Fair practices for renting a unit

[Raman Bindlish]

I am looking for some feedback on what are the fair practices / law for renting a unit when there are multiple applicants.

(Background - Sacramento, California - duplex rental unit - for local law considerations)

I have marketed the property at a particular price. I have accepted applications at that price point. Now I have another applicant coming in (who is not necessarily more qualified for the unit but in similar ball-park as other applicants) and offering me a higher rent amount to consider him. Will it be fair for other applicants if I accept this applicant?

Can you ask for a higher rent for the unit if there are multiple applicants who really wants the rental and willing to offer a higher price? Do I need to inform all the existing applicants if they want to bid for the higher rent? Is it legal / fair? 

I know there are some sites which have cropped up that allow rental bidding. But I am sure they must have figured out the law and the loopholes. I am trying to do right thing while getting market rent for my property (which seems to be higher than what I advertised for).