I ran across this coinage in a newspaper column a few days ago. It refers to a fairly common showdown at 4-way stops in our laid back city when two drivers arrive about the same time, are unsure who has the right of way, and defer so insistently to each other that no one moves, and traffic stops for a few seconds, sort of like the old Alphonse-and-Gaston routine ("After you, my dear sir.” “No, no, I insist, after you.")

Anyway, I started wondering about the origin of this family of nouns, like “bake-off”, “cook-off”, etc. How did it get started?

The OED describes “bake-off” and “cook-off” as formed in imitation of “play-off”, that noun being formed from the verb phrase.  The relevant adverbial sense of “off” is not specifically identified, but would seem to me to be “So as to exhaust or finish; so as to leave none; to the end; entirely, completely, to a finish; as to clear off, drink off, finish off, pay off, polish off, work off, etc.”

It seems to me the “-off” in the phrase “polite-off” is more likely to come from “stand-off”, as in Mexican, rather than the play-off/bake-off series, or am I missing the point?

Nice expression, though - we get the same situation in RIghtpondia when three or four people arrive simultaneously at a mini-roundabout (I have no idea what the Leftpondian expression is for one of those - micro-traffic circle?) and no one is certain who should be giving way to whom. For Rightpondians even older than me, that would be known as “After you, Claude - no, after you, Cecil”, from the Second World War BBC radio comedy programme ITMA - which brings us back to catchphrases ...

I agree with Zytho about “standoff” as the likely source.