"When you have multiple superiors handing you work, all due ASAP, how do you prioritize?"

I guess it might help to specify that for this job, I'd be one associate helping out all three of the analysts (research), so everybody's work would be theoretically "equally important"

Although it sounds counter intuitive, I think you should prioritize the work given by the person who is the bigger asshole primarily because a nicer person will more likely be understanding of your situation.

The answer is communication. Keep all superiors informed of your workload at any given time and estimated time to completion. If you have questions on prioritization (you are completely over-burdened), you need to ask the superiors and let them figure it out.

If you get three things and one takes 10 mins, one takes 30 mins and one takes 10 hours you do it in the order of how long it takes. Don't make the 10 min job guy wait 10 hours for you to complete the hardest thing then start his and the guy who has to wait 10 hours anyways can wait an extra 10 for you to get equally as important work done.

As each person gives you work, let them know what else is in the line-up and ask where to prioritize theirs. Obviously this won't work if one them always has bs work and wants it done ASAP. Maybe I am just lucky in that the senior guys on my team know where their work stands in relation to each other's.