I think you mean ethics and not morals.

Morals is definitely the right word here! The generally-recognized difference between morality and ethics is that the first implies either some kind of inner tendency in individuals (a 'moral core') or an expressly-given 'moral code'. By contrast, ethics refers to the problem of how moralities can play out in practical settings and even interact with each other - ethics does not pit "right versus wrong", but balances "right versus right", as Rushworth Kidder would put it. Although people will also use "ethics", or more properly "normative ethics", as an easy way of referencing values that are so widely shared among human societies that they can be considered near-universal - such as the values of honesty, fairness, good knowledge and a fully thriving life. But this is a derivative meaning and theoretically a less important one. (Anyway, yes, ethical argument is definitely one key way of balancing values, so you're on the right track there.)

Thus, OP's question is itself one of the key problems in ethics; on a larger scale, it also explains the origin of politics itself, as dispute resolution in complex societies becomes reliant on government-like institutions and broadly-acknowledged formal rules of 'fairness'.