What is polytropos?
The term polytropos literally means "many forms," but in practical literary use, it carries a meaning closer to "many personalities" and, more precisely, "adaptive personalities." People with adaptive personalities are able to correctly and effectively adapt their behavior to a specific situation in order to succeed--even if the personality that one adapts is very unlike one's natural personality.
We'll illustrate this in the context of Odysseus's behavior in The Odyssey. Odysseus, because he is a well-rounded, multi-faceted character has both good and bad traits that either save him and his men or jeopardize their lives. For example, during the encounter with Polyphemus, Odysseus's pride cannot keep him from ultimately disclosing his identity to Polyphemus, who then begs his father to wreak revenge on Odysseus and his men. That was an example of limited or failed polytropos.
An good example of adaptive personality working well is in Odysseus's precautions when his ship nears the Sirens. Odysseus, knowing that he cannot resist the temptation of the Sirens' calls, orders his men to tie him to the mast and ignore his orders. This is a point at which self-knowledge is important and an example of polytropos in action to insure the survival of Odysseus and his men.
Perhaps the ultimate example of polytropos in the Odyssey is at the conclusion when Odysseus finally arrives on Ithaca and discovers that his court and wife, Penelope, are inundated with would-be suitors. Correctly ascertaining that the suitors would most likely find a way to do away with Odysseus, he disguises himself, puts up with the disdain and mistreatment from his "royal" guests, and eventually defeats them. Had his personality not been adaptive to this unusual situation, Odysseus might have simply arrived, tried to reclaim his throne, and been murdered by the claimants to his throne and wife.