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Quote from https://opensource.google.com/docs/why/#engineering-economics:

While open source work may have benevolent results, it is not an act of charity. Releasing work as open source and the corresponding contribution process eventually result in a higher return on the initial investment made versus the alternative closed source process. John Nash, a famous mathematician and subject of the Oscar winning movie “A Beautiful Mind”, won the Nobel prize in economics for his work on “cooperative games”. He demonstrated that cooperating is not a zero sum game and that by working together all participants may yield higher returns than the investment they make. The best real world example of this may be open source software.

Stephen Walli in his blog post on open source motivations writes, “This wasn’t contributing back out of altruism. It was engineering economics. It was the right thing to do, and contributed back to the hardening of the compiler suite we were using ourselves. It was what makes well run open source projects work.”

One good example of this is Angular, a web application framework that is used extensively inside of Google. Angular saw rapid adoption by web developers who built extensions and tools which in turn increases the value inside Google as Google uses these extension and tools internally.

Which leads me to the question, is the above quote true from a mathematical point of view? Is it an open source software truly a cooperative game?

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My short answer is no, as the standard definition of cooperative game is one where binding contracts can be written (see the wikipedia post you cited in your question). Just because a game is called non-cooperative does not mean there is no cooperation. But if I understand open-source software (and I'm no expert) the developers find it in their interest to contribute, but they don't have a contract enforcing their work. I would say this is an example of a non-cooperative game that is not zero sum (zero sum means my gain is your loss) where both (all) parties can gain.

More broadly the question is ambiguous, as it is a question of what is the best way to model a particular feature we seen in the real world. Thus, for all I said about cooperative versus non-cooperative games, a classic way to model perfect competition is often as a cooperative game, using the concept of the Core.

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