the expression is exactly that: carrot AND stick. it is always disappointing when major news publications misuse the expression, which is not so simplistic as "reward v. punishment." what it comes from is the idea of tying a carrot to a stick, and then dangling the stick in front of a donkey, so that the donkey keeps walking forward expecting a reward that is unattainable.
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Trump: Iran deal is a disgrace and embarrassment to this country (Haaretz)
from the article: ANALYSIS / Obama trying carrot, not stick, on Netanyahu