RESEARCH ARTICLE QUALITY COMPETITION AND MARKET SEGMENTATION IN THE SECURITY SOFTWARE MARKET Debabrata Dey ad Atau Lahr Mchael G. Foster School of Busess, Uersty of Washgto, Seattle, Seattle, WA 9895 U.S.A. {ddey@uw.edu} {lahra@uw.edu} Guoyg Zhag Dllard College of Busess, Mdwester State Uersty, Wchta Falls, Wchta Falls, TX 76308 U.S.A. {grace.zhag@wsu.edu} Apped A Proofs Proof of Lea Settg u, fro Fgure, we ca fd the aret coerage of θ,,,,, as u u, whch ca be sued oer to obta u G. Substtutg ths to ( for ad otg that p 0 θ 0 0, we fd p Gθu Gθ (A We wll ow proe the lea by ducto. It s clear that (3 reduces to (A for. Let (3 hold for, plyg p G θ θ We substtute ths to ( for to obta ( θ θ p G u p G( θ G θ θ θ G( θ θ G θ θ θ θ G θ θ I other words, f (3 holds for, the t also holds for. Sce (3 holds for, the proof s ow coplete. # MIS Quarterly Vol. 38 No. Apped/Jue 04 A
Dey et al./qualty Copetto & Maret Segetato Proof of Proposto ( Sce R GH c(θ ad <, we ca use the frst order codto wth respect to to obta GH GH Gθ θ 0 θ Therefore, we get GH θ θ GH G ( H H ( ( Now, fro defto of H l, l,,,, we get l l Hl Hl l l l θ θ θ θ l ( θ θ l l l Sce θ > θ ples that >, sug the aboe oer l, we get H H ( Hl Hl ( θl θl l l l (A Now, sce θ > θ, there ust est soe l, < l <, such that θ l θ l- > 0, plyg that the rght had sde of (A s strctly greater tha zero. Thus, H H > 0 ad, hece, θ θ > 0, whch copletes the proof. ( Frst, we ote that, for all,,,, Y θ G g whch, of course, eas that Y H Y θ. Net, we also ow that ( gg ( GH GG Y H gy Y H θ θ θ θ θ θ ( θ 0 Therefore, Y > H, ad hece Y > θ, for all,,,. Sug both sdes oer, we get Y θ Y Y θ Y ( If θ θ, the for eery l, < l <, θ l θ l- 0. Therefore, fro (A, H H 0 plyg θ θ 0 or. ( Fro the proof of part (, we ow that Y > θ. Therefore, R ( Y θ θ 0 θ g A MIS Quarterly Vol. 38 No. Apped/Jue 04
Dey et al./qualty Copetto & Maret Segetato O the other had, fro the proof of part (, we ow that GH θ G ; we ca the show that R g H YH θ( H Y < 0 I other words, whe g creases, the frst order respose by a edor to ths chage s to crease qualty ad decrease aret share. Howeer, whe θ, the edor caot crease qualty ay further ad ts oly frst order respose would be to decrease ts aret share. Sce such a respose copleets other edors actos, equlbru, ust decrease. ( We proe ths part by cotradcto. Let there be a equlbru wth θ θ <, for soe <, wth <, <. We ow that edor soles the followg azato proble: Ma R G θ θ c θ θ, Sce θ < (by assupto, the frst order codto wth respect to θ ust be satsfed: ( G g θ θ c ( θ θ 0 Furtherore, sce θ θ, for all, < <, we ow fro aboe that the aret shares of these edors would be equal. We set θ θ θ ad to get c ( θ G g θ g θ G g θ g θ θ G ( g θ ( g θ ( θ G ( g θ g θ (A3 We ow cosder how the reeue of the th edor chages wth the qualty of ts ow product: G g θ g θ c ( θ θ Substtutg (A3 to the aboe epresso, we get G ( < 0, whch s a olato of the frst order codto for a teror θ soluto. # Proof of Theore We frst show that there ests a g beyod whch all edors offer a qualty leel of oe. To see ths, cosder edor. Its proft s ge by R GH c(θ. Therefore, Gg c θ ( θ ( θ MIS Quarterly Vol. 38 No. Apped/Jue 04 A3
Dey et al./qualty Copetto & Maret Segetato Fro (4, Y Σ θ ad G gy. Furtherore, fro the proof of Proposto (, we ow that Therefore, we hae GH θ G equlbru. ( gy ( Y G gθ ( gy gy The last equalty results fro the fact that H < Y; see the proof of Proposto (. Now, fro that proof, we also ow that Y ( gy g gy. Furtherore, s a creasg fucto of gy. Hece, we ca wrte gy Gg θ g ( gy ( ( g gy g ( ( g whch s clearly a creasg fucto of g. Sce cn( s bouded, for a suffcetly large g, we wll hae θ θ ( g ( ( g ( c > 0 Sce c (@ s a creasg coe fucto, the aboe eas that, equlbru, a teror soluto s ot possble ad θ. Ths, tur, ples that θ, for all,,. I other words, there ust est a threshold for g we characterze ths threshold as γ (c Theore beyod whch ertcal dfferetato would dsappear. We ow cosder what happes whe g starts decreasg below ths threshold. Of course, f the deelopet cost s eglgble, trally, all edors would cotue to offer a qualty leel of oe, rrespecte of the alue of g. Howeer, f the deelopet cost s sgfcat, soe edors would hae to drop ther qualty leel below oe, but we wll show that they ca do so oly oe edor at a te. To proe ths last cla, suppose that two edors drop the qualty leel to below oe at the sae te. At the alue of g where ths occurs, these edors ust be barely at the sae teror soluto. Howeer, fro the proof of Proposto (, t s clear that o two edors ca hae the sae teror soluto. Therefore, whe g decreases, edors would ot oly drop ther qualty leels fro oe, but would also do so oly oe at a te, whle atag the order of ther qualty leels. Equaletly, as g creases, ther qualtes would reach oe at dfferet alues of g. It s also clear fro the proof of Proposto ( that, oce a qualty leel reaches oe, t caot drop whe g creases further. Tae together, t s clear that, as g creases, the segetato leel the aret gradually decreases. #, so Proof of Theore γ To proe the estece of the erse fucto, t s suffcet to show that γ (g s a strctly ootoc fucto. It turs out that g g. To see ths, we obsere that g where ( ( γ ( g ( ( A B g 4g g ( ( ( 3 3 3 4 3 3 A 8g 8g 6 3g 0g 6 8g g 4g 3g, B C 4g g, ad C 4g 4 5g 3g g 3 ( > 0 3 3 Now A B 4g g D, where D 4g g ; hece, A > B, or A > B, as log as D > 0. If g <, D s always poste. ( g Therefore, we oly cosder the case where g >. I that case, D > 0 f ad oly f <. Suppose ot. The, there s a δ > 0 such that δ ( g. Substtutg ths to C leads to g g A4 MIS Quarterly Vol. 38 No. Apped/Jue 04
Dey et al./qualty Copetto & Maret Segetato ( g ( g ( g 3 C 3 δ 3 ( 8 ( gg 9g g6 g ( 3g g The frst ad the thrd ters are clearly egate sce g >. Furtherore, sce δ > 0, whe g >, t ca be show, after soe algebra, that the secod ter caot be poste. Therefore, C < 0, plyg B < 0. Sce A > 0 always, ths, tur, ples that A > B, whch copletes the proof of the frst part. For the secod part, we ote that the olgopoly equlbru ca be oly oe of ( regos. Let Rego I deote the rage of g alues wth the frst aret cofgurato, where all edors offer the qualty leel of oe. Slarly, let Rego II be the rage for the secod oe, where oly the lowest qualty edor, aely edor, offers a qualty leel below oe (θ <. Vertcal dfferetato wll be obsered as soo as the equlbru outcoe oes out of Rego I. Therefore, we oly eed to eae the boudary betwee Regos I ad II. I both the regos, θ θ 3 θ, ad t follows fro Proposto ( that 3. Let h deote ths coo aret share. The optzato proble of edor ca, therefore, be splfed to ( ( h ( ( h ( δ 3 ( ( ; Ma R θ g θ c θ θ, s.t. θ > 0, h The followg frst order codto ust be satsfed by the soluto of the ucostraed proble: ( θ θ( ( ( θ ( ( h g( ( h g h c 0 θ Sce, at the boudary of Regos I ad II, θ, we substtute t aboe to obta ( ( ( ( ( ( c g h h γ (A4 Now, whe θ s are all oe, frst order codtos wth respect to,,,,, result whch ca be substtuted to (A4 to obta where h ( g( 4g( g ( g( γ ( g ( g ( g ( 3 μ g 3 ( ( ( ( ( 6 4 ( ( 3 5 4 μ g g g g ad ( 3 4 ( ( ( ( ( g g g g g Therefore, the codto cn( > γ (g whch s equalet to g < γ (cn( esures that the outcoe s ot Rego I. # Proof of Proposto Sce γ ( g ( g ( g ( 3 μ g The result follows drectly fro the aboe. #, usg l Hosptal s rule twce, we get γ ( 0 lγ ( g ( 0 ( 0 ( μ g 4 0 3 MIS Quarterly Vol. 38 No. Apped/Jue 04 A5
Dey et al./qualty Copetto & Maret Segetato Proof of Theore 3 Recall that where, as before Therefore, we hae Cobg the aboe, we ca wrte ad The aboe epressos lead to R GH GH G Hl l c( θ t t t G gy, Y θ, ad H θ θ GH G H H G G θ gθ ad H f θ otherwse G H H G G H lt l H t lt ( GH Gθ gθ H Gθ gθ H Gθ gθh t lt lt lt t G H H G GH G H H G G H lt l H t lt ( GH Gθ gθ H Gθ gθ H Gθ gθ H t lt lt lt t G G θ θ θθ θ θ R R R G ( θh θh ( θ θ θ θl l θθ t t t (A5 We ow obsere θh θh θθ θθ θθ θθ ( θθ θ θ θ θ θ ( θ θ θ θ θ θ ( θ θ θ θ ( θ θ θ Substtutg ths to (A5 leads to A6 MIS Quarterly Vol. 38 No. Apped/Jue 04
Dey et al./qualty Copetto & Maret Segetato G ( ( θ θ θθ θ θ θ θ θ θ θ l l t t t Sce θ s the largest aog all the ersos proded, t s easy to see that the rght had sde of the aboe epresso s poste, whch s a olato of the codto (7. # Proof of Lea Ths proof s slar to that of Lea. We ca show that (9 ples p gθ θ θ 0 0 Substtutg 0 Σ ad θ 0 φ ad rearragg ters, we get (0. # Proof of Theore 4 It s slar to the proofs of Theores ad, wth gn g( φ. # MIS Quarterly Vol. 38 No. Apped/Jue 04 A7