TL;DR
An acquaintance of mine wants to buy the same apartment as me, and neither of us is willing to back down.
Background
In short, we discovered before going to the showing that were both very interested in the same house (we discovered the listing independently). I don't want to enter into a bidding war with this person, where we would both lose out (by ending up paying more).
Problem
I of course feel that my reasons for wanting to move to this particular flat are stronger than hers, but naturally she feels the same since she doesn't want to back down either. I am staying with my parents and they have just put their property on the market, so I am in need of place. This apartment meets all of my needs, and also allows me to stay in the same complex as my sister and brother-in-law, who I have a good relationship with. She however has told me that she needs to enter the market soon so she can have her parents over in the holidays. She also said that I should look at something else since she has perceived that I earn enough to get a bigger place (and implies that she can't buy anything else), which seems to be a guilt tactic to me. So, I don't find her reasons valid, and she doesn't accept mine.
What I have tried
I told her I was going to put in an offer, to play open cards and show my seriousness (which might have encouraged her to put in an offer herself sooner than she might have without me telling her this). Because measuring who wants it more (or who has more right to it) is a pointless exercise, we are thus leaving it up to the buyer to decide without having disclosed our offering prices (going to put in our bids blindly against each other).
We also made the estate agent aware of our situation so he will not disclose the other's bid. This is the solution we have gone with, and at least one of us will end up not getting it. I am certain things could have been less stressful for both of us if we could have come to an agreement beforehand.
Questions
How can I ask friend or acquaintance to relinquish attempts to offer to purchase property so that my own attempts will be cheaper/easier?