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January  2022, 18(1): 315-339. doi: 10.3934/jimo.2020155

Preannouncement strategy of platform-type new product for competing platforms: Technical or marketing information

 1 Center for Behavioral Decision and Control, School of Management and Engineering, Nanjing University, Hankou Road 22, Nanjing, Jiangsu 210093, China 2 School of Business, Jiangsu Open University, Jiangdong North Road 399, Nanjing, Jiangsu 210036, China

* Corresponding author: Tiaojun Xiao

Received  March 2020 Revised  July 2020 Published  January 2022 Early access  October 2020

Fund Project: This research is funded by the National Natural Science Foundation of China (No. 71871112) and Jiangsu province's "333 project" training funding project (No. BRA2019040)

What message should be released to consumers and developers is an important part of the preannouncement strategy of platforms' new product. From the perspectives of consumers and developers' information perceptions, we develop a game model of two-sided market, which can better describe the impacts of information preannouncement on consumers, developers, and platforms behavior in a competitive environment. There are two preannouncement strategies: Technical or marketing information. Our studies reveal that (i) when the development capabilities are heterogeneous enough, both platforms release technical information; (ii) both platforms preannounce marketing information when the heterogeneity of development capability is sufficiently small, even if it decreases total social welfare; (iii) the platform lacking competitive advantage is more inclined to adopt a strategy different from the competitive advantage platform, and competitive advantage platform is likely to change the preannouncement strategy constantly; (iv) the heterogeneity of platforms is the prerequisite for the asymmetric equilibrium, even if it may decrease the overall social welfare.

Citation: Ye Jiang, Tiaojun Xiao. Preannouncement strategy of platform-type new product for competing platforms: Technical or marketing information. Journal of Industrial & Management Optimization, 2022, 18 (1) : 315-339. doi: 10.3934/jimo.2020155
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References:
Game structure of the PNPP
(a) SPNE change with $t_{e}$ when $t = 15$ and $f_{e} = 1$ and (b)SPNE change with $f_{e}$ when $t = 1$ and $t_{e} = 15$
(a) SPNE distribution when $\beta_{b} = 1.01$ and (b)SPNE distribution when $\beta_{b} = 5.01$
(a) Consumer price changes with $\delta$ and (b)Developer price changes with $\delta$
(a) SPNE changes with $f_{e}$ when $\delta = 0.2$ and (b)SPNE changes with $f_{e}$ when $\delta = 1.2$
SPNE changes with $\delta$
Total social welfare changes with $\delta$
Equilibrium results under $(M, M)$
 Decisions Consumers' price $p_{bi}^{MM}=t+t_{e}+\beta_{b}(1-g)/(4g)-\beta_{b}(2v_{d} +\beta_{b}+3\beta_{d})/(8gc_{d})$ Developers' price $p_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}-\beta_{d}]/4$ Consumers' number $n_{bi}^{MM}=1/2$ Developers' number $n_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}+\beta_{d}]/(8gc_{d})$ Equilibrium profits/total utilities Platforms' profit $\Pi_{i}^{MM}=(t+t_{e})/2+(4V_{1}^2-B_{0}-2\beta_{b}\beta_{d})/(32gc_{d})-c_{M}$ Consumers' total utility $BS_{i}^{MM}=[4v_{b}-5(t+t_{e})]/8+[B_{0}+2(\beta_{b}+\beta_{d})V_{1}]/(16gc_{d})$ Developers' total utility $DS_{i}^{MM}=(2V_{1}+\beta_{b}+\beta_{d})^2/(64gc_{d})$
 Decisions Consumers' price $p_{bi}^{MM}=t+t_{e}+\beta_{b}(1-g)/(4g)-\beta_{b}(2v_{d} +\beta_{b}+3\beta_{d})/(8gc_{d})$ Developers' price $p_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}-\beta_{d}]/4$ Consumers' number $n_{bi}^{MM}=1/2$ Developers' number $n_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}+\beta_{d}]/(8gc_{d})$ Equilibrium profits/total utilities Platforms' profit $\Pi_{i}^{MM}=(t+t_{e})/2+(4V_{1}^2-B_{0}-2\beta_{b}\beta_{d})/(32gc_{d})-c_{M}$ Consumers' total utility $BS_{i}^{MM}=[4v_{b}-5(t+t_{e})]/8+[B_{0}+2(\beta_{b}+\beta_{d})V_{1}]/(16gc_{d})$ Developers' total utility $DS_{i}^{MM}=(2V_{1}+\beta_{b}+\beta_{d})^2/(64gc_{d})$
Nash equilibrium under $(T, T)$
 Decisions Consumers' price $p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d})$ Developers' price $p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4$ Consumers' number $n_{bi}^{TT}=1/2$ Developers' number $n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d})$ Equilibrium profits/total utilities Platforms' profit $\Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T}$ Consumers' total utility $BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d})$ Developers' total utility $DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d})$
 Decisions Consumers' price $p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d})$ Developers' price $p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4$ Consumers' number $n_{bi}^{TT}=1/2$ Developers' number $n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d})$ Equilibrium profits/total utilities Platforms' profit $\Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T}$ Consumers' total utility $BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d})$ Developers' total utility $DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d})$
Nash equilibrium under $(M, T)$
 Decisions Consumers' price $p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d})$ Developers' price $p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4$ Consumers' number $n_{bi}^{TT}=1/2$ Developers' number $n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d})$ Equilibrium profits/total utilities Platforms' profit $\Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T}$ Consumers' total utility $BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d})$ Developers' total utility $DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d})$
 Decisions Consumers' price $p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d})$ Developers' price $p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4$ Consumers' number $n_{bi}^{TT}=1/2$ Developers' number $n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d})$ Equilibrium profits/total utilities Platforms' profit $\Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T}$ Consumers' total utility $BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d})$ Developers' total utility $DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d})$
Factors of effect asymmetric strategy to be a SPNE
 $t=1$ $c_{d}=1$ $c_{d}$ $\hat{f}_{e1}$ $\hat{f}_{e2}$ $\hat{f}_{e2}-\hat{f}_{e1}$ $t$ $\hat{f}_{e1}$ $\hat{f}_{e2}$ $\hat{f}_{e2}-\hat{f}_{e1}$ 1 0.31 0.61 0.30 1 0.31 0.61 0.30 3 0.93 1.72 0.79 3 0.34 0.58 0.24 5 1.52 2.74 1.22 5 0.36 0.56 0.20 7 2.08 3.69 1.61 7 0.37 0.54 0.17 9 2.62 4.59 1.96 9 0.38 0.53 0.15
 $t=1$ $c_{d}=1$ $c_{d}$ $\hat{f}_{e1}$ $\hat{f}_{e2}$ $\hat{f}_{e2}-\hat{f}_{e1}$ $t$ $\hat{f}_{e1}$ $\hat{f}_{e2}$ $\hat{f}_{e2}-\hat{f}_{e1}$ 1 0.31 0.61 0.30 1 0.31 0.61 0.30 3 0.93 1.72 0.79 3 0.34 0.58 0.24 5 1.52 2.74 1.22 5 0.36 0.56 0.20 7 2.08 3.69 1.61 7 0.37 0.54 0.17 9 2.62 4.59 1.96 9 0.38 0.53 0.15
Nash equilibrium of non-identical platforms
 $(M, M)$ $(T, T)$ $p_{bi, E}^{MM}=p_{bi}^{MM}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{1})-E_{3}/(12gc_{d})]$ $p_{bi, E}^{TT}=p_{bi}^{TT}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{2})-E_{3}/(12gc_{d})]$ $p_{di, E}^{MM}=p_{di}^{MM}+F(i)E_{2}/W_{1}+(v_{di}-v_{d})/2$ $p_{di, E}^{TT}=p_{di}^{TT}+F(i)E_{2}/(4W_{2})+(v_{di}-v_{d})/2$ $n_{bi, E}^{MM}=n_{bi}^{MM}+F(i)E_{3}/W_{1}$ $n_{bi, E}^{TT}=n_{bi}^{TT}+F(i)E_{3}/W_{2}$ $n_{di, E}^{MM}=n_{di}^{MM}+F(i)E_{4}/W_{1}+(v_{di}-v_{d})/(4gc_{d})$ $n_{di, E}^{TT}=n_{di}^{TT}+F(i)E_{4}/W_{2}+(v_{di}-v_{d})/(4gc_{d})$ $\Pi_{i, E}^{MM}=\Pi_{i}^{MM}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}-\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} $\Pi_{i, E}^{TT}=\Pi_{i}^{TT}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}+\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} $(M, T)$ $p_{bi, E}^{MT}=p_{bi}^{MT}-\beta_{b}(v_{di}-v_{d})/(4gc_{d})-F(i)E_{3}(B_{2}-2W_{3})/(24gc_{d}W_{3})$ $p_{di, E}^{MT}=p_{di}^{MT}+F(i)E_{2}/(4W_{3})+(v_{di}-v_{d})/2$ $n_{bi, E}^{MT}=n_{bi}^{MT}+F(i)E_{3}/(2W_{3})$ $n_{di, E}^{MT}=n_{di}^{MT}+F(i)E_{4}/W_{3}+(v_{di}-v_{d})/(4gc_{d})$ $\Pi_{1, E}^{MT}=\Pi_{1}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}+\delta-2\phi)\delta]/(96gc_{d})$ $-(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})-(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ $\Pi_{2, E}^{MT}=\Pi_{2}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}-\delta-2\phi+2f_{e})\delta]/(96gc_{d})$ $+(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ $(T, M)$ $\Pi_{1, E}^{TM}=\Pi_{1, E}^{MT}+[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[8\delta(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2]/(24gc_{d}W_{3})$}} $\Pi_{2, E}^{TM}=\Pi_{2, E}^{MT}-[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[(\beta_{b}-\beta_{d})^2-8\delta(\beta_{b}+\beta_{d})]/(24gc_{d}W_{3})$
 $(M, M)$ $(T, T)$ $p_{bi, E}^{MM}=p_{bi}^{MM}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{1})-E_{3}/(12gc_{d})]$ $p_{bi, E}^{TT}=p_{bi}^{TT}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{2})-E_{3}/(12gc_{d})]$ $p_{di, E}^{MM}=p_{di}^{MM}+F(i)E_{2}/W_{1}+(v_{di}-v_{d})/2$ $p_{di, E}^{TT}=p_{di}^{TT}+F(i)E_{2}/(4W_{2})+(v_{di}-v_{d})/2$ $n_{bi, E}^{MM}=n_{bi}^{MM}+F(i)E_{3}/W_{1}$ $n_{bi, E}^{TT}=n_{bi}^{TT}+F(i)E_{3}/W_{2}$ $n_{di, E}^{MM}=n_{di}^{MM}+F(i)E_{4}/W_{1}+(v_{di}-v_{d})/(4gc_{d})$ $n_{di, E}^{TT}=n_{di}^{TT}+F(i)E_{4}/W_{2}+(v_{di}-v_{d})/(4gc_{d})$ $\Pi_{i, E}^{MM}=\Pi_{i}^{MM}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}-\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} $\Pi_{i, E}^{TT}=\Pi_{i}^{TT}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}+\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} $(M, T)$ $p_{bi, E}^{MT}=p_{bi}^{MT}-\beta_{b}(v_{di}-v_{d})/(4gc_{d})-F(i)E_{3}(B_{2}-2W_{3})/(24gc_{d}W_{3})$ $p_{di, E}^{MT}=p_{di}^{MT}+F(i)E_{2}/(4W_{3})+(v_{di}-v_{d})/2$ $n_{bi, E}^{MT}=n_{bi}^{MT}+F(i)E_{3}/(2W_{3})$ $n_{di, E}^{MT}=n_{di}^{MT}+F(i)E_{4}/W_{3}+(v_{di}-v_{d})/(4gc_{d})$ $\Pi_{1, E}^{MT}=\Pi_{1}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}+\delta-2\phi)\delta]/(96gc_{d})$ $-(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})-(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ $\Pi_{2, E}^{MT}=\Pi_{2}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}-\delta-2\phi+2f_{e})\delta]/(96gc_{d})$ $+(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ $(T, M)$ $\Pi_{1, E}^{TM}=\Pi_{1, E}^{MT}+[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[8\delta(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2]/(24gc_{d}W_{3})$}} $\Pi_{2, E}^{TM}=\Pi_{2, E}^{MT}-[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[(\beta_{b}-\beta_{d})^2-8\delta(\beta_{b}+\beta_{d})]/(24gc_{d}W_{3})$
Mathematical abbreviation and threshold
 $V_{1}=v_{d}-c_{d}(1-g)$ $B_{0}=\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d}$ $B_{1}=\beta_{b}^2+\beta_{d}^2+6\beta_{b}\beta_{d}$ $\phi=c_{d}(1-g)$ $R_{1}=\beta_{b}^2-2\beta_{d}^2+\beta_{b}\beta_{d}$ $R_{2}=f_{e}(\beta_{b}+\beta_{d})+2gc_{d}t_{e}$ $R_{3}=24gc_{d}(c_{T}-c_{M})-6f_{e}[v_{d}-c_{d}(1-g)]$ $+2gc_{d}t_{e}-f_{e}(3f_{e}+2\beta_{b}+2\beta_{d})$ $R_{4}=f_{e}(\beta_{b}+\beta_{d})+5gc_{d}t_{e}$ $R_{5}=6(\beta_{b}+\beta_{d})(2v_{d}-2c_{d}+2c_{d}g+f_{e})$ $+48gc_{d}v_{b}+2\beta_{b}^2+8\beta_{b}\beta_{d}+\beta_{d}^2$ $R_{6}=2v_{d}-2c_{d}(1-g)+f_{e}+\beta_{b}+\beta_{d}$ $R_{7}=f_{e}(\beta_{b}+\beta_{d})+8gc_{d}t_{e}$ $E_{2}=2\delta(\beta_{b}^2-\beta_{d}^2)$ $E_{3}=2\delta(\beta_{b}+\beta_{d})$ $E_{4}=2\delta(\beta_{b}+\beta_{d})^2/(8gc_{d})$ $E_{5}=2v_{d}+\beta_{b}+\beta_{d}$ $W_{1}=12gc_{d}(t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $W_{2}=12gc_{d}t-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $W_{3}=6gc_{d}(2t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $H_{1}=[6f_{e}V_{1}+3f_{e}^2+2f_{e}(\beta_{b}+\beta_{d})$ $-2gc_{d}t_{e}]/(24gc_{d})$} $H_{2}=R_{2}[4gc_{d}t_{e}-(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$} $H_{3}=R_{2}[4gc_{d}t_{e}+(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$} $H_{4}=R_{2}^2(\beta_{b}-\beta_{d})^2/(96gc_{d}W_{3}^2)$ Threshold $\hat{t}^{MM}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}$ $\hat{t}^{MT}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}/2$ $\hat{t}^{TT}=\beta_{b}\beta_{d}/(2gc_{d})$ $\hat{t}_{e7}=-(1+g)f_{e}(2\phi+f_{e}+E_{5})/(gc_{d})$ $\hat{f}_{e7}=\{2\phi-2v_{d}-\beta_{b}-\beta_{d}+\sqrt{(1+g)[(1+g)(E_{5}-2\phi)^2-4gc_{d}t_{e}]}/(1+g)\}/2$
 $V_{1}=v_{d}-c_{d}(1-g)$ $B_{0}=\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d}$ $B_{1}=\beta_{b}^2+\beta_{d}^2+6\beta_{b}\beta_{d}$ $\phi=c_{d}(1-g)$ $R_{1}=\beta_{b}^2-2\beta_{d}^2+\beta_{b}\beta_{d}$ $R_{2}=f_{e}(\beta_{b}+\beta_{d})+2gc_{d}t_{e}$ $R_{3}=24gc_{d}(c_{T}-c_{M})-6f_{e}[v_{d}-c_{d}(1-g)]$ $+2gc_{d}t_{e}-f_{e}(3f_{e}+2\beta_{b}+2\beta_{d})$ $R_{4}=f_{e}(\beta_{b}+\beta_{d})+5gc_{d}t_{e}$ $R_{5}=6(\beta_{b}+\beta_{d})(2v_{d}-2c_{d}+2c_{d}g+f_{e})$ $+48gc_{d}v_{b}+2\beta_{b}^2+8\beta_{b}\beta_{d}+\beta_{d}^2$ $R_{6}=2v_{d}-2c_{d}(1-g)+f_{e}+\beta_{b}+\beta_{d}$ $R_{7}=f_{e}(\beta_{b}+\beta_{d})+8gc_{d}t_{e}$ $E_{2}=2\delta(\beta_{b}^2-\beta_{d}^2)$ $E_{3}=2\delta(\beta_{b}+\beta_{d})$ $E_{4}=2\delta(\beta_{b}+\beta_{d})^2/(8gc_{d})$ $E_{5}=2v_{d}+\beta_{b}+\beta_{d}$ $W_{1}=12gc_{d}(t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $W_{2}=12gc_{d}t-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $W_{3}=6gc_{d}(2t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $H_{1}=[6f_{e}V_{1}+3f_{e}^2+2f_{e}(\beta_{b}+\beta_{d})$ $-2gc_{d}t_{e}]/(24gc_{d})$} $H_{2}=R_{2}[4gc_{d}t_{e}-(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$} $H_{3}=R_{2}[4gc_{d}t_{e}+(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$} $H_{4}=R_{2}^2(\beta_{b}-\beta_{d})^2/(96gc_{d}W_{3}^2)$ Threshold $\hat{t}^{MM}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}$ $\hat{t}^{MT}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}/2$ $\hat{t}^{TT}=\beta_{b}\beta_{d}/(2gc_{d})$ $\hat{t}_{e7}=-(1+g)f_{e}(2\phi+f_{e}+E_{5})/(gc_{d})$ $\hat{f}_{e7}=\{2\phi-2v_{d}-\beta_{b}-\beta_{d}+\sqrt{(1+g)[(1+g)(E_{5}-2\phi)^2-4gc_{d}t_{e}]}/(1+g)\}/2$
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