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Auctions -1

Auctions -1. Debasis Mishra QIP Short-Term Course on Electronic Commerce Indian Institute of Science, Bangalore February 15, 2006. Outline. Single-item auctions Models of bidder behavior Multi-item auctions References. Auctions - Introduction.

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Auctions -1

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

  2. Outline • Single-item auctions • Models of bidder behavior • Multi-item auctions • References QIP Course on E-Commerce : Auctions- 1

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

  4. Auctions - Why and Why Not? • 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 revenue-generating markets. • Why not auctions? • Overhead of time and infrastructure. • Fixed price methods are simple. • Values of bidders are almost known. QIP Course on E-Commerce : Auctions- 1

  5. Auction Settings • Forward auctions: a seller selling items to buyers (bidders). • Reverse auctions: a buyer buying items from sellers/suppliers (bidders). • Both the settings are natural transpose of each other: • 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. QIP Course on E-Commerce : Auctions- 1

  6. Auctions in Practice • Selling of flowers (Holland), tobacco, fish, tea, art objects and antique pieces (Sotheby's). • 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. • Auction of spectrum rights worldwide - US, Europe, and even India. • 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. QIP Course on E-Commerce : Auctions- 1

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

  8. Private Value Model • 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). • Most plausible when the value of the item to a bidder is derived from its use alone and the bidder knows the item well. QIP Course on E-Commerce : Auctions- 1

  9. Interdependent Value Model (1 of 2) • 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. • Formally, every bidder has a signal xi and the value of bidder i is vi(x1, x2,..., xn) QIP Course on E-Commerce : Auctions- 1

  10. Interdependent Value Model (2 of 2) • Information, such as estimates or signals, of other bidders will influence the value of a bidder. • Values are unknown to bidders at the time of bidding and may be affected by information available to other bidders. • 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. • v(x1, x2,..., xn) QIP Course on E-Commerce : Auctions- 1

  11. Single Item Auctions • Two formats: (i) sealed-bid (ii) open-cry • Sealed-bid: Bidders submit bids once (in a sealed envelope to the auctioneer) • 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. QIP Course on E-Commerce : Auctions- 1

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

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

  14. Single Item Sealed-Bid Auctions (3 of 3) • 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. • Though economically robust, Vickrey auction is less transparent to bidders – transparency in auction design is important. QIP Course on E-Commerce : Auctions- 1

  15. 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. • 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. QIP Course on E-Commerce : Auctions- 1

  16. 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. • Further, the outcome of English auction is equivalent to (as ε reaches zero) the Vickrey auction. • English auction is popular in practice – more transparent – and has similar economic properties as the Vickrey auction. QIP Course on E-Commerce : Auctions- 1

  17. Single-Item Open-Cry Auctions (3 of 3) • 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. • Dutch auction is strategically equivalent to the first-price sealed-bid auction. QIP Course on E-Commerce : Auctions- 1

  18. Strategic Considerations • 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. QIP Course on E-Commerce : Auctions- 1

  19. Dominant Strategy in Vickrey Auction • 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 b2 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 b2. • His payment will be b2 always, independent of his bid amount, if he wins. His payoff is v1 - b2, where v1 is his value. So, own bidding strategy does not influence payoff implying truthful bidding is a dominant strategy. QIP Course on E-Commerce : Auctions- 1

  20. Equivalence of Auction Forms • 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. • 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. • Dutch auction = first-price sealed-bid auction. English auction = Vickrey auction. QIP Course on E-Commerce : Auctions- 1

  21. Revenue in Auctions (1 of 3) • Values are drawn from uniform distribution with range [0,a] for n bidders. • 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). • Expected highest value = 0∫a nF(x)(n-1) x f(x) dx = a n/(n+1). • 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. QIP Course on E-Commerce : Auctions- 1

  22. Revenue in Auctions (2 of 3) • 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. QIP Course on E-Commerce : Auctions- 1

  23. 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. • 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. • We assumed risk neutral bidders: payoff=value-price. QIP Course on E-Commerce : Auctions- 1

  24. Multi-Item Auctions (1 of 3) • Number of items more than one. • Items may be of same type (homogeneous) or different type (heterogeneous). • Examples: Sale of different components of a computer, sale of 1000 memory chips etc. • 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. QIP Course on E-Commerce : Auctions- 1

  25. Multi-Item Auctions (2 of 3) • If there are n items, a bidder can have values on 2n number of bundles - exponential number of bundles. • Simultaneous sale of multiple items is also known as combinatorial auctions. • Examples of combinatorial auctions: • 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. QIP Course on E-Commerce : Auctions- 1

  26. Multi-Item Auctions (3 of 3) • Two buyers and two items (a,b). Values are: v1(a)=5, v1 (b)=7, v1(a+b)=15; v2(a)=7, v2 (b)=6, v2(a+b)=12. • 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. • This is not efficient - does not maximize total value of the system. • Does not maximize the revenue of the seller also. QIP Course on E-Commerce : Auctions- 1

  27. Design Objectives (1 of 2) • 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. QIP Course on E-Commerce : Auctions- 1

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

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

  30. Other Auction Design Issues (2 of 2) • Collusion: Bidders form groups (called bidding rings) and act as one to bid in auctions. • 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. QIP Course on E-Commerce : Auctions- 1

  31. References • Vijay Krishna, Auction Theory, Academic Press, 2002. • 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). QIP Course on E-Commerce : Auctions- 1

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