Information Management and Pricing in Platform Markets Supplementary Material

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1 Informaton Management and Prcng n Platform Markets Supplementary Materal Bruno Jullen Toulouse School of Economcs Alessandro Pavan Northwestern Unversty May 4, 208 Ths document contans addtonal results for the manuscrpt Informaton Management and Prcng n Platform Markets. All numbered tems (.e., sectons, subsectons, lemmas, condtons, propostons, and equatons) n ths document contan the prefx S. Any numbered reference wthout the prefx S refers to an tem n the man text. Please refer to the man text for notaton and defntons. Secton S contans results for the monopoly case. It formally shows that the same Condtons (M) and (Q) n the man text that guarantee exstence and unqueness of a contnuaton equlbrum n the duopoly case also guarantee exstence and unqueness of a contnuaton equlbrum n the monopoly case, as clamed (but not proved) n Secton 4. n the man text. In the same secton, we also dscuss platform desgn and nformaton management n monopolstc markets. Secton S2 dentfes condtons guaranteeng that the equlbrum allocatons (prces and partcpaton decsons) n the baselne-model reman equlbrum allocatons n an enrched model n whch agents can opt out of the market, or multhome. Secton S3 contans a dynamc extenson n whch platforms revse ther prces at the same frequency at whch agents revse ther belefs about the dstrbuton of preferences, and hence ther partcpaton decsons. Secton S4 contans an extenson to a market wth wthn-sde network effects. Fnally, Secton S5 contans a descrpton of a flexble famly of economes wth Gaussan nformaton satsfyng the condtons n Secton 5. n the man text.

2 S Monopoly S. Exstence and Unqueness of Contnuaton Equlbra n the Monopoly Case In ths secton, we formally establsh the result clamed n the man text that, n the monopoly case, the contnuaton equlbrum s unque when network effects are small. Lemma S. Assume Condtons (M) and (Q) n the man text hold. Then, for any vector of prces p = (p B, pb 2 ) set by the monopolst, there exsts one and only one soluton to the system of equatons gven by s + ˆv 2 + γ [ M j (ˆv j ˆv ) = p B, j =, 2, j. (S) In turn, ths mples that, n each equlbrum of the game, gven any vector of prces set by the platform, all agents from sde whose v exceeds ˆv jon the platform, whereas all agents whose v s less than ˆv refran from jonng, where (ˆv, ˆv 2 ) s the unque soluton to (S). Proof of Lemma S. The proof parallels the one for Lemma n the man text. To ease the readng, we re-wrte Condtons (M) and (Q) here: Condton (M): for any, j =, 2, j, any v, v 2 R, 2γ M j (v j v ) v > 0; Condton (Q): for any, j =, 2, j, any v, v 2 R, [ [ 2 γ M 2(v 2 v ) v 2 γ 2 M 2(v v 2 ) v 2 γ γ 2 < M 2 (v v 2 ) v M 2 (v 2 v ) v 2. Frst, observe that, when Condton (M) holds, for any ˆv j R, the gross payoff s + v 2 + γ [ M j (ˆv j v ) that an agent from sde =, 2 wth valuaton vl B = s + v 2 obtans from jonng platform B when he expects agents from sde j to jon the platform f and only f v j > ˆv j s strctly ncreasng n v. From the mplct functon theorem, ths n turn means that, gven the prces p = (p B, pb 2 ), for any ˆv j R, there exsts one and only one soluton ζ (ˆv j ) to the equaton s + ˆv 2 = p B γ [ M j (ˆv j ˆv ) wth ζ (ˆv j ) satsfyng ζ (ˆv j ) = γ M j (ˆv j ζ (ˆv j )) v j /2 γ M j (ˆv j ζ (ˆv j )) v (S2) Note that the denomnator n (S2) s strctly postve under Condton (M). Next, let z j (v j ) s + v j 2 + γ j [ M j (ζ (v j ) v j ) 2

3 denote the gross payoff that an agent from sde j =, 2 derves from jonng the platform when he expects all agents from sde j to jon f and only f v ζ (v j ). The functon z j (v j ) s dfferentable wth dervatve equal to z j(v j ) = { 2 γ Mj (ζ (v j ) v j ) j ζ v (v j ) + M } j (ζ (v j ) v j ) v j = 2 γ M j (ζ (v j ) v j ) j M j (ζ (v j ) v j ) M j (v j ζ (v j )) v j 2 γ M γ j(v j ζ (v j )) γ j v v j v Together, Condtons (M) and (Q) mply that the functon z j (v j ) s strctly ncreasng. Because lm vj z j (v j ) = and lm vj + z j (v j ) = +, we then have a soluton to the equaton z j (ˆv j ) = p j exsts and s unque. Ths n turn mples exstence and unqueness of a soluton to the system of equatons gven by (S). That, n each equlbrum of the game, gven any vector of prces, all agents from sde whose v exceeds ˆv jon the platform, whereas all agents whose v s less than ˆv refran from jonng then follows from the fact that the thresholds (ˆv, ˆv 2 ) defned by the unque soluton to (S) concde wth the thresholds defned by the procedure of terated deleton of nterm strctly domnated strateges. Q.E.D. S.2 Platform Desgn and Informaton Management n Monopolstc Markets Suppose a sngle platform, platform B, serves both sdes of the market and assume nformaton s Gaussan as n Secton 5 n the man text. Adaptng the analyss n Secton 4. n the man text to the possblty that agents may be mperfectly nformed about ther own stand-alone valuatons (n whch case partcpaton decsons depend on estmated stand-alone valuatons), we then have that the proft-maxmzng prces, along wth the partcpaton thresholds they nduce, must satsfy the followng optmalty condtons for, j =, 2, j, p B = Φ ( ( β vv ) 2 β vφ ( φ ) + Ω 2 β vv Ω βj vv j β vv ) γ j + Ω 2 φ ( β vv ) ( φ + Ω 2 βj vv j Ω ) β vv [ ( β ) + γ Ω φ ( β vv ) Φ v v along wth [ ( ) Φ βj vv j s + v [ ( 2 + γ Φ + Ω 2 βj vv j Ω ) β vv = p B, j =, 2, j. Note that the formulas above specalze the ones n Proposton 3 n the man text to the Gaussan envronment under examnaton. We now compare the platform s ncentves to algn preferences across sdes, and/or to help agents predct partcpaton decsons on the opposte sde, to ther counterparts n the duopoly case. 3

4 Proposton S. In general, monopoly profts can ether ncrease or decrease wth a hgher algnment n preferences across the two sdes (a hgher ρ v ) and/or wth nformaton campagns that help agents predct partcpaton decsons on the opposte sde (hgher Ω ). When the monopolst serves exactly half of the market on each sde, (.e., ˆv = 0, =, 2, as n the duopoly case), such polces have only second-order effects n the monopoly case, whle they have frst-order effects n the duopoly case. Suppose the market s symmetrc across sdes (.e., β v = β v, γ = γ > 0, and ˆv = ˆv, =, 2). When the monopolst serves less than half the market on each sde, profts ncrease wth the algnment n preferences across sdes. They ncrease wth nformaton campagns that help agents predct partcpaton decsons on the opposte sde when preferences are algned (ρ v > 0), whereas they decrease wth such polces when preferences are msalgned (ρ v < 0). In contrast, when the monopolst serves more than half the market on each sde, more algnment s detrmental to profts; furthermore, nformaton campagns that help agents predct partcpaton decsons on the opposte sde decrease profts when preferences are algned, whereas they ncrease profts when preferences are msalgned. Importantly, the margnal effects of the above polces (platform desgn and nformaton management) on the monopolst s profts are always smaller than n the duopoly case. Proof of Proposton S. Observe that ˆΠ B = =,2, p B [ ( β ) Φ v ˆv, wth p B as defned above. Holdng β v fxed, =, 2, and usng the envelope theorem, we thus have that ˆΠ B Ω =,j=,2, j ( γ φ + Ω 2 βj vˆv j Ω ) ( β vˆv Ω β v + Ω 2 j ˆv j [ ( β ) β ) vˆv Φ v ˆv. Hence, n general, monopoly profts can ether ncrease or decrease wth Ω. Because Ω s ncreasng n ρ v, ths means that monopoly profts can ether ncrease or decrease wth a hgher algnment n preferences across the two sdes (a hgher ρ v ) and/or wth nformaton campagns that help agents predct partcpaton decsons on the opposte sde (hgher Ω ). Now suppose the monopolst serves exactly half of the market on each sde (.e., v = v 2 = 0, as n the duopoly case). Usng the envelope formula above, t s easy to see that ˆΠ B / Ω = 0. Hence, n ths case, margnal varatons n platform desgn and/or n nformaton management have only second-order effects on the monopolst profts. Next, suppose the market s perfectly symmetrc across the two sdes (.e., β v = β v, γ = γ > 0, and ˆv = ˆv, =, 2). It s easy to see that ( sgn ˆΠ ) B / Ω = sgn(ˆv). Hence, when the monopolst serves less than half the market on each sde (.e., ˆv > 0), profts ncrease wth the algnment n preferences across sdes, for a hgher ρ v mples a hgher Ω. They ncrease wth 4

5 nformaton campagns that help agents predct partcpaton decsons on the opposte sde (.e., wth Ω ) when preferences are algned, for, n ths case, a hgher Ω mples a hgher Ω; nstead, profts decrease wth such polces when preferences are msalgned, for, n ths case, a hgher Ω mples a lower Ω. It s mmedate to see that the above conclusons are reversed when the monopolst serves more than half of the market on each sde (.e., ˆv < 0). Fnally, consder the last statement n the proposton. Note that, rrespectve of whether the monopolst serves less or more than half of the market on each sde (.e., of whether v > 0), ( ˆΠ B Ω = (γ + γ 2 )φ + Ω 2 β v v Ω β ( ) [ β v) v v v Ω ( +Ω 2 Φ β v v ) ( ) < 2 (γ + γ 2 ) Ω +Ω 2 = Π Ω where Π / Ω s the margnal effect of a varaton n Ω on duopoly profts (n the case of a market that s symmetrc across sdes), as one can derve from Condton (20) n the man text. Note that the nequalty follows from the fact that, for any x 0, the functon ( ) z(x) φ + Ω 2 x Ωx x [ Φ (x) takes value less than /2. The above nequalty mples that the margnal effects of the above polces (platform desgn and nformaton management) on the monopolst s profts are always smaller than n the duopoly case, as clamed n the proposton. Q.E.D. S2. Opts-out and Multhomng In ths secton, we show that, under addtonal assumptons, the equlbrum prces characterzed n the baselne model (along wth the partcpaton decsons they nduce) contnue to reman equlbrum outcomes n a rcher envronment n whch agents can (a) opt out of the market, or (b) multhome by jonng both platforms. We start by consderng robustness to partal partcpaton. The reason why, n general, the equlbrum n the game wth compulsory partcpaton need not be an equlbrum n the game n whch agents can opt out of the market s the followng. Frst, when the platforms set the same prces as n the equlbrum of the game wth compulsory partcpaton, some agents may experence a negatve payoff under the cut-off strategy profle of the game wth compulsory partcpaton and hence prefer to opt out. Because the equlbrum prces n the game wth compulsory partcpaton are nvarant n the ntensty of the ndvdual stand-alone valuatons (formally, n s and s 2 ), on-path full partcpaton n the game wth voluntary partcpaton can always be guaranteed by assumng that the margnal agents equlbrum payoffs are postve, whch amounts to assumng that s and s 2 are suffcently hgh. When ths s the case, gven the equlbrum prces, no agent fnds t optmal to opt out, gven that, under the cut-off strategy profle any agent s payoff s at least as hgh as that of the margnal agents. 5

6 The above condton, however, does not suffce. In fact, platforms may have an ncentve to rase one of ther prces above the equlbrum levels of the game wth compulsory partcpaton f they expect that, by nducng some agents to opt out, ther demand wll fall less than that of the rval platform, relatve to the case n whch partcpaton s compulsory. For ths to be the case, t must be that the ntensty of the network effects s suffcently strong to preval over the drect effect comng from the stand-alone valuatons. The proof below shows that ths s never the case when s and s 2 are suffcently large. Consder arbtrary values ŝ and ŝ 2 and, from now on, assume that s > ŝ for =, 2. For any vector of prces p = (p A, pa 2, pb, pb 2 ), any =, 2, let v A+ (p) p B p A + γ, v A (p) mn { 2ŝ 2p A ; p B p A γ } v B (p) p B p A γ, v B+ (p) max { 2p B 2ŝ ; p B p A + γ } Note that, gven the prces p, rrespectve of what other agents do, any agent from sde wth standalone dfferental v > v A+ (p) prefers jonng platform B to jonng platform A, whereas any agent wth stand-alone dfferental v < v A (p) prefers jonng platform A to ether jonng platform B or optng out of the market. Lkewse, rrespectve of what other agents do, any agent from sde wth stand-alone dfferental v < v B (p) prefers jonng platform A to jonng platform B, whereas any agent from sde wth stand-alone dfferental v > v B+ (p) prefers jonng platform B to ether jonng platform A or optng out. and Now, for any p = (p A, pa 2, pb, pb 2 ), =, 2, let { p Π A A (p) = Q A (va+ (p)) f p A 0 p A QA (va (p)) f p A < 0 Π B (p) = p B p B [ [ Q B (vb Q B (vb+ (p)) (p)) f p B 0 f p B < 0. Note that Π k (p) are platform k s profts when all agents follow the ratonalzable strategy most advantageous to platform k. We assume that the followng condton holds, whch guarantees that devatons to arbtrarly large prces are never proftable. Condton (S-P). For any vector of equlbrum prces p = ( p A, pa 2, pb, pb 2 ) n the game n whch partcpaton to one of the two platforms s compulsory there exst strctly postve scalars R k( p) R ++, =, 2, k = A, B, wth R k( p) > pk such that the followng s true for k = A, B : Π k (p k, p k 2, p k, p k 2 ) Πk ( p) for all (p k, p k 2) s.t. p k > R k ( p), for some =, 2. =,2 where Π k ( p) are the equlbrum profts n the game n whch partcpaton to one of the two platforms s compulsory. 6

7 The condton says that askng for (or offerng) an extreme prce on one, or both, sdes of the market leads to profts lower than n the equlbrum n the game n whch partcpaton to one of the two platforms s compulsory, ether when revenues are nflated, and/or losses are reduced, accordng to the majorzaton assocated wth the defnton of the Π k functons above. Recall that, n the game n whch partcpaton to one of the two platforms s compulsory, for any vector of prces p = (p A, pa 2, pb, pb 2 ), the partcpaton thresholds ˆv (p) are gven by ˆv = p B p A + γ [2M j (ˆv j ˆv ), j =, 2, j so the majorzatons are small when the network effects are small. We then have the followng result: Proposton S2. Suppose Condton (S-P) holds and assume that the game n whch the agents can opt out by not jonng any platform admts an equlbrum. Let p = ( p A, pa 2, pb, pb 2 ) be equlbrum prces n the game n whch partcpaton to one of the two platforms s compulsory. There exst fnte scalars (s ( p)) =,2 such that, for any (s ) =,2 wth s > s ( p), =, 2, the followng s true: the game n whch all agents can opt out of the market admts an equlbrum n whch (a) platforms offer the same prces p = ( p A, pa 2, pb, pb 2 ) and all agents make the same partcpaton decsons as n the game n whch partcpaton to one of the two platforms s compulsory. Proof of Proposton S2. Let p = ( p A, pa 2, pb, pb 2 ) be equlbrum prces n the game n whch partcpaton to one of the two platforms s compulsory. Observe that p s ndependent of s and s 2. Lkewse, observe that, for any vector of prces p = (p A, pa 2, pb, pb 2 ), the partcpaton thresholds ˆv (p) n the game n whch partcpaton s compulsory are ndependent of s and s 2, =, 2. Next, observe that, gven any vector of prces p = (p A, pa 2, pb, pb 2 ), when all agents from sde j,, j =, 2, follow a cut-off strategy wth cut-off ˆv j (p) meanng that all agents from sde j wth stand-alone dfferental v j < ˆv j (p) jon platform A, whereas all agents wth v j > ˆv j (p) jon platform B the payoff that each agent l [0, from sde wth stand-alone dfferental v l obtans by jonng the platform for whch hs utlty s the hghest s at least as hgh as the payoff r (p; s ) s 2 ˆv (p) + γ M j (ˆv j (p) ˆv (p)) p A that the margnal agent from sde wth stand-alone dfferental equal to ˆv (p) obtans by jonng platform A (ths follows drectly from Condton (M) n the man text). That r (p; s ) s contnuous n (p; s ), and strctly ncreasng n s, wth lm s + r (p; s ) = +, n turn mples that there exst fnte scalars (s ( p)) =,2 wth s ( p) > ŝ, =, 2, such that, for any (s ) =,2 wth s > s ( p), =, 2, r (p; s ) 0 for any p such that p k Rk ( p), =, 2, k = A, B. Next pck any (s ) =,2 wth s > s ( p), =, 2, and consder the game n whch agents can opt out of the market by not jonng any platform. Observe that, by assumpton, such a game admts an The functons ˆv ( ) are contnuous n p, =, 2, and the functons M j(v j v ) are dfferentable, and hence contnuous, n each argument. 7

8 equlbrum. Ths n turn means that, for any vector of prces p = (p A, pa 2, pb, pb 2 ), one can construct a strategy profle for the agents such that each agent s strategy s a best response to the other agents strateges n the game that starts wth the observaton of the prces p. It s also easy to see that, for any vector of prces p such that p k Rk ( p), =, 2, k = A, B, the strategy profle n whch each agent l [0, from each sde =, 2 jons platform A for v ˆv (p) and jons platform B for v > ˆv (p), s a contnuaton equlbrum, exactly as n the game wth compulsory partcpaton. Now consder the followng strategy profle for the agents n the game n whch all agents can opt out of the market. For any p such that r (p; s ) 0, =, 2, all agents follow the same cut-off strategy as n the game wth compulsory partcpaton. For any p such that, nstead, r (p; s ) < 0, for some =, 2, take any collecton of strateges such that each agent s strategy s a best response to the other agents strateges, gven the prces p (agan, that at least one such strategy profle exsts follows from the fact that the game n whch agents can opt out s assumed to have at least one equlbrum). We now show that, when the agents follow the above specfed strategy profle, each platform fnds t optmal to offer the same prces ( p k, pk 2 ) t would have offered n the game wth compulsory partcpaton, when t expects the rval platform to also offer the prces ( p k ), k, k = A, B, k k., p k 2 To see ths, consder the problem faced by platform A (the problem faced by platform B s symmetrc and hence omtted). Suppose that platform B offers the equlbrum prces ( p B, pb 2 ). Clearly any devaton by platform A to a par of prces (p A, pa 2 ) such that, gven p = (pa, pa 2, pb, pb 2 ), r (p; s ) 0, =, 2, s unproftable, for t leads to the same partcpaton decsons (and hence the same profts) as n the game n whch partcpaton to one of the two platforms s compulsory. Thus consder a devaton to a par of prces (p A, pa 2 ) such that, gven p = (pa, pa 2, pb, pb 2 ), r (p; s ) < 0 for some {, 2}. Observe that ths mples that p A j > RA j ( p) for some j {, 2} possbly dfferent than. Condton (S-P) then mples that such devaton s unproftable for platform A, rrespectve of whch partcular contnuaton equlbrum the agents play followng the observaton of p = (p A, pa 2, pb, pb 2 ). Applyng the same arguments to platform B yelds the result. Q.E.D. Next, consder the possblty that agents multhome by choosng to jon both platforms. We assume that, by dong so, each agent l [0, from each sde =, 2 obtans a gross payoff equal to (2 κ )s + γ (q A j + µb j ), where µb j s the measure of agents from sde j who jon platform B wthout jonng platform A (to avod double countng), and where κ R s a scalar that parametrzes the degree of subaddtvty (for κ > 0) or superaddtvty (for κ < 0) between the two stand-alone valuatons. 2 We then have the followng result: Proposton S3. Consder the varant of the game n whch agents from each sde of the market can multhome, as descrbed above. For any vector of prces p = (p A, pa 2, pb, pb 2 ) such that 2 Note that (2 κ )s + γ (q A j + µ B j ) = v A + v B κ s + γ (q A j + µ B j ). 8

9 p A + p B γ + 2( κ )s, =, 2, there exsts a contnuaton equlbrum n whch each agent from each sde snglehomes. Conversely, such a contnuaton equlbrum fals to exst for any vector of prces for whch p A + p B < γ + 2( κ )s, for some {, 2}. Proof of Proposton S3. jonng platform A to jonng platform B f and only f Recall that each agent l [0, from each sde =, 2 prefers v l + γ E [ q B j q A j v l p B p A. (S4) The same agent prefers jonng platform A to multhomng f and only f 2 v l + ( κ )s + γ E [ µ B j v l p B 0. (S5) Note that Condton (S5) s mpled by Condton (S4) f and only f 2( κ )s + 2γ E [ µ B j v l γ E [ q B j q A j v l p A + p B. (S6) In any contnuaton equlbrum where all agents snglehome, qj B = µ B j = qj A. In ths case, the nequalty n (S6) becomes equvalent to γ + 2( κ )s p A + p B. The same concluson apples to those agents that prefer platform B to platform A. From the results n the man text, we know that the game where multhomng s not possble always admts a contnuaton equlbrum. We then conclude that, when p A + pb γ + 2( κ )s such a contnuaton equlbrum s also a contnuaton equlbrum n the game where agents can multhome. Conversely, when p A + p B < γ + 2( κ )s, there exsts no contnuaton equlbrum where all agents snglehome, for, f such equlbrum exsted, then t would satsfy q B j = µ B j = q A j. Invertng the nequaltes above, we would then have that some agent from sde {, 2} would necessarly prefer to multhome. Q.E.D. The condton n the proposton guarantees that any agent who expects all other agents to snglehome (accordng to the same threshold rule as n the game n whch multhomng s not possble) prefers to jon hs most preferred platform to multhomng. As the proposton makes clear, the condton s also necessary, n the sense that, when t s volated, then n any contnuaton equlbrum some agents necessarly multhome. The followng corollary s then an mmedate mplcaton of the above result: Corollary S. Suppose that platforms cannot set negatve prces. Let p = ( p A, pa 2, pb, pb 2 ) be equlbrum prces n the game n whch multhomng s not possble, and assume that p k γ + 2( κ )s, =, 2, k = A, B. In the game n whch multhomng s possble, there exsts an equlbrum n whch platforms offer the same prces p and all agents make the same partcpaton decsons as n the game n whch agents can only snglehome. The result n Corollary (S) appears consstent wth the fndng n Armstrong and Wrght (2007) that strong product dfferentaton on both sdes of the market mples that agents have no ncentve 9

10 to multhome when prces are restrcted to be non-negatve (As argued n that paper, and n other contexts as well, the assumpton that prces must be non-negatve can be justfed by the fact that negatve prces can create moral hazard and adverse selecton problems). Together, the results n Proposton (S) and Corollary (S) mply that, when (a) the standalone valuatons of the margnal agents are nether too hgh nor too low (ntermedate s ), (b) the two platforms are suffcently dfferentated on both sdes of the market (so that the equlbrum prces are suffcently hgh) and (c) prces are restrcted to be postve, then the equlbrum prces and partcpaton decsons n the baselne game are also equlbrum allocatons n the more general game where agents can multhome and opt out of the market. S3. Dynamcs under Flexble Prces In ths secton, we study duopoly prcng n the same two-perod dynamc economy examned n Secton 5.3 n the man text, but assumng the platforms change prces at the same frequency at whch agents revse ther belefs about the dstrbuton of stand-alone valuatons n the cross-secton of the populaton. We contnue to denote the perod- prces by p k, and then denote the perod-2 prces n each state θ by p k (θ), k = A, B, =, 2. Because there are no swtchng costs, the perod- prces naturally concde wth those n the statc benchmark n the man text. Under these prces, the perod- partcpaton rates expected by the two platforms are Q A = E θ [Λ θ (ˆv ) and Q B = E θ [Λ θ (ˆv ), where (ˆv, ˆv 2 ) are the unque solutons to the system of equatons gven by ˆv 2γ M j (ˆv j ˆv ) + γ = p B p A, j =, 2, j and where the partcpaton rates reflect the assumpton that the two platforms share a common pror, as n Secton 5.3 n the man text. [Recall that Λ θ s the true cumulatve dstrbuton of stand-alone dfferentals on sde n state θ, wth densty λ θ. Next, consder the choce of the perod-2 prces. The observaton of the perod- partcpaton rates reveals to all agents and to the platforms the true state θ. Parallelng the analyss n the statc benchmark n the man text, but observng that, under complete nformaton, M j (v j v ) = Λ θ j (v j) all v, v j R, we then have that the perod-2 equlbrum prces, p A (θ), and the assocated partcpaton rates, q A (θ), must satsfy p A (θ) = Λθ (ˆvθ ) λ θ (ˆvθ ) 2γ jλ θ j(ˆv θ j ), q A (θ) = Λ θ (ˆv θ ),, j =, 2, j, p B (θ) = Λθ (ˆvθ ) ( ) λ θ (ˆvθ ) 2γ j Λ θ j(ˆv j θ ), q B (θ) = Λ θ (ˆv θ ),, j =, 2, j, wth the equlbrum thresholds satsfyng ˆv θ = p B (θ) p A (θ) γ ( q B j (θ) q A j (θ) ),, j =, 2, j. 0

11 Clearly, the perod-2 dfferental n the prces set by the platforms, p B (θ) p A (θ), now vares wth the state, reflectng the property that dfferent states feature a dfferent degree of relatve apprecaton for the two platforms products. Our prmary nterest s n how the perod- prces compare to the average perod-2 prces. In what follows, we focus on the case of symmetrc competton. Defnton S. We say that competton s ex-ante symmetrc n perod two f, based on the perod- pror, (a) both platforms expect to set equal prces and enjoy equal partcpaton rates on each sde n perod two (that s, E[p A (θ) = E[pB (θ) and E[qA (θ) = E[qB (θ) = /2, =, 2). When belefs are consstent wth a common pror, as assumed here, the defnton amounts to assumng that the pror dstrbuton over the actual cross-sectonal dstrbutons of stand-alone valuatons s symmetrc. Note that ths condton s satsfed n the Gaussan model. It s easy to verfy that, when perod-2 competton s ex-ante symmetrc, the expected perod-2 prces must satsfy the followng condtons [ Λ E[p A θ (θ) = E (ˆv θ) [ Λ θ λ θ (ˆvθ ) γ j = E (ˆv θ) λ θ (ˆvθ ) γ j = E[p B (θ), j =, 2, j. Comparng the expected perod-2 prces to ther perod- counterparts (see Corollary n the man text), we have that ( [ Λ E[p k (θ) p k θ = E (ˆv θ) λ θ (ˆvθ ) M j (0 0) v + γ j E[λ θ (0) ) M j (0 0) 2E [ λ θ (0) + γ v E[λ θ (0) where we used the fact that, under a common pror, ψ (0) = E [ λ θ (0). There are three effects that contrbute to the dfference between the prces that, on average, the platforms set n perod two under complete nformaton and the prces that they set n perod one under dspersed nformaton. The frst effect s the change n the (nverse) sem-elastcty of the stand-alone demand functons. Ths effect s captured by the frst round bracket n (S7) and s present also n the absence of network effects. Ths effect orgnates from the fact that the platforms learn the exact dstrbuton of the stand-alone valuatons n the second perod. The second effect s due to the fact that, under dspersed nformaton, the belefs of the margnal agent on each sde dffer from the platform s belefs. As dscussed n the man text, ths effect s only present n the frst perod, when nformaton s dspersed, and s captured by the second round bracket n (S7). The thrd effect s due to the fact that the adjustment to the sde-j s prce necessary to mantan the sde-j s partcpaton constant when the sde- s partcpaton changes depends on whether or not nformaton s dspersed. Ths effect s captured by the last round bracket n (S7). (S7)

12 As dscussed n the man text, the sgn of the second effect s negatve when preferences are algned, thus contrbutng to hgher prces n the frst than n the second perod. The sgn of the other two terms s n general ambguous. However, there are nterestng markets n whch all three effects contrbute to lower prces n the second perod, as we show next. Defnton S2. When competton s symmetrc, nformaton softens competton on sde f E [ Λ θ (ˆv θ) λ θ (ˆvθ ) whereas t strengthens t when the nequalty s reversed. 2E [ λ θ (0) Defnton S3. The perod- margnal agents belefs are more concentrated than the platforms belefs around the mean f for all, j =, 2, j, M j (0 0) v [ > E λ θ (0). Defnton S3 says that the densty of agents from sde who are expected to be ndfferent between the two platforms products by an agent from sde j who s hmself ndfferent, M j (0 0) / v, s larger than the densty expected by the platforms ex-ante. When competton s symmetrc n perod one, E[v l 0 = 0. In ths case, the condton says that agents who are ndfferent have belefs that are more concentrated around the mean than the platforms, a property that appears natural when the pror s symmetrc. In fact, the property always holds under a Gaussan nformaton structure. More generally the property holds under symmetrc competton when platforms belefs are a smple mean-preservng spread (SMPS) of the margnal agents belefs. 3 We then have the followng result (the proof follows drectly from the arguments above): Proposton S4. Suppose that competton s symmetrc n perod one (as defned n the man text) and s ex-ante symmetrc n perod two (n the sense of Defnton (S)). Further assume that the margnal agents perod- belefs are more concentrated than the platforms belefs around the mean (n the sense of Defnton (S3)). The expected perod-2 prces on sde are larger than ther perod- counterparts f () nformaton softens competton (n the sense of Defnton (S2)) and () ether preferences are msalgned or γ /γ j s small. Conversely, the expected perod-2 prces on sde are lower than ther perod- counterparts f () nformaton strengthens competton and () preferences are algned and γ /γ j s large. The dynamcs of expected prces thus combne the dynamcs of the nverse sem-elastctes of the stand-alone valuatons wth the dynamcs of the slopes of the nverse resdual demands that orgnate 3 See Damond and Stgltz (974). Under SMPS, M j (v 0) crosses E [ Λ θ (v) only once and from below. Gven that, under symmetrc competton, M j (0 0) = /2 = E [ Λ θ (0), the slope of M j (v 0) at zero, M j (0 0) / v, must be hgher than the slope of E [ Λ θ (v) at v = 0 whch s gven by ψ (0) = E [ λ θ (0). 2

13 from the ablty of the two sdes to predct the partcpaton decsons on the other sde. Whle, n general, prces may ether ncrease or decrease over tme, the proposton dentfes specal cases n whch the dynamcs of the averages prces can be sgned. S4. Wthn-Sde Network Effects Consder a market n whch agents on one or both sdes care not only about the number of agents jonng the platform from the opposte sde but also about the number of agents jonng from ther own sde. For example, advertsers may compete wth other advertsers for vewers attenton; when ths s the case, the payoff that an advertser derves from jonng a platform decreases wth the number of advertsers who also jon, whch amounts to negatve wthn-sde network effects. When, nstead, the platform s a frm producng a new smart-phone operatng system, end-users may beneft not only from a hgh adopton rate by developers from the opposte sde of the market, but also from other end-users from the same sde adoptng the same technology, whch amounts to postve wthn-sde network effects. Such possbltes can be captured by assumng that ndvdual payoffs are gven by U k l = vk l + γ q k j + ξ q k p k,, j =, 2, j, k = A, B, where the new parameter ξ R controls for the ntensty of the wthn-sde network effects on sde =, 2, and where all other terms are as n the baselne model. Now let M (ˆv v) denote the measure of agents from sde =, 2 wth dfferental n standalone valuatons smaller than or equal to ˆv, as expected by an agent from sde wth dfferental n stand-alone valuatons equal to v and then let M (v) M (v v). Below, we show that, under condtons analogous to Condtons (M) and (Q) n the baselne model, for any vector of prces there exst a unque contnuaton equlbrum n threshold strateges. Such contnuaton equlbrum s the unque contnuaton equlbrum when the wthn-sde network effects are non-negatve on both sdes, but not necessarly when they are negatve on one or both sdes. To be consstent wth the analyss n the rest of the paper, n the dscusson below, we assume that, n case there are multple contnuaton equlbra, the selected one s the one n threshold strateges. Condton (M-C). For any, j =, 2, j, any v, v 2, v l R, 2γ M j (v j v l ) v l 2ξ M (v v l ) v l > 0. Condton (Q-C). For any, j =, 2, j, any v j R, γ γ 2 < v 2γ M j (v j v ) 2ξ M (v ) s strctly ncreasng n v. Furthermore, for any v, v 2 R, [ 2 γ M 2(v 2 v ) dm v ξ (v ) dv [ 2 γ 2 M 2(v v 2 ) dm v 2 ξ 2 (v 2 ) 2 dv M 2 (v v 2 ) v M 2 (v 2 v ) v 2. 3

14 Parallelng the analyss n the baselne model, we then have the followng result: Proposton S5 Suppose Condtons (M-C) and (Q-C) hold, and competton s symmetrc, as defned n the baselne model. Then there exst scalars ψ (0), =, 2, such that belefs must satsfy the followng condtons: (a) Q k (0) = /2, and (b) dqk (0)/dv = ψ (0), k = A, B, =, 2. Furthermore, equlbrum prces are gven by M j (0 0) p k = 2ψ (0) γ j v ψ (0) k = A, B, =, 2. γ M j (0 0) v ψ (0) [ dm (0) dv ξ ψ (0) Proof of Proposton S5. The proof s n three steps. Step shows that, under condtons (M- C) and (Q-C), there exsts a unque contnuaton equlbrum n threshold strateges, for all prces. Step 2 then shows that, when platforms expect the contnuaton equlbrum to be n threshold strateges (whch, as explaned above, s always the case when ξ 0, =, 2, for, n ths case, the contnuaton equlbrum s unque), then the equlbrum prces must satsfy optmalty condtons smlar to those n the man text, but augmented by a new term. Fnally, Step 3 shows how the aforementoned optmalty condtons yeld the formulas n the proposton when competton s symmetrc. Step. We start wth the followng lemma. Lemma S2. Suppose Condtons (M-C) and (Q-C) hold. Then, for any vector of prces, there exst a unque contnuaton equlbrum n threshold strateges. Proof of Lemma S2. The proof parallels the one n the baselne model wthout wthn-sde network effects. Gven the platforms prces, each agent l [0, from each sde =, 2, chooses platform B f v l + γ E[q B j q A j v l + ξ E[q B q A v l > p B p A and platform A f the above nequalty s reversed. When Condton (Q-C) holds, for any ˆv j R, the gross payoff dfferental ˆv + γ + ξ 2γ M j (ˆv j ˆv ) 2ξ M (ˆv ) that an agent from sde =, 2 wth dfferental n stand-alone valuatons equal to ˆv obtans from jonng platform B relatve to jonng platform A, when he expects (a) all agents from sde j to jon platform B when v j > ˆv j and platform A when v j < ˆv j and (b) all agents from hs own sde to jon platform B when v ˆv and platform A when v < ˆv s strctly ncreasng n ˆv. From the ntermedate and mplct functon theorems, ths means that, gven the prces p = (p A, pa 2, pb, pb 2 ), for any ˆv j R, there exsts a unque soluton ˆv = ϱ (ˆv j ) to the equaton ˆv + γ + ξ 2γ M j (ˆv j ˆv ) 2ξ M (ˆv ) = p B p A, (S8) 4

15 wth ϱ (ˆv j ) satsfyng M 2γ j (ˆv j ϱ (ˆv j )) ϱ v (ˆv j ) = j. (S9) M 2γ j (ˆv j ϱ (ˆv j )) dm v 2ξ (ϱ (ˆv j )) dv Note that the denomnator n (S9) s strctly postve under Condton (M-C). Next observe that, under Condton (M-C), any agent from sde wth stand-alone dfferental v < ϱ (ˆv j ) strctly prefers jonng platform A to jonng platform B f he expects (a) all agents from sde j to follow a threshold strategy wth cut-off equal to ˆv j and (b) all agents from hs own sde to follow a threshold strategy wth cut-off equal to ϱ (ˆv j ). Lkewse, any agent from sde wth the same expectatons as above but wth stand-alone dfferental v > ϱ (ˆv j ) prefers jonng platform B to jonng platform A. Next, let L j (ˆv j ) = ˆv j + γ j + ξ j 2γ j M j (ϱ (ˆv j ) ˆv j ) 2ξ j M j (ˆv j ) denote the gross payoff dfferental that an agent from sde j =, 2 derves from jonng platform B relatve to jonng platform A when he expects (a) all agents from sde j to jon platform B when v ϱ (ˆv j ) and platform A when v < ϱ (ˆv j ) and (b) all agents from hs own sde j to jon platform B when v j ˆv j and platform A when v j < ˆv j. The functon L j (ˆv j ) s dfferentable wth dervatve equal to { L Mj (ϱ (v j ) v j ) j(ˆv j ) = 2γ j ϱ v (ˆv j ) + M } j (ϱ (ˆv j ) ˆv j ) dm j (ˆv j ) 2ξ j v j dv = 4γ M j (ˆv j ϱ (ˆv j )) M j (ϱ (v j ) v j ) γ j v j v M 2γ j (ˆv j ϱ (ˆv j )) dm v 2ξ (ϱ (ˆv j )) dv M j (ϱ (ˆv j ) ˆv j ) dm j (ˆv j ) 2γ j 2ξ j. v j dv Together, Condtons (M-C) and (Q-C) mply that the functon L j (ˆv j ) s strctly ncreasng. Because lm vj L j (ˆv j ) = and lm vj + L j (ˆv j ) = +, we then have a soluton to the equaton L j (ˆv j ) = p B j p A j exsts and s unque. Ths n turn mples that there exsts one and only one soluton to the system of equatons gven by (S8). Step 2. Now assume agents play threshold strateges (as explaned above, ths s always the case when ξ 0, =, 2). In ths case, the demands expected by the two platforms contnue to be gven by Q A (ˆv ) and Q B (ˆv ) but wth the thresholds now solvng the ndfference condtons ˆv + γ + ξ 2γ M j (ˆv j ˆv ) 2ξ M (ˆv ) = p B p A. (S0) Parallelng the analyss n the baselne model, now suppose that platform B ams at gettng on board Q agents from sde one and Q 2 agents from sde two. Gven ts belefs, the platform has to set prces equal to p B = p A + V B (Q ) + γ + ξ 2γ M j ( V B j (Q j ) V B (Q ) ) ( 2ξ M V B (Q ) ). Ths means that the slopes of the nverse (resdual) demand curves are now gven by ( ) p B = dv B (Q ) M j Vj B(Q j) V B ( (Q ) dv B (Q ) dm V B 2γ 2ξ (Q ) ) dv B (Q ). Q dq v dq dv dq 5

16 Note that the same condtons that guarantee exstence and unqueness of monotone contnuaton equlbra also guarantee that the above demand curves slope downwards, even n the presence of wthn-sde network effects. Expressng each platform s profts as a functon of the partcpaton thresholds, and then fxng the prces offered by the rval frm and dfferentatng profts wth respect to the partcpaton thresholds, we have that the proft-maxmzng prces, expressed as a functon of the partcpaton thresholds they nduce, must satsfy the followng condton p k = Q k (ˆv ) dq k (ˆv )/dv 2γ M j(ˆv j ˆv ) Q k (ˆv ) v dq k (ˆv )/dv 2γ j M j (ˆv ˆv j ) v Q k j (ˆv j) dq k (ˆv )/dv, 2ξ dm (ˆv ) dv Q k (ˆv ) (S), j =, 2, j, k = A, B, where the partcpaton thresholds ˆv and ˆv 2 are mplctly defned by the system of equatons gven by (S0), =, 2. Step 3. The fnal step conssts n observng that, when competton s symmetrc, the partcpaton thresholds are gven by ˆv = 0, =, 2. Furthermore, because the two platforms expect the same partcpaton rates, Ψ k (0) = /2, k = A, B. From (S), we then have that the equlbrum prces must satsfy p A = { ψ A(0) p B = ψ B (0) M j (0 0) M j (0 0) M (0) γ j ξ v v v 2 γ { 2 γ M j (0 0) M j (0 0) M (0) γ j ξ v v v Clearly, the terms n curly brackets cannot be equal to zero, for otherwse the platforms prces would be equal to zero and platforms would have a proftable devatons. For the platforms prces to concde, t must then be that ψ B (0) = ψb (0) = ψ (0), =, 2. The above results mply that, when competton s symmetrc, the equlbrum prces must satsfy the formulas n the proposton. Q.E.D. The prcng formulas n Proposton S5 are qualtatvely smlar to the one n the baselne model. The only dfference s the last term, whch captures the mpact on the sde- prce of a margnal varaton n the sde- wthn-sde network effects orgnatng from a hgher partcpaton by sde. Interestngly, the mpact of ths effect on the sde- prce combnes the drect effect of a hgher sde- partcpaton wth the varaton n the belefs of the sde- margnal agent about hs own sde s partcpaton. To see ths notce that dm (ˆv ) dv = M (ˆv v ) ˆv + v =ˆv } } M (ˆv v ) v v =ˆv (S2) The frst term on the rght hand sde of (S2) s the drect effect of ncreasng the sde- partcpaton holdng the belefs of the sde- margnal agent fxed. The second term s the ndrect effect of varyng the belefs of the sde- margnal agent for gven partcpaton threshold, and s negatve 6

17 when preferences are algned wthn sde. Clearly, ths second effect s present only under dspersed nformaton. In fact, under complete nformaton, dm (ˆv ) dv = λ θ (ˆv ) n whch case the equlbrum prces become p k = 2λ θ (0) γ j ξ,, j =, 2, j, k = A, B. Under complete nformaton, whether wthn-sde network effects contrbute to hgher or lower equlbrum prces then depends only on the sgn of the wthn-sde network effects. Under dspersed nformaton, nstead, whether wthn-sde network effects contrbute to hgher or lower equlbrum prces thus depends not only on the sgn of the wthn-sde network effects ξ (as under complete nformaton), but also on whether the drect or the ndrect effect domnates n the margnal agents belefs about the dstrbuton of valuatons on ther own sde. In partcular, under dspersed nformaton, wthn-sde network effects contrbute to hgher equlbrum prces when ether (a) the wthn-sde network externaltes are negatve (.e., ξ < 0) and the drect effects domnate the ndrect effects n the margnal agents belefs about the dstrbuton of valuatons on ther own sde (.e., dm (0) /dv > 0) or (b) the wthn-sde network externaltes are postve (.e., ξ > 0), preferences are algned wthn sdes (.e., M (ˆv v ) / v < 0), and the ndrect effects domnate the drect effects n the margnal agents belefs (.e., dm (0) /dv < 0). On the contrary, wthnsde network effects contrbute to lower equlbrum prces when ether (c) the wthn-sde network externaltes are postve (.e., ξ > 0) and the drect effects n the margnal agents belefs domnate the ndrect effects (.e., dm (0) /dv > 0), or (d) the wthn-sde network externaltes are negatve (.e., ξ < 0), preferences are algned wthn sde (.e., M (ˆv v ) / v < 0) and the ndrect effects domnate the drect effects n the margnal agents belefs (.e., dm (0) /dv < 0). To see why ths s the case, suppose that valuatons are drawn from a common pror and that nformaton s Gaussan, as n the prevous secton. In ths case, M (v) = M (v v) s naturally ncreasng n v. Postve wthn-sde network effects then contrbute to flatter nverse demands whch n turn contrbute to lower equlbrum prces. 4 As the dscusson above clarfes, ths concluson extends to more general nformaton structures nsofar as M (v) remans ncreasng n v (meanng that those agents who are most enthusastc about a platform s product are also those who expect most agents from ther own sde to have an apprecaton lower than ther own). When ths property fals to hold, platforms may, nstead, rase ther equlbrum prces (relatve to the benchmark model), despte wthn-sde network effects beng postve on both sdes. We conclude by consderng the effect of platform desgn and nformaton polces on profts, welfare, and consumer surplus n the presence of wthn-sde network effects. consder agan the Gaussan specfcaton ntroduced above. To ths purpose, As we show n Corollary S2 below, 4 The opposte s true when wthn-sde network effects are negatve, as n the case of congeston. 7

18 equlbrum prces are then gven by p = 2 β vφ(0) γ [ j + Ω 2 + γ Ω ξ + Ω 2 Ω where Ω ρ v/ (ρ v) 2 captures the ablty of the sde- agents to predct partcpaton decsons by other agents from ther own sde, wth ρ v cov(v, v )/var(v ) denotng the coeffcent of lnear correlaton between the valuatons of any par of agents from sde. 5 Equlbrum profts and equlbrum welfare can then be expressed as Π = Π 2 ξ. [ [ + Ω 2 Ω 2 ξ 2 + Ω 2 2 Ω 2 and W = W + 2ξ Pr(v 0, v 0) + 2ξ 2 Pr(v 2 0, v 2 0), where Π and W are, respectvely, equlbrum profts and equlbrum welfare n the absence of wthn-sde network effects. We then have the followng result: Corollary S2. Suppose nformaton s Gaussan, as n Secton 5 n the man text. When wthnsde network effects are postve, equlbrum prces are lower than n the benchmark wthout wthnsde network effects. Furthermore, polces that algn preferences wthn sdes (formally captured by ncreases n Ω for gven Ω and β v, =, 2) ncrease profts and welfare but reduce consumer surplus. The opposte conclusons hold when wthn-sde network effects are negatve. Proof of Corollary S2. When nformaton s Gaussan, an agent wth estmated stand alone dfferental equal to v expects any other agent from hs own sde to have an estmated stand alone dfferental v normally dstrbuted wth mean E[v v = cov(v, v ) var(v ) v = ρ vv and varance where var(v v ) = var(v )( (ρ v) 2 ) = (ρv )2 β v ρ v cov(v, v ) var(v )var(v ) = cov(v, v ) = βη var(v ) β η + βθ wth (β θ ) = δ 2 + ( δ ) 2 + 2δ ( δ )ρ ω. Lettng Ω ρ v (ρ v ) 2 5 Formally speakng, the ablty of the sde- agents to predct partcpaton decsons by agents from ther own sde s captured by Ω. However, becauseω > 0, we do not need to dstngush between Ω and Ω. 8

19 we then have that and hence M (v) = P r(v v v = v) = Φ dm (0) dv ( ) β v ( ρ ( ) v)v β = Φ v (ρ v ) 2 [ v + Ω 2 Ω = β [ vφ(0) + Ω 2 Ω. Replacng the latter nto the formula for the equlbrum prces and usng the fact that ψ (0) = β v φ(0), M j (0 0) / v = Ω β vφ(0), and M j (0 0) / v = + Ω 2 β v φ(0), we have that the equlbrum prces are gven by p = 2 β vφ(0) γ [ j + Ω 2 + γ Ω ξ + Ω 2 Ω Each platform equlbrum profts are then equal to [ Π = Π + 2 ξ Ω [ + Ω ξ 2 Ω 2 + Ω 2 2. where [ Π = + + 4φ(0) β v β v 2 2 (γ + γ 2 ) [Ω + Ω 2 are the equlbrum profts n the benchmark wthout wthn-sde network effects. Lkewse, steps smlar to those that lead to the formula for equlbrum welfare n the absence of wthn-sde network effects mply that equlbrum welfare n the presence of wthn-sde network effects s equal to W = W + 2ξ Pr(v 0, v 0) + 2ξ 2 Pr(v 2 0, v 2 0) where W = =,2 E [ V A l I(v l 0) + V B l I(v l > 0) +2(γ + γ 2 )Pr(v 0, v 2 0) s equlbrum welfare n the absence of wthn-sde network effects and Pr(v 0, v 0) = 4 + φ2 (0)arcsn(ρ v), =, 2. Consder now the margnal effect on profts, total welfare, and consumer surplus of the same polces dscussed n the baselne model. The effect of the polces dscussed n the Corollary on profts, welfare, and consumer surplus can then be read from the above formulas along wth the formulas that relate Ω and Ω to the prmtve parameters. Q.E.D. 9

20 The effect of polces algnng preferences wthn sdes s thus smlar to the effect of the polces examned n the prevous secton. The only dfference s that preferences are always algned wthn sdes. What s perhaps more nterestng s the effect of polces that affect both the agents ablty to predct partcpaton decsons from agents on the opposte sde as well as from agents on ther own sde. Consder, for example, the promoton of forums, blogs, and other nformaton polces that shft the agents attenton towards platform dmensons of nterest prmarly to agents from the opposte sde, possbly at the expenses of dmensons of nterest prmarly to agents from ther own sde. An example of such polcy s the promoton of blogs that attract agents from both sdes as opposed to specalzed blogs that target agents only from one sde. When wthn-sde network effects are postve and preferences are algned across sdes (ξ > 0, =, 2, and ρ v > 0) such polces ncrease the agents ablty to predct partcpaton decsons from the opposte sde but reduce ther ablty to predct partcpaton decsons from agents on ther own sde (formally, Ω ncreases but Ω decreases). In ths case, wthn-sde network externaltes reduce the postve effect of such polces on profts and welfare as well as the negatve effect of such polces on consumer surplus. When, nstead, wthn-sde network effects are postve but preferences are msalgned across sdes (ξ > 0, =, 2, and ρ v < 0), such polces always ncrease the agents ablty to predct partcpaton decsons by agents on ther own sde, but have ambguous effects on the agents ablty to predct partcpaton decsons from the opposte sde. S6. Gaussan economes The followng famly of Gaussan economes s an example of a class of economes satsfyng the varous condtons n Secton 5. n the man text. The aggregate state θ = (θ, θ 2 ) s drawn from a b-varate Gaussan dstrbuton wth zero mean and varance-covarance matrx ( β θ ) ρ θ ρ θ β θ β θ 2 β θ β θ 2 Each ndvdual true stand-alone dfferental s gven by V l = k [θ + ε l, whereas each agent s nformaton s summarzed n the un-dmensonal statstcs x l = δ θ + ( δ )θ j + η l, where δ and δ 2 are postve scalars, and where each par (ε l, η l ) s drawn ndependently from θ and ndependently across all agents from a b-varate Gaussan dstrbuton wth mean (0, 0) and varance-covarance matrx (βε ) ρ β ε β η ( β θ 2 ) ρ β ε β η (β η ) wth the parameter ρ 0 denotng the coeffcent of lnear correlaton between ε l and η l.. 20

21 Next observe that, n ths economy, each ndvdual estmated stand-alone dfferental s equal to v l E [V l x l = κ x l, wth κ cov[v l, x l var[x l ( ) δ β θ + ( δ ) ρ θ + ρ β θ = k β2 θ β η βε ( ) ( ) δ 2 β θ + ( δ ) 2 βj θ + (β η ) + 2δ ( δ ) ρ θ β θ β2 θ () To see how β v and Ω depend on the prmtve parameters, then let ω δ θ + ( δ )θ j, =, 2, and observe that, ex-ante, the par ω (ω, ω 2 ) s drawn from a bvarate Normal dstrbuton wth mean (0, 0) and varance-covarance matrx Σ ω = (βω ) ρ ω β ω β2 ω ρω β ω β2 ω (β ω 2 ) where (β ω ) = δ 2 ( ) ( ) β θ + ( δ ) 2 βj θ ρ θ + 2δ ( δ ) β θβθ 2 (2) and ρ ω β ω β ω 2 ( ) ( = cov(ω, ω 2 ) = δ ( δ j ) β θ + ( δ )δ j β θ j ) + [δ δ j + ( δ )( δ j ) ρ θ β θ βθ 2. (3) From an ex-ante perspectve, the dstrbuton of estmated stand-alone dfferentals on each sde =, 2 s thus Normal wth mean 0 and varance-covarance matrx (βv ) ρ v β v β2 v ρv β v β2 v (β v 2 ) where and ρ v = cov(x, x 2 ) var(x )var(x 2 ) = (β v ) = κ 2 [ (β ω ) + (β η ) ρω β ω β2 ω ((β ω ) + (β η ) ) ((β2 ω) + (β η 2 ) ) The coeffcent of mutual forecastblty can then be expressed as a functon of the prmtve parameters as follows: Ω ρ v ρ 2 v = ρ ω β ω β ω 2 ( ((β ω) + (β η ) ) ((β2 ω) + (β η 2 ) ) ρ ω β ω β2 ω ) 2 (4) 2

22 Fnally, observe that the normalzaton n the man text s obtaned by lettng the varous parameters be such that κ =. The above specfcaton has the advantage of beng tractable, whle at the same tme rch enough to capture a varety of stuatons. The pure common-value case where all agents from sde have dentcal stand-alone valuatons for the two platforms but dfferent nformaton about the stand-alone dfferental s captured as the lmt n whch β ε, n whch case, almost surely, V l = k θ all l [0,. The parameter s then a measure of dfferentaton between the two platforms, as perceved by sde. Lettng β θ β θ = βθ 2 and ρ θ = whle allowng β η βη 2 then permts us to capture stuatons where the qualty dfferental between the two platforms s the same on both sdes, but where one sde may have superor nformaton than the other. Lettng k = 0 n turn permts one to capture stuatons where agents on sde do not care about the ntrnsc qualty dfferental between the two platforms but nonetheless possess nformaton about the dstrbuton of preferences on the opposte sde (as n the case of advertsers who choose whch meda platform to place ads on entrely on the bass of ther expectaton of the platform s ablty to attract readers and vewers from the opposte sde). More generally, allowng the correlaton coeffcent ρ θ to be dfferent from one permts one to capture stuatons where the qualty dfferental between the two platforms dffers across the two sdes (ncludng stuatons where t s potentally negatvely correlated), as well as stuatons where one sde may be able to perfectly predct the behavor of each agent from that sde but not the behavor of agents from the opposte sde (whch corresponds to the lmt n whch β η ). The model can also capture stuatons n whch dfferent users from the same sde have dfferent preferences for the two platforms. Ths amounts to lettng the varance of ε l be strctly postve or, equvalently, β ε <. Dependng on the degree of correlaton ρ between ε l and η l, agents may then possess more or less accurate nformaton about ther own stand-alone valuatons. For example, the case where each agent perfectly knows hs own valuatons but s mperfectly nformed about the valuatons of other agents (from ether sde) s captured as the lmt n whch δ and ρ. The extreme case of ndependent prvate values then corresponds to the lmt n whch β θ and β ε <. Fnally, the scalars δ control for the nature of the agents nformaton, that s, the extent to whch ther nformaton correlates wth the nformaton possessed by the agents from the opposte sde, for gven dstrbuton of true stand-alone valuatons. References [ Armstrong M. and J. Wrght (2007): Two-Sded Markets, Compettve Bottlenecks and Exclusve contracts, Economc Theory 32,