It is interesting to know that in 1972 Kenneth Thomas and Ralph Kilmann, categorised 5 different styles of how to deal with conflict. Each of them works to some degree and may work very well in certain situations. But to become truly masterful at dealing with conflict you need to have the flexibility to change from one style to the next depending on the type of person you are dealing with.
The styles are: avoidance, competition, compromise, accommodation and collaboration.
The avoiding style is when you choose to avoid the conflict as a way of dealing with it. The aim would be to allow time to cool tempers or if the subject matter is so inconsequential that it really doesn't matter. Another area is where the person giving you conflict is a complete stranger and you are probably never going to see him or her again - Like the yobbo on the street. The danger however is that by continually avoiding an important matter, it could one day blow up in your face.
The competing style is when you stress your position over other points of view. Probably one of the most common methods, here we anchor ourselves in to our position and slug it out till there is a winner and a loser. If you have some authority this may work better and it might be a fast way to resolve things but it can result in resentment and often the loosing side give up due to hostility rather than conceding that the other is right.
The compromising style is the classic "give a little, take a little" or split the difference. This style is often used when both people don't agree but where the stakes are so high that neither party can afford for the deal to fall in a heap due to disagreement. A common tactic to get the deal done but may over the long term lead to an eroded position and perhaps even cynicism.
The accommodating style is when you forego your concerns in order to meet the concerns of others. Used mostly where the relationship is far more important than the disagreement. This may include a business relationship that must be kept intact for long term benefit. The danger however is that you may build a precedent that puts you in a weaker position the next time there is a disagreement and possibly eventually, other will perceive you as a push over.
The collaborating style and the last of the 5 methods is used by people looking for a win-win situation. Perhaps the most beneficial of all the 5 methods it also calls for the most imaginative of minds. In this method people bring alternate possibilities into the mix to help entice the conflicting parties to find an arrangement that suits both. Although it is possible to get a good result with this method, one must also consider that the time needed to find a solution can be much greater than any of the other methods. Also, it is possible that a person not used to this negotiating style may believe that their opponent is trying to avoid the subject.
In summary, no one style is best for all situation but by becoming fluent in each of the styles and willing to quickly swap to a different style in the middle of a disagreement when one style is not working will give you the greatest chance of resolving an argument as quickly and as amenably as possible.