Article Two pricing mechanisms in sponsored search advertising
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1 econsto De Open-Access-Publikationsseve de ZBW Leibniz-Infomationszentum Witschaft The Open Access Publication Seve of the ZBW Leibniz Infomation Cente fo Economics Yang, Wei; Feng, Youyi; Xiao, Baichun Aticle Two picing mechanisms in sponsoed seach advetising Games Povided in Coopeation with: Multidisciplinay Digital Publishing Institute (MDPI), Basel Suggested Citation: Yang, Wei; Feng, Youyi; Xiao, Baichun (2013) : Two picing mechanisms in sponsoed seach advetising, Games, ISSN , MDPI, Basel, Vol. 4, Iss. 1, pp , This Vesion is available at: Nutzungsbedingungen: Die ZBW äumt Ihnen als Nutzein/Nutze das unentgeltliche, äumlich unbeschänkte und zeitlich auf die Daue des Schutzechts beschänkte einfache Recht ein, das ausgewählte Wek im Rahmen de unte nachzulesenden vollständigen Nutzungsbedingungen zu vevielfältigen, mit denen die Nutzein/de Nutze sich duch die este Nutzung einvestanden eklät. Tems of use: The ZBW gants you, the use, the non-exclusive ight to use the selected wok fee of chage, teitoially unesticted and within the time limit of the tem of the popety ights accoding to the tems specified at By the fist use of the selected wok the use agees and declaes to comply with these tems of use. zbw Leibniz-Infomationszentum Witschaft Leibniz Infomation Cente fo Economics
2 Games 2013, 4, ; doi: /g OPEN ACCESS games ISSN Aticle Two Picing Mechanisms in Sponsoed Seach Advetising Wei Yang 1,3,, Youyi Feng 2 and Baichun Xiao 3 1 Nanjing Univesity of Finance and Economics, Nanjing, Jiangsu Povince , China 2 Zaagoza Logistics Cente, C/Bai 55, Edificio Náyade 5 (PLAZA) 50197, Zaagoza, Spain; yfeng@zlc.edu.es 3 College of Management, LIU Post, Bookville, New Yok 11548, USA; baichun.xiao@liu.edu Autho to whom coespondence should be addessed; wei.yang@liu.edu; Tel.: ; Fax: Received: 6 Octobe 2012; in evised fom: 29 Januay 2013 / Accepted: 19 Febuay 2013 / Published: 20 Mach 2013 Abstact: Sponsoed seach advetising has gown apidly since the last decade and is now a significant evenue souce fo seach engines. To amelioate evenues, seach engines often set fixed o vaiable eseve pice to in influence advetises bidding. This pape studies and compaes two picing mechanisms: the genealized second-pice auction (GSP) whee the winne at the last ad position pays the lage value between the highest losing bid and eseve pice, and the GSP with a posted eseve pice (APR) whee the winne at the last position pays the eseve pice. We show that if advetises pe-click value has an inceasing genealized failue ate, the seach engine s evenue ate is quasi-concave and hence thee exists an optimal eseve pice unde both mechanisms. While the numbe of advetises and the numbe of ad positions have no effect on the selection of eseve pice in GSP, the optimal eseve pice is affected by both factos in APR and it should be set highe than GSP. Keywods: Sponsoed seach advetising; symmetic Nash equilibium; genealized failue ate; genealized second-pice auction; eseve pice
3 Games 2013, Intoduction In sponsoed seach advetising each advetise bids fo keywods and puchases sponsoed links elevant to his business fom Google, Yahoo!, MSN, and thei patnes. When an intenet use seaches a specific tem o keywod in a seach engine, sponsoed links may be displayed in the font page in addition to seach esult links. Wheneve a sponsoed link is clicked, its sponso o advetise will pay the seach engine a fee, called cost pe-click (CPC), fo the sevice of leading consumes to his web site. Many copoations and small businesses take advantage of this affodable, tageted advetising method. Google epoted an annual evenue of $37.91 billion in 2011, and 96% of it came fom its advetising business (Google Investo Relations [1]). Sponsoed links associated with a paticula keywod ae sold though auction whee each advetise submits a bid to the seach engine fo his willingness to pay fo a CPC. In 2002 Google intoduced AdWods, which stipulates that an advetise in position i pays a CPC that equals the bidding pice of the advetise in position i 1 plus a minimum incement. Named the genealized second-pice (GSP) auction, this mechanism has become the building block in sponsoed seach advetising. The evenue ceated by each sponsoed link to the seach engine is detemined by the advetise s CPC and the click though ate (CTR), i.e., aveage numbe of clicks the link eceives pe unit time. Thee ae a limited numbe of sponsoed links that a seach engine can display on each page, and the links at the top of a page eceive significantly moe clicks than ones at the bottom (Books [2]). Fo instance, ad links on Google s fist page ae most valuable, of which up to 3 links ae on the top of seach esults and at most 8 links ae on the ight side. To sceen advetises and impove the quality of sponsoed links, seach engines set up eseve pice called minimum bid to each keywod auction. Advetises must pay at least the minimum bid pe click to maintain the active status of thei ad links. Since ealy 2008 Yahoo! has no longe fixed its minimum bid at $0.10 and stated imposing vaiable minimum bids fo some keywods (Yahoo! Seach Maketing Blog [3]). Late Google intoduced the fist page bid estimate to appoximate the minimum CPC needed to show a sponsoed link in the fist page, which is based on placement in the last position on the ight hand side of Google. This may be anywhee between positions 1 and 11, depending on how many ads ae shown fo that quey (Google Adwods Help [4]). If a link appeas in the bottom-most position among top links above seach esults, o if it is the only top link, the advetise pays the amount of the CPC theshold (Google Adwods Help [5]). Both fist page bid estimate and CPC theshold ae announced to advetises and updated egulaly. To avoid vaious nomenclatues, we use eseve pice to epesent the CPC that is chaged to the advetise winning the last ad position. Motivated by the economy in sponsoed seach advetising and intigued by the opaqueness of its pactice, we study and compae two picing mechanisms fo selling multiple heteogeneous ad positions. The fist is the GSP auction defined by Edelman et al. [6] and Vaian [7] whee the bidde winning the lowest position pays the lage value between eseve pice and the highest losing bid. The second, which we suspect being adopted in pactice, especially fo selling top sponsoed links above seach esults, is basically a hybid between an auction and a posted pice mechanism. It is simila to GSP except fo stipulating that the winne of lowest position pays the posted eseve pice egadless the numbe of positions sold. Calling it the GSP auction with posted eseve pice (APR), we attempt to addess the
4 Games 2013, following questions: (i) Does the optimal eseve pice exist fo each mechanism? Is it the same as the one that applies to single-item auctions? (ii) In addition to bidde s pivate value, ae thee othe factos that affect the selection of optimal eseve pice? (iii) Which mechanism ceates moe expected evenue fo the seach engine? Although GSP has been widely used in the liteatue to analyze sponsoed seach advetising, it is based on a static game that ignoes the dynamic natue of the auction. The seach engine detemines a eseve pice befoe the auction stats while no ad position has been sold. In the eality, howeve, the seach engine may eview peiodically as to whethe the cuent eseve pice needs to be evised given pat of o all ad positions have been occupied. In a sepaate pape (Yang et al. [8]), we will show that in the dynamic setting, the constant eseve pice, suggested by GSP, is not optimal. On the othe hand, the optimal eseve pice deived fom APR is able to captue the influence of the numbe of positions sold. The est of the pape is oganized as follows. Section 2 eviews the elated liteatue and highlights ou contibution. Section 3 stats with the discussion on equilibium stategy in GSP and APR. We then study these two picing mechanisms fo the seach engine and deive the optimal eseve pices. Compaison and analysis ae pesented in Section 4. Concluding emaks ae placed in Section Liteatue Review and Contibution of the Pape Reseve pice fo auctions has been studied long befoe sponsoed seach advetising came into pactice. Myeson [9], Riley and Samuelson [10] conside single-item auctions whee isk neutal biddes have independent pivate values. They discove that the item should be awaded to the bidde who has the highest value if and only if it exceeds a citical theshold. Maskin and Riley [11] extend the analysis to auctions with multiple identical items and show that the selle should set a fixed eseve pice independent fom the numbe of biddes. Bulow and Robets [12] use maginal evenue analysis to associate auction theoy with economic fundamentals. They let F ( ) and f( ) denote the pobability distibution and density functions of biddes willingness to pay, espectively. If the monopolistic selle sets a take-it-o-leave-it pice, then the expected quantity of sales would be q() = 1 F () = F (). They show that the selle should disqualify those biddes with negative maginal evenues, and the value that endes zeo maginal evenue is the optimal eseve pice. Explicitly, it is the solution of MR() = d dq [ q()] = F () f() = 0 (1) A complication of sponsoed seach advetising is the heteogeneity of auction items o ad positions. As the click-though ate is descending with positions, each advetise chooses a position (o none) to maximize his payoff, and equilibium becomes a natual issue. Since GSP auctions ae infinitely epeated games, the sets of equilibia can be vey lage and analysis of possible equilibium stategies becomes intactable. Edelman et al. [6] focus on a static GSP in which bids ae stabilized. They find that the GSP auction geneally does not have equilibium in dominant stategies, and tuth-telling does not lead to an equilibium. They define a locally envy-fee equilibium of the simultaneous game induced by GSP whee each advetise cannot be bette off by swapping bids with the advetise anked one position above. Vaian [7] poposes a notion of symmetic Nash equilibium (SNE) and shows that the bids at
5 Games 2013, the lowe bound of SNE ae the same as the bids unde the locally envy-fee equilibium. Vaian [7] also shows that thee is a compelling eason fo playes to bid at the lowe bound of SNE. Empiical evidence indicates that actual bidding obseved in Google s ad auction is in the neighbohood of lowe bound of SNE. While eseach on sponsoed seach auction has been popula, study on its eseve pice is spoadic. Edelman and Schwaz [13] show that (1) applies to the optimal eseve pice fo a genealized English auction. Defined in Edelman et al. [6] with moe details, the genealized English auction stats with a clock showing the cuent pice, inceasing ove time fom a eseve pice. Biddes paticipating in the auction ae willing to pay at least the eseve pice. As pice goes up biddes stat dopping out. One s bid is the pice shown on the clock when he quits. The auction ends until the last bidde emains, who wins the top ad position paying the pice shown on the clock when the next-to-last one quits. The next-to-last bidde wins the second position and pays the bid pice of the thid-highest bidde, and so on. In light of this definition, in the genealized English auction, if the numbe of qualified bidde does not exceed the numbe of positions, the one winning the last position pays the eseve pice pe click; othewise, he pays the lage value between the eseve pice and the highest losing bid. In Theoem 1 we pove the same optimality condition (1) fo a genealized English auction when the numbe of biddes is uncetain. Howeve, in the APR mechanism when the winne of the lowest position pays the eseve pice egadless of the numbe of slots sold, the optimal eseve pice does not satisfy (1). In this aticle we study two picing mechanisms, GSP and APR, in selling multiple heteogeneous ad positions. In both mechanisms advetise at position i pays the bid pice of the advetise at position i1. The only diffeence is that in APR the one occupying the last slot pays the posted eseve pice while in GSP he pays the lage value between eseve pice and the highest losing bid. Since the numbe of biddes in each peiod may vay, we assume that the bidding steam follows a Poisson distibution. The numbe of qualified biddes, hence, is also a Poisson vaiable endogenously affected by the eseve pice. Poisson aival assumption is standad in evenue management liteatue (Xiao and Yang [14], Feng and Xiao [15]), and it is often adopted by economists. Wang [16] compaes an auction with a posted-pice mechanism selling a uniquely indivisible object when buyes aive andomly accoding to a Poisson pocess. He finds that if the auctioning cost is absent, the auction always ceates moe evenue than posted-pice selling. As the empiical study by Vaian [7] at Google shows that advetises tend to follow bidding stategies at the lowe bound of symmetic Nash equilibium, we focus on the lowe bound of SNE (o equivalently locally envy-fee stategies) and maximize the seach engine s long-tem expected evenue ate with eseve pice as a contol vaiable. We pove that in eithe mechanism the expected evenue ate is quasi-concave if advetises pe-click value has an inceasing genealized failue ate function. In GSP the optimal eseve pice satisfies the condition of (1), while in APR, optimality condition suggests that the optimal eseve pice should be set highe than the one defined in GSP. It depends on the numbe of positions, the numbe of advetises, and thei pe-click values. Fom the dynamic point of view, this is easonable because when the seach engine peiodically eviews its eseve pice, the numbe of positions available fo sale is unlikely constant. As a esult, the eseve pice should eflect it and be adjusted accodingly. The minimum bids stated by seach engines (see Yahoo! Seach Maketing Help [17]) cooboate ou findings.
6 Games 2013, Picing Mechanisms The keywods auctions in Yahoo! and Google ae epeated games in which biddes gadually lean the value of thei competitos and adjust thei bids accodingly. The set of equilibia in epeated games can be vey lage and the stategies leading to such equilibia ae complex. As Edelman et al. [6] note, it may not be easonable to expect advetises capable of executing such stategies, especially when they manage thousands of keywods. Edelman and Schwaz [13] show that a static game with cetain equilibium stategy set is able to captue many impotant chaacteistics of the undelying dynamic game. Hence, we study a simultaneous-move one-shot auction assuming that (1) biddes ae likely to lean all elevant infomation about each othe s value ove time; (2) biddes submit thei stable bids in equilibium. A dawback of this simplification is it implies that the auction always begins with the same numbe of positions available pio to aivals of biddes. In eality, the auction is continuous and the numbe of positions emaining fo sale vaies fom time to time. In pactice, Google anks advetises based on the poduct of each advetise s bid and his quality scoe. The quality scoe is detemined by a numbe of factos including the CTR, the keywod s elevancy to the advetise s business and the quality of his web site. Vaian [7] and Edelman et al. [6] both show that incopoating quality scoe in the study does not povide much additional insight. Hence, in ou pape both GSP and APR detemine winnes solely based on bid pices Equilibium in GSP and APR An exogenously given k ad positions ae fo sale though auction. Thee ae n advetises endogenously affected by the eseve pice. The expected CTR eceived by ad position i is α i. Since top positions eceive highe CTR than lowe positions, we have α i > α i1 fo 1 i k 1 and α i = 0 fo i > k. Advetises ae isk-neutal possessing independent pivate infomation. They may have distinct willingness to pay fo one click, but fo each advetise, his pe-click value stays the same even if his ad appeas in diffeent positions. Let b (i), v (i) and I(i) denote the bid pice, pe-click value and identity of the ith highest bid, espectively. If b (i) is geate than the eseve pice, both mechanisms allocate the ith position to the advetise with the ith highest bid, I(i), whee i {1,..., min(n, k)}. When the ith link is clicked by a use, advetise I(i) pays the seach engine an amount that equals the next highest bid, b (i1). If advetise I(i) happens to be the winning bidde in the lowest anking, he pays the posted eseve pice in APR o the lage value between eseve pice and the highest losing bid in GSP. The evenue eceived by the seach engine fom the ith sponsoed link is α i b (i1) and the suplus to advetise I(i) is α i (v (i) b (i1) ). Definition 1 (Vaian [7]) In a symmetic Nash equilibium (SNE) bid pices satisfy α i (v (i) b (i1) ) α j (v (i) b (j1) ), i and j (2) Vaian shows that if a set of pices is an SNE, then it is a Nash equilibium (NE). He futhe shows that if the state of advetise I(i) satisfies condition (2) fo position j = i 1 and j = i 1, then it satisfies (2) fo all j. Hence, one only needs to compae each advetise s cuent position with its two adjacent slots.
7 Games 2013, Since the advetise in position i does not want to move down, we have α i (v (i) b (i1) ) α i1 (v (i) b (i2) ) (3) On the othe hand, the advetise in position i 1 has no incentive to move one slot up. Thus, α i1 (v (i1) b (i2) ) α i (v (i1) b (i1) ) (4) The above two inequalities lead to (α i α i1 )v (i1) α i1 b (i2) α i b (i1) (α i α i1 )v (i) α i1 b (i2) α i (5) Note that any b (i1) satisfying the above inequalities is an equilibium bid. Hence, thee is a ange of such equilibium stategies. Fom the pactical point of view, this seems to make the equilibium bids indeteminate. Vaian [7] examines the bounday cases by choosing the uppe and lowe bounds in (5), which yields b (i1) = (α i α i1 )v (i1) α i1 b (i2) α i, (lowe bound) (6) b (i1) = (α i α i1 )v (i) α i1 b (i2) α i, (uppe bound) (7) It is easy to see that the bidding stategy at the lowe bound belongs to the locally envy-fee equilibium (Edelman et al. [6]) and it is in fact the lowest-evenue envy-fee equilibium (Edelman and Schwaz [13]). Equations based on such an equilibium with vaious names can be found in seveal liteatues including the self-selection condition in the pioity picing mechanism studied in Hais and Raviv [18], and the indiffeence constaint in Wilson [19] that consides the myopic incentive constained picing. Questions that follow natually ae why advetises ae destined to the lowe bound of SNE and how thei bids convege to this paticula point? Vaian [7] explains that it is moe easonable fo the bidde to compae his cuent payoff with what he will end up if he outbids the advetise one slot above him. He agues that among all SNE stategies, the lowe bound stategy is the most appealing one to advetises since it defines the lowest bid pices that ende each agent the most advantageous position. Vaian also shows that the lowe bound stategy is closely elated to the two-sided matching poblem discussed in Roth and Sotomayo [20]. Edelman et al. [6] show that if the dynamic game eve conveges to a static vecto of bids, that static equilibium should coespond to a locally envy-fee equilibium of the static game induced by the GSP. In addition, both Edelman et al. [6] and Vaian [7] show that in the dominant-stategy Vickey Clake Goves equilibium, the position and CPC of each advetise coincide with those in the lowe bound of the SNE, and the outcome is the best fo advetises but wost fo the seach engine. Hence, the lowe bound stategy povides incentives to advetises and it is most likely attainable. Edelman and Schwaz [13] futhe show that the lowe bound stategy in a GSP with complete infomation, captuing behavios in an incomplete infomation game, can be viewed as a valid appoximation. The empiical analysis conducted by Vaian [7] ove a andom sample of 2425 auctions at Google also shows that thee is demonstable evidence indicating that the actual bidding in Google s ad auction conveges to the neighbohood of the equilibium stategy at the lowe bound of SNE.
8 Games 2013, In the est of the pape we use the tem of (lowest-evenue) envy-fee equilibium and lowe bound of the SNE intechangeably. Any equilibium stategy examined will satisfy the lowe bound constaint (6) Optimal Reseve Pice in Sponsoed Seach Advetising We conside n qualified advetises whose pe-click values ae epesented by an n-dimensional andom vecto (X 1, X 2,, X n ). The ealization of these n values is (x 1, x 2,, x n ). Reaanging x 1, x 2,, x n in an inceasing ode so that x (1) x (2) x (n), whee x (1) = min{x 1, x 2,, x n }, x (2) is the second smallest value in the n-tuple, and so on. The function X (i) of (X 1, X 2,, X n ) that takes on the x (i) in each possible sequence (x 1, x 2,, x n ) of values assumed by (X 1, X 2,, X n ) is known as the ith ode statistic. Let X 1, X 2, X n be identical and independent andom vaiables with common pobability density function f(x) and distibution function F (x). We appoximate the total numbe of advetises by a Poisson andom vaiable with a mean λ. When the eseve pice is equal to, the pobability fo each potential advetise to paticipate in the auction is F () = 1 F () and λ F () is the aveage numbe of qualified biddes. One can egad λ F () as the demand function fo ad positions and λ as the maximum aival ate when the eseve pice is set to zeo. Hence, the pobability of having n eligible biddes pe unit time is equal to [λ F ()] n e λ F (). Following Laiviee and Poteus [21], we define h(x) = f(x) as the F (x) failue ate of F ( ) and g(x) = xh(x) as the genealized failue ate. As g(x) = x f(x) F (x) = x d F (x) dx F (x) the genealized failue ate g(x) is the absolute value of the pice elasticity. Fo a qualified bidde j its pe-click value satisfies the following conditional distibution function F (x) = P [X j x X j ] = P [ X j x] P [X j ] Hence, the maginal pobability density function of X (i) is given by = F (x) F () F () f (i) (x) = (i 1)!(n i)! [F (x)] i 1 n i f(x) [1 F (x)] F () (8) and the expectation of X (i) can be witten as E[X (i) ] = xf (i) (x)dx = x[f (x)] i 1 n i f(x) [1 F (x)] dx (9) (i 1)!(n i)! F () Let p be the CPC the advetise at the last position pays, i.e., b (n k)1 = p, whee n k = min(n, k). Note that in GSP p is the lage value between and the highest losing bid while p = in APR. Equation (6) enables us to deive b (i) ecusively. Fo i < n k, b (i) = α ib (i1) (α i 1 α i )X (n i1) α i 1 (10)
9 Games 2013, When thee ae k positions offeed and n advetises qualified, seach engine s evenue pe unit time is given by In view of (10), fo l < n k 1, we have R(n, n k) = pα n k n α l b (l1) (11) α l b (l1) = α l1 b (l2) (α l α l1 )X (n l) = pα n k (α l α l1 )X (n l) (α n k 1 α n k )X (n n k1) which can be substituted into (11) to fom a linea function of ode statistics: R(n, n k) = p(n k)α n k n l(α l α l1 )X (n l) (12) As the tue pe-click values of advetises ae unknown to the seach engine, R(n, n k) is uncetain and its expectation can be depicted as E[R(n, n k)] = p(n k)α n k n l(α l α l1 )E[X (n l) ] Conditioning on the numbe of qualified biddes n, the expected evenue ate is Ψ() = E[R(n, n k)] [λ F ()] n n=0 e λ F () (13) The objective of the seach engine is to maximize Ψ() by choosing an optimal eseve pice GSP Lemma 1 In the lowe bound of the locally envy-fee equilibium, the expected evenue ate to the seach engine fom the static genealized second-pice auction (GSP) fo sponsoed seach advetising is Ψ A () = (α 1 α n ) [λ F ()] n e λ F ( () n ) [λ F (x)] n α l nα n [λ (α 1 α k ) F ()] n e λ F () [λ α F (x)] n l (14) The poof of Lemma 1 and all othe poofs ae placed in the appendix. In a GSP, the evenue contibutions fom n biddes consist of two pats: the minimum amount to win a position, epesented by (α 1 α n k ) [λ F ()] n e λ F () and the exta to secue each winning position, epesented by [(α 1 α n ) (α n 1 α n )] [λ F (x)] n
10 Games 2013, if n k and [λ (α 1 α k ) F (x)] n if n > k. The standad egulaity condition in the auction liteatue equies an inceasing failue ate (IFR) (Maskin and Riley [11]), while a milde condition is an inceasing genealized failue ate (IGFR). The assumption that the distibution function has an IGFR is pevalent in picing and evenue management liteatue (Ziya et al. [22] and Laiviee [23]). It ensues eithe a unimodal o quasi-concave evenue function when selling a single poduct. Ou model extends the esult to the scenaio with heteogeneous poducts. Theoem 1 If the value pe click of each advetise is a vaiable with IGFR, then (i) Ψ A () is a quasi-concave function; (ii) the optimal eseve pice is given as the solution to the equation 1 F ( ) f( ) = 0 (15) Note that the genealized failue ate is the absolute value of pice elasticity. IGFR implies that consumes espond to a pice incease moe negatively when the eseve pice is high than when it is low. In sponsoed seach advetising, it means that the demand eduction in ad links esulted fom a mak-up is inceasing in the eseve pice, which eflects the behavio of ational advetises. Thus, the assumption of IGFR in Theoem 1 is mino and easonable. In auctions of selling a single item o multiple homogeneous items, the optimal eseve pice solves Equation (1) (Bulow and Robets [12]). Theoem 1 shows that it is also the optimality condition fo the static GSP with heteogeneous items. Notice that condition (15) is equivalent to g() = 1 and g() is the absolute value of pice elasticity of demand. As g() is an inceasing function of, it makes pefect sense fo (15): incease the eseve pice when demand is inelastic until the elasticity eaches unity; futhe inceases will educe the evenue. Although English auctions with multiple heteogeneous items and endogenous enties have been studied in the liteatue, to ou knowledge, thee has not been theoetical development fo an analytical solution of the optimal eseve pice. The significance of Theoem 1, hence, is twofold. It povides the optimal eseve pice fo a static genealized English auction whee the payment schedule fo the last winning bidde is dependent on the numbe of biddes; second, it shows that heteogeneity among objects alone does not necessaily lead to a vaied optimal eseve pice APR Lemma 2 In the locally envy-fee equilibium, the expected evenue ate to the seach engine fom APR fo sponsoed seach advetising is [ n k [λ Ψ P () = α F ()] n l e λ F ( n k ) ] [λ F (x)] () n α l (n k)α n k (16) When thee ae n biddes in the auction, thei contibutions consist of two pats: the minimum amount they have to pay to win a position, epesented by (α 1 α n k ) [λ F ()] n e λ F ()
11 Games 2013, and the pemium paid fo secuing each winning position, epesented by [(α 1 α n k ) (α n k 1 α n k )] [λ F (x)] n Fo example, the advetise taking the fist position in the auction must pay α 1 [λ F ()] n e λ F () pe unit time to pass the evenue theshold defined by the eseve pice, and an exta amount (α 1 α n k ) [λ F (x)] n to be on top. In geneal, the heteogeneity among items fo sale dives each consume to select the most advantageous one fo him. If all items ae identical (with the same CTR o quality), no one will pay moe than the eseve pice. The maginal evenue function can be veified as Ψ P () = [1 f() F () ] [λ nα F ()] n n e λ F () kα k [λ F ()] n e λ F () Theoem 2 If each advetise s pe-click value is a andom vaiable with IGFR, then Ψ P () is quasi-concave and thee exists a unique eseve pice that maximizes the expected evenue ate. The optimal eseve pice is given as the solution to the equation [1 θ( )] 1 F ( ) f( ) = 0 (17) whee [λ F ( )] n θ( ) = kα k k [λ nα F ( )] n n The optimality condition (17) of eseve pice in APR clealy diffes fom (15) in GSP, which can be witten as g() = 1 θ(). The petubation facto θ() shows that it is diectly elated to the bidde population λ, the numbe of available ad positions k, and the distibution of biddes valuation function F. The poof of Theoem 2 shows that θ() is deceasing in. With the assumption of IGFR, thee is an optimal eseve pice that achieves the maximal expected evenue ate. 4. Discussions 4.1. Revenue Compaison The only diffeence between GSP and APR discussed in the peceding section is what the kth highest bidde pays when thee ae at least k qualified advetises. In APR the kth position winne pays eseve pice while in GSP, the bidde winning the kth ad spot pays the eseve pice, o the (k 1)th highest bid, whicheve comes highe. The two diffeent auction mechanisms, hence, affect bidding in thei own ways. APR biddes may peceive that they have good chance to win as long as they bid ove the eseve pice. On the contay, GSP biddes have no idea how much the one at the last position will pay, thus each has a tendency to bid moe aggessively. As a esult, GSP ceates highe evenue fo the selle.
12 Games 2013, Coollay 1 The evenue suplus of GSP ove APR is given explicitly by [λ Ψ A () Ψ P () = kα F (x)] n k As the eseve pice inceases fom zeo, both evenue ates go up while APR is affected moe since Ψ A() < Ψ P (). When the eseve pice eaches a level whee at most k biddes can affod it, the maginal evenue diffeence diminishes and Ψ P () conveges to Ψ A (). Although on aveage the GSP mechanism ceates moe evenue, APR has some pactical appeals. Fist, the sponsoed seach auction is continuous and advetises can update thei bids ove time. When a consume clicks the ith ad link, advetise I(i) pays the seach engine the bid pice b (i1) in eal-time. Without a posted eseve pice, it may take a long time fo advetises to eveal thei tue willingness to pay. The longe the bid pice lingeing at a low level, the moe evenue loss the seach engine will suffe. Second, stategic bid behavios in sponsoed seach auction ae not uncommon. In GSP it is easie fo advetises to collude and atificially depess bid pices, especially when the demand exceeds the numbe of links since the winne of the last position pays the next highest bid pice instead of the eseve pice. Imposing a stating pice is able to mitigate that isk. Thid, eseve pice has become an effective tool fo seach engines to impove advetising quality. A consume s need may not always be met afte clicking a sponsoed link. Imposing a eseve pice and sceening out ielevant links is able to educe consumes utility loss and incease the taffic fo existing ads. In addition, we have just shown that once the eseve pice is aised to a level whee thee ae at most k qualified biddes, the evenue diffeence between APR and GSP becomes negligible Elevated Reseve Pice Classical auction theoy shows that when selling identical items to a given numbe of biddes, the maginal evenue fom a bidde with value is 1. Without complete infomation to tell biddes h() apat, the selle chages only 1 to the bidde instead of extacting his entie value. Hence, the h() invese failue ate 1 h() can be undestood as the consume suplus o his infomation ent attibuted to the pivate infomation on his willingness to pay. If we compae the fist-ode conditions (17) with (1), the only diffeence is the petubation facto θ(). By specifying a eseve pice fo winning the last item, the selle eveals additional infomation to biddes with which they can use the eseve pice as a efeence fo bidding. Hence, in APR biddes etain moe infomation ent and achieve highe consume suplus that equals to 1θ(). It is woth noting that θ() is deceasing in. As eseve pice inceases, h() the additional consume suplus disappeas because the chance of undebidding is diminishing. Coollay 2 In APR mechanism selling multiple heteogeneous ad positions, the optimal eseve pice should be set highe than the one in GSP. Intuitively 1θ() implies highe consume suplus and lowe maginal evenue fo the selle with the h() same. Hence the selle will set a highe eseve pice to offset educed evenue esulting fom the evealed infomation. Example 1 Thee ae k = 5 sponsoed links fo sale. Bid aival follows a Poisson distibution with mean λ = 20. The pe-click value of each advetise is andom and unifomly distibuted within [0, 1]. The CTRs ae defined accoding to the empiical esults in Books [2].
13 Games 2013, Figue 1. Seach engine s expected evenue ates fom APR and GSP mechanisms Figue 1 depicts the seach engine s expected evenue ates vesus eseve pice in APR and GSP, espectively. GSP ceates moe evenue than APR, and its evenue advantage disappeas as the eseve pice inceases. It is clea that the optimal eseve pice in GSP is 0.5 accoding to (15). Figue 1 also veifies that the seach engine should set the eseve pice highe in APR Advetise Population Each keywod fo sale has its own maket, whose demand is gauged by both the maket size and depth. While maket size epesents the aveage numbe of potential buyes, maket depth stands fo the amount they each bid. The dynamic natue of advetising business is likely to influence the maket size and depth ove the couse of time. Hence, it is necessay fo the seach engine to eview the eseve pice fo each keywod when such changes in the maket take place. The following poposition shows how the population of potential biddes affects the optimal eseve pice. Coollay 3 In APR whee the value distibution is IGFR, the optimal eseve pice is an inceasing function of the aival ate of advetises λ. Any additional demand is beneficial to the seach engine because a new advetise not only pays moe fo a sponsoed link than the cuent eseve pice, but also inceases the competition and pompts othe advetises to aise the bids accodingly to secue thei positions. Hence, given the limited space fo displaying sponsoed esults on its font page, when the population of biddes fo a keywod inceases, the seach engine should lift eseve pice to ceate moe evenue. Recognizing the impotance of the maket size, seach engines make continuous effots to ecuit new advetises. Fo instance, Yahoo! offes a $25 of signing cedit fo any new custome. In addition, no longe using the fixed $.10 minimum
14 Games 2013, bid policy fo sponsoed seach, Yahoo! now adjusts eseve pice in a paticula keywod maket based on multiple factos including the numbe of biddes and the bid amounts (see Yahoo! Seach Maketing Help [17]). Such a shift in picing stategy cooboates ou theoetical findings. 5. Conclusion In sponsoed seach advetising advetises compete fo ad links that appea in the font page of majo seach engines including Google and Yahoo! though auctions. Seach engines often set vaiable eseve pice to influence advetise s bidding and ceate moe evenue. Howeve, as advetising capacity is peishable, abitay thesholds may advesely affect evenues. This pape compaes two picing mechanisms in sponsoed seach advetising the genealized second-pice auction (GSP) whee the last winning position is chaged the lage value between the eseve pice and the highest losing bid, and an altenative GSP with posted eseve pice (APR) whee the last winning position pays the eseve pice egadless how many positions ae sold. Although the standad GSP is extensively studied in the liteatue to deive optimal eseve pice, it fails to captue the dynamics of the auction. The continuous bidding game is unlikely to have the same numbe of unsold positions in each peiod and may equie peiodical eviews of the eseve pice. The poposed APR mechanism is an attempt to emedy the poblem. We assume that the numbe of eligible biddes is endogenously contolled by the eseve pice. Focusing on the lowe bound of the SNE ou models maximize the long-tem expected evenue ate. If the value pe click of biddes has the popety of IGFR, the expected evenue ates in both GSP and APR ae quasi-concave of the eseve pice and thee exists a unique optimize fo each mechanism. In paticula, the optimal eseve pice in GSP is poved to be the same as the one in single-item auctions that is dependent on only the pivate value distibution. In contast, the optimal eseve pice in APR, depending on both the numbes of biddes and positions, should be set highe compaed with the one in GSP. Cuent eseach can be extended in seveal diections. Fist, in ou model the CTR of each slot is exogenous. Although this is a standad assumption, it may not always be the case especially when not all slots ae occupied. Fo instance, when it is only ad showing the link on the ight of the fist page may eceive moe clicks than when it has companies. Taking this situation into account will make ou picing model much moe complex. On the othe hand, it may facilitate the analysis on the optimal numbe of sponsoed links a seach engine should sell in the fist page. Second, ou model maximizes the expected evenue ate in a static auction with an endogenous numbe of biddes. It might be inteesting to conside a dynamic model investigating how the seach engine should update the eseve pice at each peiod based on the incumbent and potential new enty. Thid, it is woth exploing the Bayes Nash equilibia of GSP and APR, and investigating whethe the Revenue Equivalence Theoem (Myeson [9]) applies to the two mechanisms.
15 Games 2013, Appendix Poof of Lemma 1 Let R A (n, n k) be the expected aveage evenue ate fom a GSP. We have R A (n, n) = nα n n 1 l(α l α l1 )X (l1), n k and R A (n, k) = l(α l α l1 )X (l1), n > k Obviously, R A (0, 0) = 0 as no ad position will be occupied. In an auction with n biddes and eseve pice, R A (n, n k) is a andom vaiable. Fo 1 n k, and fo n > k, n 1 E[R A (n, n)] = α n n x E[R A (n, k)] = x l(α l α l1 )F n l 1 (x) F (x) l f(x) (n l 1)!l! F () dx l(α l α l1 )F n l 1 (x) F (x)f(x) l (n l 1)!l! F dx () whee α k1 = 0 indicates that the seach engine does not offe moe than k ad positions. These equalities ae tue because fo n k, the last bidde will pay fo each click and fo n > k, the kth bidde will pay the (k 1)th bid pice, which may be highe than. Let Ψ A () denote the expected evenue ate of a GSP. It can be decomposed into the sum of Ψ A1 () and Ψ A2 () whee Ψ A1 () = α n n [λ F ()] n e λ F () and Ψ A2 () = = λ n=2 λ n e λ F () x n 1 x l(α l α l1 )[F (x) F ()] n l 1 F l (x) f(x)dx (n l 1)!l! l(α l α l1 )[F (x) F ()] n l 1 F l (x)f(x) λ n e λ F () dx (n l 1)!l! xf(x)l(α l α l1 ) [λ F (x)] l l! Hence, the expected evenue ate function becomes Ψ A () = α n n [λ F ()] n Applying integation by pats to the last item in Ψ A (), we have e λ F () λ xf(x)l(α l α l1 ) [λ F (x)] l l! Ψ A () = (α 1... α n ) [λ F ()] n e λ F () [λ (α 1... α n nα n ) F (x)] n [λ (α 1... α k ) F ()] n e λ F () [λ (α 1... α k ) F (x)] n
16 Games 2013, Poof of Theoem 1 Taking the deivative of Ψ A () with espect to leads to Ψ A() = [1 f() F () ] Dividing both sides by k nα n [λ F ()] n e λ F () leads to Ψ A() e λ F () k nα n [λ F ()] n [λ nα F ()] n n e λ F () = 1 f() F () If advetises willingness to pay has an inceasing genealized failue ate (IGFR), then the atio f() F () is inceasing in and the ight-hand side of the peceding equality is deceasing in. Theefoe, Ψ A() is eithe positive fo all 0 o negative fo all 0 o equals zeo at a point such that fo, Ψ A() 0 and fo >, Ψ A() < 0. That is, Ψ A () is quasi-concave in and has a sole maximum at. Poof of Lemma 2 Let n be the numbe of qualified biddes. When n = 1, the seach engine gains the expected evenue E[R P (1, 1)] = R P (1, 1) = α 1. Fo 1 < n k, the expected evenue is and fo n > k, it is n 1 E[R P (n, n)] = α n n x E[R P (n, k)] = α k k x l(α l α l1 )F n l 1 (x) F (x) l f(x) (n l 1)!l! F () dx l(α l α l1 )F n l 1 F (x) F () whee F (x) =, F (x) = 1 F 1 F () (x) and F () = 1 F (). Let Ψ P () = Ψ P 1 () Ψ P 2 () whee (x) F (x)f(x) l (n l 1)!l! F dx () and Ψ P 1 () = α n n [λ F ()] n e λ F () n=k α k k [λ F ()] n e λ F () It follows that Ψ P 2 () = n=2 λ n e λ F () x n=k n 1 x l(α l α l1 )[F (x) F ()] n l 1 F l (x) f(x)dx (n l 1)!l! l(α l α l1 )[F (x) F ()] n l 1 F l (x)f(x) λ n e λ F () dx (n l 1)!l! = Ψ P 2 () = x k 2 n=k k 2 e λ F () λ n1 e λ F () x n l(α l α l1 )[F (x) F ()] n l F l (x) f(x)dx (n l)!l! l(α l α l1 )[F (x) F ()] n l 1 F l (x)f(x) λ n e λ F () dx (n l 1)!l! k 2 x n=l λ n1 l(α l α l1 )[F (x) F ()] n l F l (x) f(x)dx (n l)!l!
17 Games 2013, l(α l α l1 )[F (x) F ()] n l 1 F l (x)f(x) x λ n e λ F () dx n=k (n l 1)!l! = e λ F k 2 () xl(α l α l1 )λ l1 eλ[f (x) F ()] {λ[f (x) F ()]} n l n=k 1 (n l)! l(α l α l1 )[F (x) F ()] n l 1 F l (x)f(x) x λ n e λ F () dx n=k (n l 1)!l! = e λ F k 2 () xl(α l α l1 )λ l1 e F λ[f (x) F ()] l (x) f(x)dx l! e λ F () x(k 1)(α k 1 α k )λ k e F λ[f (x) F ()] k 1 (x) (k 1)! f(x)dx = λ xf(x)l(α l α l1 ) [λ F (x)] l l! Applying integation by pats to the last expession leads to F l (x) f(x)dx l! Ψ P 2 () = n(α n α n1 ) l=n1 Hence, the expected evenue ate becomes [λ F ()] l e λ F () l! n(α n α n1 ) l=n1 [λ F (x)] l l! Ψ P () = Ψ P 1 () Ψ P 2 () = α n n [λ F ()] n e λ F () α k k [λ F ()] n e λ F () n=k [λ n(α n α n1 ) ()] l e λ F k 1 () [λ n(α n α n1 ) (x)] l l=n1 l! l=n1 l! = (α 1... α n ) [λ F ()] n e λ F () [λ (α 1... α k ) ()] n e λ F () (α 1... α n nα n ) [λ F (x)] n e λ F (x) [λ (α 1... α k kα k ) (x)] n e λ F (x) n=2 dx Poof of Theoem 2 Taking the deivative of Ψ P 1 () with espect to, we obtain Ψ P 1() = Ψ P 1() Similaly, n=k = Ψ P 1() α n n n[λ F ()] n 1 [ λf()] e λ F k 1 () α k k n[λ F ()] n 1 [ λf()] e λ F () n=k α n n n[λ F ()] n 1 [ λf()] e λ F k 1 () α k kλf()e λ F () [λ F ()] k 1 (k 1)! Ψ P 2() = λ λ n e λ F () n(α n α n1 ) F n () f() α n n [λ F ()] n [λf()] e λ F () α k k [λ F ()] n [λf()] e λ F () α n n [λ F ()] n [λf()] e λ F ()
18 Games 2013, = λ Adding Ψ P 1() and Ψ P 2() yields Ψ P () = Ψ P 1() e λ F () nα n[λ F ()] n f() λ α n n [λ F ()] n 1 e λ F () λf() (n 1)! α k kλf()e λ F () [λ F ()] k 1 (k 1)! = Ψ P 1() = Ψ P 1() = Ψ P 1() = α n n [λ F ()] n 1 (n 1)! λ e λ F () nα n1[λ F ()] n f() e λ F () nα n1[λ F ()] n f() e λ F k 1 () λf() λ α n (n 1) [λ F ()] n 1 e λ F () λf() (n 1)! α n λf()e λ F () [λ F ()] n 1 (n 1)! α n λf()e λ F () [λ F ()] n 1 (n 1)! α n n [λ F ()] n e λ F () = [1 f() F () ] α n λf()n [λ F ()] n 1 nα n [λ F ()] n e λ F () e λ F () nα n1[λ F ()] n f() α n1 n [λ F ()] n e λ F () λf() α k k [λ F ()] n e λ F () e λ F () α k k Dividing both sides by k nα n [λ F ()] n e λ F () leads to [λ F ()] n e λ F () Ψ P () e λ F () k nα n [λ F ()] n [λ F ()] n = 1 f() F () kα k k [λ nα F ()] n n [λ F ()] n k = 1 f() F () kα k k [λ nα F ()] n k n If advetises willingness to pay has an inceasing genealized failue ate (IGFR), then the atio f() F () is inceasing in. This implies, afte simple algeba, that the ight-hand side of the peceding equality is deceasing in. Theefoe, Ψ P () is eithe positive fo all 0 o negative fo all 0 o equals zeo at a point such that fo, Ψ P () 0 and fo >, Ψ P () < 0. That is, Ψ P () is quasi-concave in and has a sole maximum at. Poof of Coollay 1 Subtacting (16) fom (14) gives [λ Ψ A () Ψ P () = = λ xf(x)kα F (x)] k k e λ F (x) dx kα k [λ F ()] n e λ F () k! = kα k xf(x) λ[λ F (x)] k [ e λ F (x) [λ dx kα k 1 F ()] n ] e λ F () k! n=0
19 Games 2013, = kα k xf(x) λ[λ F (x)] k e λ F (x) dx kα k k! [λ = kα F (x)] n k e λ F (x) > 0 whee the thid equation comes fom Elang distibution. f(x) λ[λ F (x)] k k! Poof of Coollay 2 In light of Equation (1) the optimal eseve pice satisfies g(0) = 1. In Equation (17), θ() is clealy positive. Hence, g( ) = 1 θ( ) > 1, and we must have > 0 because of the assumption of IGFR. Poof of Coollay 3 We let θ(, λ) eplace θ() to undescoe its dependence on λ. Howeve, g() is ielevant to λ. The fist-ode condition (17) can be eaanged as 1 g((λ)) θ((λ), λ) = 0. Diffeentiating with espect to λ in the both sides of the equation establishes g ((λ)) (λ) θ((λ), λ) λ θ((λ), λ) (λ) = 0 o θ((λ),λ) λ (λ) = g ((λ)) θ((λ),λ) As θ(, λ) is deceasing in and inceasing in λ, and the valuation of biddes is IGFR, (λ) > 0. In othe wods, the optimal eseve pice is inceasing in the total population of biddes that ae inteested in the keywod ad positions. Refeences 1. Google Inc Annual Repot on Fom 10-K. Available online: (accessed on 1 Apil 2012). 2. Books, N. The Atlas Rank Repot: How Advetisement Engine Rank Impacts Taffic; Technical Repot; Atlas Institute, Univesity of Coloado: Boulde, CO, USA, July Yahoo! Seach Maketing Blog. Reseve Pices, Minimum Bids No Longe Fixed at $.10 fo Sponsoed Seach. Available online: (accessed on 1 Febuay 2009). 4. Google Adwods Help. What Ad Position Is the Fist Page Bid Related to? Available online: (accessed on 12 Febuay 2011). 5. Google Adwods Help. How Much Do I Pay fo a Click on My Ad? What If My Ad Is the Only One Showing? Available online: (accessed on 12 Febuay 2011). 6. Edelman, B.; Ostovsky, M.; Schwaz, M. Intenet advetising and the genealized second pice auction: Selling billions of dollas woth of keywods. Am. Econ. Rev. 2007, 97 (1), Vaian, H.R. Position auctions. Int. J. Ind. Ogan. 2007, 25 (6),
20 Games 2013, Yang, W.; Feng, Y.; Xiao, B. Dynamic eseve pice in sponsoed seach advetising. LIU Post, 2012 (woking pape). 9. Myeson, R.B. Optimal auction design. Math. Ope. Res. 1981, 6 (1), Riley, J.G.; Samuelson, W.F. Optimal auctions. Am. Econ. Rev. 1981, 71 (3), Maskin, E.; Riley, J. Optimal Multi-Unit Auctions; Oxfod Univesity Pess: Oxfod, UK, Bulow, J.; Robets, J. The simple economics of optimal auctions. J. Polit. Econ. 1989, 97 (5), Edelman, B.; Schwaz, M. Optimal auction design and equilibium selection in sponsoed seach auctions. Am. Econ. Rev. 2010, 100 (2), Xiao, B.; Yang, W. A evenue management model fo poducts with two capacity dimensions. Eu. J. Ope. Res. 2010, 205 (2), Feng, Y.; Xiao, B. A continuous-time yield management model with multiple pices and evesible pice changes. Manage. Sci. 2000, 46 (5), Wang, R. Auctions vesus posted-pice selling. Am. Econ. Rev. 1983, 83 (4), Yahoo! Seach Maketing Help. Picing and Minimum Bids. Available online: all/faqs.html#picing (accessed on 1 May 2009). 18. Hais, M.; Raviv, A. A theoy of monopoly picing schemes with demand uncetainty. Am. Econ. Rev. 1981, 71 (3), Wilson, R.B. Nonlinea Picing; Oxfod Univesity Pess: Oxfod, UK, Roth, A.; Sotomayo, M. Two-Sided Matching; Cambidge Univesity Pess: Cambidge, UK, Laiviee, M.A.; Poteus, E.L. Selling to the newsvendo: An analysis of pice-only contacts. M&SOM-Manuf. Sev. Op. 2001, 3 (4), Ziya, S.; Ayhan, H.; Foley, R.D. Relationships among thee assumptions in evenue management. Ope. Res. 2004, 52 (5), Laiviee, M.A. A note on pobability distibutions with inceasing genealized failue ates. Ope. Res. 2006, 54 (3), c 2013 by the authos; licensee MDPI, Basel, Switzeland. This aticle is an open access aticle distibuted unde the tems and conditions of the Ceative Commons Attibution license (
Chapter 3 Savings, Present Value and Ricardian Equivalence
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