Thank you for your interest in our contest -- and for reaching out with your question.
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TL;DR: yes and it depends.
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Yes, if they are selected as a Finalist, the Finalists submit the source code and assets along with their compiled game and a video trailer. This allows us to validate the game before we release it to the judges. We want to be sure that the game works properly for the judges and also need to confirm that the content meets the contest rules.
After judging, the winning game(s) will also undergo some additional work before publishing. These changes are based on the critiques by the judges. The game devs are welcome to do the additional work if they'd prefer or we do it ourselves.
That said, it is not strictly necessary to use a commercially available engine. However we would need to have access to the necessary components to replicate that custom engine in order to fully test the game and then, if it is a winning game, prepare it for publishing. We have done things like this before in a limited way, but it would depend on the difficulty of using that custom engine.
After the game is published we occasionally do have to make updates. Some of them are based on policy requirements at the app stores (currently Apple wants us to update the games we have in their AppStore) or other technical changes (when we were using Google Play they were regularly updating target API level requirements). And sometimes the dialogue includes out-dated cultural references that we'd like to update.
When we began this competition in 2008 most of the games were developed with ActionScript for Flash. Most of those games have not been readily accessible for a few years. Although we've converted one of those games into a downloadable Windows executable (jag.itch.io/graces-diary) and made a few content updates whle doing so, we are trying to avoid running into similar problems as we did with Flash.
We are a very small charity and have very limited resources. But we are open to new ideas and appreciate innovation. We'd be happy to talk further about your custom engine and see if we could find a way to work with it.
Also: entering the contest only requires submitting a game pitch by March 19th. If the pitch is selected then you would become a Finalist and develop that pitch into a game from March 30th until June 30th.
Please let us know if we can provide further insight. And thank you again for your interest and your question!