Economics Alumni Working Paper Series

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1 Economics Alumni Woking Pape Seies Auctioning with Aspiations: Keep Them Low (Enough) Nicholas Shunda Univesity of Redlands Woking Pape July Mansfield Road, Unit 1063 Stos, CT Phone: (860) Fax: (860) This woking pape is indexed on RePEc,

2 Abstact In an auction with a buy pice, a selle offes biddes the oppotunity to fogo competing in an auction by tansacting immediately at a pe-specified fixed pice. If a selle has aspiations in the fom of a efeence pice that depends upon the auction s eseve pice and buy pice, she does best to keep he aspiations sufficiently low by designing a no-eseve auction with a buy pice low enough that some bidde types would execise it with positive pobability in equilibium. The selle is indiffeent between the auction component of he mechanism being a fist- o second-pice auction. Jounal of Economic Liteatue Classification: D44, D82, L86, C72 Keywods: Auction, Aspiation, Buy pice, Intenet, Refeence-dependence I would like to thank Loenzo Gabo and Johannes Moenius fo helpful suggestions.

3 1 Intoduction Fom the gowth of auctions on the Intenet emeged a new selling mechanism, an auction with a buy pice. In an auction with a buy pice, a selle offes biddes the oppotunity to fogo competing in an auction by tansacting immediately at a pe-specified fixed pice. The Buy-It-Now featue of industy leade ebay is a leading example of an auction with a buy pice. 1 In the fist quate of 2009, ebay s sales (Goss Mechandise Volume) wee $12.8 billion, 49% of which wee though buy pice tansactions. 2 Thus, auctions with buy pices ae popula among buyes and selles in online auctions and ae economically significant. That auctions with buy pices ae so popula poses somewhat of a puzzle fom the vantage point of standad auction theoy. Namely, when demand is uncetain and makets ae thin, the maket pice is unclea and auctioning dominates fixed pice selling fom a selle s pespective (Milgom, 1989; Wang, 1993). Thus, when a selle has an auction mechanism at he disposal online, the option to augment he auction with a fixed pice seems unnecessay. A gowing theoetical liteatue studies this hybid mechanism and seeks to undestand unde what conditions a selle would benefit fom augmenting he auction with a buy pice. Existing ationales fo auctions with buy pices include: Bidde o selle isk avesion (Budish & Takeyama, 2001; Hidvégi, Wang, & Whinston, 2006; Mathews & Katzman, 2006; Reynolds & Woodes, 2009); bidde o selle impatience (Mathews, 2004); auction tansaction costs fo biddes (Wang, Montgomey, & Sinivasan, 2008); selle competition and multi-unit demands (Kikegaad & Ovegaad, 2008); pice discimination when the set of bidde types is not connected (Bose & Daipa, 2009); and bidde efeence-dependence (Shunda, 2009). This pape contibutes to this liteatue by exploing a model of auctioning with aspiations. I model a selle with aspiations as one who evaluates outcomes in elation to some fixed efeence outcome; such a selle has efeence-dependent pefeences ove evenue (see, e.g., Kahneman and Tvesky (1979) and Köszegi and Rabin (2006) fo geneal models of efeence-dependence). Thus, a selle with aspiations caes about evenue and (possibly vey slightly) how he sales pice compaes to a efeence pice. I demonstate that a iskneutal selle with aspiations does best to conduct he sale via a no-eseve auction with a buy pice in contast to a pue auction and is indiffeent between the auction component of he mechanism being a fist- o second-pice auction. 1 The Buy Pice on Yahoo!, Take-It on Amazon, and ubuy It on ubid auctions ae othe examples of online auctions with buy pices. 2 Figues ae fom 2

4 2 Model A selle conducts the sale of a single unit of an indivisible good fo which she deives value v 0 = 0 though an auction with a buy pice B and, if no bidde accepts the buy pice, holds a second-pice sealed-bid auction with eseve pice. The auction s buy pice is tempoay in that it disappeas if no bidde is willing to execise it. 3 Thee ae n 2 isk-neutal biddes, each of whose valuation v i is pivate infomation and is an independent daw fom the common distibution F that is continuous on its suppot [0, v] with a density f that is finite on its suppot and bounded away fom zeo. A bidde with valuation v i who wins the good at a pice p eans suplus of v i p and eans 0 othewise. The numbe of biddes, buy pice, eseve pice, and the distibution of valuations ae common knowledge. Thus, the model is within the symmetic independent pivate values famewok. To demonstate that a selle would set a buy pice that some bidde types would execise with positive pobability in equilibium, I additionally equie that f df exists and is finite on its suppot. dv The novel featue of the model is the addition of a selle with aspiations. Such a selle caes about evenue and (possibly vey slightly) how he sales pice compaes to a fixed efeence pice (which one can conceptualize as he aspied selling pice). 4 Thus, if the good sells fo pice x 0, the selle s utility is x + δ(x σ). The selle s efeence pice is σ and δ is a small positive numbe. I assume that the selle s efeence pice is σ = µ + (1 µ)b with µ (0, 1). Thus, a selle with aspiations caes about evenue and expeiences a utility (disutility) of δ(x σ) if x > (<)σ. isk-neutal selle as a special case. Setting δ = 0 epoduces the standad model of a 3 Results The model I descibe in the pevious section induces a two-stage game of incomplete infomation among the biddes and selle. Given the selle s choice of B and, biddes decide whethe o not to execise the buy pice in stage one and, if no bidde execised the buy pice, submit a bid to the auction in stage two. Ties ae boken andomly. To analyze the model, I seach fo a symmetic Bayes-Nash equilibium. Since a standad second-pice sealed-bid auction occus if no bidde execises the buy pice, it is well-known that each bidde has a weakly dominant stategy to bid accoding to 3 The Buy-It-Now pice is a leading example of a tempoay buy pice. Buy pices on Yahoo! and Amazon auctions ae examples of pemanent buy pices available fo the auction s entie duation. Fo analyses of auctions with pemanent buy pices, see Hidvégi et al. (2006) and Reynolds and Woodes (2009). 4 My conceptualization of efeence-dependence follows closely Rosenkanz and Schmitz (2007) who intoduce a simila fomulation into auction theoy via bidde utilities. 3

5 β S (v i ) = v i if v i and to not paticipate othewise. Suppose a bidde execises the buy pice if and only if v i v and ejects it othewise. Fo a bidde with v i, expected suplus fom execising the buy pice given that ivals with v v execise it is π B (v i, v ) = = n 1 ( n 1 k k=0 ( 1 F (v ) n n(1 F (v )) ) F (v ) n k 1 (1 F (v )) k k (v i B ) ) (v i B ). (1) On the othe hand, fo such a bidde, expected suplus fom the auction is π A (v i, v ) = (v i )F () n 1 + ˆ min{vi,v } = (v i min{v i, v })F (min{v i, v }) n 1 + (whee the second equality follows fom integation by pats). (v i x)(n 1)F (x) n 2 f(x)dx ˆ min{vi,v } F (x) n 1 dx (2) Following Mathews and Katzman (2006), define as B(v i, ) the buy pice that makes a bidde of type v i indiffeent between the buy pice and the auction. This theshold buy pice solves ˆ vi ( ) 1 F F (x) n 1 (vi ) n dx = (v i B(v i, )). (3) n(1 F (v i )) Staightfowad diffeentiation eveals that B(v i, ) is inceasing in both v i and. Define v as the unique v [0, v] such that B(v, ) = B and define v = v if B(v, ) < B fo all v [0, v]. I obtain the following esult on bidde behavio. Poposition 1. Thee exists a symmetic equilibium in which a bidde of type v i accepts the buy pice if and only if B < B(v i, ) and submits a bid of β S (v i ) = v i in the auction. This is the unique symmetic equilibium. Poof. Equilibium bidding accoding to β S (v i ) = v i follows fom a standad dominance agument. A necessay condition fo v to be pat of an equilibium is that π A (v, v ) = π B (v, v ). Since v satisfies B(v, ) = B, it is immediate that v i = v equates (1) and (2). Note that 4

6 dπ A dv i = F (min{v i, v }) n 1 F (v ) n 1. Theefoe, dπ A dv i dπb dv i F (v ) n 1 1 F (v ) n n(1 F (v )) = (1 nf (v ) n 1 + (n 1)F (v ) n ) n(1 F (v )) < 0 because 1 nf (v ) n 1 + (n 1)F (v ) n > 0 fo F (v ) < 1. Thus, dπ B dv i > dπa dv i. Since π A (v i, v ) = π B (v i, v ) at v i = v, we must have π A (v i, v ) > π B (v i, v ) fo v i < v and π B (v i, v ) > π A (v i, v ) fo v i > v. To see that this symmetic equilibium is unique, note that because B(v i, ) is inceasing in v i, the existence of some ˆv v satisfying B(ˆv, ) = B would lead to a contadiction. Denote the highest valuation x and the second-highest valuation y. Given the bidde behavio Poposition 1 descibes, the selle s expected utility takes the fom ˆ v ˆ U(, v ) = F () n (0 + δ(0 σ)) + ( + δ( σ))n(n 1)F (y) n 2 f(y)dyf(x)dx 0 ˆ v ˆ x + (y + δ(y σ))n(n 1)F (y) n 2 f(y)dyf(x)dx +(1 F (v ) n )(B(v, ) + δ(b(v, ) σ)) = F () n δσ + nf () n 1 (F (v ) F ())( + δ( σ)) (4) + ˆ v (y + δ(y σ))n(n 1)F (y) n 2 f(y)(f (v ) F (y))dy +(1 F (v ) n )(B(v, ) + δ(b(v, ) σ)) (whee the second equality follows fom exchanging the ode of integation of the second double integal). Because B(v i, ) is inceasing in v i, we can analyze the selle s choice of B as he choice of v. Maximizing U(, v ) with espect to [0, v] and v [0, v] yields the following esult. Poposition 2. Given biddes equilibium stategies, a selle with aspiations maximizes he expected utility by conducting a no-eseve auction with a buy pice low enough that some bidde types would execise it with positive pobability in equilibium. 5

7 Poof. Diffeentiating U(, v ) yields U = nf () n 1 f()(1 + δ) + nf () n 1 F (v )(1 + δ) nf () n (1 + δ) +(1 F (v ) n )(1 + δ) B δµ δ(1 µ) B and ˆ U v = nf (v ) n 1 f(v )(1 + δ)(v B(v, )) (1 + δ) nf (y) n 1 f(v )dy v +(1 F (v ) n )(1 + δ) B B δ(1 µ) v v. Because B/ = n(1 F (v ))F () n 1 /(1 F (v ) n ), U = nf ()n 1 f()(1 + δ) δµ δ(1 µ)n(1 F (v ))F () n 1 1 F (v ) n < 0 fo all v [0, v]. Theefoe, it is optimal to set = 0, a no-eseve auction. Note that if the selle sets v = v, a buy pice no bidde type would execise with positive pobability in equilibium, we have U v v =v ˆ v = nf(v)(1 + δ)(v B(v, )) (1 + δ) nf (y) n 1 f(v)dy δ(1 µ) B v = 0 using (3) and the fact that B/ v v =v = 0. Howeve, diffeentiating again and evaluating at v = v, we have 2 U = n(n 1)f(v) 2 (1 + δ)(v B(v, )) + 2nf(v)(1 + δ) > 0 v =v v 2 v =v fo all [0, v] using again the fact that B/ v v =v = 0 and that 2 B/ v 2 v =v = 0. This implies that U/ v < 0 to the left of v and, thus, that it is optimal to set v < v, a buy pice that some bidde types would execise with positive pobability in equilibium. Finally, a selle with aspiations is indiffeent between fist- and second-pice auction ules of he auction with a buy pice. Poposition 3. Fist- and second-pice auctions with a buy pice ae utility equivalent to a selle with aspiations. 6

8 Poof. Conside a fist-pice sealed-bid auction with a buy pice. If a bidde execises the buy pice if and only if v i v and ejects it othewise, expected suplus fom execising the buy pice is as in (1). Since a standad fist-pice auction occus if no bidde execises the buy pice, it is well-known that biddes with v i bid accoding to β F (v i ) = F ˆ ()n 1 vi F (v i ) + (n 1)xF (x) n 2 f(x) dx (5) n 1 F (v i ) n 1 in a symmetic equilibium. Thus, expected suplus fom the auction is π A (v i, v ) = (v i β F (min{v i, v }))F (min{v i, v }) n 1 = (v i min{v i, v })F (min{v i, v }) n 1 + ˆ min{vi,v } F (x) n 1 dx, (6) which, substituting (5) and again using integation by pats to obtain the second equality, is identical to (2). Theefoe, a bidde s theshold buy pice satisfies (3) as in a secondpice auction with a buy pice. Aguments simila to those in the poof of Poposition 1 establish the existence of a (unique) symmetic equilibium in which a bidde with valuation v i execises the buy pice if and only if B < B(v i, ) and bids accoding to (5) in the auction. Given bidde behavio, the selle s expected utility takes the fom U(, v ) = F () n (0 + δ(0 σ)) + ˆ v +(1 F (v ) n )(B(v, ) + δ(b(v, ) σ)) (β F (x) + δ(β F (x) σ)) nf (x) n 1 f(x)dx = F () n δσ + nf () n 1 (F (v ) F ())( + δ( σ)) (7) + ˆ v (y + δ(y σ))n(n 1)F (y) n 2 f(y)(f (v ) F (y))dy +(1 F (v ) n )(B(v, ) + δ(b(v, ) σ)), which, substituting (5) and again exchanging the ode of integation of the double integal to obtain the second equality, is identical to (4). 4 Conclusion I constuct and study a model of an auction with a buy pice fo a selle with aspiations in the fom of a efeence pice that depends upon the auction s eseve pice and buy pice. Such a selle does best to keep he aspiations sufficiently low by designing a no-eseve auction with a buy pice low enough that some bidde types would execise it with positive 7

9 pobability in equilibium. Futhe, the selle is indiffeent between the auction component of he mechanism being a fist- o second-pice auction. 8

10 Refeences Bose, S., & Daipa, A. (2009). Optimal Sale acoss Venues and Auctions with a Buy-Now Option. Economic Theoy, 38 (1), Budish, E. B., & Takeyama, L. N. (2001). Buy Pices in Online Auctions: Iationality on the Intenet?. Economics Lettes, 72 (3), Hidvégi, Z., Wang, W., & Whinston, A. B. (2006). Buy-Pice English Auction. Jounal of Economic Theoy, 129 (1), Kahneman, D., & Tvesky, A. (1979). Pospect Theoy: An Analysis of Decision unde Risk. Econometica, 47 (2), Kikegaad, R., & Ovegaad, P. B. (2008). Buy-Out Pices in Auctions: Selle Competition and Multi-Unit Demands. RAND Jounal of Economics, 39 (3), Köszegi, B., & Rabin, M. (2006). A Model of Refeence-Dependent Pefeences. Quately Jounal of Economics, 121 (4), Mathews, T. (2004). The Impact of Discounting on an Auction with a Buyout Option: a Theoetical Analysis Motivated by ebay s Buy-It-Now Featue. Jounal of Economics (Zeitschift fü Nationalökonomie), 81 (1), Mathews, T., & Katzman, B. (2006). The Role of Vaying Risk Attitudes in an Auction with a Buyout Option. Economic Theoy, 27 (3), Milgom, P. (1989). Auctions and Bidding: A Pime. Jounal of Economic Pespectives, 3 (3), Reynolds, S. S., & Woodes, J. (2009). Auctions with a Buy Pice. Economic Theoy, 38 (1), Rosenkanz, S., & Schmitz, P. W. (2007). Reseve Pices in Auctions as Refeence Points. Economic Jounal, 117 (520), Shunda, N. (2009). Auctions with a Buy Pice: The Case of Refeence-Dependent Pefeences. Games and Economic Behavio, fothcoming. Wang, R. (1993). Auctions vesus Posted-Pice Selling. Ameican Economic Review, 83 (4),

11 Wang, X., Montgomey, A., & Sinivasan, K. (2008). When Auction Meets Fixed Pice: A Theoetical and Empiical Examination of Buy-It-Now Auctions. Quantitative Maketing and Economics, 6 (4),

Unobserved Correlation in Ascending Auctions: Example And Extensions

Unobserved Correlation in Ascending Auctions: Example And Extensions Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay