The transliterated Greek title of Sophocles famous play is given, because it is important to be aware that the meaning of kingship is at stake, and that Oedipus is defined as the less legitimate kind of king, tyrannus rather than basilieus. The very negative meaning tyrant has now was not so clear for the Ancient Greeks, though the word was certainly used in a negative way. Plato and Aristotle certainly gave a negative meaning to tyrant as opposed to the king who rules through moderation and justice. However, Plato tried to educate the tyrant of Syracuse Dionysius II, in becoming a proper philosopher-king, and other tyrants attended Plato’s school the academy. Of course, Plato’s intentions in contacts with such people was to persuade them to abandon tyrannical behaviour, I can’t see any reason to think that Plato wanted them to give up the power they had.
Going back to 6th century Athens, the tyrant Peisistratos seized power, using it against the aristocracy. He appears to have had the ability tı converse with the lower class Athenians and is treated with some respect by ancient writers, who usually leaned towards the aristocratic in politics. There is a case for saying that he forced through political and social changes which allowed the flourishing of democracy and individual liberty in 5th century Athens. Peisistratus is perhaps relevant to Sophocles’s play, it is at least worth thinking about how that might work. The dominating figure of 5th century Athenian democracy, Pericles is also worth thinking about here, and scholars have done so. Pericles died in a plague in the city of Athens during the Peloponnesian War with Sparta, which does suggest a possible parallel with Oedipus who experiences plague in Thebes. He does not die of the plague, but his attempt to end it, by looking for the murderer of King Laius ends in self-blinding and exile, when he realises that he murdered Laius, that Laius was his father, and that he has married his own mother. None of these horrors apply to Pericles, but he was a kind of elected king.
Unlike the more traditional form of elected king who was appointed for life, Pericles had to stand for his office (of military commander in theory, political leader in practice) every year. Nevertheless he could be taken as a king-tyrant like figure in his power. That his power rested on his personal abilities and public charisma makes him like the tyrant in the meaning of the man who seizes power, the term is also applied to the descendants of such a person who hold power after him, or more traditionally established kings who behave badly. Pericles was in some ways like Oedipus, as the seeker of knowledge. He had a very educated (and aristocratic) background, his friends included the philosopher Anaxagoras, and his power rested on his intelligence, as part of his general force of personality. Oedipus came to power in Thebes, as an apparent foreigner, because he overcame the riddling sphinx, and is then destroyed by the search for knowledge of who killed Laius.
After a rather long detour through possible personal and political references in Oedipus Tyrannus, we should consider the play, which amongst other things is concerned with what kind of ruler Oedipus was. The presentation switched between the leader who listens to the people and is their representative in the opening of the play to the paranoiac tyrant who thinks the Prophet Tiresias is conspiring with Oedipus’ brother in law Creon to overthrow him. Oedipus is even identified as a tyrant by the chorus during these conflicts. He threatens torture and cannot listen to inconvenient opinions without accusations of sedition. When he does fall Creon does become king, and himself seems a bit of a tyrant, ordering Oedipus to hurry up towards his exile and ordering him to leave his children behind. Oedipus has become revealed as something monstrous, an anti-human who cannot live in the city. That status comes from being the exile who was offered a throne he did not inherit, and who found himself to be a native who had the right of inheritance to the throne. That discovery can only come with the knowledge that condemns him to be an outcast monster. In the opening scene, the Thebans are identified by Oedipus as descendants of Cadmus, not knowing that he is one of them. That already introduces the theme of incest, it is the city where every generation has intermarried, reflecting the general greek unwillingness to accept foreigners as citizens, or even the children of marriage between a foreigner and a citizen. The right of inheritance self-destructs in the play, through the outsider discovering he is legitimate heir, which brings the whole idea of kingship into question. All notions of political authority and citizenship disintegrate, more so than the ending in which power has passed smoothly to Creon suggests.