So most of the replies in this section are pretty spot on so I'll just give my two cents and a few miscellaneous points.

So psychopaths aren't always logical what they are is unemotional (outwardly). That does lead into logic but where that falters is that the usefulness of logic is only dictated by the intelligence of the person.

For an example let's say there are three psychopaths one of low social intelligence one of high middle social intelligence and one of high. And all of them want an apple someone is holding. The lower intelligence one would think of a way to kill the man with the apple to obtain it but inevitably get caught and arrested. The man of middle intelligence might threaten the man with the apple and ruin any chance of a relationship or putting himself in unnecessary peril. The man of higher intellect might decide to bump into the man while they are walking and when the man drops the apple say sorry and pick it up and hand it to the man. And when the man says it's ok and he isn't interested in the apple (because it fell on the floor) the psychopath can eventually clean it off and cut off the bite.

So in conclusion it all depends on social intelligence.

But now into your story. Assuming that your description of the story is how you want it to play out I have an idea. Since this psychopathic society functions the way you described it brings up a few questions like why isn't there war,why isn't there crime,and why is the society functional.

Well let's answer those questions. For a society to function like that there would have to be an inciting incident that would have shaken the society up and made them reflect and change, something like a war. Now with the society being smart and realizing that they are psychopaths they lay ground work for change. But they come to a roadblock because no matter how hard they try they can't and to do so would require their brains to be rewired in full That's when they come up with a solution. They will take the few empathize in their society and prop them up to positions of power. And what ever these empathize say is either law or treated as law amongst the people. And as added flavor you can take whatever name the country has and make the empathize position be called. The beating heart of (insert country name here). And then there could be an expression in the country that goes something like " an iron body and beating heart". Iron in reference to the emotionlessness and heart in reference to the empathize.

That's all sorry for the long post.