The ability of negotiators to psychologically use gender as a tactical advantage is a real phenomenon, and is worthy of some study. Using gender strengths and eschewing weakness to persuade is a vast topic and could fill volumes.
Many studies have shown statistically significant differences between the genders in a negotiation setting. For example, women who negotiate in person and use non-verbal cues (e.g., eye contact) are much more successful than men. Some studies show men should negotiate via email[viii].
The way to deal with this in your interactions is to study up on gender strengths and weaknesses. E.g. email favors some men as does phone; face-to-face interaction favors women who control personal cues, and/or professional female negotiators[ix].