Solomon was a son of King David (of Goliath fame). After outmaneuvering challengers to the throne (which is fairly page-turning, for the Old Testament), Solomon takes his position as the new king. The Bible explains Solomon's legendary wisdom:

 ¹ Now Solomon made a treaty with Pharaoh king of Egypt, and married Pharaoh’s daughter; then he brought her to the City of David until he had finished building his own house, and the house of the LORD, and the wall all around Jerusalem. ² Meanwhile the people sacrificed at the high places, because there was no house built for the name of the LORD until those days. ³ And Solomon loved the LORD, walking in the statutes of his father David, except that he sacrificed and burned incense at the high places.
⁴ Now the king went to Gibeon to sacrifice there, for that was the great high place: Solomon offered a thousand burnt offerings on that altar. ⁵ At Gibeon the LORD appeared to Solomon in a dream by night; and God said, “Ask! What shall I give you?”
⁶ And Solomon said: “You have shown great mercy to Your servant David my father, because he walked before You in truth, in righteousness, and in uprightness of heart with You; You have continued this great kindness for him, and You have given him a son to sit on his throne, as it is this day. ⁷ Now, O LORD my God, You have made Your servant king instead of my father David, but I am a little child; I do not know how to go out or come in. ⁸ And Your servant is in the midst of Your people whom You have chosen, a great people, too numerous to be numbered or counted. ⁹ Therefore give to Your servant an understanding heart to judge Your people, that I may discern between good and evil. For who is able to judge this great people of Yours?”
¹⁰ The speech pleased the Lord, that Solomon had asked this thing. ¹¹ Then God said to him: “Because you have asked this thing, and have not asked long life for yourself, nor have asked riches for yourself, nor have asked the life of your enemies, but have asked for yourself understanding to discern justice, ¹² behold, I have done according to your words; see, I have given you a wise and understanding heart, so that there has not been anyone like you before you, nor shall any like you arise after you. ¹³ And I have also given you what you have not asked: both riches and honor, so that there shall not be anyone like you among the kings all your days. ¹⁴ So if you walk in My ways, to keep My statutes and My commandments, as your father David walked, then I will lengthen your days.”
¹⁵ Then Solomon awoke; and indeed it had been a dream. And he came to Jerusalem and stood before the ark of the covenant of the LORD, offered up burnt offerings, offered peace offerings, and made a feast for all his servants. (1 Kings 3:1-15, New King James Version)

So that's pretty neat. You could say that Solomon's true wisdom consisted in his asking for wisdom.

OK it's one thing to just say, "This guy in our story was wise." But the Bible then gives a specific example, and I think all will admit that this is pretty neat as well:

   
¹⁶ Now two women who were harlots came to the king, and stood before him. ¹⁷ And one woman said, “O my lord, this woman and I dwell in the same house; and I gave birth while she was in the house. ¹⁸ Then it happened, the third day after I had given birth, that this woman also gave birth. And we were together; no one was with us in the house, except the two of us in the house. ¹⁹ And this woman’s son died in the night, because she lay on him. ²⁰ So she arose in the middle of the night and took my son from my side, while your maidservant slept, and laid him in her bosom, and laid her dead child in my bosom. ²¹ And when I rose in the morning to nurse my son, there he was, dead. But when I had examined him in the morning, indeed, he was not my son whom I had borne.”
²² Then the other woman said, “No! But the living one is my son, and the dead one is your son.”
And the first woman said, “No! But the dead one is your son, and the living one is my son.”
Thus they spoke before the king.
²³ And the king said, “The one says, ‘This is my son, who lives, and your son is the dead one’; and the other says, ‘No! But your son is the dead one, and my son is the living one.’|” ²⁴ Then the king said, “Bring me a sword.” So they brought a sword before the king. ²⁵ And the king said, “Divide the living child in two, and give half to one, and half to the other.”
²⁶ Then the woman whose son was living spoke to the king, for she yearned with compassion for her son; and she said, “O my lord, give her the living child, and by no means kill him!”
But the other said, “Let him be neither mine nor yours, but divide him.”
²⁷ So the king answered and said, “Give the first woman the living child, and by no means kill him; she is his mother.”
²⁸ And all Israel heard of the judgment which the king had rendered; and they feared the king, for they saw that the wisdom of God was in him to administer justice. (1 Kings 3:16-28, New King James Version)

Now note that even though Solomon relied on (what game theorists would call) the self-revelation principle, it actually doesn't constitute a Nash equilibrium. In other words, if two different harlots (who had heard of the first encounter) approached Solomon with the same problem, then the liar would know to say, "Don't kill the baby! Give him to her!"

But so what? Only tenured game theorists prefer to study (subgame perfect) equilibria, rather than concentrate on what actual people do when they have to play a game. And note, I'm not talking endorsing "experimental economics" here; you can go ahead and theorize in an ivory tower. My point is, don't restrict your theorizing to equilibrium cases, when (say) every chess game that has ever been played has been in "disequilibrium."

BTW, if you want an example of someone using the self-revelation principle where it's also self-enforcing, try this story (which I think I got from either Steve Landsburg or David Friedman): There is a bad storm tossing a ship to and fro. A safe containing the personal savings of the crew gets bounced around, so that all the gold and silver coins in the safe fall out of their bins. The captain also loses the piece of paper that recorded how much each sailor had deposited.

So the captain makes an announcement: "I want each man to write on a piece of paper which coins are his. After I get all the slips of paper, I will tally up the numbers. If the total you indicate is either too high or too low, I will throw the whole chest overboard."

Extra credit for the geeks in the crowd: Imagine that the captain will only throw the chest overboard if there are more coins claimed than exist. If there are fewer coins claimed, then the captain distributes them as claimed, and pockets the remainder. What happens then to the optimal bid?