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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
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##### References:
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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