Game Theory and Solomon’s Wisdom

There is a very famous story about Solomon’s wisdom. N the story there are two new mothers, one of which who had smothered her baby during sleep and claimed the other woman’s baby as her own. The problem is presented to King Solomon, who proposes the baby be split in half, each woman receiving one half of the child. The woman who was lying agrees to the compromise, while the real mother immediately feels sympathy for her offspring. Rather than see her child killed, she says the baby belongs to the other woman. Solomon instantly gives the baby to the real mother, realizing that a true mother would compromise to see her offspring survive.
After learning game theory in class, let’s look at this story again. If both of the two women are experts in game theory, Solomon’s wisdom will fail. Assume woman A is the real mother and woman B is the liar. When Solomon asks to cut the baby into halves, A will definitely say no because she loves the baby and does not want to hurt it. On the other hand, if B understands game theory and think rationally, she will make the decision of giving the baby to A also. Now the situation is that both women A and B would like to give the baby to the other. We come back to the origin. Whether Solomon will cut the baby into halves or not, he cannot judge which woman is the baby’s real mother now. Solomon’s wisdom failed after learning game theory.
Is there any other method to solve the problem Solomon faces? Or is it still a puzzle?