The problem I would have with such a concept is this: what would be the "point"? Would you, at the end of The Quest, receive some particular useful benefit - a special weapon mod, access to a new ship type, an NPC pilot with extra skills, or anything else really that you could then use in-game forever afterwards in the game, in any mode?
If the answer is "yes", then it really is a pay-to-win scenario.
If the answer is "no", then I get back to the "what's the point" question. You've basically paid FD some money to play a separate mini-game for a while. Why not pay FD, or some other software company, just as much money to play a completely separate game for a little while instead?
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I think you already outlined the point, although not mentioning it: Yes, it would be a separate game. Just on the same game engine and in the same universe. But as long as the other game would be fun, it would be great. I would buy and play it. I really enjoyed games like Wing Commander, Starlancer, I-Wars2, etc. in the past. Having something like that in the ED universe would definitely get the money from me.
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At the same time, it'd have a massive other advantage: it'd be less of a time eater. The mere size of ED and the time it takes to get to a bigger ship was enough for several of my friends to not even try ED. You can spend hundreds of hours here to buy, upgrade and especially engineer ships. It got better, but it's still a looong path for the new player.
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For those people, a campaign version of this game, with like 30 hours of playtime, looks much more interesting. Between work and family, that'd still last them for several weeks, but they'd actually appreciate it much more than the base game. So it could be just the way onto those peoples computers, who were scared away be the mere size of ED.
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Next to that, just when writing this, I realized another aspect: among my friends, we already agreed to get MechWarrior 5 when it comes out, and play through it together. After all, it promises that you can do all the campaign in cooperative mode. ED already has net code and everything. The P2P system, while still not being my favorite, is even perfect for that. So the "narrative mode" version of ED could very well come with coop. Being able to play through some story together would be awesome.
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Of course, that's all just dreaming and painting a picture of how great it could be. But hey, from a mechanical point of view, most of what a campaign mode of ED would need is already there. It would "only" require some good story writing and a lot of fine tuning in the presentation, voice acting, etc. (Whereas "only" probably would still require a year or more of dedicated work of a good team. But that's not my problem. I just dream...

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