Appendix A: Optimal value of Theorem 3.1
The optimal coverage development rate and quality level of first mode are as follows.
$${{r}_{A}}^{*,1}=\frac{an{k}_{1}({c}_{A}{\eta }_{A}-{p}_{A}){h}_{C}}{{m}_{A}{\mathrm{r}}_{0}\left(-2{\mathrm{nk}}_{1}{h}_{C}+{\beta }^{2}\left(1+{u}_{C}\right)\right)}$$
(15)
$${{q}_{A}}^{*,1}=\frac{dn{k}_{1}({c}_{A}{\eta }_{A}-{p}_{A}){h}_{C}}{{s}_{A}\left(-2{\mathrm{nk}}_{1}{h}_{C}+{\beta }^{2}\left(1+{u}_{C}\right)\right)}$$
(16)
Appendix B: Optimal value of Theorem 3.2
The optimal performance and selling price of the services are calculated as follows, respectively.
$${{\mathrm{f}}_{\mathrm{C}}}^{*,1}=\frac{\upbeta {\mathrm{T}}_{1} -{\mathrm{nk}}_{1}{\mathrm{h}}_{\mathrm{C}}{\mathrm{T}}_{2}}{\left(-2{\mathrm{nk}}_{1}{\mathrm{h}}_{\mathrm{C}}+{\upbeta }^{2}\left(1+{\mathrm{u}}_{\mathrm{C}}\right)\right) \left({\mathrm{m}}_{\mathrm{A}}{\mathrm{s}}_{\mathrm{A}} \left(-3{\mathrm{nk}}_{1}\mathrm{h}+{\upbeta }^{2} \left(1 +{\mathrm{u}}_{\mathrm{C}}\right)\right)\right)}$$
(17)
$${{p}_{C}}^{*,1}=\left({\beta }^{4} {m}_{A}{s}_{A}{c}_{C}{\eta }_{C}{\left(1 +{u}_{C}\right)}^{2}+b{k}_{1}{m}_{A}{p}_{A}{s}_{A}{h}_{C}\left(-2{\mathrm{nk}}_{1}{h}_{C}+{\beta }^{2}\left(1+{u}_{C}\right)\right) +{\beta }^{2} {m}_{A}{s}_{A}h\left(1 +{u}_{C}\right)\left(-4n{c}_{C}{\eta }_{C}+\left(1 +{u}_{C}\right)\left({c}_{M}{\eta }_{M}+\gamma {p}_{A,sim}+\mathrm{L}+W\right)\theta \right) +n {{k}_{1}}^{2}{{h}_{C} }^{2}{T}_{4}\right)/\left(-2{\mathrm{nk}}_{1}{h}_{C}+{\beta }^{2}\left(1+{u}_{C}\right)\right) (1 +{u}_{C})\left({m}_{A}{s}_{A} \left(-3n{k}_{1}h+{\beta }^{2} \left(1 +{u}_{C}\right)\right)\right)$$
(18)
With:
$$T_{1} = bm_{A} p_{A} s_{A} \left( {1{ } + u_{C} } \right){ }\left( { - 2nk_{1} h + \beta^{2} { }\left( {1{ } + u_{C} } \right)} \right){ } + \beta^{2} m_{A} s_{A} \left( {1{ } + u_{C} } \right)c_{C} \eta_{C} + \gamma p_{A,sim} { }\left( {1{ } + u_{C} } \right){ } + \left( {1{ } + u_{C} } \right)\left( {c_{M} \eta_{M} + {\text{L}} + W} \right)\theta$$
(19)
$$T_{2} = - d^{2} m_{A} p_{A} \left( {1{ } + u_{C} } \right){ }\left( {1{ } + u_{C} } \right){ }{-}a^{2} { }p_{A} s_{A} \left( {1{ } + u_{C} } \right){ } + c_{C} \eta_{C} \left( {d^{2} m_{A} + a^{2} s_{A} } \right){ }\left( {1{ } + u_{C} } \right){ } + 2m_{A} s_{A} \left( {nc_{C} \eta_{C} + \gamma p_{A,sim} \left( {1{ } + u_{C} } \right){ } + \left( {1{ } + u_{C} } \right){ }\left( {c_{M} \eta_{M} + {\text{L}} + W} \right)\theta } \right)$$
(20)
$$T_{3} = d^{2} m_{A} p_{A} \left( {1{ } + u_{C} } \right){ } + a^{2} { }p_{A} s_{A} \left( {1{ } + u_{C} } \right) - c_{A} \eta_{A} \left( {d^{2} m_{A} + a^{2} s_{A} } \right){ }\left( {1{ } + u_{C} } \right){ } - 2m_{A} s_{A} \left( { - 2nc_{C} \eta_{C} + \gamma p_{A,sim} \left( {1 + u_{C} } \right){ } + \left( {1 + u_{C} } \right)\left( {c_{M} \eta_{M} + {\text{L}} + W} \right)\theta } \right)$$
(21)
Appendix C: Optimal value of Theorem 3.3
The optimal selling price of mobile phone is calculated as follows, respectively.
$${{p}_{M}}^{*,1}=\left(bn{k}_{1}{m}_{A}{p}_{A}{s}_{A}{h}_{C}(1 +{u}_{C})(-2n{k}_{1}{h}_{C}+{\beta }^{2}(1 +{u}_{C}))+{\beta }^{4}{m}_{A}{s}_{A}{(1 +{u}_{C})}^{3}({c}_{M}{\eta }_{M}+\mathrm{L}+W)\theta +{\beta }^{2}n{k}_{1}{m}_{A}{s}_{A}{h}_{C}(1 +{u}_{C})\left(n{c}_{C}{\eta }_{C}+\gamma {p}_{A,sim} (1 +{u}_{C}) -4(1 +{u}_{C})({c}_{M}{\eta }_{M}+\mathrm{L}+W)\theta \right) +{n}^{2}{{k}_{1}}^{2}{{h}_{C}}^{2}{T}_{4}\right)/\left(-2{\mathrm{nk}}_{1}{h}_{C}+{\beta }^{2}\left(1+{u}_{C}\right)\right) (1 +{u}_{C}) \left({m}_{A}{s}_{A} \left(-3n{k}_{1}h+{\beta }^{2} \left(1 +{u}_{C}\right)\right)\right)$$
(22)
With
$$T_{4} = d^{2} m_{A} p_{A} \left( {1 + u_{C} } \right) + a^{2} p_{A} s_{A} \left( {1 + u_{C} } \right) - c_{A} \eta_{A} \left( {d^{2} m_{A} + a^{2} s_{A} } \right)\left( {1 + u_{C} } \right) - 2m_{A} s_{A} \left( {nc_{C} \eta_{C} + \gamma p_{A,sim} \left( {1 + u_{C} } \right) - 2\left( {1 + u_{C} } \right)\left( {c_{M} \eta_{M} + {\text{L}} + W} \right)\theta } \right)$$
(23)
Appendix D: Optimal value of Theorem 3.4
The optimal and lonely \(\left({{f}_{D}}^{*,2},{{p}_{D}}^{*,2}\right)\) can be achieved to optimize mobile applications developer \(D\)'s expected profit.
$${{f}_{D}}^{*,2}=\frac{1}{{T}_{8}}\left((\beta +\sigma )\left({T}_{6}+b{p}_{A}(1+{u}_{D})(-(2n+3\sigma ){k }_{1}{h}_{D}+{\beta }^{2}(1+{u}_{D}) +{T}_{7}+\beta \mu \left(1+{u}_{D}\right)\left(-2\mathrm{a}{r}_{A}{r}_{0}+\mathrm{a}\alpha {r}_{A}{r}_{0}+3 n{c}_{D}{\eta }_{D}-2\mathrm{a}{r}_{A}{r}_{0}{u}_{D}+\mathrm{a}\alpha {r}_{A}{r}_{0}{u}_{D}+3\upgamma {p}_{A,sim}(1+{u}_{D}) -3d{q}_{A}(1+{u}_{D}) +3{p}_{m}\theta +3{p}_{m}\theta {u}_{D}\right)\right)\right)$$
(24)
$${{p}_{D}}^{*,2}=\frac{1}{{T}_{12}}\left({\beta }^{4}{c}_{D}{\eta }_{D}{\left(1+{u}_{D}\right)}^{2}+{T}_{9}+\beta \mu (1+{u}_{D}){T}_{10}-{k }_{1}{h}_{D}\left({T}_{11}+d{q}_{A}\left(1+{u}_{D}\right)(-(2n+3\sigma ){k }_{1}{h}_{D}+2{\mu }^{2}(1+{u}_{D}) )-2{\mu }^{2}{p}_{m}\theta +2n{k }_{1}{h}_{D}{p}_{m}\theta +3\sigma {k }_{1}{h}_{D}{p}_{m}\theta -4{\mu }^{2}{p}_{m}\theta {u}_{D}+2n{k }_{1}{h}_{D}{p}_{m}\theta {u}_{D}+3\sigma {k }_{1}{h}_{D}{p}_{m}\theta {u}_{D} -2{\mu }^{2}{p}_{m}\theta {{u}_{D}}^{2}\right)\right)$$
(25)
With
$${T}_{6}=2{\mu }^{2}\left(\upgamma {p}_{A,sim}-{q}_{A}d\right)-\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}-\left(2n+3\sigma \right)\upgamma {p}_{A,sim}{k }_{1}{h}_{D} +\left(2n+3\sigma \right)d{k }_{1}{q}_{A}{h}_{D}+2\mathrm{a}{k }_{1}{r}_{A}{r}_{0}{h}_{D}\left(n-n\alpha +\sigma \right)-\mathrm{a}\alpha \sigma {k }_{1}{r}_{A}{r}_{0}{h}_{D}+2n{\mu }^{2}{c}_{D}{\eta }_{D}-\left(2n+3\sigma \right){k }_{1}{h}_{D}{c}_{D}{\eta }_{D}+4\left(\upgamma {p}_{A,sim}-{q}_{A}{h}_{D}\right){\mu }^{2}{u}_{D}-2\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}{u}_{D} -2\upgamma {p}_{A,sim}n{k }_{1}{h}_{D}{u}_{D}-3\upgamma {p}_{A,sim}\sigma {k }_{1}{h}_{D}{u}_{D}+\left(2n+3\sigma \right)d{k }_{1}{q}_{A}{h}_{D}{u}_{D}+2\mathrm{a}n{k }_{1}\sigma {h}_{D}{u}_{D}-2\mathrm{a}\left(n\alpha -\sigma \right){k }_{1}{r}_{A}{r}_{0}{h}_{D}{u}_{D}-\mathrm{a}\alpha {k }_{1}{r}_{A}{r}_{0}{h}_{D}{u}_{D}+2n{\mu }^{2}{c}_{D}{\eta }_{D}{u}_{D}+2\upgamma {p}_{A,sim}{\mu }^{2}{{u}_{D}}^{2}-2\mathrm{ d}{\mu }^{2}\mathrm{q }{{u}_{D}}^{2}-\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}{{u}_{D}}^{2}$$
(26)
$${T}_{7}=3\beta \mu (1+{u}_{D})+2{\mu }^{2}(1+{u}_{D}))+2{\mu }^{2}{p}_{m}\theta -2n{k }_{1}{h}_{D}{p}_{m}\theta -3\sigma {k }_{1}{h}_{D}{p}_{m}\theta +4{\mu }^{2}{p}_{m}\theta {u}_{D} -2n{k }_{1}{h}_{D}{p}_{m}\theta {u}_{D}-3\sigma {k }_{1}{h}_{D}{p}_{m}\theta {u}_{D}+2{\mu }^{2}{p}_{m}\theta {{u}_{D}}^{2}+{\beta }^{2}(1+{u}_{D}) (-\mathrm{a}{r}_{A}{r}_{0}+\mathrm{a}\alpha {r}_{A}{r}_{0} +n{c}_{D}{\eta }_{D}-\mathrm{a}{r}_{A}{r}_{0}{u}_{D}+\mathrm{a}\alpha {r}_{A}{r}_{0}{u}_{D}+\upgamma {p}_{A,sim}(1+{u}_{D})-d{q}_{A}(1+{u}_{D})+{p}_{m}\theta +{p}_{m}\theta {u}_{D})$$
(27)
$${T}_{8}=({\beta }^{4}({1+{u}_{D})}^{2}+4{\beta }^{3}\mu ({1+{u}_{D})}^{2}+(2n+\sigma ){k }_{1}{h}_{D}((2n+3\sigma ){k }_{1}{h}_{D}-2{\mu }^{2}(1+{u}_{D}))+2\beta \mu (1+{u}_{D}) (-(4n+3\sigma ){k }_{1}{h}_{D}+{\mu }^{2}(1+{u}_{D}))+{\beta }^{2}(1+{u}_{D}) (-4 (n+\sigma ){k }_{1}{h}_{D}+5{\mu }^{2}(1+{u}_{D})))$$
(28)
$${T}_{9}=4{\beta }^{3}\mu {c}_{D}{\eta }_{D}{\left(1+{u}_{D}\right)}^{2}+b{k }_{1}{p}_{A}{h}_{D}(1+{u}_{D}) (-(2n+3\sigma ){k }_{1}{h}_{D} +{\beta }^{2}(1+{u}_{D})+3\beta \mu (1+{u}_{D}) +2{\mu }^{2}(1+{u}_{D}) )+{\beta }^{2}(1+{u}_{D})(-\mathrm{a}{r}_{A}{r}_{0}{h}_{D}+\mathrm{a}\alpha {r}_{A}{r}_{0}{h}_{D}+5{\mu }^{2}{c}_{D}{\eta }_{D}-3n{k }_{1}{h}_{D}{c}_{D}{\eta }_{D}-4\sigma {k }_{1}{h}_{D}{c}_{D}{\eta }_{D}-\mathrm{a}{r}_{A}{r}_{0}{u}_{D}+\mathrm{a}\alpha {r}_{A}{r}_{0}{u}_{D}+5{\mu }^{2}{c}_{D}{\eta }_{D}{u}_{D}+\upgamma {p}_{A,sim}{k }_{1}{h}_{D}(1+{u}_{D}) -d{k }_{1}{q}_{A}{h}_{D}(1+{u}_{D}) +{k }_{1}{h}_{D}{p}_{m}\theta +{k }_{1}{h}_{D}{p}_{m}\theta {u}_{D})$$
(29)
$${T}_{10}=-2\mathrm{a}{r}_{A}{r}_{0}{h}_{D}+\mathrm{a}\alpha {r}_{A}{r}_{0}{h}_{D}+2{\mu }^{2}{c}_{D}{\eta }_{D}-5n{k }_{1}{h}_{D}{c}_{D}{\eta }_{D}-6\sigma {k }_{1}{h}_{D}{c}_{D}{\eta }_{D}-2\mathrm{a}{k }_{1}{r}_{A}{r}_{0}{u}_{D}{h}_{D}+\mathrm{a}\alpha {k }_{1}{r}_{A}{r}_{0}{u}_{D}{h}_{D}+2{\mu }^{2}{c}_{D}{\eta }_{D}{u}_{D}+3\upgamma {p}_{A,sim}{k }_{1}{h}_{D}(1+{u}_{D}) -3d{k }_{1}{q}_{A}{h}_{D}(1+{u}_{D}) +3{k }_{1}{h}_{D}{p}_{m}\theta +3{u}_{D}{k }_{1}{h}_{D}{p}_{m}\theta$$
(30)
$${T}_{11}=\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}-2\mathrm{a}{k }_{1}{r}_{A}{r}_{0}{h}_{D}\left(n-n\alpha +\sigma \right)+\mathrm{a}\alpha \sigma {k }_{1}{r}_{A}{r}_{0}{h}_{D}+2n{\mu }^{2}{c}_{D}{\eta }_{D}+2\sigma {\mu }^{2}{c}_{D}{\eta }_{D}-2{n}^{2}{k }_{1}{h}_{D}{c}_{D}{\eta }_{D}-5n{k }_{1}{h}_{D}{c}_{D}{\eta }_{D}\sigma -3{\sigma }^{2}{k }_{1}{h}_{D}{c}_{D}{\eta }_{D}+2{\mu }^{2}{r}_{A}{r}_{0}{u}_{D}-2\mathrm{a}{k }_{1}{r}_{A}{r}_{0}{u}_{D}{h}_{D}\left(n-n\alpha +\sigma \right)+\mathrm{a}\alpha \sigma {k }_{1}{r}_{A}{r}_{0}{u}_{D}{h}_{D}+2{\mu }^{2}{c}_{D}{\eta }_{D}{u}_{D}\left(n+\sigma \right) +\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}{{u}_{D}}^{2}+\upgamma {p}_{A,sim}(1+{u}_{D})(-(2n+3\sigma ){k }_{1}{h}_{D}+2{\mu }^{2}(1+{u}_{D}) )$$
(31)
$$T_{12} = \left( {1 + u_{D} } \right){ }\left( {\beta^{4} \left( {1 + u_{D} } \right){ }^{2} + 4\beta^{3} \mu \left( {1 + u_{D} } \right){ }^{2} + { }\left( {2n + \sigma } \right)k _{1} h_{D} \left( {\left( {2n + 3\sigma } \right)k _{1} h_{D} - 2\mu^{2} \left( {1 + u_{D} } \right){ }} \right){ } + 2\beta \mu \left( {1 + u_{D} } \right)\left( { - \left( {4n + 3\sigma } \right)k _{1} h_{D} + { }\mu^{2} \left( {1 + u_{D} } \right)} \right) + \beta^{2} \left( {1 + u_{D} } \right)\left( { - 4{ }\left( {n + \sigma } \right)k _{1} h_{D} + 5\mu^{2} \left( {1 + u_{D} } \right)} \right)} \right)$$
(32)
Appendix E: Optimal value of Theorem 3.5
The optimal and lonely \(\left({{r}_{A}}^{*2}, {{q}_{A}}^{*,2}\right)\) can be achieved to optimize the operator \(A\)’s expected profit. The optimal coverage development rate and quality level are calculated as follows, respectively.
$${{r}_{A}}^{*,2}=\frac{1}{{\mathrm{m}}_{A}{{r}_{0}}^{2}}\left(-\frac{\left({c}_{A}{\eta }_{A}\left(n+\sigma \right){k }_{1}{h}_{D}\left(-\mathrm{a}{r}_{0}-\mathrm{a}{r}_{0}{u}_{D}\right)\right)}{\left(1+{u}_{D}\right) \left(\left(-2n-\sigma \right){k }_{1}{h}_{D}+{\beta }^{2}\left(1 +{u}_{C}\right)+\beta \mu \left(1 +{u}_{C}\right)\right)}+\frac{\left(n+\sigma \right){k }_{1}{h}_{D}{p}_{A}\left(-\mathrm{a}{r}_{0}-\mathrm{a}{r}_{0}{u}_{C}\right)}{\left(1 +{u}_{C}\right)}-\left(\left(-2n-\sigma \right){k }_{1}{h}_{C}+\beta \mu (1 +{u}_{D})+\beta \mu (1 +{u}_{D})\right)\right)-{F}_{A}$$
(33)
$${{q}_{A}}^{*,2}=\frac{1}{{s}_{A}}\left(\frac{-{c}_{A}{\eta }_{A}(n+\sigma ){k }_{1}{h}_{D} \left(-2d-2d{u}_{C}\right)}{(1 +{u}_{C}) \left((-2n-\sigma ){k }_{1}{h}_{D}+\left({\beta }^{2}+\beta \mu \right)\left(1 +{u}_{C}\right)\right)} - \frac{(n+\sigma ){k }_{1}{h}_{D}{p}_{A}(-2d-2d{u}_{D})}{(1 +{u}_{D})\left((-2n-\sigma ){k }_{1}{h}_{D}+{\beta }^{2}(1 +{u}_{C})+\beta \mu \left(1 +{u}_{D}\right)\right)}\right)$$
(34)
Appendix F: Optimal value of Theorem 3.6
Thus, the optimal and lonely \(\left({{f}_{C}}^{*,2},{{p}_{C}}^{*,2}\right)\) can be achieved to optimize application developer \(C\)'s expected profit. The optimal performance and selling price of the services are calculated as follows, respectively.
$${{f}_{C}}^{*,2}=\frac{1}{{T}_{15}}\left((\beta +\mu ) ({T}_{13}+\mathrm{b}{p}_{A}(1+{u}_{C})\left(-(2n+3\sigma ){k}_{1}{h}_{C} +{\beta }^{2}(1+{u}_{C}) +3\beta \mu (1+{u}_{C})+2{\mu }^{2}\left(1+{u}_{C}\right)\right) +{T}_{14}+{\beta }^{2}(1+{u}_{C})\left(-\mathrm{a}\alpha{r}_{A}{r}_{0}+n{c}_{C}{\eta }_{C} -\mathrm{a}\alpha{r}_{A}{r}_{0}{u}_{C}+\upgamma {p}_{A,sim}(1+{u}_{C})-d{q}_{A}(1+{u}_{C})+{p}_{m}\theta +{p}_{m}\theta {u}_{C}\right)\right)$$
(35)
$${{p}_{C}}^{*,2}=\frac{1}{{T}_{15}}\left({\beta }^{3}{\left(1+{u}_{C}\right)}^{2}{c}_{C}{\eta }_{C}(\beta +4\mu )+b{k}_{1}{p}_{A}{h}_{C}(1+{u}_{C})\left(-(2n +3\sigma ){k}_{1}{h}_{C}+{\beta }^{2}(1+{u}_{C})(\beta +3\mu )+2 {\mu }^{2}\left(1+{u}_{C}\right)\right)+{T}_{16}+{k}_{1}{h}_{C}{T}_{17}\right)$$
(36)
with
$${T}_{13}=\upgamma {p}_{A,sim}{\mu }^{2}-\mathrm{a}{\mu }^{2}\left({r}_{A}{r}_{0}+d{q}_{A}\right)- \left(2n+3\sigma \right){k}_{1}{h}_{C}\left(\upgamma {p}_{A,sim}+d{q}_{A}\right) +\mathrm{a}{k}_{1}{r}_{A}{r}_{0}{h}_{C}\left(2{\alpha }n+\sigma +{\alpha }\sigma \right)\left(1+{u}_{C}\right)+2n{\mu }^{2}{c}_{C}{\eta }_{C} -\left(2n+3\sigma \right){k}_{1}{h}_{C}{c}_{C}{\eta }_{C} +4\upgamma {p}_{A,sim}{\mu }^{2}{u}_{C}\left(\upgamma {p}_{A,sim}-d{q}_{A}\right)-2\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}{u}_{C}-\left(2n+3\sigma \right)\upgamma {p}_{A,sim}{k}_{1}{h}_{C}{u}_{C}\left(-\upgamma {p}_{A,sim}-d{q}_{A}\right)+2n{\mu }^{2}{c}_{C}{\eta }_{C}{u}_{C}+ 2\upgamma {p}_{A,sim}{\mu }^{2}{{u}_{C}}^{2} -2d{\mu }^{2}{q}_{A}{{u}_{C}}^{2}-\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}{{u}_{C}}^{2}$$
(37)
$${T}_{14}=2{\mu }^{2}{p}_{m}\theta -\left(2n+3\sigma \right){\left(1+{u}_{C}\right)k}_{1}{p}_{m}{h}_{C}\theta +4{\mu }^{2}{p}_{m}{u}_{C}\theta +2{\mu }^{2}{p}_{m}{{u}_{C}}^{2}\theta -\beta \mu (1+{u}_{C})\left(\left(\mathrm{a}{r}_{A}{r}_{0}+\mathrm{a}\alpha{r}_{A}{r}_{0}\right)(1+{u}_{C})-3n{c}_{C}{\eta }_{C}-3\upgamma {p}_{A,sim}(1+{u}_{C}) +\left(3d{q}_{A}-3{p}_{m}\theta \right)(1+{u}_{C})\right)$$
(38)
$${T}_{15}={\beta }^{4}{\left(1+{u}_{C}\right)}^{2}+4{\beta }^{3}\mu {(1+{u}_{C})}^{2}+ (2n +\sigma ){k}_{1}{h}_{C}((2 n+3\sigma ){k}_{1}{h}_{C}-2{\mu }^{2}(1+{u}_{C}))+2\beta \mu (1+{u}_{C}) (-(4n+3\sigma ){k}_{1}{h}_{C}+{\mu }^{2}(1+{u}_{C})) +{\beta }^{2}(1+{u}_{C})(-4 (n+\sigma ){k}_{1}{h}_{C}+5{\mu }^{2}(1+{u}_{C}))$$
(39)
$${{T}_{16}=\beta }^{2}(1+{u}_{C})\left(-\mathrm{a}\alpha{k}_{1}{r}_{A}{r}_{0}{h}_{C}+ 5{\mu }^{2}{c}_{C}{\eta }_{C}(1+{u}_{C})-(3n{k}_{1}+4\sigma ){h}_{C}{c}_{C}-\mathrm{a}\alpha{k}_{1}{r}_{A}{r}_{0}{h}_{C}{u}_{C}+\upgamma {p}_{A,sim}{k}_{1}{h}_{C}(1+{u}_{C}) -d{k}_{1}{q}_{A}{h}_{C}(1+{u}_{C}){k}_{1}{p}_{m}{h}_{C}\theta +{k}_{1}{p}_{m}{h}_{C}\theta {u}_{C}\right)+\beta \mu (1+{u}_{C})\left(-\mathrm{a}{k}_{1}{r}_{A}{r}_{0}{h}_{C}(1+{\alpha })+2{\mu }^{2}{c}_{C}{\eta }_{C} -(5n+6\sigma {k}_{1}){h}_{C}{c}_{C}{\eta }_{C}-\mathrm{a}{k}_{1}{r}_{A}{r}_{0}{u}_{C}(1+{h}_{C})+2{\mu }^{2}{c}_{C}{\eta }_{C}{u}_{C}+3\upgamma {p}_{A,sim}{k}_{1}{h}_{C}(1+{u}_{C}) -d{k}_{1}{q}_{A}{h}_{C}(1+{u}_{C})+3{k}_{1}{p}_{m}{h}_{C}\theta (1+{u}_{C})\right)$$
(40)
$${T}_{17}=-\mathrm{a}{\mu}^{2}{r}_{A}{r}_{0}+ 2\mathrm{a}\alpha {k}_{1}{r}_{A}{r}_{0}{h}_{C} +\mathrm{a}\alpha{k}_{1}{r}_{A}{r}_{0}{h}_{C}+\mathrm{a}\alpha{k}_{1}{r}_{A}{r}_{0}{h}_{C}\sigma -2n{c}_{C}{\eta }_{C}-2\sigma {\mu }^{2}{c}_{C}{\eta }_{C} +2{n}^{2}{k}_{1}{h}_{C}{c}_{C}{\eta }_{C}+5n\sigma {k}_{1}{h}_{C}{c}_{C}{\eta }_{C}+3{\sigma }^{2}{k}_{1}{h}_{C}{c}_{C}{\eta }_{C}-2\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}{u}_{C}+2\mathrm{a}\alpha{k}_{1}{r}_{A}{r}_{0}{h}_{C}{u}_{C}n+\mathrm{a}\sigma {k}_{1}{h}_{C}{r}_{A}{r}_{0}+\mathrm{a}\alpha{k}_{1}{r}_{A}{r}_{0}{h}_{C}\sigma {\eta }_{C}{u}_{C} -2n{\mu }^{2}{c}_{C}{\eta }_{C}{u}_{C}-2\sigma {\mu }^{2}{c}_{C}{\eta }_{C}{u}_{C}-\mathrm{a}{\mu }^{2}{r}_{A}{r}_{0}{{u}_{C} }^{2}+\upgamma {p}_{A,sim}(1+{u}_{C})(-(2n+3\sigma ){k}_{1}{h}_{C}+2{\mu }^{2}(1+{u}_{C}))-d{q}_{A}(1+{u}_{C})(-(2n+3\sigma ){k}_{1}{h}_{C}+2{\mu }^{2}(1+{u}_{C}))+2{\mu }^{2}{p}_{m}\theta -2n{k}_{1}{p}_{m}{h}_{C}\theta -3\sigma {k}_{1}{p}_{m}{h}_{C}\theta +4{\mu }^{2}{p}_{m}\theta {u}_{C}-2n{k}_{1}{p}_{m}{h}_{C}\theta {u}_{C} -3\sigma {k}_{1}{p}_{m}{h}_{C}\theta {u}_{C} +2{\mu }^{2}{p}_{m}\theta {{u}_{C} }^{2})$$
(41)
Appendix G: Optimal value of Theorem 3.7
The optimal and exclusive \({{(p}_{M}}^{*,2})\) can be achieved to optimize mobile phones manufacturer \(M\)'s expected profit. The optimal selling price of mobile phone is calculated as follows, respectively.
$${{p}_{M}}^{*,2}= \frac{1}{4\left(1+{u}_{D}\right)\theta }(2d{q}_{A}+\mathrm{a}{r}_{0}{r}_{A}-2n{c}_{C}{\eta }_{C}-2n{c}_{D}{\eta }_{D}+2d{q}_{A}{u}_{C}+\mathrm{a}{r}_{0}{r}_{A}{u}_{D}-2b{p}_{A}(1 +{u}_{C})-2\upgamma {p}_{A,sim}(1+{u}_{D})+2{c}_{M}{\eta }_{M}\theta +2\mathrm{L}\theta +2{c}_{M}{\eta }_{M}\theta {u}_{C}+2{c}_{M}{\eta }_{M}\theta {u}_{D}+2\mathrm{L}\theta {u}_{C}+2\mathrm{L}\theta {u}_{D}+2W\theta +2W\theta {u}_{C}+2W\theta {u}_{D})$$
(42)
Appendix H: Optimal value of Theorem 3.8
The optimal and exclusive \(\left({{r}_{B}}^{*,2},{{q}_{B}}^{*,2}\right)\) can be achieved to optimize the mobile operator \(B\)'s expected profit.
$${{r}_{B}}^{*,3}=\frac{1}{4({m}_{B}){{r}_{0}}^{2}\left(-n{k }_{1}{h}_{C}+{\beta }^{2}(1+{u}_{C})\right)}\left(n{k }_{1}{h}_{C}(4 (-1+\alpha ){p}_{B,sim}-D({c}_{B}{\eta }_{B}-{p}_{B})(3\mathrm{a}(-1+\alpha ){r}_{0}-4\varepsilon ))+2{\beta }^{2}(1+{u}_{C})\left(-2 (-1+\alpha ){p}_{B,sim}+D({c}_{B}{\eta }_{B}-{p}_{B})(\mathrm{a }(-1+\alpha ){r}_{0} -2\varepsilon )\right)\right)$$
(43)
$${{q}_{B}}^{*,3}=-\left(\frac{\left({c}_{B}{\eta }_{B}-{p}_{B}\right)\left(-n{k }_{1}\left(3d+4\varepsilon \right){h}_{C}+2{\beta }^{2}\left(d+2\varepsilon \right)\left(1+{u}_{C}\right)\right)}{4{s}_{B}\left(-n{k }_{1}{h}_{C}+ {\beta }^{2}\left(1+{u}_{C}\right)\right)}\right)$$
(44)
Appendix I: Optimal value of Theorem 3.9
The optimal coverage development rate and quality level are calculated as follows, respectively.
$${{r}_{A}}^{*,3}=\frac{1}{4({m}_{A}){{r}_{0}}^{2}}\left(\alpha \left(\frac{4 {p}_{A,sim}}{D}+2\mathrm{a}\left(-{c}_{A}{\eta }_{A}+{p}_{A}\right){r}_{0}+\frac{\mathrm{a}n{k }_{1}\left(-{c}_{B}{\eta }_{B}+{p}_{B}\right){r}_{0}{h}_{C}}{n{k }_{1}{h}_{C}-{\beta }^{2}(1+{u}_{C})}\right)+4\left(-{c}_{A}{\eta }_{A}+{p}_{A}\right)\varepsilon \right)$$
(45)
$${{q}_{A}}^{*,3}=\frac{1}{{s}_{A}}\left(\frac{-{c}_{A}{\eta }_{A}(n+\sigma ){k }_{1}{h}_{D} \left(-2d-2d{u}_{C}\right)}{(1 +{u}_{C}) \left((-2n-\sigma ){k }_{1}{h}_{D}+\left({\beta }^{2}+\beta \mu \right)\left(1 +{u}_{C}\right)\right)} - \frac{(n+\sigma ){k }_{1}{h}_{D}{p}_{A}(-2d-2d{u}_{D})}{(1 +{u}_{D})\left((-2n-\sigma ){k }_{1}{h}_{D}+{\beta }^{2}(1 +{u}_{C})+\beta \mu \left(1 +{u}_{D}\right)\right)}\right)$$
(46)
Appendix J: Optimal value of Theorem 3.10
The optimal and exclusive \(\left({{f}_{C}}^{*,3},{{p}_{C}}^{*,3}\right)\) can be obtained to optimize application developer \(C\)'s expected profit. The optimal performance and selling price of the services are calculated as follows, respectively.
$${{f}_{C}}^{*,3}= \frac{1}{(16D{m}_{A}{s}_{A}{r}_{0}+8D{m}_{B}{p}_{B}{r}_{0}){(n{k }_{1}{h}_{C}-{ \beta }^{2}(1+{u}_{C}))}^{2}}\left(2{ \beta }^{3}(1 +{u}_{C}){T}_{18}-\beta n{k }_{1}{h}_{C}(-6{d}^{2}D{m}_{B}{p}_{B}{r}_{0}(1+{u}_{C}) -8dD{m}_{B}\updelta {p}_{B}{r}_{0}(1+{u}_{C})+{c}_{B}{\eta }_{B}D(1+{u}_{C}){T}_{19}\right)$$
(47)
$${{p}_{C}}^{*,3}=\frac{1}{32D{m}_{A}{s}_{A}{r}_{0}(1+{u}_{C}){(n{k }_{1}{h}_{C}- {\beta }^{2}(1+{u}_{C}))}^{2}}\left(32{ \beta }^{4}D{m}_{A}{s}_{A}{r}_{0}{c}_{C}{\eta }_{C}{\left(1+{u}_{C}\right)}^{2}+2{ \beta }^{2}{k }_{1}{h}_{C}(1+{u}_{C}) {T}_{20}+n{{k }_{1}}^{2}{{h}_{C}}^{2}\left(6{ d }^{2}{m}_{A}{p}_{A}{r}_{0}D(1+{u}_{C}) +8dD{m}_{B}\updelta {p}_{B}{r}_{0}(1+{u}_{C})-{c}_{B}{\eta }_{B}D(1+{u}_{C})\left(6{ d }^{2}{m}_{B}{r}_{0}+ 8D{m}_{B}\updelta {r}_{0}+\mathrm{a}{s}_{B}(3\mathrm{a}(1+2((-1+\alpha )\alpha ){r}_{0}+4\varepsilon ))+{s}_{B}{T}_{21}\right)\right)\right)$$
(48)
With
$${T}_{18}=-2{d}^{2}D{m}_{A}{p}_{A}{r}_{0}(1+{u}_{C})-4dD{m}_{A}\updelta {p}_{A}{r}_{0}(1+{u}_{C}) +{c}_{A}{\eta }_{A}D(1+{u}_{C})(2d{m}_{A}(d+2\updelta ){r}_{0}+{a}^{2}(1+2 (-1+\alpha )\alpha ){r}_{0}{\mathrm{s}}_{A}+2{\mathrm{as}}_{A}\varepsilon ) +{\mathrm{s}}_{B}(-{a}^{2}(1 + 2 (-1+\alpha )\alpha )D{p}_{B}{r}_{0}(1+{u}_{C})-2\mathrm{a}(1+{u}_{C})({p}_{A,sim}+{p}_{B,sim}+2(-1+\alpha )\alpha ({p}_{A,sim}+{p}_{B,sim})+D{p}_{B}\varepsilon ) +4D{m}_{B}{p}_{B}{r}_{0}((n{c}_{C}{\eta }_{C}+(d{p}_{A}+d\alpha {p}_{B})(1+{u}_{C}) +(\gamma {p}_{A,sim}+\gamma \alpha {p}_{B,sim})(1+{u}_{C}))+(1+{u}_{C})({c}_{M}{\eta }_{M}+\mathrm{L}+W)\theta ))$$
(49)
$${T}_{19}=6{d}^{2}{m}_{A}{r}_{0}+8d{m}_{B}\updelta {r}_{0}+{\mathrm{as}}_{B}(3\mathrm{a }(1 + 2 (-1+\alpha )\alpha ){r}_{0}+4\varepsilon )) +{\mathrm{s}}_{B}\left(-3{a}^{2}(1 + 2 (-1+\alpha )\alpha )D{p}_{B}{r}_{0}(1+{u}_{C}) -4\mathrm{a}(1+{u}_{C})({p}_{A,sim} +2(-1+\alpha )\alpha {p}_{B,sim}+D{p}_{B}\varepsilon ) +D{m}_{B}{r}_{0}\left(\left(n{c}_{C}{\eta }_{C}+b{p}_{A}(1+{u}_{C}) +(\gamma {p}_{A,sim}+\gamma \alpha {p}_{B,sim})(1+{u}_{C})\right) + (1+{u}_{C})\left({c}_{M}{\eta }_{M}+\mathrm{L}+W\right)\theta \right)\right)$$
(50)
$${T}_{20}=-2{ d }^{4}{m}_{A}{p}_{A}{r}_{0}D(1+{u}_{C}) -4dD{m}_{A}\updelta {p}_{A}{r}_{0}(1+{u}_{C})+{c}_{A}{\eta }_{A}D(1+{u}_{C})(2dD{m}_{B}(d+2\updelta ){r}_{0}+{ a }^{2}(1 +2(-1+\alpha )\alpha ){r}_{0}{s}_{B}+2\mathrm{a}{s}_{B}\varepsilon ) -{s}_{B}({ a }^{2}(1 + 2 ((-1+\alpha )\alpha ){p}_{B}{r}_{0}D(1+{u}_{C})+2a(1+{u}_{C})({p}_{B,sim}+{p}_{A,sim}+ 2 (-1+\alpha )\alpha {p}_{A,sim}+{p}_{B}{r}_{0}D\varepsilon ) +4D{m}_{B}{r}_{0}(-(-7n{c}_{C}{\eta }_{C}+b{p}_{B}(1+{u}_{C})+(\gamma {p}_{A,sim}+\gamma \alpha {p}_{B,sim})(1+{u}_{C})) -(1+{u}_{C})({c}_{M}{\eta }_{M}+\mathrm{L}+W)\theta ))$$
(51)
$$T_{21} = 3{ }a{ }^{2} \left( {1{ } + { }2\left( { - 1 + \alpha } \right)\alpha } \right)Dp_{B} r_{0} \left( {1 + u_{C} } \right) + 4{\text{a}}\left( {1 + u_{C} } \right){ }\left( {p_{B,sim} + p_{A,sim} + 2{ }\left( { - 1 + \alpha } \right)\alpha \left( {p_{B,sim} + p_{A,sim} } \right) + Dp_{B} \varepsilon } \right){ } + 8m_{A} r_{0} \left( { - \left( { - 3c_{C} \eta_{C} + bp_{B} \left( {1 + u_{C} } \right) + \gamma \alpha p_{B,sim} \left( {1 + u_{C} } \right)} \right) - \left( {1 + u_{C} } \right)\left( {c_{M} \eta_{M} + {\text{L}} + W} \right)\theta } \right)$$
(52)
Appendix K: Optimal value of Theorem 3.11
The optimal selling price of mobile phone is calculated as follows, respectively.
$${{p}_{M}}^{*,3}=\frac{1}{16D{m}_{B}{s}_{B}{r}_{0}(1+{u}_{C})\left(-n{k }_{1}{h}_{C}+ {\beta }^{2}\left(1+{u}_{C}\right)\right)\theta }\left(-2{\beta }^{2}(1+{u}_{C}){T}_{22}+n{k }_{1}{h}_{C}(-6{d }^{2}D{m}_{B}{p}_{B}{r}_{0}(1+{u}_{C})-8dD{m}_{A}\updelta {p}_{B}{r}_{0}(1+{u}_{C})+{c}_{B}{\eta }_{B}D(1+{u}_{C})\left(6{d }^{2}{m}_{A}{r}_{0} +8dD{m}_{A}\updelta {p}_{B}{r}_{0} +\mathrm{a}{s}_{B}(3\mathrm{a}(1 + 2 (-1+\alpha )\alpha ){r}_{0}+4\varepsilon ))-{s}_{B}{T}_{23}\right)\right)$$
(53)
with
$${T}_{22}=-2 {d }^{2}D{m}_{A}{p}_{A}{r}_{0}(1+{u}_{C}) -4dD{m}_{A}{p}_{A}{r}_{0}\varepsilon (1+{u}_{C})+{c}_{A}{\eta }_{A}D(1+{u}_{C})(2d{m}_{B}(d+2\varepsilon ){r}_{0}+{a }^{2}(1 +2 (-1+\alpha )\alpha ){r}_{0}{s}_{B}+2\mathrm{a}{s}_{A}\updelta )-\mathrm{s}({a }^{2}(1 + 2 (-1+\alpha )\alpha )D{p}_{B}{r}_{0}(1+{u}_{C})+2\mathrm{a}(1+{u}_{C}) ({p}_{A,sim}+2(-1+\alpha )\alpha {p}_{B,sim}+D{p}_{A}\upvarepsilon )+4D{m}_{A}{r}_{0}(-(n{c}_{C}{\eta }_{C}+\mathrm{b}{p}_{B}(1+{u}_{C})+\gamma {p}_{A,sim}(1+{u}_{C}))+(1+{u}_{C})({c}_{M}{\eta }_{M}+\mathrm{L}+W)\theta ))$$
(54)
$${T}_{23}=3{a }^{2}(1 + 2 (-1+\alpha )\alpha )D{p}_{A}{r}_{0}(1+{u}_{C})+4\mathrm{a }(1+{u}_{C}) ({p}_{A,sim}+2(-1+\alpha )\alpha {p}_{B,sim} +D{p}_{B}\varepsilon ) +8D{m}_{A}{r}_{0}(-(n{c}_{C}{\eta }_{C}+\mathrm{b}{p}_{A}(1+{u}_{C})+\gamma {p}_{B,sim}(1+{u}_{C}))+(1+{u}_{C})({c}_{M}{\eta }_{M}+\mathrm{L}+W)\theta )$$
(55)
