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One of the most basic conflict managment techniques is to stop arguing and concede. By conceding the point, the other party no longer has anything to argue about -- they already won. When the conflict is causing more damage than would be caused by conceding, it makes sense to concede rather than continue for the sake of some lofty principle.
Some personality types (the MeyersBriggsTypeIndicator Feeling and Judging types) are particularly likely to carry on an argument even after the other side has conceded, because it takes time for them to internalize the agreement.