Gam of Platforms: Stratgc Expanson nto Rval (Onln) Trrtory Sagt Bar-Gll Ϯ Abstract Onln platforms, such as Googl, Facbook, or Amazon, ar constantly xpandng thr actvts, whl ncrasng th ovrlap n thr srvc offrng. Motvatd by ths, w study an xpanson gam btwn two onln platforms, offrng dffrnt srvcs to usrs for fr, whl sllng usr clcks to advrtsrs. Platforms dcd whthr or not to xpand by addng th srvc alrady offrd by th rval. Expanson dcsons affct th partton of usrs n th markt, whch, n turn, affcts platform prcs and profts. W dmonstrat that, n qulbrum, platforms may choos not to xpand, vn though xpanson s costlss. Such stratgc "no xpanson" dcsons ar du to quantty and prc ffcts of changs n th usr partton, and spcfcally changs n th dgr of usr multhomng, brought on by xpanson. Kywords: Mda conomcs, ntry, onln platforms, two-sdd markts. JEL classfcaton: D43, L10, L41 I thank Cham Frshtman, Andr Hagu, Yaron Yhzkl, partcpants at th BU Platform Stratgy Rsarch Symposum, th 7 th Pars ICT confrnc, and smnar partcpants at Tl Avv Unvrsty for hlpful commnts and dscussons. Fnancal support from th NET nsttut (www.ntnst.org) and from th Polack foundaton s gratfully acknowldgd. Ϯ Th Sloan School of Managmnt, MIT. Emal: sbargll@mt.du
1. Introducton Googl Plus, Googl's socal ntworkng srvc, was ntroducd n 2011, n th wak of Buzz, Googl's prvous and qut unsuccssful attmpt to xpand nto socal ntworkng. Wth 25 mllon usrs n on month, ndustry analysts ntally dubbd Googl Plus th "Facbook Kllr", xpctng Googl Plus to b th nxt Facbook or Twttr. Ths narratv was quck to chang. Most Googl Plus usrs wr not actv, and n 20 analysts and bloggrs largly rfrrd to Googl Plus as a "Ghost Town", compard to ts vry actv and lvly countrpart, Facbook. Today, Googl Plus s dfnd as a "socal layr" on top of Googl. Its actv usrs ar prdomnantly from th tch communty, and us t manly for aggrgatng, sharng and dscussng nws tms rlatd to thr common ntrsts. Stll, Googl Plus s far bhnd Facbook, wth 28 mllon unqu monthly vstors spndng around 7 mnuts on avrag on st, compard to Facbook's 142 mllon unqus, avragng almost 7 hours spnt on th socal ntwork. 1 In th onln world, whr traffc and tm spnt qual mony, Googl Plus s hardly a succss story. 2 Googl Plus s just on xampl of xpanson by an onln platform nto a trrtory alrady occupd by a rval platform. Googl, Facbook, Amazon, and Appl - th "bg four" of th tch ndustry - ar constantly xpandng thr actvts, whl ncrasng thr ovrlap. Googl and Facbook now compt both n socal ntworkng, whr Facbook was th ncumbnt, and - snc th rcnt ntroducton of Facbook Graph Sarch - also n sarch, Googl s stronghold. Appl and Amazon compt n sllng dgtal mda and dvcs. Mor ovrlap btwn ths platforms s found n cloud srvcs, opratng systms, smartphons, -commrc, and th lst gos on. Wth nw srvcs and products addd ach month, th ovrlap n ths gants actvts wll contnu to ncras, as thy strv to provd a on-stop shop for thr usrs. 3 1 Ths fgurs from Nlsn ar for dsktop vstors n th US, n March 2013 (Wassrman 2013). 2 Tassy 2011, Evans 2013, and Warrn 2013 dscuss Googl Plus's voluton and currnt stat. 3 Also s Th Economst, Dcmbr 1st 20. 2
And to what nd? A major drvr of ths platforms xpanson s cultvatng xclusv and ntmat rlatonshps wth usrs that translat nto larg advrtsng rvnus 45 (.g., through mprovd ad targtng; s Kang 20). Indd, both Facbook and Googl's rvnu coms prdomnantly from advrtsng, and n 2013 Amazon launchd ts ad xchang, whch was stmatd to gnrat $835 mllon n 2014, by allowng rtargtng of shopprs aftr thy lav Amazon.com (Edwards 20, Grffth 20, and Taub 2013). Motvatd by ths, w study optmal xpanson stratgs of onln ad-fnancd platforms, thus addng to th growng ltratur on varous stratgc bhavors n platform markts. 6 Rcnt work has xamnd platform stratgs such as opnnss and dvlopr proprty rghts (Parkr and Van Alstyn 2008, Esnmann t al. 2008, Boudrau 2010), compatblty wth rval platforms (Casadsus-Masanll and Ruz-Alsda 2009), th choc of xclusv contracts vrsus multhomng (Hagu and L 2011), tyng (Cho 2010, Amlo and Julln 20), xcluson of som usr typs (Hagu 2011), and platform nvstmnt n proprtary contnt (Hagu and Spulbr 2013), to nam but a fw. 7 In ths work, w focus on xpanson nto srvcs alrady offrd by rval platforms, such that xpanson may affct th ovrlap n platforms usr bass, or th dgr of multhomng. W modl a markt wth two platforms, through whch advrtsrs may rach potntal buyrs, also rfrrd to as platform usrs. Th platforms provd fr srvcs to usrs, gnratng 4 Evans 2009 survys th voluton of onln advrtsng mthods, provds som ndustry numbrs, and dscusss prvacy concrns. 5 Googl's ad rvnu s now largr than that of th ntr US prnt mda, accordng to Edwards 2013. 6 Th ltratur on mult-sdd platform markts has largly focusd on platforms prcng stratgs, undr varyng assumptons rgardng markt charactrstcs (s, for xampl, Callaud and Julln 2001 and 2003, Roch and Trol 2002, 2003, and 2006, Parkr and Van Alstyn 2005, Armstrong 2006, Hagu 2006 and 2009b, Julln 2008, Wyl 2010, Cabral 2011, Halaburda and Yhzkl 2013a and 2013b). Mor rcntly, th platforms ltratur has bn volvng to consdr stratgs othr than prcng. 7 Also s Esnmann t al. 2006, Boudrau and Hagu 2008, and Hagu 2009a, who analyz cas studs n thr dscussons of dffrnt stratgs n platform markts. 3
rvnus by sllng usr clcks to advrtsrs. 8 At th outst, ach platform offrs on srvc typ, and buyrs optmally choos whch platform(s) to us, nducng th ntal buyr partton. Platforms ngag n an xpanson gam, whr ach platform stratgcally chooss whthr or not to costlssly xpand by addng th typ of srvc alrady offrd by ts rval. Th gam procds n two stags. In th frst stag, platforms mak thr xpanson dcsons, whr ths affct buyrs platform choc, dtrmnng th fnal buyr partton. In th scond stag, platforms st prcs pr usr clck chargd to advrtsrs, gvn th buyr partton. Advrtsrs thn obsrv platform prcs and th buyr partton, and choos whr to plac thr ads, whch, n turn, dtrmns platforms xpctd profts. Th xpanson gam s solvd by backward nducton. For ach par of xpanson dcsons, th buyr partton s drvd, and platform prcng and profts follow. Platforms optmal xpanson dcsons ar thn dtrmnd. To allow for a tractabl analyss of ths two-stag gam, w smplfy by assumng that th only ntwork ffct s a postv ffct xrtd by buyrs on advrtsrs, whras no ntwork ffct s xrtd by advrtsrs on th buyr populaton,.., platform usrs nthr suffr nor bnft from ads dsplayd on th platforms. Endognty of th buyr partton s an mportant fatur n our sttng. Th spcfcs of ths ndognty follow from th buyr modl, whch dfns buyrs platform prfrncs and optmal choc bhavor, gvn platforms xpanson dcsons. Th buyr modl s mantand as a sparat unt of analyss wthn our gnral framwork, such that our sttng s modular and can accommodat dffrnt buyr modls samlssly. Th papr thus studs th ffcts of ndognty of th buyr partton on optmal platform xpanson dcsons for a gnral buyr partton, consdrng dffrnt forms of ndognty. Followng th gnral analyss, a buyr modl s proposd, analyzd and usd to provd xampls for th cass consdrd n th gnral framwork. In th proposd modl, ndognty of th usr partton rsults from an ndognous lvl of compatblty btwn th platforms; whr both ncrass and dcrass n compatblty followng xpanson ar consdrd. 8 Th advrtsrs' sd of th markt fully subsdzs th buyrs' sd. Ths prc structur s commonly assumd n th mda platforms ltratur (.g. Andrson and Coat 2005, Andrson t al. 20, Rsngr 20, Ambrus t al. 2013). 4
Th basc motvaton for xpanson n our sttng s mprovd ad targtng, modlld as an ncras n usrs ad-clck probablty, rsultng from ad xposur on an addtonal srvc. Whn th buyr partton s xognous, xpanson ncrass clck probablty, thrby ncrasng th xpctd quantty of clcks sold to advrtsrs, ladng to hghr xpctd profts. Howvr, whn th buyr partton s ndognous, ths bnft of xpanson must b wghd aganst th ffcts of th chang n th usr partton. W dstngush btwn two ffcts assocatd wth th ndognty of th partton of buyrs a quantty and a prc ffct. Th formr s a drct ffct rsultng from th chang n usr partton that may thr ncras or dcras th total numbr of potntal buyrs rachd through th platform 9 followng xpanson, thrby affctng xpctd profts. Th lattr s an ndrct ffct, as qulbrum prcs pr clck dcras n th dgr of multhomng. Ths s bcaus prcs ar optmally st such that advrtsrs plac ads on both platforms, but do not pay doubl for rachng multhomrs twc. Consdrng th abov ffcts of xpanson w show that xpanson s a domnant stratgy whnvr t dcrass multhomng, whl no-xpanson may b optmal whn usr multhomng ncrass wth xpanson,.. whn th prc ffct s ngatv. Ths allows us to charactrz xpanson qulbra for dffrnt typs of ndognty of th usr partton. Spcfcally, whn multhomng monotoncally dcrass wth xpanson th only qulbrum s symmtrc xpanson, but othrws dffrnt typs of qulbrum may ars dpndng on th magntuds of th ffcts dscussd. Notably, asymmtrc xpanson may b qulbrum vn for symmtrc platforms, and symmtrc no-xpanson may also b optmal, whn th bnft of mprovd ad targtng for xclusv subscrbrs s small. Th followng scton 2 rvws th rlatd ltratur. Th gnral modl s thn st up n scton 3, and analyzd n scton 4. Scton 5 proposs a buyr modl, whch s ncorporatd nto th gnral framwork n scton 6, and usd to dmonstrat th gnral rsults. Concludng rmarks and a dscusson of managral mplcatons ar offrd n scton 7. 9 Not that both xclusv and multhomng usrs ar rachd, yt th two usr typs dffr n thr ad clck probablty. 5
2. Rlatd Ltratur Studyng advrtsng-fnancd platforms, w rlat to th ltratur on mda platforms. Ths ltratur has xamnd qulbrum ad prcs, lvls of advrtsng, contnt dffrntaton, and platform ntry (.g., Andrson and Coat 2005, Cramps t al. 2009, Andrson t al. 2011 and 20, Rsngr 20, and Ambrus t al. 2013). Compard to ths paprs, w smplfy by assumng that advrtsrs ar homognous, and that th only ntwork ffct s xrtd by potntal consumrs on advrtsrs, so as to allow for tractablty n solvng th platform xpanson gam. Whl smplfyng, th man faturs of our modl ar consstnt wth prvous work n th mda platforms ltratur. Advrtsrs n our modl may multhom, as s commonly assumd n th ltratur (an xcpton s Rsngr 20). W furthr allow for usr multhomng, as n Andrson t al. 2011 and 20, Ambrus t al. 2013, and Athy t al. 2013. Usr multhomng crats som rdundancy for multhomng advrtsrs n our modl, and thus gvs rs to prcng whch follows th "prncpl of ncrmntal prcng" dfnd n Andrson t al. 2011. Namly, prcs ar dtrmnd accordng to th ncrmntal bnft of placng ads on an addtonal platform, thrby ntrnalzng th rdundancy for multhomng advrtsrs. A smlar noton of rdundancy n advrtsng arss n Athy t al. 2013 as a rsult of consumr swtchng. 10 Not that n our modl thr s only partal rdundancy from rachng multhomng usrs twc, as usrs clck-probablty ncrass wth ad xposur on an addtonal srvc. As a rsult, th dgr of multhomng xrts a ngatv ffct on prcs. Ths s n contrast to Andrson t al. 2011, whr th rdundancy s full and th dgr of multhomng dos not drctly affct ad prcng. On th othr hand, and qut naturally, usr xclusvty has a postv ffct on prcs n both paprs. 11 10 Whn trackng tchnologs ar mprfct. 11 In Athy t al. 2013 prcs may thr ncras or dcras wth consumr swtchng, as a rsult of th spcfc modllng of th swtchng procss. 6
Wthn th mda platforms ltratur, svral paprs hav focusd on platforms ndognous dffrntaton (.g., Gabszwcz t al. 2002, Duks and Gal-Or 2003, Bhrngr and Flstrucch 2009). Platforms n our modl choos whthr or not to xpand by addng a scond srvc alrady offrd by thr rval - our sttng ndognzs both th numbr of srvcs offrd and thr ovrlap. W thrfor analyz a dffrnt noton of ndognous dffrntaton, charactrstc of onln platforms xpanson bhavor. Closly rlatd, Esnmann t al. 2011 study platform "nvlopmnt", dfnd as "ntry by on platform provdr nto anothr s markt by bundlng ts own platform s functonalty wth that of th targt s". Envlopmnt s thus ssntally th sam as th xpanson bhavor w study, yt th spcfc rsarch qustons and approach n th two paprs ar qut dffrnt. Namly, Esnmann t al. buld a typology of nvlopmnt attacks basd on th lvl of complmntarty btwn th attackr and targt platforms, drvng condtons for succss of nvlopmnt attacks. W, on th othr hand, modl a stratgc xpanson gam btwn two platforms, so thr s no "attackr" and "targt" - both platforms may or may not xpand, and condtons for dffrnt typs of xpanson qulbra ar drvd. Furthrmor, whl Esnmann t al. tak a vry broad prspctv, w focus on advrtsng-fnancd onln platforms, whch allows for a tractabl modl. 3. Th Modl W modl a markt wth two platforms, offrng onln srvcs to buyrs or platform usrs, n ordr to attract advrtsrs who would lk usrs to clck on thr ads. Platforms ar ad-fnancd, and advrtsng rvnus ar collctd on a pr clck bass. Th focus of th modl s platforms stratgc xpanson bhavor whn th buyr partton s ndognous, and thus changs wth xpanson dcsons. W analyz an ntry or xpanson gam btwn th two platforms, whr ach platform may or may not xpand by addng th srvc ntally offrd by th rval platform. Followng xpanson dcsons, platforms st prcs pr usr clck chargd to advrtsrs. Advrtsrs obsrv th partton of buyrs rsultng from platforms xpanson dcsons, as wll as platform prcs, and choos thr advrtsng stratgy - placng ads on both platforms, on on of th platforms, or not advrtsng at all. 7
Th modl s constructd n a modular fashon. In ths and th followng scton w consdr an ndognous buyr partton wth a gnral structur and charactrstcs, abstractng from th undrlyng buyr modl whch dtrmns how xpanson affcts th partton of buyrs. A possbl buyr modl whch ylds an ndognous usr partton s prsntd n scton 5. 3.1 Platforms - basc assumptons and notaton Thr ar two platforms n th markt and lt {1,2} dnot th platform ndx. At th outst, ach platform provds on srvc, dffrnt from th on provdd by ts rval. Platform srvcs ar provdd to usrs for fr. A stratgy for platform s a coupl (, p ), whr {E, E } rprsnts th platform s xpanson dcson thr xpanson dnotd E or noxpanson dnotd E, and p [0, ) s th prc pr usr clck chargd to advrtsrs on th platform. Lt ( 1, 2 ) dnot a par of xpanson dcsons. W assum that platforms xpand only by addng th srvc alrady offrd by thr rval. Expanson s costlss n th modl. In ralty, nvstmnt costs rlatd wth xpanson ar clarly postv and affct xpanson dcsons. W abstract away from such costs to focus on th ffcts of usr partton ndognty on platform xpanson, dntfyng crcumstancs undr whch platforms wll not xpand, vn whn xpanson s costlss. Platform profts ar drvd from usr clcks sold to advrtsrs. W thus turn to ntroduc assumptons rgardng buyrs and advrtsrs n th markt. 3.2 Buyrs Thr s a unt mass of buyrs n th markt. Buyrs choos whch platform(s) thy subscrb to, and ths ylds th buyr partton - a partton of th buyr populaton nto thr groups: xclusv usrs of platform 1 and 2, and multhomrs, who subscrb to both platforms, usng ach platform s cor srvc. Ths partton s an qulbrum of th buyr modl, rgardd n ths and th followng sctons as a black box modl, and prsntd n scton 5. Th buyr partton s a functon of platforms xpanson dcsons ( 1, 2 ), dnotd B 1 2 {b 1 2 1, b 1 2 2, b 1 2 }, whr b 1 2 s th group of platform s xclusv subscrbrs and ts mass, 8
and b 1 2 s th group of multhomrs and ts mass, for ( 1, 2 ). W assum that th markt s covrd for any ( 1, 2 ),.., b 1 2 1 + b 1 2 2 + b 1 2 = 1. W wll consdr both th ntal and fnal buyr partton, as buyrs choos platforms twc. Thr ntal choc s mad bfor platforms dcd on xpanson, and wthout antcpatng possbl xpanson. Ths rsults n th ntal buyr partton - B E E. Followng xpanson dcsons, buyrs may swtch away from thr ntal subscrpton chocs, whch rsults n th fnal buyr partton - B 1 2, whr 1, 2 {E, E}. To llustrat th ffct of platform xpanson on th partton of buyrs, w turn our attnton to fgur 1. Th fgur dpcts th cas of ncrasd multhomng followng platform xpanson dcsons. Ths s rprsntd by th dashd llps, whch shows th growth of th group of multhomrs, at th xpns of th groups of xclusv usrs of ach platform. Nw srvcs addd by th platforms (whn = E) ar njoyd by thr rmanng xclusv usrs. Fgur 1: Endognous buyr partton - th cas of ncrasd multhomng followng xpanson. Buyrs ad clck probablty. Th xpctd numbr of clcks gnratd by ach group of buyrs dpnds on ts mass and clck-through rat (hncforth, CTR). CTR s th probablty that a usr clcks on an ad, whr ad xposur occurs va platform srvcs. Lt ρ (0,1) dnot usrs ad clck probablty, or CTR, for ad xposur on on srvc. W assum that clcks ar ndpndnt across srvcs, such that a usr xposd to an ad on two srvcs wll clck xactly onc wth probablty 2ρ(1 ρ), twc wth probablty ρ 2, and wll not clck at all wth probablty (1 ρ) 2. Ths rprsnts th basc motvaton for platform xpanson mprovng usrs Ths assumpton wll allow us to gan tracton n th analyss, wth vry lttl assumptons on th charactrstcs of th ndognous buyr partton. Howvr, th assumpton s not ncssary for obtanng our man rsults. 9
ovrall clck probablty by ncrasng th numbr of contact ponts wth th platform s usr bas. 3.3 Advrtsrs Thr s a unt mass of homognous advrtsrs n th markt. Advrtsrs stratgy s a choc of platform or platforms on whch to plac ads, dnotd α A {{1}, {2}, {1,2}, }. Th xpctd bnft of α. Advrtsrs bnft from unqu usr clcks on thr ads,.., scond clcks by usrs ar consdrd rdundant. Ths s rprsntd by a constant valu for a usr s frst ad clck, normalzd to 1, whras th valu of a usr s scond clck s zro. W ntroduc th notaton ρ 2ρ ρ 2, whch s th probablty of at last on usr clck, whn rachng th sam usr on two srvcs. ρ s thus th rlvant clck probablty whn consdrng th bnft of ad xposur on two srvcs. Th xpctd bnft of α s th product of th rlvant clck probablty and th mass of usrs rachd, whr platform provds accss to both ts xclusv subscrbrs and multhomng usrs, who subscrb to on srvc from ach platform. For α = {}, th xpctd bnft s smply ρ b 1 2 + ρb 1 2, whr ρ ρ ( ) rprsnts th probablty of a unqu clck and dpnds on th platform s xpanson dcson, such that ρ (E ) = ρ and ρ (E) = ρ. For α = {1,2} th xpctd bnft s ρ 1 b 1 2 1 + ρ 2 b 1 2 2 + ρ b 1 2, as multhomrs ar now rachd through two srvcs (and only unqu clcks mattr). Th xpctd cost of α. Th xpctd cost s th amount chargd by th platforms n α, whch s, for ach α, th product of prc pr clck and th xpctd numbr of clcks provdd. Consdrng th xpctd numbr of clcks provdd by, w assum that th platform prfctly tracks ts xclusv usrs, and thus chargs advrtsrs only for unqu clcks by ths usrs, ρ b 1 2. On th othr hand, multhomrs subscrb to only on srvc by th platform, and ar not trackd outsd of th platform. Each platform thus provds ρb 1 2 xpctd clcks by ths usrs. 13 Summarzng, ach platform α chargs ts advrtsrs a sum of p [ρ b 1 2 + ρb 1 2 ]. 13 Multhomng advrtsrs ar thus rqurd to pay for rdundant clcks. An altrnatv modllng choc s to assum that platforms dntfy multhomng usrs, and thrfor th xpctd cost of thr clcks s computd 10
Th xpctd valu of choc α s dfnd as th dffrnc btwn th xpctd bnft and cost of α, and dnotd V α V(α (, p ) =1,2, B 1 2 ): (1) V α = (1 p )[ρ b 1 2 + ρb 1 2 ] for α = {} {[ρ 1 b 1 2 1 + ρ 2 b 1 2 2 + ρ b 1 2 ] p 1 [ρ 1 b 1 2 1 + ρb 1 2 ] p 2 [ρ 2 b 1 2 2 + ρb 1 2 ] for α = {1,2} 0 for α = Th mportant fatur of V α s that advrtsr multhomng, or α = {1,2}, ntals som rdundancy, as multhomng usrs ar rachd twc, and advrtsrs pay both platforms for thr clcks. Spcfcally, whn α = {1,2}, accss to multhomng usrs provds an xpctd bnft of ρ b 1 2, at an xpctd cost of (p 1 + p 2 )ρb 1 2, whr ρ < ρ < 2ρ. Thrfor, V < V 1 + V 2. 3.4 Platforms - profts Each par of platform xpanson dcsons ( 1, 2 ) dfns an xpanson subgam, and dtrmns B 1 2 n th subgam. Platform s xpctd proft n an xpanson subgam ( 1, 2 ), for prc p, gvn th rval s prc p j, ar dnotd π 1 2 (p p j ), and gvn by: (2) π 1 2 (p p j ) = { p [ρ b 1 2 + ρb 1 2 ] α 0 α Whr th xpctd numbr of clcks, [ρ b 1 2 + ρb 1 2 ], s comprsd of ρ b 1 2 clcks by xclusv subscrbrs, and ρb 1 2 clcks by multhomrs, who us only th platform s cor srvc, rgardlss of. 3.5 Tmln Th tmln of th modl s as follows: 1. Platforms mak xpanson dcsons 1 and 2. accordng to a clck probablty of 0.5ρ. Our man rsults contnu to hold undr ths altrnatv assumpton, as V < V 1 + V 2 contnus to hold. 11
2. Th fnal buyr partton B 1 2 s dtrmnd (B 1 2 s th qulbrum of th buyr modl). 3. Platforms st prcs pr usr clck p 1 and p 2. 4. Advrtsrs choos th platform(s) on whch thy plac thr ads, α, and ths dtrmns platform profts. 3.6 Markt qulbrum W dfn markt qulbrum for th smultanous mov xpanson gam, whch rprsnts an nvronmnt whr platforms dvlopmnt fforts ar kpt scrt untl nw srvcs ar launchd. Dfnton 1: Markt qulbrum s a coupl α, (, p ) =1,2, such that: 1. Advrtsrs platform choc s optmal, gvn platforms xpanson and prcng dcsons, and th rsultng B 1 2 : α = argmax α A {V(α (, p ) =1,2, B 1 2 )}. 2. Platform prcng s Nash qulbrum, gvn thr xpanson dcsons, B 1 2, and α : p = argmaxπ 1 2 (p p j ). Subgam qulbrum profts: Π ( 1, 2 ) π 1 2 (p p j ). 3. Platforms xpanson dcsons ar Nash qulbrum n th xpanson gam: = argmaxπ (, j ). Th squntal mov vrson of th platform xpanson gam s also consdrd, as t rprsnts a markt whr platforms dvlopmnt fforts ar known. Th squntal vrson s furthr usd as a mans of qulbrum slcton, whnvr multpl qulbra ars n th smultanous mov gam. Th dfnton of markt qulbrum for th squntal gam wll b smlar, dffrng only n th qulbrum concpt for optmal xpanson dcsons (tm 3), whch wll b SGPE. 4. Analyss W bgn by drvng platform prcng and profts n a gvn xpanson subgam, thn procd to drv optmal xpanson ruls, and charactrz xpanson qulbra.
4.1 Prcng Equlbrum W show that qulbrum prcs and profts ar constrand by th dgr of usr multhomng n th fnal partton. Th ntuton for ths rsult s as follows. In qulbrum, platforms st prcs that nduc advrtsr multhomng (.., α = {1,2}), 14 and ntrnalz th rsultng rdundancy. Namly, multhomng advrtsrs suffr partal rdundancy from rachng multhomng usrs through both platforms (rcall that V < V 1 + V 2 ). Platforms ntrnalz ths rdundancy by sttng qulbrum prcs accordng to th ncrmntal bnft of advrtsng on an addtonal platform. 15 Equlbrum prcs thus dcras n th mass of multhomrs, b 1 2, and ncras n th mass of xclusv usrs, b 1 2. In othr words, markt powr n th modl stms from th dgr of xclusvty, and dcrass n th dgr of multhomng. Th followng proposton 1 provds a charactrzaton of th prcng qulbrum, and th rsultng platform profts, n a gvn subgam. (Throughout th proof, th suprscrpt 1 2 s omttd.) Proposton 1: Gvn ( 1, 2 ) and th rsultng B 1 2, platform sts ts prc at p, whr: And ts profts ar gvn by: (3) p = 1 ρ 2 b 12 ρ b 12 +ρb 12 (4) π 1 2 = ρ b 1 2 + ρ(1 ρ)b 1 2 Equlbrum prcs dcras n th dgr of multhomng, b 1 2, and ncras n th dgr of xclusvty, b 1 2. Proof: Gvn ( 1, 2 ) and B, advrtsrs plac ads on both platforms whnvr V V 1, V 2, 0, and choos a sngl platform whnvr V > V, V j, 0. 16 Solvng V V j w fnd that α = {1,2} whnvr p p. Furthrmor, not that V V j f and only f p p (p j ), whr 14 Othrws, thr wll b on platform wth no advrtsrs and zro profts that wll dvat to a lowr prc. 15 Ths s n th sprt of th "prncpl of ncrmntal prcng" (dfnd n Andrson t al. 2011). 16 WLOG w assum that ndffrnc btwn α = {1,2} and α = {} s rsolvd n favor of α = {1,2}. 13
p (p j ) [ρ b ρ j b j ]+p j [ρ j b j +ρb ] ρ b +ρb. It s asly vrfd that p = p (p j ), thus V 1 = V 2 = V = ρ 2 b for p = p, = 1,2. W show that prcng at p s proft maxmzng. Frst not that α for all p p, and th proft maxmzng prc n ths rgon s clarly p = p. W now consdr p > p. Prcng at p > 1 lads to α and zro profts, and s not proft maxmzng. Thrfor assum p (p, 1): f p j (p j, 1) thn only on platform s chosn by advrtsrs - assum that s chosn,.. V V j. Ths mpls zro profts for j, and a proftabl dvaton to p j. Altrnatvly, f p j (p j, 1) and p j = p j thn V j > V, thus α and has a proftabl dvaton to p = p. W hav thus shown that for any prc p p thr xsts a proftabl dvaton to p = p. Nash qulbrum prcs n a gvn subgam ar thus p = p for = 1,2. Substtutng for p n (2) ylds th xprsson n (4) (4) for platforms profts n subgam ( 1, 2 ). To s that p drvatvs: dcrass n b and ncrass n b w xamn th followng frst ordr (5) p = ρ2 ρ b < 0 b (ρ b +ρb ) 2 (6) p b = ρ2 ρ b (ρ b +ρb ) 2 > 0 Ths ntrm rsult wll play an mportant rol n th analyss of platform xpanson dcsons. 4.2 Charactrzng Expanson Equlbra Th nxt stp of our analyss s th drvaton of an optmal xpanson rul basd on th xpctd profts n ach subgam, as gvn by xprsson (4). But frst, w add notaton for xpanson ffcts on th buyr partton that wll b usful down th road. W ntroduc th notaton Δb k j b k E j b k E j for k {, j, }, whch rprsnts th chang n th mass of group k, rsultng from s xpanson, gvn j {E, E}. Platforms ar assumd to xrt symmtrc ffcts whn thy xpand, such that Δb k E = Δb k j E and Δb k E = Δb k j E. 14
W hncforth focus on symmtrc platforms,.. b 1 1 2 = b 2 1 2 for 1 = 2, and symmtrc xpanson ffcts. Clarly, ths may not b th cas n most ral world stuatons. But rathr than rstrct our analyss, ths wll dmonstrat that platform asymmtry s not rqurd to obtan nontrval qulbrum outcoms (ncludng qulbra wth asymmtrc xpanson). W procd to drv an optmal xpanson rul for platform, gvn j. Solvng π E j π E j (usng (4)) ylds th followng condton: E (7) j = E ρ(1 ρ)b j + ρδb j + ρ(1 ρ)δb j 0 Th abov condton rprsnts th thr ffcts of xpanson: th CTR ffct, and th quantty and prc ffcts assocatd wth changs n th usr partton. Platform xpanson dcsons ar thus basd on wghng ths ffcts aganst ach othr. Furthr dscusson and ntuton on ths ffcts s hrby provdd. Th CTR ffct. Th CTR ffct rfrs to th ncras n xclusv usrs clck probablty, brought on by xpanson. Spcfcally, xpanson ncrass ths usrs clck probablty from ρ to 2ρ ρ 2 - an ncras of ρ(1 ρ) n clck probablty for th mass of b E j 15 xclusv usrs. Th CTR ffct s thus rprsntd by th frst trm n (7), and s a postv ffct, drvng towards platform xpanson. It mmdatly follows that whn th usr partton s xognous,.. th partton dos not chang as a rsult of platform xpanson, platforms wll always xpand and th qulbrum s (E, E). Lastly not that th CTR ffct s non-monotonc n ρ, ncrasng n magntud as ρ ncrass for ρ < 0.5, thn dcrasng n ρ for ρ > 0.5. Th quantty and prc ffcts. Th quantty ffct s th drct ffct of platform xpanson - th chang n usr partton, whch s ndognous n our sttng. As xpanson changs th mass of xclusv and multhomng usrs, t changs th xpctd numbr of clcks sold by th platform. Th prc ffct of xpanson s th ndrct ffct of th chang n usr partton, as changs n multhomng and xclusvty affct prcs (s proposton 1). Th quantty and prc ffcts ar jontly rprsntd n th scond and thrd trm n (7). It s mportant to not that changs n th mass of multhomrs xrt opposng quantty and prc ffcts. Namly, an ncras (dcras) n multhomng ntals a postv (ngatv) quantty ffct as th platform provds accss to mor (lss) multhomrs. At th sam tm, ncrasd
(dcrasd) multhomng lads to lowr (hghr) prcs, mplyng a ngatv (postv) prc ffct. Furthrmor, our covrd markt assumpton mpls that changs n on group s mass wll always b accompand by corrspondng changs to on or both of th othr two usr groups. Thrfor th ovrall quantty and prc ffcts of xpanson wll tak nto account changs n both xclusvty and multhomng. W can now charactrz platform xpanson qulbra n th markt. Our analyss wll mploy E th followng condton (8), obtand by arrangng th nqualty n (7) and substtutng b j = E b j + Δb j : (8) j = E 2 ρ Δb E 1 ρ j + b j + Δb j 0 Thr ar two man cass to consdr, dpndng on th ffct of a sngl platform s xpanson on th lvl of multhomng n th markt, ths ar: (1) Δb j 0 and (2) Δb j > 0. Such ncrass or dcrass n multhomng may b th rsult of ndognous compatblty costs ncurrd by multhomng usrs, as ths ar lkly to b affctd by platform xpanson. Th buyr modl prsntd n scton 5 ncluds ndognous compatblty costs as th drvr of ndognty of th usr partton. Th analyss hrby procds consdrng xpanson ffcts n gnral, allowng applcaton of th gnral modl to any framwork of usr prfrncs and choc. To gan tracton n th analyss of th gnral modl w consdr xpanson as provdng an advantag ovr a nonxpandd rval, at last wakly. Ths advantag mpls that an xpandd platform has a (wakly) largr mass of xclusv usrs than ts rval, whn th rval platform has not xpandd. Ths s a rsultng fatur of th buyr modl of scton 5, takn hr as an assumpton. Th prvously mntond assumptons - covrd markt and symmtry - wll also play an mportant rol n th upcomng analyss of optmal xpanson stratgs n our gnral sttng. 17 17 Ths assumptons allow for th analyss of xpanson stratgs wthout spcfyng th buyr modl. Gvn a spcfc buyr modl ths assumpton ar no longr ndd, as th structur of ndognty of B 1 2 follows from th modl, and optmal xpanson stratgs ar drvd usng condton (8). 16
Th followng Lmmas 1 and 2 consdr optmal xpanson stratgs sparatly for th cass of dcrasd and ncrasd multhomng followng xpanson, rspctvly. Lmma 1 stats that whnvr multhomng dcrass wth xpanson, xpanson s a domnant stratgy. Intutvly, ths rsult follows from th advantag assocatd wth xpanson, whch mpls that th dcras n multhomng s accompand by an ncras n xclusvty for th xpandng platform (whn th markt s covrd). In ths cas, th prc ffct s postv and larg nough, such that th prc ffct and th always-postv CTR ffct wll always lad to xpanson, vn whn th total quantty ffct s ngatv (.., whn th dcras n multhomng s largr than th ncras n xclusvty). Lmma 1: Expanson s a domnant stratgy whnvr t dcrass multhomng,.., Δb j 0 mpls j E. Proof: Δb j 0 and th covrd markt assumpton mply that Δb j + Δb j j 0. Snc an xpandd platform has an advantag (at last wakly) ovr a non-xpandd rval, Δb j Δb j j and thrfor t must b that Δb j 0, whras both Δb j j 0 and Δb j j < 0 ar possbl. For Δb j j < 0, a covrd markt mpls Δb j = Δb j + Δb j j, thrfor Δb j + Δb j 0 such that th nqualty n (8) always holds. For Δb j j 0, a covrd markt mpls Δb j + Δb j j = Δb j. W substtut ths nto th LHS of th nqualty n (8): LHS = 2 ρ Δb E 1 ρ j + b j (Δb j + Δb j j ) = 1 Δb E 1 ρ j + b j Δb j j 0. Th nqualty always holds snc Δb j Δb j j. W hav thus shown that j E whnvr Δb j 0. W turn to consdr th cas of ncrasd multhomng followng xpanson. Whn both multhomng and xclusvty ncras, th ovrall prc ffct may b thr ngatv or postv, but th ovrall quantty ffct s postv and larg. Th postv quantty and CTR ffcts domnat and platforms wll optmally xpand. On th othr hand, whn xclusvty dcrass, 17
th ovrall prc ffct s ngatv. Th ovrall quantty ffct s postv but small, as th dcras n xclusvty s smallr than th ncras n multhomng, du to th xpanson advantag. Hnc, th xpanson stratgy wll dpnd on th rlatv magntuds of ths and th CTR ffct. Namly, whn th CTR ffct s rlatvly larg (ρ s small), th quantty ffct s larg and th prc ffct small (whn th dcras n xclusvty s small), thn xpanson s optmal, and othrws qulbrum no-xpanson may ars. Ths s summarzd n Lmma 2. Lmma 2: Whn s xpanson ncrass multhomng,.., Δb j > 0-1. If s xclusvty ncrass thn xpanson s a domnant stratgy: Δb j 0 mpls j E. 2. If s xclusvty dcrass thn xpanson s optmal whn th dcras n s xclusvty s rlatvly small compard to j s, and th CTR s rlatvly small. Formally, for Δb j > 0 and Δb j < 0: j = E f and only f Δb j (1 ρ) [b E j Δb j j ]. Proof: Δb j > 0 and th covrd markt assumpton mply Δb j + Δb j j < 0. Snc an xpandd platform njoys a wak advantag ovr a non-xpandd rval, w hav Δb j Δb j j. It follows that Δb j j < 0, whras both Δb j < 0 and Δb j 0 ar possbl. If Δb j 0 thn th nqualty n (8) trvally holds. Othrws, f Δb j < 0, w substtut Δb j + Δb j j = Δb j nto th LHS of th nqualty n (8): LHS = 2 ρ Δb E 1 ρ j + b j Δb j Δb j j = 1 Δb E 1 ρ j + b j Δb j j, and thrfor - (9) j = E Δb j (1 ρ) [b E j Δb j j ] Proposton 2 mploys Lmmas 1 and 2 to provd a full charactrzaton of xpanson qulbra, whn th buyr partton s ndognous, consdrng dffrnt forms of ndognty. Expanson 18
s sad to monotoncally ncras multhomng whn b EE EE E E > b > b dcras multhomng whn b EE EE E E EE < b < b (rcall that b = b E E ). and to monotoncally Proposton 2: Expanson qulbra dpnd on th ndognty of th buyr partton: 1. Whn xpanson monotoncally dcrass multhomng th qulbrum s (E, E). 2. Whn xpanson monotoncally ncrass multhomng possbl qulbra ar: (E, E), (E, E ), both (E, E) and (E, E ), or both (E, E) and (E, E ). EE E E 3. Whn b > b and b EE EE < b th qulbrum s thr (E, E) or both (E, E) and (E, E ). EE E E 4. Whn b < b and b EE EE > b th qulbrum s thr (E, E) or both (E, E) and (E, E ). Proof: Follows mmdatly from Lmmas 1 and 2. W hav thus shown that platforms may optmally choos not to xpand, whn xpanson ncrass multhomng. Ths s bcaus ncrasd multhomng dcrass markt powr, and ncrass prc comptton n our modl. Intrstngly, asymmtrc xpanson qulbra may ars vn for symmtrc platforms. Whnvr asymmtrc qulbra ars n th smultanous xpanson gam, th frst movr wll xpand n th squntal gam. Ths s bcaus, n an asymmtrc qulbrum, th xpandd platform has a (wak) advantag and thus hghr profts. Whn both (E, E) and (E, E ) ar qulbra n th smultanous gam, th SGPE of th squntal gam s drvd by comparng π EE and π E E and wll thus dpnd on th rlatv magntuds of th ffcts of xpanson. Exstnc of th dffrnt qulbra mntond n proposton 2 s shown n scton 6, as th buyr modl s analyzd for dffrnt cass of ndognous compatblty costs (ncurrd by multhomrs), and th xpanson qulbrum s drvd for svral numrcal xampls. W conclud by consdrng th ffcts of changs n paramtrs, whn xpanson ncrass multhomng and dcrass xclusvty. In ths cas, th xpanson stratgy dpnds on th rlatv magntuds of xpanson ffcts, and thrfor paramtr changs may rsult n a shft from qulbrum xpanson to no-xpanson. Ths s statd n th followng corollary, whch follows drctly from Lmma 2. 19
Corollary 1: Whn xpanson ncrass multhomng and dcrass xclusvty for th xpandd platform, th followng changs n paramtrs may rsult n a swtch from optmal xpanson to no-xpanson: Incrass n th CTR, ρ. Incrass n th xpanson ffct on own mass, Δb j. Dcrass n th xpanson ffct on th rval s mass, Δb j j. E Dcrass n th pr-xpanson mass b j. As xpctd, ncrass n CTR lowr th magntud of th CTR ffct whn ρ > 0.5, and may thrfor rsult n a swtch to no-xpanson. Such a swtch may also rsult from largr dcrass n own mass followng xpanson, as thy dcras th magntud of th postv quantty ffct, and furthr strngthn th alrady-ngatv prc ffct. Furthrmor, whn th rval suffrs a smallr dcras followng xpanson, thn th xpanson advantag s waknd, and mor xclusv usrs ar lost by th xpandng platform, for a gvn ncras n multhomng. Thrfor a smallr usr loss for th rval may rsult n a swtch to no-xpanson. Lastly, a smallr mass pror to xpanson ncrass th rlatv magntud of th ffcts assocatd wth loss of xclusv usrs, such that dcrass n th pr-xpanson mass may lad to optmal noxpanson. 5. Th buyr modl In ths scton w propos a buyr prfrnc and choc modl that ylds th proprty of ndognous partton, cntral n our sttng. Ths scton thus provds a possbl foundaton for B 1 2, and a tst cas for our gnral modl. W consdr usrs charactrzd by prfrncs for platform srvc qualty, who ncur a compatblty cost whn multhomng, and a swtchng cost whn swtchng away from thr ntal platform choc followng xpanson dcsons. Th compatblty cost vars across th usr populaton, and ts dstrbuton may chang wth platform xpanson. Th dtals of th buyr modl ar hrby provdd. 20
Platform {1,2} offrs srvc(s) of total qualty q. At th outst ach platform offrs on srvc of qualty q, such that q = q, and xpanson mpls addng a scond srvc of qualty Δq (0, q). Thrfor, q = q for = E, and q = q + Δq for = E. Platforms ntal or cor srvcs ar thus of hghr qualty than nwly addd srvcs. Multhomng usrs subscrb to platforms cor srvcs, thus njoyng an aggrgat qualty of 2q. Multhomrs ncur htrognous and ndognously dtrmnd compatblty costs, such that th compatblty cost for usr b B s c 1 2 b ~F 1 2 [c 1 2 L, c 1 2 H ], whr F 1 2 s th PDF of compatblty costs. Not that both F 1 2 and ts doman may chang wth platform xpanson. Ths rprsnts possbl changs n platform compatblty rsultng from th addton of th rval's cor srvc, whr compatblty may thr ncras or dcras wth xpanson, and th chang n compatblty may vary across usrs. Utlty for usr b B s wrttn as: (10) u b = q b = 2q c 1 2 b u Whr u b b s b s utlty from subscrpton to a sngl platform and u s b s utlty from multhomng. Usrs do not antcpat platform xpanson, and thrfor choos platforms twc. Thy mak an ntal subscrpton choc bfor xpanson dcsons ar mad, and may thn swtch to anothr choc followng platform xpanson. Swtchng to a nw choc s costly, such that th utlty for b a swtchng usr b B from a nw choc nw {1,2,} s u nw n (10) and s 0 s th swtchng cost ncurrd. 21 b s, whr u nw s dfnd To summarz, a usr s ntal platform choc s mad gvn hs ndvdual ralzaton of c b E E, and hs fnal platform choc (followng xpanson dcsons) s mad gvn hs ralzaton of c b 1 2 and th swtchng cost s. Usrs platform chocs for ach par ( 1, 2 ) ar mad so as to maxmz utlty, as dfnd abov. W mak two t-brakng assumptons. Frst, w assum that ndffrnc btwn th two platforms s rsolvd by a far con flp,.. usrs ndffrnt btwn platform 1 and 2 wll choos ach platform wth probablty 0.5. Ths rprsnts an dosyncratc platform prfrnc,
whn platforms offr xactly th sam qualty lvl. Scond, w assum that ndffrnc btwn multhomng and choc of a sngl platform s rsolvd n favor of multhomng. W procd to dfn buyr qulbrum. Dfnton 2: Buyr qulbrum s a choc of platform(s) for ach buyr b B gvn ( 1, 2 ), such that ach buyr s choc s utlty maxmzng, gvn c 1 2 b and s. A partton B 1 2 thus consttuts buyr qulbrum gvn ( 1, 2 ). Endognty of B 1 2 s th rsult of th ndognty of usrs compatblty costs, whch, n turn, rprsnts ndognous ntr-platform compatblty lvls. 6. Applyng th gnral framwork: ndognous compatblty costs In ths scton w us th proposd buyr modl to provd a tst cas wth numrcal xampls and furthr ntuton for th gnral modl. W thrfor solv for buyr qulbrum bfor and aftr platform xpanson, undr a crtan spcfcaton for F 1 2, chosn for ts smplcty and ralsm. For th followng analyss w assum that F E E s a unform dstrbuton ovr [0,1], and that t shfts thr lft or rght wth xpanson dcsons. Spcfcally, w consdr usrs who draw a compatblty cost onc at th outst from F E E, and may thn xprnc an ncras or dcras n ths cost, whch dpnds on th par of xpanson dcsons. All usrs xprnc th sam ncras or dcras n compatblty cost, such that ach usr mantans th sam rlatv cost compard to hs prs. Our symmtry assumpton mpls that F EE = F E E. For as of notaton lt n {1,2} dnot th numbr of xpandd platforms, and lt F n ~U[Δc n, 1 + Δc n ] b th dstrbuton of compatblty costs whn n platforms xpand, whr Δc n s thr a postv or ngatv constant. Nxt, solvng for B E E, group masss n th ntal usr partton ar b 1 E E = b 2 E E = 1 q E E b = q. 2 and Whn a sngl platform xpands (WLOG, lt 1 = E, 2 = E ) c b EE ~U[Δc 1, 1 + Δc 1 ]. If Δc 1 > 0 and s s not too hgh, ntal multhomrs wth rlatvly hgh compatblty costs wll swtch to th xpandd platform. On th othr hand, f Δc 1 < 0 and s s not too hgh, ntal subscrbrs of 22
both platforms wth a rlatvly low compatblty cost wll swtch to multhomng. For both Δc 1 > 0 and Δc 1 < 0, usrs from th non-xpandd platform 2 wth a hgh compatblty cost wll swtch to th xpandd platform whnvr s Δq, and wll othrws rman captv usrs of platform 2. Solvng for B EE, w focus on th followng cass, for Δc 1 > Δq: 18 (a) For Δc 1 > 0: EE For s Δq, usrs wth c b (q Δq + s, q + Δc 1 ] wll swtch from multhomng to th xpandd platform 1, and all usrs of platform 2 wll swtch to 1. For s (Δq, Δq + Δc 1 EE ], usrs wth c b (q Δq + s, q + Δc 1 ] wll swtch from multhomng to th xpandd platform 1, and all usrs of platform 2 rman captv. (b) For Δc 1 < 0, s mn{δq, Δc 1 EE Δq}: ntal platform 1 subscrbrs wth c b (q Δc 1 EE, q Δq s] and ntal platform 2 subscrbrs wth c b (q Δc 1, q Δq] wll swtch to multhomng; th rmanng platform 2 usrs wll swtch to platform 1. Whn both platforms xpand c EE b ~U[Δc 2, 1 + Δc 2 ], and thr s symmtrc swtchng away from multhomng whn Δc 2 > 0 and towards multhomng whn Δc 2 < 0. Solvng for B EE, w focus on th cas of s mn{δq, Δc 2 Δq}, for Δc 2 > Δq: (a) For Δc 2 > 0: usrs wth c EE b (q Δq + s, q + Δc 2 ] wll swtch from multhomng to th xpandd platforms, dvdng qually btwn th two. (b) For Δc 2 < 0: ntal subscrbrs of platforms 1 and 2 for whom c EE b (q Δc 2, q Δq s] wll swtch from th xpandd platforms to multhomng. Th rsultng B EE and B EE ar summarzd n th followng tabl 1 (for Δc n > Δq and s mn{δq, Δc n Δq}, unlss othrws statd): B EE (n = 1) B EE (n = 2) (a) Δc n > 0 b 1 EE = 1 q + Δq + Δc 1 s b 2 EE = 0 b 1 EE = b 2 EE = 1 q+δq+δc2 s 2 b EE = q Δq Δc 2 + s 18 W do not conduct a full analyss of th buyr modl for all possbl paramtr valus, as our goal s to provd xampls for th cass dscussd n proposton 2. 23
(b) Δc n < 0 EE b = q Δq Δc 1 + s For s (Δq, Δq + Δc 1 ]: b 1 EE = 1 q 2 + Δq + Δc1 s b 2 EE = 1 q 2 EE b = q Δq Δc 1 + s b 1 EE = 1 q + Δq Δc 1 + 0.5s b 2 EE = 0 EE b = q Δq + Δc 1 0.5s b 1 EE = b 2 EE = 1 q+δq Δc2 +s 2 b EE = q Δq + Δc 2 s Tabl 1: Buyr parttons B EE and B EE for Δc n > Δq and s mn{δq, Δc n Δq} (unlss othrws statd). W now provd xampls for th qulbrum outcoms dscussd n proposton 2. Th followng subsctons 6.1-6.4 corrspond to cass 1-4 of proposton 2. For cass 1-4 w us s mn{δq, Δc n Δq} and for cas 4 w addtonally consdr s (Δq, Δq + Δc 1 ]. 6.1 Expanson monotoncally dcrass multhomng Our buyr modl has monotoncally dcrasng multhomng for Δc 2 > Δc 1 > 0. Usng B EE, B EE from th abov tabl n condton (8), w fnd that th nqualty always holds for j {E, E} and thrfor xpanson s a domnant stratgy. Ths also follows from Lmma 1. Th qulbrum s thrfor (E, E). 6.2 Expanson monotoncally ncrass multhomng Th cas of monotoncally ncrasng multhomng corrsponds to Δc 1, Δc 2 < 0 and Δc 2 Δc 1 > 0.5s n th buyr modl. W us th abov B EE, B EE, and condton (8) to fnd: E = E whnvr ρ ρ E, whr ρ E = 3 + Δq Δc1 + 1 2 s. 2 1 q E = E whnvr ρ ρ E, whr ρ E = 2(1 q+δq Δc1 )+s 1 q+δq+ Δc 2 2 Δc 1. 24
Not that th thrsholds ρ E, ρ E may b smallr than 1. Spcfcally, ρ E < 1 for Δc 1 > 1 q+2δq+s 2, and ρ E < 1 for Δc 2 > 1 q + Δq + s. W thrfor show xstnc of th dffrnt qulbrum typs mntond n proposton 4, usng a numrcal xampl. Numrcal xampl: For paramtr valus Δc 1 = 0.2, Δc 2 = 0.25, Δq = 0.1 and s = 0, th rlatv sz of th xpanson thrsholds ρ E and ρ E dpnds on q: (a) For q = 0.89: ρ E = 0.59 and ρ E = 0.33, such that ρ E < ρ E and th qulbrum s (E, E) for ρ ρ E, both (E, E ) and (E, E) for ρ (ρ E, ρ E ], and (E, E ) for ρ > ρ E (s panl (a) of fgur 2); (b) For q = 0.83: ρ E = 0.91 and ρ E > 1, such that ρ E < ρ E and th qulbrum s (E, E) for ρ ρ E, and both (E, E ) and (E, E) for ρ (ρ E, 1] (s panl (b) of fgur 2). (a) (b) Fgur 2. Equlbrum charactrzaton for Δc 1 = 0.2, Δc 2 = 0.25, Δq = 0.1, s = 0: (a) q = 0.89 and (b) q = 0.83. 6.3 Expanson frst ncrass and thn dcrass multhomng Ths cas corrsponds to thr Δc 1, Δc 2 < 0 and Δc 2 Δc 1 < 0.5s, or to Δc 1 < 0 and Δc 2 > 0 n th buyr modl. To show xstnc of th typs of qulbrum dscussd n proposton 2, w focus on th cas of Δc 1 < 0 and Δc 2 > 0. Usng B EE, B EE, and condton (8) w fnd: E = E whnvr ρ ρ E, whr ρ E s dfnd n 6.2 abov. E E, as th nqualty n (8) always holds for j = E (also follows from Lmma 1). Rcall that ρ E < 1 for Δc 1 > 1 q+2δq+s. Onc agan, a numrcal xampl s provdd to show 2 xstnc of th qulbra dscussd n proposton 2. 25
Numrcal xampl: For paramtr valus Δc 1 = 0.35, Δc 2 = 0.2, Δq = 0.1, s = 0, and q = 0.8, th xpanson thrshold s ρ E = 0.25, such that th qulbrum s (E, E) for ρ ρ E, and both (E, E ) and (E, E) for ρ (ρ E, 1] (th llustraton s smlar to th on n panl (b) of fgur 2). 6.4 Expanson frst dcrass and thn ncrass multhomng Ths cas corrsponds to thr Δc 1, Δc 2 > 0 and Δc 1 > Δc 2, or to Δc 1 > 0 and Δc 2 < 0 n th buyr modl. To show xstnc of th rlvant qulbrum typs dscussd n proposton 2, w focus on th cas of Δc 1 > 0 and Δc 2 < 0, wth Δq < s < Δq + Δc 1. Usng B EE, B EE, and condton (8) w fnd: E E, as th nqualty n (8) always holds for j = E (also follows from Lmma 1). E = E whnvr ρ ρ E, whr ρ E = 1 q+2δq+2δc1 2s 1 q+δq+ Δc 2 +2Δc 1 3s. Not that ρ E < 1, snc w assumd s < Δc 2 Δq. Th qulbrum s thrfor (E, E) for ρ ρ E, and both (E, E ) and (E, E) ar qulbrum for ρ (ρ E, 1] (s fgur 3). Numrcal xampl: For paramtr valus Δc 1 = 0.1, Δc 2 = 0.3, Δq = 0.1, s = 0.15, and q = 0.1, w hav ρ E 0.95, and ρ E dcrass n q. Fgur 3. Equlbrum charactrzaton for Δc 1 = 0.1, Δc 2 = 0.3, Δq = 0.1, s = 0.15, q = 0.1. 7. Dscusson and managral mplcatons W hav prsntd a gam thortc framwork for analyss of onln platforms xpanson dcsons, whn th buyr partton s ndognous and thus rsponds to xpanson. Th analyss dmonstrats that xpanson may not b optmal whn t ncrass th dgr of usr multhomng, as ncrasd multhomng lowrs platforms markt powr. 26
Whn wll xpanson ncras usr multhomng? Dffrnt mchansms or buyr modls may b usd to consdr possbl xpanson ffcts on buyrs platform choc and multhomng bhavor. Th proposd buyr modl suggsts changs n ntr-platform compatblty lvls as th undrlyng caus for ndognty of th buyr partton. Othr mchansms may consdr possbl ffcts of changs n buyrs choc sts, or xpanson ffcts on usrs prcptons of platform dntty and srvc dffrntaton. Th gnral framwork may b furthr adaptd to consdr xpanson nto srvcs that ar substtuts or complmnts to th rval s srvc offrng, as ths ar also xpctd to affct th partton of usrs and thr multhomng bhavor. Our gnral framwork s thus amnabl to dffrnt markt sttngs, and may b appld by managrs vn wthout fully spcfyng a buyr modl, gvn forcasts on th partton of usrs followng xpanson. Whnvr multhomng may ncras cauton s advsd, as xpanson may dcras, rathr than ncras, profts. Rturnng to th xampl of Googl Plus - Dos th modl mply that Googl should not hav xpandd nto socal ntworkng? Th short answr s no. Whl not succdng n stalng away (most) Facbook usrs, Googl Plus dd provd th platform wth valuabl socal ntwork data usd to ncras ad targtablty. It sms that, for Googl, th bnft of mprovd targtng was much largr than any ffcts of changs n th usr partton. Ths may not b th cas for many platforms currntly xpandng nto contnt stramng srvcs. It s plausbl that xpanson nto a vdo stramng srvc rsults n an ncrasd apptt for stramd contnt, thrby ncrasng usrs multhomng wth compttors. For xampl, Amazon usrs who bgn usng Amazon Instant Vdo may ncras thr us of Ntflx, Hulu, and othr compttors. Expanson nto contnt stramng should thrfor tak nto account such stratgc ffcts. Rfrncs Ambrus, A., Calvano, E., and Rsngr, M., 2013, Ethr or Both Comptton: A "Two-sdd" Thory of Advrtsng wth Ovrlappng Vwrshps, Duk workng papr. 27
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