I believe you're overthinking it. The guy has something he wants to sell and it happens to be something you want to buy. He has an advantage because the property is next to yours and he probably knows that the value of an adjoining property is usually higher (known as the concept of assemblage in real estate). Basically, you'll be willing to pay more for the property than anyone else because it can expand your space and also block anyone undesirable from purchasing it. With that in mind, you do have to consider those same things in crafting your offer. You can stick to your guns at 15 and hope someone else won't give him 18, but if they do you have lost the opportunity to have an adjoining lot for the foreseeable future and may get a neighbor you don't want. Also, have the additional lot may increase the value of your existing property because if you go to sell you are going to be able to offer something more unique that, since it appeals to you, will also probably appeal to other buyers down the road.
I guess my advice is that you have to decide what the extra lot is really worth to you, make that value be the maximum you'll pay, and then negotiate the best you can to come in under that number. I would say look, I know other lots in here are going for 15, but I would like to just wrap this up quick so I'll give you 16 and we can close in "x" days. If he sticks to 20 don't counter back immediately, tell him if he reconsiders to give you a call. Wait a few days, tell him you spoke it over with your wife and that she agreed to go to 17, but that's all you can do. If he turns it down, tell him thanks and to keep you in mind and then just let it go for a week. After a week call him back and let him know your 17 offer still stands.
