Highest and best offer...on a rental property? Is this a thing?

[Mauricio Quintana]

Agents who work with rental clients,

We have all heard of bidding wars on home sales.   However, does this also happen on rentals?  I recently put up a place for rent and the response has been overwhelming.  It's priced just under the market average but it's probably one of the nicer units.  Is it normal for you to ask tenants for highest and best offer as well?  I've never heard of this for rental properties but it could also have to do with my limited experience in properties with so many inquiries.

Thanks in advance