You have it completely backwards. A psychopath would likely claim that she/he cares about some "larger" common good while backstabbing all of the closest partners.
It's by the definition of psychopath as someone who lacks empathy that my argument follows. It is not a normal human condition to feel empathy for large/abstract groups of people. Most people I know do not feel deeply and profoundly sad for millions who died during a war or a famine, while most of them do experience intense sadness when a person close to them or one they identify with dies. Any empathy expressed for large groups of people is typically of an intellectual kind, a political/machiavellian tactic undertaken only when observed by others, something quite close to what a psychopath would do.
