What to do with multiple interested parties?

[Jeffrey Gignac]

I am currently listing a property for $1,200.00/ monthly and have received a TON of qualified applicants that have already paid for credit/background screening. One potential tenant suggested that he be allowed to pay $1,400.00/ monthly so that he will be selected to occupy my property. Are there any violations with allowing the highest bidder to have the house as would be the case in the sale of a home??? 

I want to maximize value on the property without doing anything illegal or questionable. Your thoughts and suggestions are much appreciated.

JDG