It's hard to keep incentives aligned during negotiations. Building trust to find what your counter-parter really wants is delicate dance. You need to probe, but you can't be too pushy. Without this understanding, it's difficult to make an attractive offer.
Information asymmetries provide the biggest advantages in negotiations. For example, if you sell a house that you know has significant dry-wrought and the buyer doesn't, you have a better understanding of the underlying value. As a result, you can sell the house for a higher price than if everyone had the same information.
Aside from due diligence, my professor suggests following the Moscow rules to protect yourself from being taken advantage of. Most importantly, have a healthy dose of skepticism about motivation, challenge assumptions, and remember that you're always on stage. You never know who may overhear internal team discussions.