But I have been puzzling for a few days now over a curious post by Daniel Hamermesh about game theory in the Bible. Maybe someone who is more knowledgeable about religion can help me out with this.
Hamermesh analyzes a story from Genesis 20:1-18:
Genesis 20:1-18 tells of Abraham visiting Avimelech and offering him Sarah (who, so Abraham tells Avimelech, is his sister, when she is also his wife). Despite Avimelech’s ignorance of the true relation between Abraham and Sarah, and despite the fact that he hasn’t slept with Sarah, G-d is angry at him and renders the women of his household sterile. G-d asks Avimelech to free Sarah. Presumably, the men in Avimelech’s household are not happy about their wives’ infertility, which causes Avimelech trouble.Hamermesh summarizes Avimelech's and God's preferences in the following payoff matrix:
Although both free/fertile and keep/sterile are Nash equilibria, Hamermesh suggests we'll end up at free/fertile equilibrium — why, I don't quite understand.
But even more puzzling is Avimelech's payoffs. Regardless of whether he keeps or frees Sarah, Avimelech would prefer everyone be fertile; this makes sense. If everyone has to be sterile, Avimelech would rather keep Sarah than free her; this makes sense — presumably he wouldn't have accepted Sarah as his wife if he didn't want her. So why, then, if everyone is fertile, would Avimelech prefer to free Sarah rather than keep her?
Given the way Hamermesh describes the story, this doesn't make any sense to me. A few commenters on the original post noted this as well. It seems to me that we should reverse the keep/fertile and free/fertile payoffs for Avimelech. This would make keeping Sarah a dominant strategy for Avimelech, and since God knows Avimelech will always choose to keep Sarah, His best option is to make everyone sterile. This leads to an outcome inconsistent with the Bible, which is no good.
But after reading the Bible passage directly, it seems to me like Hamermesh is telling the story wrong. Hamermesh's version seems to suggest that Avimelech would like to keep Sarah, but it is God rendering everyone sterile that changes his mind. Reading the Bible passage, it sounds more to me like Avimelech doesn't realize Sarah is married to Abraham, God tells Avimelech the truth, and Avimelech, knowing Sarah's true identity, is no longer interested in keeping her — regardless of whether or not God renders everyone sterile. If this interpretation is correct, we get a clear Nash equilibrium of free/fertile (consistent with the Bible — yay!), but it makes the whole game theory exercise a bit of a moot point.
Perhaps someone who understands the Bible a little better than me can explain what's going on in this Genesis story, and what a proper payoff matrix should look like, so I can stop puzzling over this.