Conflict is a normal part of life. In any conflict there are three choices. You are right, you are wrong, or you are a little bit of both.
Assuming the other person is rational, try to listen to them to see if you might be wrong. If you are wrong, the sooner you can resolve the issue the better. The best way to resolve an issue when you are wrong, is to first admit you are wrong.
If you are right you will understand the other side better, and be able to state your point better.
If the issue does not matter enough to put that much effort into it, just say you are wrong and move on.
Admitting Fault in Large Organizations
I have tried to admit as soon as I can if I am wrong. When I worked in a call center this was not good practice. It would make you look weak, and it is not a good office politics play.
On the other hand issues that should have been easily resolved in the call center, took weeks because no one wanted to admit fault.
Passing the Buck Costs your Consumer (and Hence your Business)
Sadly passing the buck is an effective way to survive large organizations, but it decreases productivity.
I think in this day and age with lawsuits, people just do not want to take any responsibility for anything that has gone wrong. It is sad, because it often involves the issue taking much longer to be resolved, and the customer getting very annoyed. The customer should never have to pay for office politics, but they do.
Remember, the office exists to serve the customer, not the other way around.
Being wrong is the fastest way to grow and learn, but it can be a risky play in an office.
I personally live my life to grow and learn, and avoid office politics. That is perhaps why I am self employed, but this is a decision we must all make. I like focusing on value adding activities. Improving processes, pleasing the customer, etc. None of this can happen if no one ever admits to being wrong.In the end, you should do what makes you a better stronger person. And that is admitting you are wrong as soon as you realize it.