Supply chains of mobile apps: competition, private labels and bypassing when the app’s quality is co-created

Abstract

This work deals with issues characterizing the mobile app industry. In particular, we focus on an app developer who competes against either a rival app developer or a private label of the distribution platform, and who can bypass the platform’s distribution and billing systems. The innovation of this work lies in considering the co-creation of app quality by the app developer and the distribution platform while using a revenue-sharing contract, which is popular in the mobile app industry. We investigate both tactical equilibrium (the parties’ operational decisions for a given market structure) and strategic equilibrium (for the developer—whether or not to bypass; for the platform—whether or not to discourage bypassing by improving the contract terms). Our research provides answers to the following strategic questions: Is it beneficial for the app developer to bypass the distribution platform and offer the app to users directly in a competitive environment? Is it beneficial for the distribution platform to introduce a private label to compete with the app developer? Can bypassing also be beneficial for the platform and/or the rival developer? In response to these questions, several counter-intuitive results are revealed.

Appendices

Appendix A: Proofs

Proof of Proposition 1

In order to find the developer’s best response, $q_{1} \left( {\eta ,q} \right)$, we solve the optimization problem given in the constraint of (4). Since $\pi_{1} (\eta ,q,q_{1} )$ is a concave parabolic function of $q_{1}$, then it is maximized for $q_{1} \left( {\eta ,q} \right) = qr\alpha \left( {1 - \eta } \right)/\gamma_{1}$.

Investigating the objective function of (4), the gradient of $\pi_{0} \left( {\eta ,q_{0} ,q_{1} (\eta ,q_{0} )} \right)$ is $\vec{\nabla } = \left[ {\begin{array}{*{20}c} {\eta r - qc + 2\eta (1 - \eta )qr^{2} \alpha^{2} /\gamma_{1} } \\ {rq\left( {1 - \left( {1 - 2\eta } \right)qr\alpha^{2} /\gamma_{1} } \right)} \\ \end{array} } \right]$. Solving necessary condition $\vec{\nabla } = \vec{0}$, we obtain two solutions: the degenerate solution $\left( {\eta = 0,q = 0} \right)$, which results in zero profits for the parties, and $\left( {\eta^{*} = \frac{{\gamma_{1} c}}{{r^{2} \alpha^{2} }},q^{S*} = \frac{{\gamma_{1} r}}{{2\gamma_{1} c - r^{2} \alpha^{2} }}} \right)$. The Hessian matrix of $\pi_{0} \left( {\eta ,q,q_{1} (\eta^{*} ,q^{*} )} \right)$ is $H = \left[ {\begin{array}{*{20}c} { - \frac{{c\left( {2\gamma_{1} c - r^{2} \alpha^{2} } \right)}}{{r^{2} \alpha^{2} }}} & { - r} \\ { - r} & { - \frac{{2r^{4} \gamma_{1} \alpha^{2} }}{{\left( {2\gamma_{1} c - r^{2} \alpha^{2} } \right)^{2} }}} \\ \end{array} } \right]$ with both $H_{11} < 0$ and $Det\left( H \right) = \frac{{r^{4} \alpha^{2} }}{{2\gamma_{1} c - r^{2} \alpha^{2} }} > 0$ for $c > r^{2} \alpha^{2} /(2\gamma_{1} )$. Under the latter condition, $H$ is negatively definite, so $\left( {\eta^{*} ,q^{*} } \right)$ maximizes the objective function of (4). The constraint $\eta^{*} < 1$ results in $c < r^{2} \alpha^{2} /\gamma_{1}$, so the assumption $r^{2} \alpha^{2} /(2\gamma_{1} ) < c < r^{2} \alpha^{2} /\gamma_{1}$ is required to ensure the existence of the interior maximum point. Under this assumption we obtain $\left( {\eta^{S} = \eta^{*} ,q^{S} = q^{*} } \right)$.

By substituting $\left( {\eta^{S} ,q^{S} } \right)$ into $q_{1} \left( {\eta ,q} \right)$, $\pi_{0} \left( {\eta ,q,q_{1} (\eta ,q)} \right)$ and $\pi_{1} \left( {\eta ,q,q_{1} (\eta ,q)} \right)$, we obtain $q_{1}^{S} = \frac{{r^{2} \alpha^{2} - \gamma_{1} c}}{{\alpha \left( {2\gamma_{1} c - r^{2} \alpha^{2} } \right)}}$, $\pi_{0}^{S} = \frac{{\gamma_{1}^{2} c}}{{2\alpha^{2} \left( {2\gamma_{1} c - r^{2} \alpha^{2} } \right)}}$ and $\pi_{1}^{S} = \frac{{\gamma_{1} \left( {(r^{2} \alpha^{2} - \gamma_{1} c)(3\gamma_{1} c - r^{2} \alpha^{2} ) - r^{2} \alpha^{2} \gamma_{1} c_{1} } \right)}}{{2\alpha^{2} \left( {2\gamma_{1} c - r^{2} \alpha^{2} } \right)^{2} }}$. Note that the developer agrees to play when $\pi_{1}^{S} \ge 0$, so in order to avoid a solution in which the profit of the developer at equilibrium equals zero, we assume that $c_{1} < \frac{{(r^{2} \alpha^{2} - \gamma_{1} c)(3\gamma_{1} c - r^{2} \alpha^{2} )}}{{r^{2} \alpha^{2} \gamma_{1} }}$. □

Proof of Proposition 2

In order to find the developers’ best responses, we solve the set of two optimization problems represented by the constraints of (9) regarding $q_{i} \left( {\eta ,q} \right), i = 1,2$. The necessary conditions are $\frac{{d\pi_{i} }}{{dq_{i} }} = \left( {1 - \eta } \right)qr\alpha /2 - q_{i} \gamma_{i} = 0, i = 1,2$, which are also the sufficient conditions for a maximum, since $\pi_{i}$ is a concave parabolic function of $q_{i} .$ Since $\frac{{d\pi_{i} }}{{dq_{i} }}$ is independent of $q_{3 - i}$, we conclude that the developers’ best responses remain the same for any order in which the two developers make their extra-quality decisions (simultaneously or sequentially). Thus, the developers’ best responses $q_{i} \left( {\eta ,q} \right) = qr\alpha \left( {1 - \eta } \right)/(2\gamma_{i} ), i = 1,2$ constitute a conditional Nash equilibrium for a given $(\eta ,q)$.

Investigating the objective function of (9), and solving the necessary condition $\vec{\nabla }\left( {\pi_{0} \left( {\eta ,q,q_{1} (\eta ,q),q_{2} (\eta ,q)} \right)} \right) = \vec{0}$, we obtain two solutions: the degenerate solution $\left( {\eta = 0,q = 0} \right)$, which results in zero profits of the parties, and $\left( {\eta^{CD} = \frac{4c}{{{\Lambda }r^{2} }},q^{CD} = \frac{4r}{{8c - {\Lambda }r^{2} }}} \right)$. The Hessian matrix of $\pi_{0} \left( {\eta ,q,q_{1} \left( {\eta^{CD} ,q^{CD} } \right),q_{2} (\eta^{CD} ,q^{CD} )} \right)$ is $H = \left[ {\begin{array}{*{20}c} { - c\left( {\frac{8c}{{{\Lambda }r^{2} }} - 1} \right)} & { - r} \\ { - r} & { - \frac{8}{{{\Lambda }\left( {\frac{8c}{{{\Lambda }r^{2} }} - 1} \right)^{2} }}} \\ \end{array} } \right]$ with $H_{11} < 0$ and $Det\left( H \right) = \frac{{r^{2} }}{{\frac{8c}{{{\Lambda }r^{2} }} - 1}} > 0$ for $c > {\Lambda }r^{2} /8$. In such a case, $H$ is negatively definite and $\left( {\eta^{CD} ,q^{CD} } \right)$ maximizes the objective function of (9). The constraint $\eta < 1$ results in $c < {\Lambda }r^{2} /4$, so the assumption ${\Lambda }r^{2} /8 < c < {\Lambda }r^{2} /4$ is required to ensure the existence of the interior maximum point.

By substituting $\left( {\eta^{CD} ,q^{CD} } \right)$ into $q_{i} \left( {\eta ,q} \right)$ and $\pi_{0} \left( {\eta ,q,q_{1} (\eta ,q),q_{2} (\eta ,q)} \right)$, we obtain $q_{i}^{CD} = \frac{{2\alpha ({\Lambda }r^{2} - 4c)}}{{\gamma_{i} (8c - {\Lambda }r^{2} )}}$ and $\pi_{0}^{CD} = \frac{8c}{{{\Lambda }(8c - {\Lambda }r^{2} )}}$. Note that developer i agrees to play when $\pi_{i}^{CD} \ge 0$, so in order to avoid a solution in which the profit of the developer at equilibrium equals zero, we assume that $c_{i} < 4\left( {\frac{{{\Lambda }r^{2} }}{4} - c} \right)\left( {\frac{1}{{{\Lambda }r^{2} }}\left( {\frac{{2\gamma_{3 - i} }}{{\gamma_{1} + \gamma_{2} }} - \frac{{\alpha^{2} (\gamma_{3 - i} - 2\gamma_{i} )}}{{{\Lambda }\gamma_{1} \gamma_{2} }}} \right)\left( {\frac{{{\Lambda }r^{2} }}{4} - c} \right) - \frac{1}{4}} \right),{ }i = 1,2$. □

Proof of Proposition 3

In order to find the developers’ best responses, we solve the set of two optimization problems given in the constraints of (15) regarding $q_{1} \left( q \right)$ and $q_{2} \left( {\eta ,q} \right)$. The necessary conditions are $\frac{{d\pi_{1} }}{{dq_{1} }} = qr\alpha /2 - q_{1} \gamma_{1} = 0$ and $\frac{{d\pi_{2} }}{{dq_{2} }} = \left( {1 - \eta } \right)qr\alpha /2 - q_{2} \gamma_{2} = 0$, which are also the sufficient conditions for a maximum, since $\pi_{i}$ is a concave parabolic function of $q_{i} .$ Since $\frac{{d\pi_{i} }}{{dq_{i} }}$ is independent of $q_{3 - i}$, we conclude that the developers’ best responses remain the same for any order in which the two developers make their extra-quality decisions (simultaneously or sequentially). Thus, the developers’ best responses $q_{1} \left( {\eta ,q} \right) = qr\alpha /(2\gamma_{1} )$ and $q_{2} \left( q \right) = qr\alpha \left( {1 - \eta } \right)/(2\gamma_{2} )$ constitute a conditional Nash equilibrium for a given $(\eta ,q)$.

Investigating the objective function of (15), and solving the necessary condition $\vec{\nabla }\left( {\pi_{0} \left( {\eta ,q,q_{1} (\eta ,q),q_{2} (q)} \right)} \right) = \vec{0}$, we obtain two solutions: the degenerate solution $\left( {\eta = 0,q = 0} \right)$, which results in zero profits for the parties, and $\left( {\eta^{B} = \frac{4c}{{{\Psi }r^{2} }},q^{B} = \frac{{2 {\Psi }\gamma_{2} r(1 + \theta )}}{{8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} }}} \right)$. The Hessian matrix of $\pi_{0} \left( {\eta ,q,q_{1} \left( {\eta^{B} ,q^{B} } \right),q_{2} (q^{B} )} \right)$ is $H = \left[ {\begin{array}{*{20}c} { - c\left( {\frac{{8c\alpha^{2} }}{{\gamma_{2} {\Psi }^{2} r^{2} }} - 1} \right)} & {\frac{ - r(1 + \theta )}{2}} \\ {\frac{ - r(1 + \theta )}{2}} & { - \frac{{2\alpha^{2} (1 + \theta )^{2} }}{{\gamma_{2} {\Psi }^{2} \left( {\frac{{8c\alpha^{2} }}{{\gamma_{2} {\Psi }^{2} r^{2} }} - 1} \right)^{2} }}} \\ \end{array} } \right]$ with $H_{11} < 0$ and $Det\left( H \right) = \frac{{c(1 + \theta )^{2} r^{2} }}{{4\left( {\frac{{8c\alpha^{2} }}{{\gamma_{2} {\Psi }^{2} r^{2} }} - 1} \right)}} > 0$ for $c > \gamma_{2} {\Psi }^{2} r^{2} /(8\alpha^{2} )$. Under the latter condition, $H$ is negatively definite and $\left( {\eta^{B} ,q^{B} } \right)$ maximizes the objective function of (15). The constraint $\eta < 1$ results in $c < {\Psi }r^{2} /4$, so the assumption $\gamma_{2} {\Psi }^{2} r^{2} /(8\alpha^{2} ){ } < c < {\Psi }r^{2} /4$ is required to ensure the existence of the interior maximum point. Since $\gamma_{2} {\Psi }^{2} r^{2} /(8\alpha^{2} ){ } < {\Psi }r^{2} /4$ is equivalent to $\frac{\alpha }{{\gamma_{2} }} + \frac{\beta }{{\gamma_{1} }} > 0$, the domain of c always exists.

By substituting $\left( {\eta^{B} ,q^{B} } \right)$ into $q_{i} \left( {\eta ,q} \right)$ and $\pi_{0} \left( {\eta ,q,q_{1} (q),q_{2} (\eta ,q)} \right),$ we obtain $q_{2}^{B} = \frac{{\alpha (1 + \theta )\left( {{\Psi }r^{2} - 4c} \right)}}{{8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} }}$ and $q_{1}^{B} = \frac{{\alpha (1 + \theta ){\Psi }\gamma_{2} r^{2} }}{{\gamma_{1} (8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} )}}$, $\pi_{0}^{B} = \frac{{2c(1 + \theta )^{2} \gamma_{2} }}{{8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} }}$, $\pi_{2}^{B} = \frac{{(1 + \theta )^{2} \gamma_{2} \left( {\alpha^{2} \left( {12c - {\Psi }r^{2} } \right)\left( {{\Psi }r^{2} - 4c} \right) - 4c_{2} \gamma_{2} {\Psi }^{2} r^{2} } \right)}}{{2(8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} )^{2} }}$ and $\pi_{1}^{B} = \frac{{r^{2} (1 + \theta ){\Psi }\left( {\gamma_{1}^{2} \left( {1 + \theta } \right)\left( {{\Psi }r^{2} - 4c} \right)\left( {{\Psi }\gamma_{2} - \alpha^{2} } \right) + \gamma_{1} \gamma_{2} \left( {(1 - \theta )\left( {8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} } \right) - 2c_{1} \left( {1 + \theta } \right){\Psi }\gamma_{2} } \right) + {\Psi }r^{2} \alpha^{2} \gamma_{2}^{2} \left( {1 + \theta } \right)/2} \right)}}{{\gamma_{1} (8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} )^{2} }} - G$. Note that developer i agrees to play when $\pi_{i}^{B} \ge 0$, so in order to avoid a solution in which the profit of Developer 1 at equilibrium equals zero, we assume that $c_{2} < \frac{{\alpha^{2} \left( {12c - {\Psi }r^{2} } \right)\left( {{\Psi }r^{2} - 4c} \right)}}{{4{\Psi }^{2} r^{2} \gamma_{2} }}$ and $c_{1} + \frac{G}{2}\left( {\frac{{8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} }}{{r\gamma_{2} (1 + \theta ){\Psi }}}} \right)^{2} < \frac{{\gamma_{1} \left( {{\Psi }\gamma_{2} - \alpha^{2} } \right)\left( {{\Psi }r^{2} - 4c} \right)}}{{2{\Psi }\gamma_{2}^{2} }} + \frac{{(1 - \theta )(8\alpha^{2} c - \gamma_{2} {\Psi }^{2} r^{2} )}}{{2(1 + \theta ){\Psi }\gamma_{2} }} + \frac{{\alpha^{2} r^{2} }}{{4\gamma_{2} }}$. □

Proof of Proposition 4

In order to find $q_{0} \left( {\eta ,q} \right)$ and $q_{1} \left( {\eta ,q} \right)$, we solve the set of two optimization problems given in the two constraints of (21). The necessary conditions are $\frac{{d\pi_{0} }}{{dq_{0} }} = \left( {\alpha - \beta \eta } \right)qr\alpha /2 - q_{0} \gamma_{0} = 0$ and $\frac{{d\pi_{1} }}{{dq_{i1} }} = \left( {1 - \eta } \right)qr\alpha /2 - q_{1} \gamma_{1} = 0,$ which are also the sufficient conditions for a maximum, since $\pi_{i}$ is a concave parabolic function of $q_{i} .$ Since $\frac{{d\pi_{i} }}{{dq_{i} }}$ is independent of $q_{1 - i}$, we conclude that $q_{0} \left( {\eta ,q} \right)$ and $q_{1} \left( {\eta ,q} \right)$ remain the same for any order of the extra-quality decisions of the parties (simultaneous or sequential). Thus, $q_{0} \left( {\eta ,q} \right) = qr\left( {\alpha - \beta \eta } \right)/(2\gamma_{0} )$ and $q_{1} \left( {\eta ,q} \right) = qr\alpha \left( {1 - \eta } \right)/(2\gamma_{1} )$ constitute a conditional Nash equilibrium for a given $(\eta ,q)$.

Investigating the objective function of (21), and solving the necessary condition $\vec{\nabla }\pi_{0} (\eta ,q,q_{0} \left( {\eta ,q} \right),q_{1} \left( {\eta ,q} \right)) = \vec{0}$, we obtain two solutions: the degenerate solution $\left( {\eta = - 1,q = 0} \right)$, which results in zero profits of the parties, and $\left( {\eta^{PL} = \frac{{\alpha^{2} r^{2} \left( {\gamma_{1} (\alpha + \beta ) - \gamma_{0} (\alpha + 3\beta )} \right) - 4\gamma_{1} \gamma_{0} c_{p} }}{{r^{2} (\beta \left( {\alpha + \beta } \right)\gamma_{1} - \alpha \left( {3\alpha + \beta } \right)\gamma_{0} )}},q^{PL} = \frac{{2 \gamma_{0} \gamma_{1} r(1 + {\upeta }^{PL} )}}{{4\gamma_{0} \gamma_{1} c_{p} + \left( {2\alpha \gamma_{0} (1 - {\upeta }^{PL} )(\beta - {{\alpha \eta }}^{PL} ) - \gamma_{1} (\alpha - {{\beta \eta }}^{PL} )^{2} } \right)r^{2} }}} \right)$.

Let $A \equiv \frac{{r^{2} (\alpha + \beta )\left( {\left( {\alpha + \beta } \right)\gamma_{1} - 4\alpha \gamma_{0} } \right)}}{{4\gamma_{0} \gamma_{1} }}$, $B \equiv \frac{{r^{2} {\upalpha }(\alpha - \beta )^{2} \left( {2\left( {\alpha + \beta } \right)\gamma_{1} + \alpha \gamma_{0} } \right)}}{{4\gamma_{1} (2\alpha^{2} \gamma_{0} - \beta^{2} \gamma_{1)} }}$, $C \equiv \frac{{\alpha r^{2} \left( {\left( {\alpha + \beta } \right)\gamma_{1} - (\alpha + 3\beta )\gamma_{0} } \right)}}{{4\gamma_{0} \gamma_{1} }}$ and $D \equiv \frac{{(\alpha - \beta )r^{2} \left( {\left( {\alpha + \beta } \right)\gamma_{1} + 2\alpha \gamma_{0} } \right)}}{{4\gamma_{0} \gamma_{1} }}$. In order to ensure that the Hessian matrix of $\pi_{0} \left( {\eta ,q,q_{0} \left( {\eta^{PL} ,q^{PL} } \right),q_{1} (\eta^{PL} ,q^{PL} )} \right)$ is negatively definite, the following two restrictions are required:

$$ c_{p} > max \left( {A, B} \right)\;{\text{and}}\;2\alpha^{2} \gamma_{0} - \beta^{2} \gamma_{1} > 0. $$

(A.1)

In such a case, $\left( {\eta^{PL} ,q^{PL} } \right)$ maximizes the objective function of (21). The constraint $0 < \eta < 1$ results in the following two restrictions:

$$ C < c_{p} < D\;{\text{when}}\;\alpha \left( {3\alpha + \beta } \right)\gamma_{0} - \beta \left( {\alpha + \beta } \right)\gamma_{1} > 0; $$

(A.2)

$$ D < c_{p} < C\;{\text{when}}\;\alpha \left( {3\alpha + \beta } \right)\gamma_{0} - \beta \left( {\alpha + \beta } \right)\gamma_{1} < 0. $$

(A.3)

Note: $\alpha \left( {3\alpha + \beta } \right)\gamma_{0} - \beta \left( {\alpha + \beta } \right)\gamma_{1} > 0$ implies $A < C$, so restrictions (A.1) and (A.2) can be unified to

$$ {\text{max}}(B,C) < c_{p} < D,\; \beta \left( {\alpha + \beta } \right)\gamma_{1} - \alpha \left( {3\alpha + \beta } \right)\gamma_{0} < 0\;{\text{and}}\;2\alpha^{2} \gamma_{0} - \beta^{2} \gamma_{1} > 0. $$

(A.4)

On the other hand, since $\alpha \left( {3\alpha + \beta } \right)\gamma_{0} - \beta \left( {\alpha + \beta } \right)\gamma_{1} < 0$ implies $A > C$, the restriction $D < c_{p} < C$ in (A.3) does not hold.

Consider the restrictions $\alpha \left( {3\alpha + \beta } \right)\gamma_{0} - \beta \left( {\alpha + \beta } \right)\gamma_{1} > 0$ and $2\alpha^{2} \gamma_{0} - \beta^{2} \gamma_{1} > 0$ in (A.4). They can be rewritten as $\frac{{\gamma_{1} }}{{\gamma_{0} }} < \frac{{\alpha \left( {3\alpha + \beta } \right)}}{{\beta \left( {\alpha + \beta } \right)}}$ and $\frac{{\gamma_{1} }}{{\gamma_{0} }} < \frac{{2\alpha^{2} }}{{\beta^{2} }}$, respectively. Since $\frac{{\alpha \left( {3\alpha + \beta } \right)}}{{\beta \left( {\alpha + \beta } \right)}}/\frac{{2\alpha^{2} }}{{\beta^{2} }}$ is an increasing function of $\beta$ for any $\alpha > \beta > 0$, its maximum is obtained when $\beta$ approaches $\alpha$ and equals 1. Thus, the restriction $2\alpha^{2} \gamma_{0} - \beta^{2} \gamma_{1} > 0$ is redundant, and (A.4) can be written as in Assumption 4.1:

$$ {\text{max}}(B,C) < c_{p} < D\;{\text{and}}\frac{{\gamma_{1} }}{{\gamma_{0} }} < \frac{{\alpha \left( {3\alpha + \beta } \right)}}{{\beta \left( {\alpha + \beta } \right)}}. $$

By substituting $\left( {\eta^{PL} ,q^{PL} } \right)$ into $q_{i} \left( {\eta ,q} \right)$ and $\pi_{0} \left( {\eta ,q,q_{0} (\eta ,q),q_{1} (\eta ,q)} \right),$ we obtain $q_{i}^{PL}$ and $\pi_{0}^{PL}$ as given in the proposition. Note that the developer agrees to play when $\pi_{1}^{PL} \ge 0$, so in order to avoid a solution in which the profit of the developer at equilibrium equals zero, Assumption 4.2 is required. □

Proof of Proposition 5

In order to find the developers’ best responses, we solve the set of two optimization problems given in the constraints of (26) regarding $q_{0} \left( q \right)$ and $q_{1} \left( q \right)$. The necessary conditions are $\frac{{d\pi_{i} }}{{dq_{i} }} = qr\alpha /2 - q_{i} \gamma_{i} = 0$, $i = 0,1$, which are also the sufficient conditions for a maximum, since $\pi_{i}$ is a concave parabolic function of $q_{i} .$ Since $\frac{{d\pi_{i} }}{{dq_{i} }}$ is independent of $q_{1 - i}$, we conclude that $q_{0} \left( q \right)$ and $q_{1} \left( q \right)$ remain the same for any order of the extra-quality decisions of the parties (simultaneously or sequentially). Thus, $q_{0} \left( q \right) = qr\alpha /(2\gamma_{0} )$ and $q_{1} \left( q \right) = qr\alpha /(2\gamma_{1} )$ constitute a conditional Nash equilibrium for a given $q$.

The objective function of (26) is a concave parabolic function of q when $c_{p} > \frac{{\alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)}}{{4\gamma_{0} \gamma_{1} }}$, which has a maximum at $q^{PLB} = \frac{{2r(1 + \theta )\gamma_{0} \gamma_{1} }}{{4c_{p} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)}}$.

By substituting $\left( {\eta^{PLB} ,q^{PLB} } \right)$ into $q_{i} \left( q \right)$ and into $\pi_{0} \left( {q,q_{0} (q),q_{1} (q)} \right)$, we obtain $q_{i}^{PLB} = \frac{{\alpha r^{2} (1 + \theta )\gamma_{i} }}{{4c_{p} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)}}$, $i = 0,1$, $\pi_{0}^{PLB} = \frac{{r^{2} (1 + \theta )^{2} \gamma_{0} \gamma_{1} }}{{2\left( {4c_{p} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)} \right)}}$, and $\pi_{1}^{PLB} = \frac{{r^{2} (1 + \theta )\gamma_{0} \gamma_{1} \left( {2\left( {4c_{p} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)} \right)\left( {1 - \theta } \right) - \left( {4c_{1} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{0} - 2\beta \gamma_{1} } \right)} \right)\left( {1 + \theta } \right)} \right)}}{{2\left( {4c_{p} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)} \right)^{2} }} - G$. Note that the developer agrees to play when $\pi_{1}^{PLB} \ge 0$, so in order to avoid a solution in which the profit of the developer at equilibrium equals zero, we assume that $c_{1} + \frac{G}{2}\left( {\frac{{4c_{p} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)}}{{r\left( {1 + \theta } \right)\gamma_{0} \gamma_{1} }}} \right)^{2} \le \frac{{\left( {4c_{p} \gamma_{0} \gamma_{1} - \alpha r^{2} \left( {\alpha \gamma_{1} - 2\beta \gamma_{0} } \right)} \right)\left( {1 - \theta } \right) + \alpha r^{2} \left( {\alpha \gamma_{0} - 2\beta \gamma_{1} } \right)\left( {1 + \theta } \right)}}{{2\left( {1 + \theta } \right)\gamma_{0} \gamma_{1} }}$. □

Appendix B: Strategic equilibrium for the case of a single developer who competes with a platform that launches a private label

To obtain the strategic equilibrium, we propose taking the steps listed in Procedure 2, which can be expounded upon as follows. First, we assume that the platform is interested in dissuading the developer from bypassing (if this assumption is not valid, then it follows that the platform will benefit from the developer bypassing its distribution and billing systems). Following this assumption, we formulate optimization problem (B.1), which is based on optimization problem (21) with two additional participation constraints. In particular, the first constraint (D1IR) ensures that the offered contract terms are not less profitable to the developer than bypassing the platform, whereas the second constraint (PIR) ensures that the contract terms are not less profitable to the platform than the bypassing option. The formulation of this problem is as follows:

$$ \begin{aligned} & \mathop {\max }\limits_{{0 \le \eta \le 1,{ }q \ge 0}} \left\{ {\pi_{0} (\eta ,q,q_{0} (\eta ,q),q_{1} (\eta ,q))} \right\} \\ & {\text{S.t}}. q_{i} \left( {\eta ,q} \right) = \mathop {{\text{argmax}}}\limits_{{q_{i} \ge 0}} \left\{ {\pi_{i} (\eta ,q,q_{0} ,q_{1} )} \right\},\quad i = 0,1 \\ & \pi_{1} \left( {\eta ,q,q_{0} (\eta ,q),q_{1} (\eta ,q)} \right) \ge \pi_{1}^{PLB} \quad \left( {{\text{D1IR}}} \right) \\ & \pi_{0} (\eta ,q,q_{0} (\eta ,q),q_{1} (\eta ,q)) \ge \pi_{0}^{PLB} \quad \left( {{\text{PIR}}} \right) \\ \end{aligned} $$

(B.1)

Based on (B.1), the procedure used to obtain strategic equilibrium, denoted by the superscript “PLS”, is summarized in the following.

Procedure 2

Step 0 Based on the results of Propositions 4 and 5, calculate $\pi_{0}^{PL} , \pi_{0}^{PLB} ,$ $\pi_{1}^{PL}$ and $\pi_{1}^{PLB}$.

Step 1 If $\pi_{1}^{PL} > \pi_{1}^{PLB}$, then the platform offers regular contract terms $\left\{ {\eta^{PLS} ,q^{PLS} } \right\} = \left\{ {\eta^{PL} ,q^{PL} } \right\}$ and the developer accepts the contract (the expressions of the equilibrium are given in Proposition 4). Go to Step 6.

Step 2 If $\pi_{1}^{PL} < \pi_{1}^{PLB}$ and $\pi_{0}^{PL} < \pi_{0}^{PLB}$, then the platform sets basic quality level $q^{PLS} = q^{PLB}$, and the developer bypasses the platform’s distribution and billing systems (the expressions of the equilibrium are given in Proposition 5). Go to Step 6.

Step 3 If $\pi_{1}^{PL} < \pi_{1}^{PLB}$ and $\pi_{0}^{PL} > \pi_{0}^{PLB}$, then solve (B.1) and obtain optimal solution $\{ \eta^{*} ,q^{*} \}$ and calculate $\pi_{0}^{*} \equiv \pi_{0} (\eta^{*} ,q^{*} ,q_{0} (\eta^{*} ,q^{*} ),q_{1} (\eta^{*} ,q^{*} ))$.

Step 4 If the problem in Step 3 has a feasible solution, then the platform offers improved contract terms $\left\{ {\eta^{PLS} ,q^{PLS} } \right\} = \left\{ {\eta^{*} ,q^{*} } \right\}$ and the developer accepts the contract (and concedes the bypassing option). Go to Step 6.

Step 5 If the problem in Step 3 has no feasible solution, then the platform sets basic quality level $q^{PLS} = q^{PLB}$ and the developer bypasses the platform’s distribution and billing systems (the expressions of the equilibrium are given in Proposition 5).

Step 6 End.

A flowchart depicting the procedure above is shown in Fig. 11.

Fig. 11

Flowchart of Procedure 2