What kind of argument? There are different kinds. The rules are different for different kinds. I get the impression this is about interpersonal arguments - domestic argument, though not limited to spouses etc.
There is also political argument. A totally different beast.
Scientific argument - which today is often more political than scientific, so its several different scopes of argument.
Legal argument - which I only grasp enough to know when I need a lawyer.
Formal debate - which is different from all the above.
Public relations - argument with agenda.
Advertising ... this is a form of argument, most closely related to public relations.
Etc ... I am bound to be missing a few on my list.
Interpersonal/domestic arguments also depend on the purpose of the argument, there are subtypes.
Sometimes argument is darn hard, particularly with ME but even without it.
My most basic rule, which I fail to adhere to on occasion, is to show respect. I think this should be number one. It doesn't matter if you are talking to a gang member on the street, a small child who thinks they have done something wrong, or a friend or family member. Respect is number one.
People who abuse that and put other people down create problems. Sure they can gain power, or satisfaction, but it is a power based relationship. It will not persuade, it will not discover, it only creates obedience or rebellion. Someone who does this may be feared, they will not be respected.