etc. ). Google's Adword auctions. Procurement auctions - freemarkets. com (now Ariba), GM and IBM's sourcing solutions. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 6

Valuations Valuation: The maximum amount a bidder is willing to pay. n In procurement auctions, the value is negative of cost of procurement - the minimum price a bidder is demanding. n Auctions are used mainly because the auctioneer is unsure about the valuations (or simply, values) of bidders. n Two models: (i) private values (ii) common or interdependent value. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 7

Private Value Model n Each bidder knows his own value of the item exactly at the time of bidding, but knows nothing about the values of other bidders. – Value of other bidders do not influence his own value. – Suitable for: auctions for paintings, stamps etc. (a bidder knows the value of a painting exactly), procurement auction settings (a supplier's cost depends only on his own production technology). n Most plausible when the value of the item to a bidder is derived from its use alone and the bidder knows the item well. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 8

Interdependent Value Model (1 of 2) n Worth of an item unknown at the time of bidding to bidders. – Examples: oil field (depth of oil wells not well known), second-hand products (quality of the product is not known). In such cases, a bidder will have an estimate or a privately known signal (an expert's opinion or a test result) that is correlated with the true value. n Formally, every bidder has a signal xi and the value of bidder i is vi(x 1, x 2, . . . , xn) n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 9

Interdependent Value Model (2 of 2) Information, such as estimates or signals, of other bidders will influence the value of a bidder. n Values are unknown to bidders at the time of bidding and may be affected by information available to other bidders. n A special case is common values - every bidder has the same value ex post (i. e. , once they know everyone’s signals). Example: oil field auction. n – v(x 1, x 2, . . . , xn) February 15, 2006 QIP Course on E-Commerce : Auctions- 1 10

Single Item Auctions n Two formats: (i) sealed-bid (ii) open-cry n Sealed-bid: Bidders submit bids once (in a sealed envelope to the auctioneer) n Open-cry: Bidders submit bids in rounds, bids result in increase in prices (commonly termed as iterative auctions) – Bids reflect if bidders are willing to participate further in the auction. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 11

Single Item Sealed-Bid Auctions (1 of 3) First-price sealed-bid: Every bidder submits a bid; the highest bidder wins and pays his bid amount. n Second-price sealed-bid (Vickrey auction): Every bidder submits a bid; the highest bidder wins but pays an amount equal to the second highest bid. n First-price auctions are common in practice. n Second-price auctions are rare: but see examples of stamp auctions and others in http: //www. u. arizona. edu/~dreiley/pa pers/Vickrey. History. pdf n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 12

Single Item Sealed-Bid Auctions (2 of 3) Example: Four bidders with values 10, 8, 6, and 4. n First-price: Bidders bid 8, 6, 5, and 3 respectively (bid value not equal to valuation). Highest bidder wins and pays 8. n Second-price/Vickrey: Bidders bid 10, 8, 6, and 4 (bid value equals valuation). Highest bidder wins but pays 8. n Neither the revenue equivalence in the two auctions nor the bid=value in Vickrey auction in this example is a coincidence. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 13

Single Item Sealed-Bid Auctions (3 of 3) n The best strategy for a bidder, irrespective of what other bidders have bid, is to bid his value. This is also called a dominant strategy equilibrium in game theory. n Though economically robust, Vickrey auction is less transparent to bidders – transparency in auction design is important. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 14

Single-Item Open-Cry Auctions (1 of 3) English auction: Auction starts from low price. A bidder bids by indicating if he is willing to buy the item at the current price. If more than one bidder bids, then the price is raised by a finite amount ε (bid increment), else the auction stops. The last bidder to bid wins at the final price. n Consider the same example (values 10, 8, 6, 4). Let the starting price be 0 and bid increment ε. At price < 4+ ε, only 3 bidders will be interested … at price < 8+ ε, only 1 bidder will be interested. Auction stops at price < 8+ ε. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 15

Single-Item Open-Cry Auctions (2 of 3) It can be shown that staying in the auction till price reaches value is the best strategy for bidders. n Further, the outcome of English auction is equivalent to (as ε reaches zero) the Vickrey auction. n English auction is popular in practice – more transparent – and has similar economic properties as the Vickrey auction. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 16

Single-Item Open-Cry Auctions (3 of 3) n Dutch auction (popular in Holland to sell flowers): The auction starts from high price where there is no demand for the item; bidders bid indicating if they are interested in the item at the current price; if no bidder bids then the price is decreased by ε (bid decrement), else the auction stops. The only bidder to bid wins at the final price. – In case, more than one bidder bids, then the item is allocated at random to either of them. n Dutch auction is strategically equivalent to the first-price sealed-bid auction. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 17

Strategic Considerations n n Strong requirement: dominant strategy Irrespective of the bidding strategy of other bidders, a bidder's best strategy (one that maximizes utility over all strategies) is to be truthful. Weak requirement: (ex post) Nash equilibrium Given that all bidders bid truthfully, a bidder's best strategy is to be truthful. Given an auction design, is bidding truthfully the best strategy? Design an auction in which truthful bidding is the best strategy. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 18

Dominant Strategy in Vickrey Auction n n Consider bidder 1. Let the bid amount of any bidder i (not 1) be bi (need not equal value). What is the best amount to bid for 1? Without loss of generality, assume b 2 to be the highest bid among bids of bidders other than 1. Losing the auction by bidding untruthfully gives zero payoff. To win the auction and make positive payoff, bidder 1 should bid more than b 2. His payment will be b 2 always, independent of his bid amount, if he wins. His payoff is v 1 - b 2, where v 1 is his value. So, own bidding strategy does not influence payoff implying truthful bidding is a dominant strategy. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 19

Equivalence of Auction Forms n Dutch auction - Where should a bidder respond? That price is the payment. First-price sealed-bid auction What bid should a bidder submit? That bid price is the payment. So, same decision in both auctions. n English auction - best strategy is to remain interested till price reaches value. This terminates the auction (approximately) at the second-highest value. This is the outcome in the Vickrey auction. n Dutch auction = first-price sealed-bid auction. English auction = Vickrey auction. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 20

Revenue in Auctions (1 of 3) Values are drawn from uniform distribution with range [0, a] for n bidders. n Expected revenue in the Vickrey auction = 0∫a n(n-1)F(x)(n-2) [1 -F(x)] x f(x) dx = a(n-1)/(n+1). n Expected highest value = 0∫a n. F(x)(n-1) x f(x) dx = a n/(n+1). n In the first-price sealed-bid auction, we will find an equilibrium in which every bidder bids k times his value (0 <= k <= 1). Such an equilibrium is called a symmetric equilibrium. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 21

Revenue in Auctions (2 of 3) n n n Let b be the bid amount. Expected profit for a bid b with value v is (v-b)b(n-1)/(ka)(n-1). Maximizing expected profit, -b(n-1)+(n-1)(v-b)b(n-2)=0. We get b=v(n-1)/n. So, if every bidder except i bids a fraction (n-1)/n of his value, then the best strategy for i is to bid a fraction (n -1)/n of his value. So expected revenue (in a symmetric equilibrium) from a first-price auction = a (n-1)/(n+1)= expected revenue from Vickrey auction (revenue equivalence theorem). In fact, this is the highest possible revenue in ANY auction for single-item private values model. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 22

Revenue in Auctions (3 of 3) So expected revenue (in a symmetric equilibrium) from a first-price auction = a (n 1)/(n+1)= expected revenue from Vickrey auction (revenue equivalence theorem). In fact, this is the highest possible revenue in ANY auction for single-item private values model. n In fact, we can say more: with independently and identically distributed private values, the expected revenue in a first-price auction is the same as the expected revenue in a second-price auction. n We assumed risk neutral bidders: payoff=valueprice. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 23

Multi-Item Auctions (1 of 3) Number of items more than one. n Items may be of same type (homogeneous) or different type (heterogeneous). n Examples: Sale of different components of a computer, sale of 1000 memory chips etc. n Bidders may have value on bundles: value for 10 memory chips need not equal 10 times value of a single memory chip; value of a monitor and a keyboard may be more than their combined value. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 24

Multi-Item Auctions (2 of 3) If there are n items, a bidder can have values on 2 n number of bundles - exponential number of bundles. n Simultaneous sale of multiple items is also known as combinatorial auctions. n Examples of combinatorial auctions: n – Sale of airport slots: a bidder will be interested in Mumbai 6 AM to 7 AM slot together with Bangalore 8 AM to 9 AM slot; but less interested in Mumbai 6 AM to 7 AM slot with Bangalore 1 PM to 2 PM slot. – Sale of train tracks in Europe, spectrum rights in different countries. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 25

Multi-Item Auctions (3 of 3) Two buyers and two items (a, b). Values are: v 1(a)=5, v 1 (b)=7, v 1(a+b)=15; v 2(a)=7, v 2 (b)=6, v 2(a+b)=12. n Assuming truthful bidding and conducting a sequential auction (selling one item after another) using the Vickrey auction yields: item 1 is awarded to buyer 2 and item 2 to buyer 1. n This is not efficient - does not maximize total value of the system. n Does not maximize the revenue of the seller also. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 26

Design Objectives (1 of 2) n Efficiency: Maximize the total value of bidders and the seller. These are called efficient auctions. – If p is the price paid by a bidder, then v-p is his payoff and the seller gets a payoff of p. – Thus, total payoff of the system (buyers and seller) due to that buyer is v-p+p=v. – So, total payoff of the system is maximized by maximizing the total value. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 27

Design Objectives Revenue: Maximize the total revenue of the seller. n These are called optimal auctions. n – Generally, have to assume some distributions on valuations. – Much difficult than designing efficient auctions. – Analysis is intractable for many practical multiple items settings. n Note: Optimal auctions maximize the payoff of seller only, whereas efficient auctions maximize the total payoff of the seller and the buyers. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 28

Other Auction Design Issues (1 of 2) Reserve price: Sellers generally set a minimum price below which they do not sell items. n Bundling issues: Sellers generally do not allow for exponential number of bundles but decide on bundles before the auction. n Information feedback in iterative auctions: What bid information should be communicated to bidders? n Bid increments: Tradeoff between length of auction and efficiency/revenue loss. n February 15, 2006 QIP Course on E-Commerce : Auctions- 1 29

Other Auction Design Issues (2 of 2) n Collusion: Bidders form groups (called bidding rings) and act as one to bid in auctions. n Privacy: Depending on the information released by the auctioneer to the bidders, the privacy of bidders can be at stake. – Example: In English auction, by bidding truthfully, all losing bidders reveal their value. February 15, 2006 QIP Course on E-Commerce : Auctions- 1 30

References n Vijay Krishna, Auction Theory, Academic Press, 2002. n Paul Klemperer, Auctions: Theory and Practice, Online book http: //www. paulklemperer. org/, Also Princeton University Press, 2004 (gives outlines for undergraduate and graduate courses – in economics and management departments). February 15, 2006 QIP Course on E-Commerce : Auctions- 1 31