Be smart and move on. When a choice has been made by another to be with someone else why fight it? Competing would become a nuisance. Stay friends if you wish.
What I think the OP is asking is if you like someone, but they haven't committed fully to you, do you just walk away or compete with another that this person is also "toying" with.
Since when do we ever begin a relationship when there isn't somebody else possibly trying to be there at the same time. Once in a while you meet someone and no one else is in their life. Most times the numbers are paired down pretty quickly to allow the two to see if the relationship will work. If most people are honest you will admit that you are writing to multiple people at the same time. You might be talking to more than one person. You are possibly dating more than one person at the same time.
So to answer the OP Thread question I believe you never give up on a person you are interested in because there is alway a chance it will develop into what you are wishing for...
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If you find yourself an option do you drop it or compete?(Vote Below)