Early on in The Republic, Socrates contends with Thrasymachus's argument that "justice is the interest of the stronger". Socrates gives some analogies, such as the relationship that doctors have with their patients, and captains with their sailors. In these analogies, the superior serves the interest of his or her inferiors. Eventually he rounds off by saying that rulers likewise serve the interest of their subjects, and only the interest of their subjects.
Wikipedia reproduces a definition of imperium, supposedly given by A.H.M Jones, i.e. "the power vested by the state in a person to do what he considers to be in the best interests of the state". Is this definition accurate? If so, how seriously did Romans take the idea that imperium should serve the best interest of the state, and not the man wielding it? Lastly, how much was this idea influenced by Plato's point that rulers should serve only the interest of their subjects?
I am particularly interested in this last question (although the previous ones are also necessary to reach it) as I'm curious to know how much influence Plato's thoughts on ruling had on the Romans.