So, how much is the “other buyer” offering?
In the end, you have to decide what it is worth to you. Blue Books, Edmunds, NADA, people here, etc... all have opinions, and that is all they are. If the seller thinks it is worth more than you or another buyer thinks it is worth more, then it IS worth more. You just have to learn to walk away, but if you think it is worth more to you (and want it) you have to be willing to pay more.
The key is to really know what your value limit is and walk if it gets to that point.