What if there is no such thing as character?

The entire eudaimonic tradition in philosophy — which includes Plato, Aristotle, the Stoics, and others in the west, as well as Confucianism in the east — is based on the notion of character, which my desktop dictionary defines as: “the mental and moral qualities distinctive to an individual.”

But what if there is no such thing as character? What if people don’t behave in certain ways because they are, say, courageous or cowardly, temperate or intemperate, but rather simply because they are responding in idiosyncratic ways to local situations?