I partly agree with EP. I'd still give a deadline, for your own protection (I do this in any market), but I think you need to make yourselves come across as a bit more savvy and serious:
1) Don't make an unusual offer as an opening bid, at least not down to hundreds! Over $500K, I would make an initial offer that's rounded to at least a $5K increment. There's no reason why an unusual figure isn't any good as an offer,
except that it makes them wonder what you're up to, and whether you're playing games (or going to play games). I wouldn't be surprised if this strategy cost you this deal; many vendors just don't want to deal with people who don't appear straightforward.
2)
Do NOT say you're making a final offer unless you absolutely mean it.
3) Be decisive by making confident bids - eg reasonable size increases. It doesn't make you look desperate; it makes you look like you know what you're doing! Just because you throw in $5K more quite willingly, doesn't mean that you're desperate, it means you're still in the game and you know how it's played.
If it's your final offer, make that clear to the agent, and reinforce it by saying that you definitely need a response by tomorrow at noon (for example), because if they don't take this offer, you need to resume your search. In other words,
desperation comes across to the agent in how you conduct yourself and the language you use when making offers, not from the numbers themselves. These are the things that smell of desperation:
"Just our luck, another bidder! Tell the vendor that our bid's better because ...."
"I really hope they accept this offer"
etc - ie anything emotional
This sounds confident and serious:
"Look, we're obviously close to an agreement, so in order to try and close this deal, we're willing to offer $5K more, but that's our final offer. We believe that we're approaching the point of offering more than the property is worth and we really can't go any higher." (May I stress again that you should only say this if you absolutely
mean it!)
Or if you're not ready to commit to a final offer:
"We want to purchase this property, but we don't wish to engage in endless back-and-forth. We can offer a little more but we're not sure whether it's going to be enough. Please ask the vendor what figure would make them happy to sign a contract with us and take this property off the market, and we'll see whether we can stretch that far." (Note you're still not committing to paying that figure; you're just putting the ball back in their court.
)
Bear in mind that this is just my preferred style; I'm not a negotiating expert. Others may have totally different strategies! But I think that it's important to be confident and decisive, yet unemotional. Never make threats (ie "final offer") that you're not willing to carry through. Never lie.
Good luck!