Strategic Intellectual Property Sharing: Competition on an Open Technology Platform Under Network Effects

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1 Online Appendix for ISR Manuscript Strategic Intellectual Property Sharing: Competition on an Open Technology Platform Under Network Effects Marius F. Niculescu, D. J. Wu, and Lizhen Xu Scheller College of Business, Georgia Institute of Technology, Atlanta, GA {marius.niculescu, dj.wu, A Proofs of Main and Supporting Results A. Proof of Monopolistic Pricing ( 4. Proof. When the incumbent is a monopoly in the market, the boundary consumer who is indifferent between purchasing from the incumbent and not purchasing at all, θ, can be formulated as ( θ + γn p = 0. Therefore, θ = p γn. In a rational expectations euilibrium (REE, consumers rationally anticipate the network size, so N and θ need to be solved simultaneously. Suppose 0 < θ <, then N = N = θ. As a result, θ canbesolvedas θ = p. As we can easily check, p > 0 if and only if p >γ; p < if and only if p <. Therefore, we can summarize the demand function for the monopolistic incumbent as follows., p γ; N (p = p, γ γ,sop =ˆp = ;ifγ,ˆp γ, sop takes the corner solution, i.e., p = γ.

2 A. Proof of Proposition Proof. To derive the pricing euilibrium, we need to examine three stages of strategic decisions (i.e., the last three stages of the whole model. Along the line of backward induction, we derive the pricing euilibrium in the following three steps: ( first, determine the demands for both firms products as functions of the prices, N i (p,p (i =, ; ( next, determine the incumbent s best response in pricing as a function of the entrant s price, p (p ; (3 finally, determine the optimal price of the entrant, p. Throughout the analysis for this proposition, we take the entrant s product uality and the strength of network effects γ as given, and discuss the euilibrium outcomes when these parameters take different values. ( We first derive the demand functions N i (p,p (i =, given p and p. Consumers choose among three options: purchasing from the incumbent, purchasing from the entrant, and not purchasing at all. The boundary consumer who is indifferent between purchasing from the incumbent and purchasing from the entrant, θ, can be derived by solving θ +γn p = ( θ + γn p, which yields θ = p p γn, wheren = N + N. Similarly, the boundary consumer indifferent between purchasing from the entrant and not purchasing at all is θ = p γn; the boundary consumer indifferent between purchasing from the incumbent and not purchasing at all is θ = p γn. As we can easily show, all consumers with θ> θ prefer purchasing from the incumbent to purchasing from the entrant, and vice versa; likewise, all consumers with θ< θ (or θ prefer purchasing nothing to purchasing from the entrant (or incumbent, and vice versa. In REE, consumers form rational expectations about the total network size N when making purchase decisions. Therefore, { θ, θ, θ } need to be simultaneously solved with N. Note that {N,N } and hence N all depend on the relative magnitude of θ, θ, θ, and the bounds of θ s range [0, ]. Comparing these relative magnitudes and solving { θ, θ, θ,n} simultaneously lead to different demand cases under different parameter conditions. For example, suppose 0 < θ ( < θ < θ <, then N = θ, N = θ θ, and as a result, N = N + N = θ. Substituting N = θ into θ = p γn, we can solve θ as θ = p γ (. Conseuently, N = p (,and θ = p p γ p (, θ = p γ p (. We then need to verify under what conditions 0 < θ ( < θ < θ < holds. Solving the ineualities after substituting the solutions of { θ, θ, θ }, we arrive at the conditions: γ < p <,and p <p < ( γp +( (.

3 In a similar way to the case of 0 < θ ( < θ < θ < analyzed above, we can exhaust all cases of different relative magnitudes of θ, θ, θ and the bounds [0, ], which gives us the demands in different regions, as summarized in Table A. Table A: Demands Given Both Firms Prices p and p Cases Conditions N (p,p N (p,p N (p,p (A 0 <p γ: (A 0 <p p +( γ; 0 (A p +( γ<p <p +( (+γ; +γ p p p p γ (A3 p +( (+γ p < +γ 0 (B γ < p <: (B 0 <p γ; 0 (B γ<p p ; (B3 (B4 (C p p 0 <p < ( γp +( ( ; p p + γ p ( ( γp +( ( p < +γ 0 p <( + γ: p (< p p p p (< ( p ( p (< ( (C 0 <p γ; 0 p (C γ<p ; p 0 (< (C3 <p < +γ ( We next derive the incumbent s best response function p (p. Based on the demands derived in Table A, we can formulate the profit function of the incumbent given the entrant s price, π (p ; p. For example, according to Case (A in Table A, when 0 p +( (+γ cannot be the optimal price for the incumbent. Therefore, we only need to focus on the second segment in (A.. The first order condition yields the solution ˆp = [( (+γ+p ]. We then need to compare ˆp against the two bounds of that segment, p +( γ and p +( (+γ. Note that ˆp p +( γ if and only if p < 3

4 ( ( γ. Because 0 . Altogether, we have the following three subcases: (a if > and 0 and ( ( γ <p γ, then p (p =p +( γ, and the demands fall into Case (A as in Table A (in fact, the intersecting bound between Cases (A and (A; (c if γ and 0 <p γ ( ( ( γ, then p (p = [( (+γ+p ], and the demands fall into Case (A as in Table A. Table A: The Incumbent s Best Response Function p (p Cases Conditions p (p Demand Cases (as in Table A ( 0 <γ<,0< γ (< γ: (a 0 <p γ; (b γ < p < ( ( ( ; γ (c ( ( ( p γ < ( 0 <γ<, γ< : (a (b (c ( (+γ+p (A ( γp +( ( 0 <p γ; γ < p ( ; γ { ( γ <p < min p, (3 0 <γ<,(γ< γ<<: (B3 (B ( (+γ+p (A ( γp +( ( } (B3 (B / (B3 ( (+γ+p (3a 0 <p ( ( γ; (A (3b ( ( γ <p γ; p +( γ (A / (A { } p (3c γ < p < min, (B / (B3 (4 (4a (4b (4c (5 γ<, 0 << ( γ: 0 <p γ; γ < p < ( ( ( γ (γ ; ( (+γ+p (A ( γp +( ( (B3 ( ( ( γ (γ p < γ (B / (B γ<, γ <: ( (+γ+p (5a 0 <p ( ( γ; (A (5b ( ( γ <p <γ; p +( γ (A / (A (5c γ p < γ (B / (B Following a similar manner, we can analyze the incumbent s best response corresponding to the demand case (B in Table A. Note that demand case (C is irrelevant because N 0inthis case, which means the entrant will not be able to make any profit if it prices within these regions. As a result, p will not fall into this region in euilibrium, and hence the demand case (C will not 4

5 appear in euilibrium. Altogether, we summarize the incumbent s best response p (p intable A. (3 We finally solve the entrant s optimal price p. Based on the incumbent s best response functions in Table A, we can formulate the entrant s profit function when anticipating the incumbent s best response in pricing, π (p ; p (p. For example, consider Case ( in Table A, that is, when 0 <γ< and 0 < γ. ( p p (p p π (p ; p (p = ( ( p γ = p (, 0 ( ( ( γ cannot be optimal for the entrant. Therefore, we only need to focus on the first two segments of (A.3. The first order conditions for the first and the second segments yield ˆp = ( ( γ andˆp = ( ( γ, respectively. In order to determine the optimal p, we need to compare ˆp and ˆp against the bounds 0, γ, and ( ( ( γ. For example, if 0 <γ< 4,thenˆp >γand γ < ˆp <( ( ( γ. As a result, p =ˆp. In total, there are 6 subcases under Case (. Table A3 shows the detailed parameter conditions with their respective euilibrium prices. Following the same approach, we examine Cases ( through (5 in Table A and derive the euilibrium prices within various sub-regions of parameter values, as we summarize in Table A3. Combining the cases with the same euilibrium outcome, we arrive at the euilibrium outcomes described in Proposition and Table (as we indicate in the rightmost column in Table A3. A.3 Proof of Corollary Proof. Given that lim γ +γ = 0, we can see that for any fixed <, there exists a threshold γ ( (0, such that when γ>γ (, we enter region (iii. Then, it immediately follows that lim γ π (,γ = <lim γ π M (γ =. Conseuently, there exists γ 0( (γ (, such that for all γ (γ 0 (, we have π (, γ <πm (γ. 5

6 Table A3: Optimal Pricing for the Entrant, p Cases Conditions p ( 0 <γ<,0< γ: p (p Cases Demand Cases Euilibrium Cases (as in Table A (as in Table A (as in Table 0 <γ 4,0< γ; ( ( γ (b (B3 (i 4 <γ<,0< γ 4γ+ ; γ ( ( γ (b (B3 (i 4 <γ<, γ 4γ+ < γ; γ (a / (b (A / (B3 (ii γ γ, 0 < γ; γ (a / (b (A / (B3 (ii <γ<,0< ; +γ γ (a / (b (A / (B3 (ii <γ<, +γ < γ ( ( γ (a (A (iii ( 0 <γ<, γ< : 0 <γ< 4, γ< γ 4γ+ γ ; 0 <γ< 4, γ 4γ+ γ ( ( γ (b (B3 (i < ; γ (a / (b (B3 / (A (ii +γ 0 <γ< 4, +γ < γ; ( ( γ (a (A (iii 4 γ<, γ< ; γ (a / (b (B3 / (A (ii +γ 4 γ<, +γ < γ; ( ( γ (a (A (iii γ<, γ< ( ( γ (a (A (iii (3 0 <γ<,<< ( ( γ (3a (A (iii (4 (5 γ<, 0 <<: γ<, 0 < ; γ (4a / (4b (A / (B3 (ii +γ γ<, +γ <<; ( ( γ (4a (A (iii γ<, γ < ( ( γ (5a (A (iii A.4 Proof of Proposition Proof. According to Proposition, the entrant s euilibrium profit by choosing uality can be written as ( 8(( γ c, π ( = ( γ ( c, max 8 ( ( γ c, { } 0 < max γ 4γ+ γ, 0 ; { } γ 4γ+ γ, 0 < +γ ; +γ <<. (A.4 Solving the first-order condition for the first segment of (A.4, we have = γ ( γ( 8c( ; the solution to the first-order condition for the second segment of (A.4 yields = γ γ c. For the third segment of (A.4, because d d π ( = 8 ( γ c<0, any > +γ cannot be optimal. 6

7 Define ˆγ (c as the uniue solution to 4γ 3 +4(c +3γ 8(c +γ +8c + = 0 for ( ( γ. As we can verify, ˆγ (c is well defined for c 0,. Note that (ˆγ = 0, 4 (ˆγ = ˆγ 4ˆγ+ ˆγ. Consider Case (a when 0 <c 6. We examine it in the following subcases. (i When 0 <γ ˆγ (c, as we can show, γ 4γ+ γ > 0, 0 < γ 4γ+ γ,and γ 4γ+ γ. Therefore, π ( reaches its peak within the first segment of (A.4 at = 4γ+,andany>γ γ is suboptimal. As a result, the entrant s optimal uality choice is = = γ ( γ( 8c(. (ii When ˆγ (c <γ<,aswecanshow,0< γ 4γ+ γ < and γ 4γ+ γ < < +γ. Therefore, π ( reaches its peak within the second segment of (A.4 at =,andany< γ 4γ+ γ or > +γ is suboptimal. As a result, = = γ γ c. (ii When γ< + 8c, because γ 4γ+ γ 0, the first segment of (A.4 no longer applies. As we can show, 0 +γ within this region. Therefore, π ( reaches its peak within the second segment of (A.4 at =, and any > +γ is suboptimal. As a result, = = γ γ c. (iii When + 8c γ<, < 0. Therefore, π ( is decreasing in for [0, ], and the optimal uality choice =0. Combining (i through (iii, we have the optimal uality choice of the entrant for Case (a when 0 <c 6. The other cases can be proven in a similar fashion. A.5 Proof of Corollary Proof. Follows immediately from Proposition by setting (γ,c > 0. A.6 Proof of Lemma Proof. Comparing the euilibrium profits of the incumbent, π, in Proposition and euation ( over various parameter regions, we have: ( If γ<, according to euation(, the incumbent s monopoly profit π = 4(. ( (4 3γ (i In region (i of Proposition, > 6( ( γ 4( if and only if (γ (4γ ( 3γ+( 5+8 > 6( ( γ 0. Note that 4γ ( 3 γ+ ( 5 +8 > 0 always holds because ( 3 6 ( 5 +8 = 7( < 0. Therefore, ( (4 3γ 6( ( γ > 4( if and only if <γ. 7

8 (ii In region (ii of Proposition, (+γ 4( > 4( if and only if (γ ( (+ γ 4( ( > 0. As we can show, because < +γ in this region, (+γ 4( > 4( if and only if <γ. (iii In region (iii of Proposition, 6 ( (3+γ > 4( if and only if < 4. ((3+γ Note that γ = +γ = 4 when γ =. ((3+γ ( If γ, according to euation (, the incumbent s monopoly profit π = γ. (i Region (i of Proposition does not apply when γ. (ii In region (ii of Proposition, (+γ 4( >γif and only if ( γ 4( > 0, which always holds. (iii In region (iii of Proposition, 6 ( (3+γ >γif and only if < 6γ. Note (3+γ that 6γ 4 = when γ = (3+γ ((3+γ. Altogether, we can conclude that sharing its IP leads to higher euilibrium profit than remaining a monopoly when < (γ, where (γ is defined in (6 A.7 Proof of Proposition 3 Proof. First note that the euilibrium profit of the incumbent in a duopoly market, π (, γ as derived in Proposition, is decreasing in the entrant s product uality. Toseethis,wetakethe d first derivative with respect to. In region (i of Table, d π (,γ = (4 3γ [( +3( ] < 0; 6( ( γ ] 3 d in region (ii, d π (, γ = 4 [ γ < 0 because < ( +γ < γ in this region; in region d (iii, d π (, γ = 6 (3 + γ < 0. We prove the results summarized in Table 3 by deviding the value range of γ into five cases: (a 0 <γ 4 ;(b 4 <γ ;(c <γ γ; (d γ<γ 3 5 ;(e 3 5 <γ<. We analyze case (a (0 <γ 4 in detail below. The other cases can be analyzed in a similar fashion. ( Recall the analysis and results for Proposition and Lemma. For 0 <γ 4, (γ,c = γ ( γ( 8c(,and (γ =γ. Solving (γ,c = (γ forc, wehavec = 3γ. Recall that 3( (γ,c is decreasing in c. Therefore, if c< 3γ, (γ,c > (γ, so sharing its IP would lead 3( to less profit for the incumbent than not sharing and remaining a monopoly. If k> 3( 3γ,with either basic sharing (i.e., ρ = or advanced sharing (i.e., ρ =,c = ρk < 3γ 3(. As a result, either basic or advanced sharing is dominated by remaining a monopoly, so the optimal strategy for the incumbent is no sharing (i.e., ρ = 0. Thus, this case constitutes a part of region ( in Table 3. 8

9 ( Solving (γ,c = γ ( γ( 8c( =0forc, wehavec = 8( γ(. Therefore, if 3γ 3( <c< 8( γ( (note that 3γ 3( < 8( γ( for γ ( 0, 4, 0 < (γ,c < (γ, so sharing its IP leads to more profit for the incumbent than not sharing, and meanwhile, the entrant is willing to enter the market by achieving a strictly positive level of net profit with > 0. For this reason, if 8 ( γ( γ <k< 3( 3γ, the optimal strategy for the incumbent is basic sharing (i.e., ρ =sothatc = k. Note that advanced sharing (i.e., ρ =sothatc = k cannot beoptimalinthiscasebecauseaswehaveshown,π (,γ (as in Proposition is decreasing in, so helping the entrant reduce development cost (which would increase (γ,c would hurt the incumbent s euilibrium profit. Altogether, this case constitutes a part of region ( in Table 3. (3 When 6( 3γ <k<8( γ( γ (note that 6( 3γ < 8( γ( γ for γ ( 0, 4, if the incumbent chooses basic sharing (i.e., ρ =, then c = k > 8( γ(. By the above analysis in (, (γ,c = 0. In other words, with basic sharing, the entrant would be unable to achieve a positive level of net profit, and hence = 0 would be its optimal uality choice; as a result, the entrant would not enter the market in the first place. If the incumbent chooses ( advanced sharing (i.e., ρ =, then c = k < 3γ < 3( 8( γ(. By the above analysis in (, (γ,c > (γ(> 0, indicating that the incumbent could achieve more profit by remaining a monopoly than advanced sharing. Altogether, neither basic nor advanced sharing can outperform the monopoly profit. Therefore, the optimal strategy for the incumbent is no sharing (i.e., ρ =0. This case hence constitutes a part of region (3 in Table 3. (4 When 4 ( γ( γ <k< 6( 3γ (< 8( γ( γ, if the incumbent chooses basic sharing (i.e., ρ =, then c = k > 8( γ(. By the above analysis in (, (γ,c = 0. As a result, the entrant would not enter the market in the first place, and the incumbent would maintain the monopoly profit. If the incumbent chooses advanced sharing (i.e., ρ =, then c = k, 3γ so <c< 3( 8( γ(. By the above analysis in ( and (, 0 < (γ,c < (γ. As a result, the entrant is willing to enter the market by achieving a strictly positive level of net profit with > 0, and meanwhile, the incumbent can achieve more profit than remaining a monopoly. Therefore, the optimal strategy for the incumbent is advanced sharing (i.e., ρ =. Thiscase hence constitutes a part of region (4 in Table 3. (5 When k<4( γ( γ, even with advanced sharing (i.e., ρ =,c = k > 8( γ(, 9

10 so (γ,c = 0. In other words, even with advanced sharing, the entrant would still be unable to achieve a positive level of net profit, and hence would not enter the market in the first place. As a result, the incumbent remains a monopoly in euilibrium. This case hence constitutes a part of region (5 in Table 3. 0

Haiyang Feng College of Management and Economics, Tianjin University, Tianjin , CHINA

RESEARCH ARTICLE QUALITY, PRICING, AND RELEASE TIME: OPTIMAL MARKET ENTRY STRATEGY FOR SOFTWARE-AS-A-SERVICE VENDORS Haiyang Feng College of Management and Economics, Tianjin University, Tianjin 300072,