Multiple interested tenants

[Tony Otis]

I'm looking for advice on the best way to proceed when multiple tenants are interested.  My rental is in Minneapolis, MN.  If I have a lot of people interested, could I ask the interested parties if they're willing to pay more rent to set their application apart?

I believe that I am priced near market rent, there are definitely cheaper, similar alternatives on Craigslist.  I think the reason my listing has so much attention is it is one of the few 4BR apartments in the area.  On the other hand, because the 4BR units are rare, perhaps it is worth it to one my prospective groups to pay more.

The other alternative is to do first come first served, and forego the potential extra rent.

Thanks,

Tony