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One of the ways to answer this very good question is to look at the Greek concept of hubris. Both characters are some of the proudest characters in all of Greek literature. In a word, they are characterized by hubris.
Achilles shows this by not bending to the command of his king, Agamemnon. Also Achilles is so proud that he will not enter the fight, even though his fellow Greeks are dying at the hands of the Trojans. We can say that his pride and sense of honor is too great.
The same can be said of Oedipus. He is the great ruler of Thebes and he cannot see or even entertain the thought that the plague on his city is due to his crimes. In a sense, he is very blind. His pride blinds him.
In light of these two examples, we can say that both characters show the universal characteristic in man - his pride.
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