Electronic Auctions. Brent Hickman University of Iowa. March 3, Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

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1 Electronic Auctions Brent Hickman University of Iowa March 3, 2009 Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

2 Introduction This paper is an attempt to better understand the determinants of bidder behavior in auctions conducted over the internet on sites like ebay, Amazon and Yahoo!. Why is this an interesting topic? Internet auction houses are known for extraordinary breadth, volume and growth. One example: ebay Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

3 Introduction This paper is an attempt to better understand the determinants of bidder behavior in auctions conducted over the internet on sites like ebay, Amazon and Yahoo!. Why is this an interesting topic? Internet auction houses are known for extraordinary breadth, volume and growth. One example: ebay In million items listed for sale in 18,000 unique categories Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

4 Introduction This paper is an attempt to better understand the determinants of bidder behavior in auctions conducted over the internet on sites like ebay, Amazon and Yahoo!. Why is this an interesting topic? Internet auction houses are known for extraordinary breadth, volume and growth. One example: ebay In million items listed for sale in 18,000 unique categories Breadth: laptop computers, automobiles, collectible coins, used bleacher seats from Fenway Park Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

5 Introduction This paper is an attempt to better understand the determinants of bidder behavior in auctions conducted over the internet on sites like ebay, Amazon and Yahoo!. Why is this an interesting topic? Internet auction houses are known for extraordinary breadth, volume and growth. One example: ebay In million items listed for sale in 18,000 unique categories Breadth: laptop computers, automobiles, collectible coins, used bleacher seats from Fenway Park Volume: in 2006 over $32 billion of goods sold at auction Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

6 Introduction This paper is an attempt to better understand the determinants of bidder behavior in auctions conducted over the internet on sites like ebay, Amazon and Yahoo!. Why is this an interesting topic? Internet auction houses are known for extraordinary breadth, volume and growth. One example: ebay In million items listed for sale in 18,000 unique categories Breadth: laptop computers, automobiles, collectible coins, used bleacher seats from Fenway Park Volume: in 2006 over $32 billion of goods sold at auction Growth: from 2001 to 2006 average annual growth was 47% Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

7 Objective To help clear up misunderstandings about how players choose their bids in internet auctions Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

8 Objective To help clear up misunderstandings about how players choose their bids in internet auctions 1 Take a new look at the strategic incentives involved Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

9 Objective To help clear up misunderstandings about how players choose their bids in internet auctions 1 Take a new look at the strategic incentives involved 2 Demonstrate important differences between electronic auctions and their commonly assumed twin, the Vickrey/English format Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

10 Private Values and Traditional Auction Formats: Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

11 Private Values and Traditional Auction Formats: Independent private values paradigm: Bidders have private values v. Private values are viewed as random variables: V [0, v]. Bidders observe their own value, and have a common prior on opponents values, F(v). Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

12 Private Values and Traditional Auction Formats: Independent private values paradigm: Bidders have private values v. Private values are viewed as random variables: V [0, v]. Bidders observe their own value, and have a common prior on opponents values, F(v). English or Vickrey auctions (aka second price auctions) Dominant strategy equilibrium: β II (v) = v i Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

13 Private Values and Traditional Auction Formats: Independent private values paradigm: Bidders have private values v. Private values are viewed as random variables: V [0, v]. Bidders observe their own value, and have a common prior on opponents values, F(v). English or Vickrey auctions (aka second price auctions) Dominant strategy equilibrium: β II (v) = v i Picture First-price, sealed-bid auctions Unique Bayes-Nash equilibrium (assuming bidders have N opponents): β I (v) = E [ V (N:N) V (N:N) v ] Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

14 Electronic Auctions (EAs) Typically viewed by researchers and experts as either Vickrey or English auctions Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

15 Electronic Auctions (EAs) Typically viewed by researchers and experts as either Vickrey or English auctions Reason: proxy bidding Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

16 Electronic Auctions (EAs) Typically viewed by researchers and experts as either Vickrey or English auctions Reason: proxy bidding Bidders report a maximal authorized bid to the online server Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

17 Electronic Auctions (EAs) Typically viewed by researchers and experts as either Vickrey or English auctions Reason: proxy bidding Bidders report a maximal authorized bid to the online server The server then competitively bids on their behalf up to the maximum Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

18 Electronic Auctions (EAs) Typically viewed by researchers and experts as either Vickrey or English auctions Reason: proxy bidding Bidders report a maximal authorized bid to the online server The server then competitively bids on their behalf up to the maximum Advice supplied to bidders by online auction houses reflects the English/Vickrey view as well: Quote Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

19 Electronic Auctions (EAs) HOWEVER... Pricing rules are less straightforward than English/Vickrey rules due to the presence of bid increments,. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

20 Electronic Auctions (EAs) HOWEVER... Pricing rules are less straightforward than English/Vickrey rules due to the presence of bid increments,. Whenever possible, the online server forces bidders to outbid their nearest competitor by. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

21 Electronic Auctions (EAs) HOWEVER... Pricing rules are less straightforward than English/Vickrey rules due to the presence of bid increments,. Whenever possible, the online server forces bidders to outbid their nearest competitor by. When two bids are closer than, the price becomes exactly the high bid. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

22 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

23 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Player 1 bids b 1 =$10 Price=$3; 1 is current high bidder Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

24 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Player 1 bids b 1 =$10 Price=$3; 1 is current high bidder 1 Case 1: Player 2 underbids 1 with $5 b 2 <$10 Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

25 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Player 1 bids b 1 =$10 Price=$3; 1 is current high bidder 1 Case 1: Player 2 underbids 1 with $5 b 2 <$10 b 2 =$7: b 1 b 2 > P = b 2 + = $9. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

26 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Player 1 bids b 1 =$10 Price=$3; 1 is current high bidder 1 Case 1: Player 2 underbids 1 with $5 b 2 <$10 b 2 =$9: b 1 b 2 < P = b 1 = $10. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

27 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Player 1 bids b 1 =$10 Price=$3; 1 is current high bidder 2 Case 2: Player 2 outbids 1 with b 2 $10 Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

28 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Player 1 bids b 1 =$10 Price=$3; 1 is current high bidder 2 Case 2: Player 2 outbids 1 with b 2 $10 b 2 =$15: b 2 b 1 > P = b 1 + = $12. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

29 An Illustrative Example Consider two bidders in a widget auction with a reserve price of $3 and =$2: Player 1 bids b 1 =$10 Price=$3; 1 is current high bidder 2 Case 2: Player 2 outbids 1 with b 2 $10 b 2 =$11: b 2 b 1 < P = b 2 = $11. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

30 What is the practical significance if is small? Bid Increment Schedule on ebay, Amazon and Yahoo!: $100 P $249.99: = $2.50 $250 P $499.99: = $5 Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

31 What is the practical significance if is small? Bid Increment Schedule on ebay, Amazon and Yahoo!: $100 P $249.99: = $2.50 $250 P $499.99: = $5 QUESTION: If is only 1%-2% of the current price, how likely is it that price adjustments follow a first-price rule? Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

32 ebay Laptop Auctions Data Table: CSR TECHNOLOGIES, INC. Sample Processor Speed RAM Hard Drive Optical Drive Low-End, Used 1.4GHz 512MB 30GB DVD-ROM Low-End, Refurbished 1.4GHz 512MB 30GB DVD-ROM Mid-Line 1.6GHz 512MB 40GB DVD-ROM High-End 1.8GHz 1GB 40GB DVD-ROM/CD-R Other Characteristics: Screen Size: 14 OS: Windows XP Professional Duration: 24hrs Shipping cost: $36 Reserve price: $0 Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

33 ebay Data DATA: bids, bidder identities, timing, final price Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

34 ebay Data DATA: bids, bidder identities, timing, final price Table: Summary Statistics for Laptop Computer Sale Prices Sample # of Obs Mean StDev Min Max % FP Rule All Data Low-End, Used Low-End, Refurbished Mid-Line High-End Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

35 Equilibrium Analysis N bidders with independent private values for each player, v [0, v] F(v) F is atomless and absolutely cts, with cts derivative f Let M N 1 denote the number of opponents each bidder faces For each player, G(v) = F(v) M is the distribution of the highest opponent private value and g(v) = G (v) Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

36 Equilibrium Analysis N bidders with independent private values for each player, v [0, v] F(v) F is atomless and absolutely cts, with cts derivative f Let M N 1 denote the number of opponents each bidder faces For each player, G(v) = F(v) M is the distribution of the highest opponent private value and g(v) = G (v) STATIC GAME: Bidders submit one sealed bid b R + to the auctioneer, who allocates the object to the highest bidder and sets a price. I concentrate solely on differences arising from the non standard pricing rule. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

37 Equilibrium Analysis N bidders with independent private values for each player, v [0, v] F(v) F is atomless and absolutely cts, with cts derivative f Let M N 1 denote the number of opponents each bidder faces For each player, G(v) = F(v) M is the distribution of the highest opponent private value and g(v) = G (v) STATIC GAME: Bidders submit one sealed bid b R + to the auctioneer, who allocates the object to the highest bidder and sets a price. I concentrate solely on differences arising from the non standard pricing rule. Constant bid increment:. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

38 Equilibrium Analysis N bidders with independent private values for each player, v [0, v] F(v) F is atomless and absolutely cts, with cts derivative f Let M N 1 denote the number of opponents each bidder faces For each player, G(v) = F(v) M is the distribution of the highest opponent private value and g(v) = G (v) STATIC GAME: Bidders submit one sealed bid b R + to the auctioneer, who allocates the object to the highest bidder and sets a price. I concentrate solely on differences arising from the non standard pricing rule. Constant bid increment:. Pricing rule: P(b (N:N),b (N 1:N) ) = min { b (N:N),b (N 1:N) + } Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

39 Equilibrium Properties A symmetric equilibrium in pure strategies is a function β : [0,v] R + such that for each i {1,2,... N}, a bid of b i = β(v i ) maximizes i s expected payoff, given b j = β(v j ) j i. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

40 Equilibrium Properties A symmetric equilibrium in pure strategies is a function β : [0,v] R + such that for each i {1,2,... N}, a bid of b i = β(v i ) maximizes i s expected payoff, given b j = β(v j ) j i. 1 Existence and Monotonicity (Proposition 1): using results proven by Jackson and Swinkels (2005) and Jackson, Simon, Swinkels and Zame (2002) it can be shown that such an equilibrium exists and it is monotonic in bidders types. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

41 Equilibrium Properties A symmetric equilibrium in pure strategies is a function β : [0,v] R + such that for each i {1,2,... N}, a bid of b i = β(v i ) maximizes i s expected payoff, given b j = β(v j ) j i. 1 Existence and Monotonicity (Proposition 1): using results proven by Jackson and Swinkels (2005) and Jackson, Simon, Swinkels and Zame (2002) it can be shown that such an equilibrium exists and it is monotonic in bidders types. 2 Differentiability (Proposition 2): It can also be shown that β is differentiable, meaning that the first-order conditions of bidders maximization problem are well-defined. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

42 Derivation - CASE 1: v [0, v ) From Player 1 s Perspective Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

43 Derivation - CASE 1: v [0, v ) From Player 1 s Perspective Given v V (M:M), player 1 always pays his own bid; thus, the decision problem is identical to the first-price auction: { max (v b) G(β 1 (b)) }. b β(v) = β I (v) = E [ V (M:M) V (M:M) v ] v 0 = xg(x)dx G(v) (1) Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

44 Derivation - CASE 1: v [0, v ) From Player 1 s Perspective Given v V (M:M), player 1 always pays his own bid; thus, the decision problem is identical to the first-price auction: { max (v b) G(β 1 (b)) }. b β(v) = β I (v) = E [ V (M:M) V (M:M) v ] v 0 = xg(x)dx (1) G(v) Note: (1) above solves v = (β I ) 1 ( ), the cutoff between player types who bid less than and those who bid more. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

45 Derivation - CASE 2: v v Player 1 s decision problem: BIDS β(v) (equilibrium bid) { g(v (M:M) v v (M:M) ) (conditional density of highest competitor) II I β 1 (β(v) ) v PRIVATE VALUES Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

46 Derivation - CASE 2: v v Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

47 Derivation - CASE 2: v v { max (v b)pr ( β 1 (b ) V (M:M) v ) b ( + v E [ β(v (M:M) ) V (M:M) β 1 (b ) ] ) Pr ( V (M:M) β 1 (b ) ) } Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

48 Derivation - CASE 2: v v { max (v b)pr ( β 1 (b ) V (M:M) v ) b ( + v E [ β(v (M:M) ) V (M:M) β 1 (b ) ] ) Pr ( V (M:M) β 1 (b ) ) } Pr ( β 1 (b ) V (M:M) v ) = G(v) G ( β 1 (b ) ) Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

49 Derivation - CASE 2: v v { max (v b)pr ( β 1 (b ) V (M:M) v ) b ( + v E [ β(v (M:M) ) V (M:M) β 1 (b ) ] ) Pr ( V (M:M) β 1 (b ) ) } Pr ( β 1 (b ) V (M:M) v ) = G(v) G ( β 1 (b ) ) Pr ( V (M:M) β 1 (b ) ) = G ( β 1 (b ) ) Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

50 Derivation - CASE 2: v v { max (v b)pr ( β 1 (b ) V (M:M) v ) b ( + v E [ β(v (M:M) ) V (M:M) β 1 (b ) ] ) Pr ( V (M:M) β 1 (b ) ) } Pr ( β 1 (b ) V (M:M) v ) = G(v) G ( β 1 (b ) ) Pr ( V (M:M) β 1 (b ) ) = G ( β 1 (b ) ) E [ β(v (M:M) ) V (M:M) β 1 (b ) ] = β 1 (b ) 0 β(v (M:M) )g(v (M:M) )dv (M:M) G(β 1 (b )) Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

51 Derivation - CASE 2: v v Hence, the decision problem can be rewritten as max b { (v b) [ G(β 1 (b)) G(β 1 (b )) ] + (v )G(β 1 (b )) β 1 (b ) 0 β(v (M:M) )g(v (M:M) )dv (M:M) } (2) Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

52 Derivation - CASE 2: v v Taking FOCs and rearranging terms, we get the following differential equation: β (v) = (v β(v)) g(v) G(v) G (β 1 (β(v) )) (3) Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

53 Derivation - CASE 2: v v Taking FOCs and rearranging terms, we get the following differential equation: β (v) = (v β(v)) g(v) G(v) G (β 1 (β(v) )) (3) Equation (3) and boundary condition β(v ) = (4) (by continuity) characterize the unique solution for the equilibrium on the rest of the interval. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

54 Observations By examining the FOC, β (v) = (v β(v)) g(v) G(v) G (β 1 (β(v) )), we can make some interesting observations about β: Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

55 Observations By examining the FOC, β (v) = (v β(v)) g(v) G(v) G (β 1 (β(v) )), we can make some interesting observations about β: 1 β I (v) < β(v) < β II (v) for each v (v,v] Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

56 Observations By examining the FOC, β (v) = (v β(v)) g(v) G(v) G (β 1 (β(v) )), we can make some interesting observations about β: 1 β I (v) < β(v) < β II (v) for each v (v,v] 2 As 0, β converges uniformly to β II Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

57 Observations By examining the FOC, β (v) = (v β(v)) g(v) G(v) G (β 1 (β(v) )), we can make some interesting observations about β: 1 β I (v) < β(v) < β II (v) for each v (v,v] 2 As 0, β converges uniformly to β II 3 As v, β converges uniformly to β I Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

58 Numerical Example: Exponential and Weibull EQUILIBRIUM BIDS β II β I β EQUILIBRIUM BIDS β II β I β PRIVATE VALUES PRIVATE VALUES g(v) λ(v) 1 g(v) λ(v) EXPONENTIAL(1); 5 BIDDERS; = WEIBULL(1,2); 5 BIDDERS; =0.23 Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

59 Increment Sensitivity EQUILIBRIUM BIDS β II β I β; =0.25 β; =0.5 β; =0.75 β; =1 EQUILIBRIUM BIDS β II β I β; =0.1 β; =0.2 β; =0.3 β; = PRIVATE VALUES PRIVATE VALUES g(v) λ(v); =0.25 λ(v); =0.5 λ(v); =0.75 λ(v); = EXPONENTIAL(1); 5 BIDDERS; [0.25,1] g(v) λ(v); =0.1 λ(v); =0.2 λ(v); =0.3 λ(v); = WEIBULL(1,2); 5 BIDDERS; [0.1,0.4] Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

60 Conclusion Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

61 Conclusion 1 The strategic environment of EAs differs significantly from that of second price auctions due to bid increments. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

62 Conclusion 1 The strategic environment of EAs differs significantly from that of second price auctions due to bid increments. 2 Rather than variants of second price auctions, EAs are a hybrid of these and first price auctions. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

63 Conclusion 1 The strategic environment of EAs differs significantly from that of second price auctions due to bid increments. 2 Rather than variants of second price auctions, EAs are a hybrid of these and first price auctions. 3 This gives rise to an equilibrium in which bidders condition their bids on their opponents values and bid strictly less than their value. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

64 Conclusion 1 The strategic environment of EAs differs significantly from that of second price auctions due to bid increments. 2 Rather than variants of second price auctions, EAs are a hybrid of these and first price auctions. 3 This gives rise to an equilibrium in which bidders condition their bids on their opponents values and bid strictly less than their value. 4 Correctly characterizing the processes which which generate bids and prices plays a pivotal role in EA research. Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

65 A Quote from the ebay Web Site Return The following quote was taken from To help avoid disappointment, ensure that the maximum bid you enter on the item page is the highest price that you re willing to pay. The ebay bidding system automatically increases your bid up to the maximum price you specify... Brent Hickman University of Iowa () Electronic Auctions March 3, / 23

66 Equilibrium Bidding Return EQUILIBRIA IN TRADITONAL AUCTIONS WITH 5 BIDDERS AND EXPONENTIAL PRIVATE VALUES β II β I EQUILIBRIUM BIDS PRIVATE VALUES Brent Hickman University of Iowa () Electronic Auctions March 3, / 23