the.com/carrot and stick

means A method of motivating someone by combining a reward for compliance with a threat of punishment for failure.

from From the old image of a cart-driver dangling a carrot in front of a mule while holding a stick behind it — the animal moves either from desire or fear, and either way it moves. The phrase became fixed in English by the mid-20th century, though the underlying picture (reward out front, threat behind) is older, drawn from simple farm-cart logic rather than any single traceable source.