• If there are multiple good answers to your question, accept the best one (i.e. the one you find most useful or the one you think others will find most useful if they have the same question in the future, as Ilya was saying). If none of the answers quite answer your question, don't accept them. If two or more answers are equally awesome and completely answer your question (the case Harry was thinking of), just accept one of them and leave a comment saying you wish you could have accepted both; you shouldn't feel bad about this at all.
• If you have a question about etiquette (or any other workings of MO), search meta. If you don't find a discussion on your topic, feel free to open a new discussion. If it's a duplicate after all (like this one was), somebody will direct you to the main thread (like I did here).

Seriously, I don't think we should choose the answers based on 'fairness'. I tend to think that our primary criterion should be the following: what answer will bring more benefit to people reading it? I don't think the speed is all that important.

In the second case, you can encourage some third person to combine two answers and, perhaps, provide more depth.

Also,

Then next time you can just give it to the other guy, if you feel bad or something.

I think Harry provides a wonderful example of how not to do things (thanks, and I really mean it). That is, we are fortunate to have a "society" (site) where "benefits" (in this case, reputation, votes, accepted answers) can be always distributed locally: you see the texts by people and you can make your decision about their usefulness on the spot.

Out of this "market" the global picture emerges naturally; sure sometimes you're lucky to get something, sometimes you're unlucky, but if the thing bothering us is that the reputation count becomes "unfair" I would suggest simply stop considering it to be so important.

Returning to your suggestion, what would happen if people follow it is that for some questions, like your hypothetical followup, a less-useful answer will be chosen about 50% of time. Why would this be an improvement?

The software doesn't have to do everything for you. If in addition to upvoting the answers (and perhaps accepting one of them), you leave a comment explicitly saying that you wish you could accept both answers, you can be pretty sure that somebody will give them an extra upvote because of it. Moreover, leaving such a comment completes the social loop in the same way that accepting an answer does.

I can't think of a coherent way to implement multiple acceptances (that doesn't invite abuse). Given that there's such a nice workaround, leaving it alone is the best option. Of course, I'm curious to hear explicit implementation ideas if you've got them.

Or, I had asked another question in two parts, and two people gave answers for one part each.

It will be unfair to choose either answer over the other, in each of these cases. What do I do?