Figure 1.
Game structure of the PNPP
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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 |
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.
Keywords:
New product preannouncement strategy,
two-sided market,
developers' capability,
platform heterogeneity,
game theory.
Mathematics Subject Classification:
Primary: 91A10; Secondary: 90B60.
Citation:
Ye Jiang, Tiaojun Xiao. Preannouncement strategy of platform-type new product for competing platforms: Technical or marketing information. Journal of Industrial and Management Optimization,
2022, 18
(1)
: 315-339.
doi: 10.3934/jimo.2020155
![]()
References:
| [1] |
M. Ali, G. Alireza and Q. Mojtaba,
Prediction of air travel demand using a hybrid artificial neural network (ANN) with Bat and Firefly algorithms: A case study, The Journal of Supercomputing, 74 (2018), 5461-5484.
|
| [2] |
M. Armstrong,
Competition in two-sided markets, The RAND Journal of Economics, 37 (2006), 668-691.
doi: 10.1007/s00199-006-0114-6. |
| [3] |
F. M. Bass, A. Krishnamoorthy, A. Prasad and S. P. Sethi,
Advertising competition with market expansion for finite horizon firms, Journal of Industrial and Management Optimization, 1 (2005), 1-19.
doi: 10.3934/jimo.2005.1.1. |
| [4] |
B. L. Bayus, S. Jain and A. G. Rao,
Truth or consequences: An analysis of vporware and new product announcements, Journal of Marketing Research, 38 (2001), 3-13.
|
| [5] |
H. K. Bhargava,
Platform technologies and network goods: Insights on product launch and management, Information Technology and Management, 15 (2014), 199-209.
|
| [6] |
R. K. Chellappa and R. Mukherjee, Platform preannouncement strategies: The strategic role of information in two-sided markets competition, Social Science Electronic Publishing, (2015), 55 pp. |
| [7] |
C.-W. Chen and V. Wong,
Design and delivery of new product preannouncement messages, Journal of Marketing Theory and Practice, 20 (2014), 203-222.
doi: 10.2753/MTP1069-6679200206. |
| [8] |
J. P. Choi, E. G. Kristiansen and J. Nahm,
Strategic product pre-announcements in markets with network effects, Hitotsubashi Journal of Economics, 60 (2005), 1-20.
doi: 10.2139/ssrn.847986. |
| [9] |
T. Doganoglu and J. Wright,
Multihoming and compatibility, International Journal of Industrial Organization, 24 (2006), 45-67.
|
| [10] |
D. Dranove and N. Gandal,
The DVD-vs.-DIVX standard war: Empirical evidence of network effects and preannouncement effects, Journal of Economics & Management Strategy, 12 (2003), 363-386.
|
| [11] |
N. Economides and J. Tag,
Network neutrality on the Internet: A two-sided market analysis, Information Economics and Policy, 24 (2012), 91-104.
|
| [12] |
J. Eliashberg and T. S. Robertson,
New product preannouncing behavior: A market signaling study, Journal of Marketing Research, 25 (1988), 282-292.
|
| [13] |
M. Gao,
Platform pricing in mixed two-sided markets, International Economic Review, 59 (2018), 1103-1129.
doi: 10.1111/iere.12298. |
| [14] |
H. A. Gerlach,
Announcement, entry, and preemption when consumers have switching costs, The RAND Journal of Economics, 35 (2004), 184-202.
doi: 10.2307/1593736. |
| [15] |
G. Alireza, Z. H. Khademi, T. Reza and A. Sadegheih,
Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34.
|
| [16] |
A. Hagiu and H. Halaburda,
Information and two-sided platform profits, International Journal of Industrial Organization, 34 (2014), 25-35.
|
| [17] |
J. A. Hoxmeier,
Software preannouncements and their impact on customers' perceptions and vendor reputation, Journal of Management Information Systems, 17 (2000), 115-139.
|
| [18] |
H. Jung,
Signaling quality with new product preannouncements: Vaporware and the role of reference quality, Journal of Business Research, 64 (2011), 1251-1258.
doi: 10.1016/j.jbusres.2011.06.032. |
| [19] |
D. Ke, H. Zhang and Y. Tu, The cannibalization effects of new product preannouncement and launch on the C2C marketplace, 15th workshop on E-business, (2016), 84–94.
doi: 10.1007/978-3-319-69644-7_8. |
| [20] |
T. Klastorin, H. Mamani and Y.-P. Zhou,
Production and Operations Management, 15th workshop on E-business, 25 (2016), 2051-2064.
|
| [21] |
E. Le Nagard-Assayag and D. Manceau,
Modeling the impact of product preannouncements in the context of indirect network externalities, International Journal of Research in Marketing, 18 (2001), 203-219.
doi: 10.1016/S0167-8116(00)00031-8. |
| [22] |
S. Lee,
Factors influencing the social networking service user's value perception and word of mouth decision of corporate post with special reference to the emotional attachment, Information Technology and Management, 17 (2016), 15-27.
doi: 10.1007/s10799-015-0227-3. |
| [23] |
Y. Lee and G. C. Oconnor,
The impact of communication strategy on launching new products: The moderating role of product innovativeness, Journal of Product Innovation Management, 20 (2003), 4-21.
doi: 10.1111/1540-5885.t01-1-201002. |
| [24] |
B. Lilly and R. Walters,
Toward a model of new product preannouncement timing, Journal of Product Innovation Management, 14 (1997), 4-20.
doi: 10.1111/1540-5885.1410004. |
| [25] |
M. J. Mccabe and C. M. Snyder,
Academic journal prices in a digital age: A two-sided market model, Topics in Economic Analysis & Policy, 7 (2007), 1-39.
|
| [26] |
E. Ofek and O. Turut,
Vaporware, suddenware, and trueware: New product preannouncements under market uncertainty, Marketing Science, 32 (2013), 342-355.
doi: 10.1287/mksc.1120.0762. |
| [27] |
J. Rochet and J. Tirole,
Platform competition in two-sided markets, Journal of the European Economic Association, 1 (2003), 990-1029.
doi: 10.1162/154247603322493212. |
| [28] |
J. Rochet and J. Tirole,
Two-sided markets: A progress report, The RAND Journal of Economics, 37 (2006), 645-667.
doi: 10.1111/j.1756-2171.2006.tb00036.x. |
| [29] |
K. Schatzel and R. Calantone,
Creating market anticipation: An exploratory examination of the effect of preannouncement behavior on a new product's launch, Journal of the Academy of Marketing Science, 34 (2006), 357-366.
doi: 10.1177/0092070304270737. |
| [30] |
L. C. Soffer, S. R. Thiagarajan and B. R. Walther,
Earnings preannouncement strategies, Review of Accounting Studies, 5 (2000), 5-26.
|
| [31] |
M. Su and V. R. Rao,
New product preannouncement as a signaling strategy: An audience-specific review and analysis, Journal of Product Innovation Management, 27 (2010), 658-672.
doi: 10.1111/j.1540-5885.2010.00743.x. |
| [32] |
M. Su and V. R. Rao,
Timing decisions of new product preannouncement and launch with competition, International Journal of Production Economics, 129 (2011), 51-64.
doi: 10.1016/j.ijpe.2010.09.001. |
| [33] |
H. Sun and S. Kumar,
A manufacturer's new product preannouncement decision and the supplier's response, Production and Operations Management, 29 (2020), 2289-2306.
doi: 10.1111/poms.12900. |
| [34] |
Y. Trope and N. Liberman,
Construal-level theory of psychological distance, Psychological Review, 117.2 (2010), 440-463.
doi: 10.1037/a0018963. |
| [35] |
M. J. Villena and M. Contreras,
Global and local advertising strategies: A dynamic multi-market optimal control model, Journal of Industrial and Management Optimization, 15 (2017), 1017-1048.
doi: 10.3934/jimo.2018084. |
| [36] |
E. G. Weyl,
A price theory of multi-sided platforms, American Economic Review, 100 (2010), 1642-1672.
doi: 10.1257/aer.100.4.1642. |
| [37] |
H. Zhang and Y. K. Choi,
Preannouncement messages: Impetus for electronic word-of-mouth, International Journal of Advertising, 37 (2018), 54-70.
doi: 10.1080/02650487.2017.1391679. |
| [38] |
J. Zhang, Q. Liang and J. Huang,
Forward advertising: A competitive analysis of new product preannouncement, Information Economics and Policy, 37 (2016), 3-12.
doi: 10.1016/j.infoecopol.2016.10.004. |
show all references
References:
| [1] |
M. Ali, G. Alireza and Q. Mojtaba,
Prediction of air travel demand using a hybrid artificial neural network (ANN) with Bat and Firefly algorithms: A case study, The Journal of Supercomputing, 74 (2018), 5461-5484.
|
| [2] |
M. Armstrong,
Competition in two-sided markets, The RAND Journal of Economics, 37 (2006), 668-691.
doi: 10.1007/s00199-006-0114-6. |
| [3] |
F. M. Bass, A. Krishnamoorthy, A. Prasad and S. P. Sethi,
Advertising competition with market expansion for finite horizon firms, Journal of Industrial and Management Optimization, 1 (2005), 1-19.
doi: 10.3934/jimo.2005.1.1. |
| [4] |
B. L. Bayus, S. Jain and A. G. Rao,
Truth or consequences: An analysis of vporware and new product announcements, Journal of Marketing Research, 38 (2001), 3-13.
|
| [5] |
H. K. Bhargava,
Platform technologies and network goods: Insights on product launch and management, Information Technology and Management, 15 (2014), 199-209.
|
| [6] |
R. K. Chellappa and R. Mukherjee, Platform preannouncement strategies: The strategic role of information in two-sided markets competition, Social Science Electronic Publishing, (2015), 55 pp. |
| [7] |
C.-W. Chen and V. Wong,
Design and delivery of new product preannouncement messages, Journal of Marketing Theory and Practice, 20 (2014), 203-222.
doi: 10.2753/MTP1069-6679200206. |
| [8] |
J. P. Choi, E. G. Kristiansen and J. Nahm,
Strategic product pre-announcements in markets with network effects, Hitotsubashi Journal of Economics, 60 (2005), 1-20.
doi: 10.2139/ssrn.847986. |
| [9] |
T. Doganoglu and J. Wright,
Multihoming and compatibility, International Journal of Industrial Organization, 24 (2006), 45-67.
|
| [10] |
D. Dranove and N. Gandal,
The DVD-vs.-DIVX standard war: Empirical evidence of network effects and preannouncement effects, Journal of Economics & Management Strategy, 12 (2003), 363-386.
|
| [11] |
N. Economides and J. Tag,
Network neutrality on the Internet: A two-sided market analysis, Information Economics and Policy, 24 (2012), 91-104.
|
| [12] |
J. Eliashberg and T. S. Robertson,
New product preannouncing behavior: A market signaling study, Journal of Marketing Research, 25 (1988), 282-292.
|
| [13] |
M. Gao,
Platform pricing in mixed two-sided markets, International Economic Review, 59 (2018), 1103-1129.
doi: 10.1111/iere.12298. |
| [14] |
H. A. Gerlach,
Announcement, entry, and preemption when consumers have switching costs, The RAND Journal of Economics, 35 (2004), 184-202.
doi: 10.2307/1593736. |
| [15] |
G. Alireza, Z. H. Khademi, T. Reza and A. Sadegheih,
Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34.
|
| [16] |
A. Hagiu and H. Halaburda,
Information and two-sided platform profits, International Journal of Industrial Organization, 34 (2014), 25-35.
|
| [17] |
J. A. Hoxmeier,
Software preannouncements and their impact on customers' perceptions and vendor reputation, Journal of Management Information Systems, 17 (2000), 115-139.
|
| [18] |
H. Jung,
Signaling quality with new product preannouncements: Vaporware and the role of reference quality, Journal of Business Research, 64 (2011), 1251-1258.
doi: 10.1016/j.jbusres.2011.06.032. |
| [19] |
D. Ke, H. Zhang and Y. Tu, The cannibalization effects of new product preannouncement and launch on the C2C marketplace, 15th workshop on E-business, (2016), 84–94.
doi: 10.1007/978-3-319-69644-7_8. |
| [20] |
T. Klastorin, H. Mamani and Y.-P. Zhou,
Production and Operations Management, 15th workshop on E-business, 25 (2016), 2051-2064.
|
| [21] |
E. Le Nagard-Assayag and D. Manceau,
Modeling the impact of product preannouncements in the context of indirect network externalities, International Journal of Research in Marketing, 18 (2001), 203-219.
doi: 10.1016/S0167-8116(00)00031-8. |
| [22] |
S. Lee,
Factors influencing the social networking service user's value perception and word of mouth decision of corporate post with special reference to the emotional attachment, Information Technology and Management, 17 (2016), 15-27.
doi: 10.1007/s10799-015-0227-3. |
| [23] |
Y. Lee and G. C. Oconnor,
The impact of communication strategy on launching new products: The moderating role of product innovativeness, Journal of Product Innovation Management, 20 (2003), 4-21.
doi: 10.1111/1540-5885.t01-1-201002. |
| [24] |
B. Lilly and R. Walters,
Toward a model of new product preannouncement timing, Journal of Product Innovation Management, 14 (1997), 4-20.
doi: 10.1111/1540-5885.1410004. |
| [25] |
M. J. Mccabe and C. M. Snyder,
Academic journal prices in a digital age: A two-sided market model, Topics in Economic Analysis & Policy, 7 (2007), 1-39.
|
| [26] |
E. Ofek and O. Turut,
Vaporware, suddenware, and trueware: New product preannouncements under market uncertainty, Marketing Science, 32 (2013), 342-355.
doi: 10.1287/mksc.1120.0762. |
| [27] |
J. Rochet and J. Tirole,
Platform competition in two-sided markets, Journal of the European Economic Association, 1 (2003), 990-1029.
doi: 10.1162/154247603322493212. |
| [28] |
J. Rochet and J. Tirole,
Two-sided markets: A progress report, The RAND Journal of Economics, 37 (2006), 645-667.
doi: 10.1111/j.1756-2171.2006.tb00036.x. |
| [29] |
K. Schatzel and R. Calantone,
Creating market anticipation: An exploratory examination of the effect of preannouncement behavior on a new product's launch, Journal of the Academy of Marketing Science, 34 (2006), 357-366.
doi: 10.1177/0092070304270737. |
| [30] |
L. C. Soffer, S. R. Thiagarajan and B. R. Walther,
Earnings preannouncement strategies, Review of Accounting Studies, 5 (2000), 5-26.
|
| [31] |
M. Su and V. R. Rao,
New product preannouncement as a signaling strategy: An audience-specific review and analysis, Journal of Product Innovation Management, 27 (2010), 658-672.
doi: 10.1111/j.1540-5885.2010.00743.x. |
| [32] |
M. Su and V. R. Rao,
Timing decisions of new product preannouncement and launch with competition, International Journal of Production Economics, 129 (2011), 51-64.
doi: 10.1016/j.ijpe.2010.09.001. |
| [33] |
H. Sun and S. Kumar,
A manufacturer's new product preannouncement decision and the supplier's response, Production and Operations Management, 29 (2020), 2289-2306.
doi: 10.1111/poms.12900. |
| [34] |
Y. Trope and N. Liberman,
Construal-level theory of psychological distance, Psychological Review, 117.2 (2010), 440-463.
doi: 10.1037/a0018963. |
| [35] |
M. J. Villena and M. Contreras,
Global and local advertising strategies: A dynamic multi-market optimal control model, Journal of Industrial and Management Optimization, 15 (2017), 1017-1048.
doi: 10.3934/jimo.2018084. |
| [36] |
E. G. Weyl,
A price theory of multi-sided platforms, American Economic Review, 100 (2010), 1642-1672.
doi: 10.1257/aer.100.4.1642. |
| [37] |
H. Zhang and Y. K. Choi,
Preannouncement messages: Impetus for electronic word-of-mouth, International Journal of Advertising, 37 (2018), 54-70.
doi: 10.1080/02650487.2017.1391679. |
| [38] |
J. Zhang, Q. Liang and J. Huang,
Forward advertising: A competitive analysis of new product preannouncement, Information Economics and Policy, 37 (2016), 3-12.
doi: 10.1016/j.infoecopol.2016.10.004. |
Figure 2.
(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 $
Figure 3.
(a) SPNE distribution when $ \beta_{b} = 1.01 $ and (b)SPNE distribution when $ \beta_{b} = 5.01 $
Figure 5.
(a) SPNE changes with $ f_{e} $ when $ \delta = 0.2 $ and (b)SPNE changes with $ f_{e} $ when $ \delta = 1.2 $
Figure 6.
SPNE changes with $ \delta $
Figure 7.
Total social welfare changes with $ \delta $
Table 1.
Equilibrium results under $ (M, M) $
| Decisions | |
| Consumers' price | |
| Developers' price | |
| Consumers' number | |
| Developers' number | |
| Equilibrium profits/total utilities | |
| Platforms' profit | |
| Consumers' total utility | |
| Developers' total utility | |
| Decisions | |
| Consumers' price | |
| Developers' price | |
| Consumers' number | |
| Developers' number | |
| Equilibrium profits/total utilities | |
| Platforms' profit | |
| Consumers' total utility | |
| Developers' total utility | |
Table 2.
Nash equilibrium under $ (T, T) $
| Decisions | |
| Consumers' price | |
| Developers' price | |
| Consumers' number | |
| Developers' number | |
| Equilibrium profits/total utilities | |
| Platforms' profit | |
| Consumers' total utility | |
| Developers' total utility | |
| Decisions | |
| Consumers' price | |
| Developers' price | |
| Consumers' number | |
| Developers' number | |
| Equilibrium profits/total utilities | |
| Platforms' profit | |
| Consumers' total utility | |
| Developers' total utility | |
Table 3.
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 | $DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d})$ |
Table 4.
Factors of effect asymmetric strategy to be a SPNE
| 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 |
| 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 |
Table 5.
Nash equilibrium of non-identical platforms
| $p_{bi, E}^{MM}=p_{bi}^{MM}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{1})-E_{3}/(12gc_{d})]$ | $p_{bi, E}^{TT}=p_{bi}^{TT}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{2})-E_{3}/(12gc_{d})]$ |
| $p_{di, E}^{MM}=p_{di}^{MM}+F(i)E_{2}/W_{1}+(v_{di}-v_{d})/2$ | $p_{di, E}^{TT}=p_{di}^{TT}+F(i)E_{2}/(4W_{2})+(v_{di}-v_{d})/2$ |
| $n_{bi, E}^{MM}=n_{bi}^{MM}+F(i)E_{3}/W_{1}$ | $n_{bi, E}^{TT}=n_{bi}^{TT}+F(i)E_{3}/W_{2}$ |
| $n_{di, E}^{MM}=n_{di}^{MM}+F(i)E_{4}/W_{1}+(v_{di}-v_{d})/(4gc_{d})$ | $n_{di, E}^{TT}=n_{di}^{TT}+F(i)E_{4}/W_{2}+(v_{di}-v_{d})/(4gc_{d})$ |
| $\Pi_{i, E}^{MM}=\Pi_{i}^{MM}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}-\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} | $\Pi_{i, E}^{TT}=\Pi_{i}^{TT}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}+\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} |
| $(M, T)$ | |
| $p_{bi, E}^{MT}=p_{bi}^{MT}-\beta_{b}(v_{di}-v_{d})/(4gc_{d})-F(i)E_{3}(B_{2}-2W_{3})/(24gc_{d}W_{3})$ | |
| $p_{di, E}^{MT}=p_{di}^{MT}+F(i)E_{2}/(4W_{3})+(v_{di}-v_{d})/2$ | |
| $n_{bi, E}^{MT}=n_{bi}^{MT}+F(i)E_{3}/(2W_{3})$ | |
| $n_{di, E}^{MT}=n_{di}^{MT}+F(i)E_{4}/W_{3}+(v_{di}-v_{d})/(4gc_{d})$ | |
| $\Pi_{1, E}^{MT}=\Pi_{1}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}+\delta-2\phi)\delta]/(96gc_{d})$ $-(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})-(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ | |
| $\Pi_{2, E}^{MT}=\Pi_{2}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}-\delta-2\phi+2f_{e})\delta]/(96gc_{d})$ $+(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ | |
| $(T, M)$ | |
| $\Pi_{1, E}^{TM}=\Pi_{1, E}^{MT}+[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[8\delta(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2]/(24gc_{d}W_{3})$}} | |
| $\Pi_{2, E}^{TM}=\Pi_{2, E}^{MT}-[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[(\beta_{b}-\beta_{d})^2-8\delta(\beta_{b}+\beta_{d})]/(24gc_{d}W_{3})$ | |
| $p_{bi, E}^{MM}=p_{bi}^{MM}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{1})-E_{3}/(12gc_{d})]$ | $p_{bi, E}^{TT}=p_{bi}^{TT}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{2})-E_{3}/(12gc_{d})]$ |
| $p_{di, E}^{MM}=p_{di}^{MM}+F(i)E_{2}/W_{1}+(v_{di}-v_{d})/2$ | $p_{di, E}^{TT}=p_{di}^{TT}+F(i)E_{2}/(4W_{2})+(v_{di}-v_{d})/2$ |
| $n_{bi, E}^{MM}=n_{bi}^{MM}+F(i)E_{3}/W_{1}$ | $n_{bi, E}^{TT}=n_{bi}^{TT}+F(i)E_{3}/W_{2}$ |
| $n_{di, E}^{MM}=n_{di}^{MM}+F(i)E_{4}/W_{1}+(v_{di}-v_{d})/(4gc_{d})$ | $n_{di, E}^{TT}=n_{di}^{TT}+F(i)E_{4}/W_{2}+(v_{di}-v_{d})/(4gc_{d})$ |
| $\Pi_{i, E}^{MM}=\Pi_{i}^{MM}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}-\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} | $\Pi_{i, E}^{TT}=\Pi_{i}^{TT}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}+\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} |
| $(M, T)$ | |
| $p_{bi, E}^{MT}=p_{bi}^{MT}-\beta_{b}(v_{di}-v_{d})/(4gc_{d})-F(i)E_{3}(B_{2}-2W_{3})/(24gc_{d}W_{3})$ | |
| $p_{di, E}^{MT}=p_{di}^{MT}+F(i)E_{2}/(4W_{3})+(v_{di}-v_{d})/2$ | |
| $n_{bi, E}^{MT}=n_{bi}^{MT}+F(i)E_{3}/(2W_{3})$ | |
| $n_{di, E}^{MT}=n_{di}^{MT}+F(i)E_{4}/W_{3}+(v_{di}-v_{d})/(4gc_{d})$ | |
| $\Pi_{1, E}^{MT}=\Pi_{1}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}+\delta-2\phi)\delta]/(96gc_{d})$ $-(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})-(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ | |
| $\Pi_{2, E}^{MT}=\Pi_{2}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}-\delta-2\phi+2f_{e})\delta]/(96gc_{d})$ $+(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$ | |
| $(T, M)$ | |
| $\Pi_{1, E}^{TM}=\Pi_{1, E}^{MT}+[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[8\delta(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2]/(24gc_{d}W_{3})$}} | |
| $\Pi_{2, E}^{TM}=\Pi_{2, E}^{MT}-[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[(\beta_{b}-\beta_{d})^2-8\delta(\beta_{b}+\beta_{d})]/(24gc_{d}W_{3})$ | |
Table 1A.
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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