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doi: 10.3934/jdg.2022010
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## Opening the source code: The threat of forking

 1 University of Vienna, Department of Business Decisions and Analytics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria 2 International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, 2361 Laxenburg, Austria

*Corresponding author: Stefan Wrzaczek

Received  December 2021 Early access May 2022

Making software open source can have substantial positive effects on the quality and diffusion of a software and strengthen the sales of complementary products. However, it is a large concern of firms that a competitor might use the very same source code to start its own competitive project, a so-called fork. This paper analyzes whether the threat of forking prevents a firm to open its source code. We consider three different regimes: In the first regime a firm develops and sells software under a proprietary license, in the second regime, it uses an open source business model. The third regime is characterized by the competition of two related open source projects. The switching times between the regimes are optimally determined. We find that the optimal strategy substantially depends on the initial state value and the extent to which a competitor can make use of the firm's software quality. A small initial software quality can prevent a firm to open the code when it cannot afford competition, only with a competitive advantage open source is attractive then. For a large initial software quality, a firm would never open the code immediately, it would either wait or keep it proprietary forever.

Citation: Andrea Seidl, Stefan Wrzaczek. Opening the source code: The threat of forking. Journal of Dynamics and Games, doi: 10.3934/jdg.2022010
##### References:
 [1] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, Mathematics in Science and Engineering, 160, Academic Press, Inc., New York-London, 1982. [2] K. Blind, S. Pätsch, S. Muto, M. Böhm and T. Schubert, et al., The Impact of Open Source Software and Hardware on Technological Independence, Competitiveness and Innovation in the EU Economy, Final Study Report, European Commission, Directorate-General for Communications Networks, Content and Technology, 2021. doi: 10.2759/430161. [3] A. Buratto and S. Wrzaczek, Advertising a product to face a competitor entry: A differential game approach, Decis. Econ. Finance, 41 (2018), 463-487.  doi: 10.1007/s10203-018-0210-7. [4] J. P. Caulkins, G. Feichtinger, D. Grass, R. F. Hartl, P. M. Kort and A. Seidl, When to make proprietary software open source, J. Econom. Dynam. Control, 37 (2013), 1182-1194.  doi: 10.1016/j.jedc.2013.02.009. [5] J. P. Caulkins, D. Grass, G. Feichtinger, R. F. Hartl and P. M. Kort, et al., The optimal lockdown intensity for COVID-19, J. Math. Econom., 93 (2021), 18pp. doi: 10.1016/j.jmateco.2021.102489. [6] R. Cellini and L. Lambertini, Dynamic R & D with spillovers: Competition vs cooperation, J. Econom. Dynam. Control, 33 (2009), 568-582.  doi: 10.1016/j.jedc.2008.08.006. [7] E. J. Dockner, S. Jørgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511805127. [8] N. Economides and E. Katsamakas, Two-sided competition of proprietary vs. open source technology platforms and the implications for the software industry, Mgmt. Sci., 52 (2006), 1057-1071.  doi: 10.1287/mnsc.1060.0549. [9] D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, Optimal Control of Nonlinear Processes. With Applications in Drugs, Corruption and Terror, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77647-5. [10] E. V. Gromova and J. D. López-Barrientos, A differential game model for the extraction of nonrenewable resources with random initial times: The cooperative and competitive cases, Int. Game Theory Rev., 18 (2016), 19pp. doi: 10.1142/S0219198916400041. [11] E. Haruvy, A. Prasad and S. P. Sethi, Harvesting altruism in open-source software development, J. Optim. Theory Appl., 118 (2003), 381-416.  doi: 10.1023/A:1025455523489. [12] E. Haruvy, A. Prasad, S. P. Sethi and R. Zhang, Competition with open source as a public good, J. Ind. Manag. Optim., 4 (2008), 199-211.  doi: 10.3934/jimo.2008.4.199. [13] E. Haruvy, S. Sethi and J. Zhou, Open source development with a commercial complementary product or service, Prod. Oper. Mgmt., 17 (2008), 29–43. Available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1089843. [14] A. Haurie, J. B. Krawczyk and G. Zaccour, Games and Dynamic Games, World Scientific—Now Publishers Series in Business, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8442. [15] S. Jørgensen and G. Zaccour, Differential Games in Marketing, International Series In Quantitative Marketing, 15, Springer, Boston, MA, 2004. doi: 10.1007/978-1-4419-8929-1. [16] T. Kiseleva and F. O. O. Wagener, Bifurcations of optimal vector fields in the shallow lake model, J. Econom. Dynam. Control, 34 (2010), 825-843.  doi: 10.1016/j.jedc.2009.11.008. [17] P. M. Kort and S. Wrzaczek, Optimal firm growth under the threat of entry, European J. Oper. Res., 246 (2015), 281-292.  doi: 10.1016/j.ejor.2015.04.030. [18] P. M. Kort and G. Zaccour, When should a firm open its source code: A strategic analysis, Prod. Oper. Mgmt., 20 (2011), 877-888.  doi: 10.1111/j.1937-5956.2011.01233.x. [19] P. B. de Laat, Copyright or copyleft?: An analysis of property regimes for software development, Research Policy, 34 (2005), 1511-1532.  doi: 10.1016/j.respol.2005.07.003. [20] P. B. de Laat, Governance of open source software: State of the art, J. Mgmt. Gov., 11 (2007), 165-177.  doi: 10.1007/s10997-007-9022-9. [21] J. Lerner and M. Schankerman, The Comingled Code: Open Source and Economic Development, The MIT Press, Massachusetts, 2010.  doi: 10.7551/mitpress/9780262014632.001.0001. [22] J. Lerner and J. Tirole, The scope of open source licensing, J. Law Econ. Org., 21 (2005), 20-56.  doi: 10.3386/w9363. [23] J. Lerner and J. Tirole, Some simple economics of open source, J. Industrial Econ., 50 (2003), 197-234.  doi: 10.1111/1467-6451.00174. [24] N. V. Long, A Survey of Dynamic Games in Economics, Surveys on Theories in Economics and Business Administration, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/9789814293044. [25] M. Mustonen, Copyleft–The economics of Linux and other open source software, Info. Econ. Policy, 15 (2003), 99-121.  doi: 10.1016/S0167-6245(02)00090-2. [26] E. S. Raymond, The cathedral and the bazaar, Know. Techn. Pol., 12 (1999), 23-49.  doi: 10.1007/s12130-999-1026-0. [27] G. Robles and J. M. González-Barahona, A comprehensive study of software forks: Dates, reasons and outcomes, IFIP Advances in Information and Communication Technology, 378, Springer, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-33442-9_1. [28] M. Schaarschmidt, G. Walsh and H. F. O. von Kortzfleisch, How do firms influence open source software communities? A framework and empirical analysis of different governance modes, Info. Org., 25 (2015), 99-114.  doi: 10.1016/j.infoandorg.2015.03.001. [29] A. Seidl, Zeno points in optimal control models with endogenous regime switching, J. Econom. Dynam. Control, 100 (2019), 353-368.  doi: 10.1016/j.jedc.2018.09.010. [30] A. Seidl, J. P. Caulkins, R. F. Hartl and P. M. Kort, Serious strategy for the makers of fun: Analyzing the option to switch from pay-to-play to free-to-play in a two-stage optimal control model with quadratic costs, European J. Oper. Res., 267 (2018), 700-715.  doi: 10.1016/j.ejor.2017.11.071. [31] R. Sen, C. Subramaniam and M. L. Nelson, Open source software licenses: Strong-copyleft, non-copyleft, or somewhere in between?, Decision Support System, 52 (2011), 199-206.  doi: 10.1016/j.dss.2011.07.004. [32] S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics, 2nd edition, Kluwer Academic Publishers, Boston, MA, 2000. [33] C. Shapiro and H. R. Varian, Information Rules: A Strategic Guide to the Network Economy, Harvard Business Review Press, Boston, 1999. [34] E. von Hippel and G. von Krogh, Open source software and the "private-collective" innovation model: Issues for organization science, Org. Sci., 14 (2003), 107-225.  doi: 10.1287/orsc.14.2.209.14992. [35] G. von Krogh and E. von Hippel, The promise of research on open source software, Mgmt. Sci., 52 (2006), 975-983. [36] F. O. O. Wagener, Skiba points and heteroclinic bifurcations, with applications to the shallow lake system, J. Econom. Dynam. Control, 27 (2003), 1533-1561.  doi: 10.1016/S0165-1889(02)00070-2. [37] S. Wrzaczek, M. Kuhn and I. Frankovic, Using age structure for a multi-stage optimal control model with random switching time, J. Optim. Theory Appl., 184 (2020), 1065-1082.  doi: 10.1007/s10957-019-01598-5.

show all references

##### References:
 [1] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, Mathematics in Science and Engineering, 160, Academic Press, Inc., New York-London, 1982. [2] K. Blind, S. Pätsch, S. Muto, M. Böhm and T. Schubert, et al., The Impact of Open Source Software and Hardware on Technological Independence, Competitiveness and Innovation in the EU Economy, Final Study Report, European Commission, Directorate-General for Communications Networks, Content and Technology, 2021. doi: 10.2759/430161. [3] A. Buratto and S. Wrzaczek, Advertising a product to face a competitor entry: A differential game approach, Decis. Econ. Finance, 41 (2018), 463-487.  doi: 10.1007/s10203-018-0210-7. [4] J. P. Caulkins, G. Feichtinger, D. Grass, R. F. Hartl, P. M. Kort and A. Seidl, When to make proprietary software open source, J. Econom. Dynam. Control, 37 (2013), 1182-1194.  doi: 10.1016/j.jedc.2013.02.009. [5] J. P. Caulkins, D. Grass, G. Feichtinger, R. F. Hartl and P. M. Kort, et al., The optimal lockdown intensity for COVID-19, J. Math. Econom., 93 (2021), 18pp. doi: 10.1016/j.jmateco.2021.102489. [6] R. Cellini and L. Lambertini, Dynamic R & D with spillovers: Competition vs cooperation, J. Econom. Dynam. Control, 33 (2009), 568-582.  doi: 10.1016/j.jedc.2008.08.006. [7] E. J. Dockner, S. Jørgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511805127. [8] N. Economides and E. Katsamakas, Two-sided competition of proprietary vs. open source technology platforms and the implications for the software industry, Mgmt. Sci., 52 (2006), 1057-1071.  doi: 10.1287/mnsc.1060.0549. [9] D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, Optimal Control of Nonlinear Processes. With Applications in Drugs, Corruption and Terror, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77647-5. [10] E. V. Gromova and J. D. López-Barrientos, A differential game model for the extraction of nonrenewable resources with random initial times: The cooperative and competitive cases, Int. Game Theory Rev., 18 (2016), 19pp. doi: 10.1142/S0219198916400041. [11] E. Haruvy, A. Prasad and S. P. Sethi, Harvesting altruism in open-source software development, J. Optim. Theory Appl., 118 (2003), 381-416.  doi: 10.1023/A:1025455523489. [12] E. Haruvy, A. Prasad, S. P. Sethi and R. Zhang, Competition with open source as a public good, J. Ind. Manag. Optim., 4 (2008), 199-211.  doi: 10.3934/jimo.2008.4.199. [13] E. Haruvy, S. Sethi and J. Zhou, Open source development with a commercial complementary product or service, Prod. Oper. Mgmt., 17 (2008), 29–43. Available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1089843. [14] A. Haurie, J. B. Krawczyk and G. Zaccour, Games and Dynamic Games, World Scientific—Now Publishers Series in Business, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8442. [15] S. Jørgensen and G. Zaccour, Differential Games in Marketing, International Series In Quantitative Marketing, 15, Springer, Boston, MA, 2004. doi: 10.1007/978-1-4419-8929-1. [16] T. Kiseleva and F. O. O. Wagener, Bifurcations of optimal vector fields in the shallow lake model, J. Econom. Dynam. Control, 34 (2010), 825-843.  doi: 10.1016/j.jedc.2009.11.008. [17] P. M. Kort and S. Wrzaczek, Optimal firm growth under the threat of entry, European J. Oper. Res., 246 (2015), 281-292.  doi: 10.1016/j.ejor.2015.04.030. [18] P. M. Kort and G. Zaccour, When should a firm open its source code: A strategic analysis, Prod. Oper. Mgmt., 20 (2011), 877-888.  doi: 10.1111/j.1937-5956.2011.01233.x. [19] P. B. de Laat, Copyright or copyleft?: An analysis of property regimes for software development, Research Policy, 34 (2005), 1511-1532.  doi: 10.1016/j.respol.2005.07.003. [20] P. B. de Laat, Governance of open source software: State of the art, J. Mgmt. Gov., 11 (2007), 165-177.  doi: 10.1007/s10997-007-9022-9. [21] J. Lerner and M. Schankerman, The Comingled Code: Open Source and Economic Development, The MIT Press, Massachusetts, 2010.  doi: 10.7551/mitpress/9780262014632.001.0001. [22] J. Lerner and J. Tirole, The scope of open source licensing, J. Law Econ. Org., 21 (2005), 20-56.  doi: 10.3386/w9363. [23] J. Lerner and J. Tirole, Some simple economics of open source, J. Industrial Econ., 50 (2003), 197-234.  doi: 10.1111/1467-6451.00174. [24] N. V. Long, A Survey of Dynamic Games in Economics, Surveys on Theories in Economics and Business Administration, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/9789814293044. [25] M. Mustonen, Copyleft–The economics of Linux and other open source software, Info. Econ. Policy, 15 (2003), 99-121.  doi: 10.1016/S0167-6245(02)00090-2. [26] E. S. Raymond, The cathedral and the bazaar, Know. Techn. Pol., 12 (1999), 23-49.  doi: 10.1007/s12130-999-1026-0. [27] G. Robles and J. M. González-Barahona, A comprehensive study of software forks: Dates, reasons and outcomes, IFIP Advances in Information and Communication Technology, 378, Springer, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-33442-9_1. [28] M. Schaarschmidt, G. Walsh and H. F. O. von Kortzfleisch, How do firms influence open source software communities? A framework and empirical analysis of different governance modes, Info. Org., 25 (2015), 99-114.  doi: 10.1016/j.infoandorg.2015.03.001. [29] A. Seidl, Zeno points in optimal control models with endogenous regime switching, J. Econom. Dynam. Control, 100 (2019), 353-368.  doi: 10.1016/j.jedc.2018.09.010. [30] A. Seidl, J. P. Caulkins, R. F. Hartl and P. M. Kort, Serious strategy for the makers of fun: Analyzing the option to switch from pay-to-play to free-to-play in a two-stage optimal control model with quadratic costs, European J. Oper. Res., 267 (2018), 700-715.  doi: 10.1016/j.ejor.2017.11.071. [31] R. Sen, C. Subramaniam and M. L. Nelson, Open source software licenses: Strong-copyleft, non-copyleft, or somewhere in between?, Decision Support System, 52 (2011), 199-206.  doi: 10.1016/j.dss.2011.07.004. [32] S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics, 2nd edition, Kluwer Academic Publishers, Boston, MA, 2000. [33] C. Shapiro and H. R. Varian, Information Rules: A Strategic Guide to the Network Economy, Harvard Business Review Press, Boston, 1999. [34] E. von Hippel and G. von Krogh, Open source software and the "private-collective" innovation model: Issues for organization science, Org. Sci., 14 (2003), 107-225.  doi: 10.1287/orsc.14.2.209.14992. [35] G. von Krogh and E. von Hippel, The promise of research on open source software, Mgmt. Sci., 52 (2006), 975-983. [36] F. O. O. Wagener, Skiba points and heteroclinic bifurcations, with applications to the shallow lake system, J. Econom. Dynam. Control, 27 (2003), 1533-1561.  doi: 10.1016/S0165-1889(02)00070-2. [37] S. Wrzaczek, M. Kuhn and I. Frankovic, Using age structure for a multi-stage optimal control model with random switching time, J. Optim. Theory Appl., 184 (2020), 1065-1082.  doi: 10.1007/s10957-019-01598-5.
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) in the symmetric case with $\epsilon < \bar{\epsilon}$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) in the symmetric case with $\epsilon = \bar{\epsilon}$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) in the symmetric case with $\bar{\epsilon} < \epsilon < \epsilon^{crit}$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) in the symmetric case with $\epsilon = \epsilon^{crit}$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$ and $\epsilon$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) for the symmetric case with high $\epsilon$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) for the asymmetric case with $\epsilon < \bar{\epsilon}$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) for the asymmetric case with $\epsilon = \bar{\epsilon}$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) for the asymmetric case with $\bar{\epsilon} < \epsilon < \epsilon^{crit}$
Optimal switching times $\tau _1$ and $\tau _2$ depending on the initial software quality $K_{10}$(left panel) and illustrative time paths for different values of $K_{10}$ (right panel) for the asymmetric case with high $\epsilon$
Model variables and parameters
 Firm 1 Firm 2 Control variables Price closed-source software $p_{s}$ Price complementary products $p_{1a}$ $p_{2a}$ Investment software quality $v_{1}$ $v_{2}$ Switching times $\tau_{i}$, $i=1,2$ $\tau_{1}$ $\tau_{2}$ $\tau^{crit}$ switch to $K^{crit}$ State variables Software quality $K_{1}$ $K_{2}$ Parameters Discount rate $r_{1}$ $r_{2}$ Weighting parameter (impact of price on demand) $\varphi$ Initial similarity between the two software projects $\epsilon$ Max. possible demand for complementary product $\alpha_{1}$ $\alpha_{2}$ Cost parameter $c_{1}$ $c_{2}$ Technical obsolescence rate $\delta$ $\delta$ OS community contribution rate $m_{1}$ $m_{2}$ Impact of other firm on demand $\eta_{1}$ $\eta_{2}$ Spillover of software development $n_{1}$ $n_{2}$ Value function param. (stage 3) $\beta_{1}$, $\gamma_{1}$, $\pi_{1}$ $\beta_{2}$, $\gamma_{2}$, $\pi_{2}$
 Firm 1 Firm 2 Control variables Price closed-source software $p_{s}$ Price complementary products $p_{1a}$ $p_{2a}$ Investment software quality $v_{1}$ $v_{2}$ Switching times $\tau_{i}$, $i=1,2$ $\tau_{1}$ $\tau_{2}$ $\tau^{crit}$ switch to $K^{crit}$ State variables Software quality $K_{1}$ $K_{2}$ Parameters Discount rate $r_{1}$ $r_{2}$ Weighting parameter (impact of price on demand) $\varphi$ Initial similarity between the two software projects $\epsilon$ Max. possible demand for complementary product $\alpha_{1}$ $\alpha_{2}$ Cost parameter $c_{1}$ $c_{2}$ Technical obsolescence rate $\delta$ $\delta$ OS community contribution rate $m_{1}$ $m_{2}$ Impact of other firm on demand $\eta_{1}$ $\eta_{2}$ Spillover of software development $n_{1}$ $n_{2}$ Value function param. (stage 3) $\beta_{1}$, $\gamma_{1}$, $\pi_{1}$ $\beta_{2}$, $\gamma_{2}$, $\pi_{2}$
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