I offer a brief observation on a long-running debate: “carrot on a stick” and “the carrot or the stick” are two different metaphors, with two very different meanings. Mostly I have encountered folks who use “the carrot or the stick” denying that the other phrase makes any sense, and claiming that it’s a mistake. This only proves that they don’t understand it.
“The carrot or the stick,” as is widely observed, is a simple metaphor for reward vs. punishment. It is useful for those who want to talk about situations where either reward or punishment can be offered as strategic alternatives. If we bring a donkey into the picture: a person has a donkey pulling a load, and can use the carrot to lure the donkey forward, or the stick to goad the donkey.
“Carrot on a stick” represents a contraption of never-satisfied desire. It is useful to talk about situations where deferred satisfaction is structural or by-design. Again, enter donkey: a carrot is dangled in front of a donkey from a stick projecting forward, from a cart the donkey pulls; as the donkey moves forward to get the carrot, the carrot moves forward also, because its purpose is not to satisfy but to keep the donkey moving.
“The carrot or the stick” is used to convey varied strategies of cooperation vs. coercion. This is supposed to teach you how to move the donkey.
“Carrot on a stick” is used to expose social constructs that offer never-delivered rewards by design. This is supposed to make you realize that you’re the donkey in a system that will never satisfy you.
Of the two metaphors, I think “carrot on a stick” is the more useful, as a simple metaphor for what are often very complex structures of perpetual, deceptive motivation. There’s nothing very subtle or insightful about reward-vs.-punishment, and I’m generally on team cooperation for getting actual, sustainable work done.