Most importantly, is there any other area of debate where we use different arguments to convince women? It would be bizarre.

You seem to be ignorant of what values are. From the point of view of a rationalist, they are axioms, and slippery ones at that as they are axioms elucidated by the individual introspecting his (or her) own emotional reactions to various theoretical situations.

Arguments to convince someone to DO something are tailored to fit the individual being convinced.

Trivial examples of using different arguments to convince women vs men (on average) include arguments to see a particular movie (chick flick vs boobsploitation or violence).