CutiePie,
Technically, yes the woman could've accepted the bid from Gamma Phi if she had not yet signed a bid card.
I just did bid matching at a campus where nine women out of 45 went bidless (20%) because they all wanted the same two houses, and those two houses reached quota by the time they got to these women. And in most cases, these women either intentionally single pref'd or only put down the two houses. Now, with that being said, quota was 7 and only the two "popular" houses made quota. So instead of opening up the possibility of snap bidding, they are letting all those bidless women COB, which means the top two houses will take all those bidless women through COB. If that isn't a prescription for dirty rushing, I don't know what is.
On my campus, we had a big problem with sororities telling women to intentionally single pref their house, and told them they wouldn't get a bid through formal rush, and that then the women could COB. And this worked for awhile. Then, finally, the Greek Advisor listened to the complaints of the smaller groups who thought this was the case, and they made a rule that any woman who intentionally single prefs a house and doesn't get a bid cannot COB at that house. This did cut down on intentional single preferencing, and it also stopped a little of the dirty rushing.
It's a shame that women go bidless. What's a bigger shame is that sororities give women the idea that they have a chance to get a bid by inviting them to preference, with no intention of offering a bid in the first place.
But that's all just my opinion
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