As opposed to a war of attrition, wherein two or more parties wear each other down over a period of time, a war of contrition is a situation where two or more parties won't stop apologising to each other.
Usually this happens because everyone is eager to avoid confrontation but if often leads to further disagreement.
Motorist #1: Oh, God - I scraped your door with my bumper! Let me pay for that...
Motorist #2: No, no, it's my fault, I shouldn't have parked there.
Motorist #1: Well that's hardly fair, I should have been looking where I was going.
Motorist #2: Look, there's no need to get upset, I've said it's my fault, and I'll deal with it.
Motorist #1: Mate, this was clearly my doing. Let me pay for it.
Motorist #2: Listen here - it's my car, I parked it badly on this corner, I accept full responsibility!
Motorist #1: FINE THEN! *rips wing mirrors off motorist #2's car*
Eyewitness: That was a real war of contrition right there.