We've made the first offer and the other side's response was very, very low. They have a controlling, win-lose mentality. How do we get movement? Do we come back and bid against ourselves and lower our original offer, or try to get our attorneys to negotiate?
It is always a difficult task for a negotiator to deal with people who approach negotiation as if it was a competition, with only one possible win-lose outcome. However, difficult does not mean impossible. As in any negotiation process, one of the most important tasks for you would be to discover and identify the other sides values and underlying needs. If you are aware of the reason behind the other sides behaviour, you are more likely to be able to effect changes in his behaviour and direct the negotiation process towards a more desirable solution.
One of the most important factors in negotiation is visibility. Besides using your attorney it might be useful to organise a face to face meeting. Personal meetings are great for finding common ground, common or different points of view around which you might be able to build up a proactive joint problem solving approach. If you are not sure that the other side is ready for collaboration, bidding against yourself is not a good move. As a starting point you would ideally want a gesture from the other side for a commitment or an obligation to work towards mutual satisfaction. If you make a concession, in order to manifest good will, you may end up giving away everything you have for free.
Look to your BATNA (Best Alternative to a Negotiated Agreement) before you make a concession. If your BATNA is strong, a concession may be unnecessary for closing the deal. If you don't have a BATNA it is advisable to develop one. Sometimes, it may so happen that even though you are focused on interests, and working towards developing possible agreements, the other side might still just want to maximise their own goals.
What all too often matters is who wields the power in negotiation. So: what are the sources of power, how can we minimise theirs and increase our own? The trap, while dealing with positional bargainers is our natural tendency for reaction. Usually when one side adopts a positional bargaining approach it starts a cycle of action and reaction. Be careful about pushing forward if the other side wields the power. It is important not to push back too hard, as you don't want to lock them in. Instead of pushing back and attacking their position, look for the interests behind it. Don't defend your ideas, rather invite criticism and advice. Ask whats wrong with your offer instead of pushing it forward aggressively. Questioning will help you to create possible alternatives to overrule objections and achieve a satisfactory solution.
The other thing that might help you to convince your counterpart to cooperate is to show him the consequences of a no-agreement deadlock. What will he lose, if you walk away from the table? How difficult/costly will it be to replace your offer or develop what they want internally? You must remember that costs are not only a financial issue. The issues of wasted time, emotional energy expended in disputing, the resources consumed and destroyed, the opportunities lost, and the recurrence of the conflict, are also there. It is always important to frame negotiation in terms of loss, as numerous studies have proven that we humans are more loss-averse than gain-focused. This is to say that we are far more likely to act when threatened with an impending loss as compared with a bright sparkling goal. Suggest alternative options, ask which they prefer and why. Be willing to walk away and strengthen your BATNA.
As a final option, consider inviting a third party to the table, to mediate between you and the other side. Make a gains-and-costs analysis and choose an option that will serve your needs the best.