"Agree to disagree" is a declaration of face saving justification.
When two or more parties or individuals after an argumentative discussion to resolve issues fail to agree, in all civility the agreement results in agree-to-disagree compromise. With that declaration egos remain intact, but time wasted. Nothing is lost, nothing is gained either.
India and Pakistan have spent over 68 years in this agree-to-disagree parleys, and still continuing.
There is a joke by a comedian from Canada named Norm Macdonald regarding the statement agree-to-disagree. One day he visits his doctor to do a health check up. The doctor says "you are very overweight, unhealthy, and out of shape". The comedian says "I am a healthy person, I don't smoke or drink and I'm in good shape". The doctor continues to say "you are not in shape at all you need to lose weight". So finally after some back and forth arguments, the comedian says "agree to disagree". The doctor's immediate response is "no I will not agree to disagree".
This brief episode suggests when two parties disagree on something, it is useless to say "agree to disagree".
