URGENT QUESTION: What would you do if you have two buyers who want the same house?

Maybe you can help me figure out what's the best way for me to do. I am representing a buyer and an investor who happens to be my mother. For a moment, let's say that the investor is not my mom, can I submit 2 offers at the same time for the same property? I was advised 2 different ways:

1. To choose which client to submit offer from, if the offer falls off, then submit the other offer.

2. As long as I disclose to each of the party that I represent both of them, then submitting two offers for one property should not be a problem...bottomline, may the best offer win.

I am in California and would certainly appreciate advices esp. from RE Attorneys in California if I would really violate any ethical law in this case. My own thinking and honest feeling is...I can submit 2 offers at the same time because in reality being an agent of each of them, I have the responsibility to represent them. I cannot tell one or the other that.."Sorry, can we get my other client's offer in first and if it falls off, I'll submit yours?"

I am facing this dilemma as I write this...I will be waiting for your insights! Thanks!