OBO or "Asking Price" for Rental Units?

[Calvin Kwan]

I living in the bay area where there is a shortage of good rental units.  I rent, and the last couple of units I've gone to see have always had over a dozen applicants.  What would the drawbacks be to setting your rental price with an OBO or "asking price"?  Worst case, you get your asking price.  Best case, you get someone willing to pay more, or even a bidding war for your rental unit.

This would be similar to ways property is sold.  Thoughts?  Is this event legal?