The question comes in how much do you like the house? If you like it, and can see yourself living there for the next 25 years you may consider writing a letter to the owners about how you see yourself living in the home. This would might include raising children, and decorating ideas. There are some cases, that I read about, where lower bids win because of such a letter. This may not work if you are dealing with institutional ownership of the property, but can work if you can make a human connection.

Another thing to play in your favor is the financial position you are in. I would include that with any letter you write. While you won't win against a cash buyer, you may win against someone with 10% down and a slightly higher bid.

In your agent's estimation how serious are they about best and final? You may put in a bid with the understanding that you can go a bit higher. They may come back to you with, "well if you can go 1,888 higher the house is yours".

If this house is not perfect that do not feel obligated to go max price. There will be other homes that come up for sale.

My Update: If you really believe that the agent and seller is being this dishonest you'd probably do best by walking away. However, in general, I would temper my editorial comments. The seller wants two things that can be competing interest. The first is a quick smooth sale, the second as much money as possible. They probably have an eye on their new house and they may also be up against competing offers. If they can get 3K more for their house, then they can up their offer for the new house as well.

This kind of thing is not unheard when housing is in high demand. Currently the low mortgage rates is increasing demand.

The previous answers ignore the way estate agents operate in the UK. Of course this answer assumes the agents are reputable and honest - otherwise all bets are off on what might happen.

First, having appointed an agent, the seller has agreed that ALL communication with potential buyers is carried out through the agent. If a buyer tries to make direct contact, either they will be politely told to to contact the agent, or ignored. It is a high risk strategy for the seller to attempt to go behind the agent's back. The agent may simply terminate the relationship, and will probably warn all other agents in the area not to do business with them, should the seller's private attempt to find a buyer falls through.

Second, the "last and final offer" is normally a legally binding procedure: the seller and the highest offerer are both required to accept this as the basis for the sale, in exactly the same way that the final bid in an auction is legally binding on both parties.

So refusing to submit a final offer is simply walking away from the sale. The agent may assume that the potential buyer doesn't understand the rules of the game and ask whether they really want to resubmit their previous offer, but the agent isn't under any obligation to do that.

The entire purpose of the "last and final offer" is to put an end to a series of offers, counter-offers, re-negotiatons of exactly what is being sold, etc and bring the process to a conclusion. Unless somebody is being dishonest, there is no way to game the process: the highest offer wins, and the selling process is terminated.

If you haven't already, consider making an offer with an escalation clause. Basically, you say, for example: "I'm offering $100,000 and I'm willing to beat any other legitimate offer by $1,000, up to a maximum of $120,000."