Why do you think Plato end the republic by invoking the myth of ER? Or what is the myth of ER?

In Book X, Plato at last pits philosophy-based education in confrontation with traditional poetry-based education. Plato has justified philosophy and the philosopher and now he displays them in relation to their rivals—the people who are currently thought most wise and knowledgeable—the poets.

The myth, in appealing to reward and punishment, represents an argument based on motivations Plato earlier dismissed. Glaucon and Adeimantus had specifically asked him to praise justice without appealing to these factors. Why is he now doing exactly that?

Allan Bloom suggests that the inclusion of this myth is connected to the distinction between philosophical virtue and civic virtue. Philosophical virtue is the kind of virtue the philosopher possesses, and this kind of virtue differs from the virtue of the normal citizen. So far, says Bloom, Plato has only shown that philosophical virtue is worthy in itself. He has not shown that civic virtue is worthy. Since Glaucon and Adeimantus and countless others are not capable of philosophical virtue, he must provide them with some reason to pursue their own sort of virtue. With the contrast between philosophical and civic virtue in mind, Plato describes the thousand year cycles of reward and punishment that follow just and unjust lives.

Yet on our understanding of what makes any virtue worthwhile—its connection to the Forms—Plato has sufficiently demonstrated the worth of both sorts of virtue. Philosophical virtue might be more worthwhile because it not only imitates the Forms, but aims at and consorts with them, but civic virtue is worthwhile as well because it involves bringing the Forms into your life by instituting order and harmony in your soul. Bloom, though, also has another plausible hypothesis for why Plato included the myth of Er, and this one coheres well with our understanding of justice’s worth. The myth of Er, Bloom explains, illustrates once again the necessity of philosophy. The civic virtues alone are not enough. Only the philosophers know how to choose the right new life, because only they understand the soul and understand what makes for a good life and a bad one. The others, who lack this understanding, sometimes choose right and sometimes wrong. They fluctuate back and forth between good lives and miserable ones. Since every soul is responsible for choosing his own life, every person must take full responsibility for being just or unjust. We willingly choose to be unjust because of our ignorance of what makes for a just or unjust soul. Ignorance, then, is the only true sin, and philosophy the only cure.