Wednesday, October 27

Freakonomics takes on the Bible

Those of you who have looked at my links to favourite blogs will know I am an avid reader of the Freakonomics blog. They produce a large quantity of posts, many of which I never read (I have yet to understand what the Yale Book of Quotations has to do with "the hidden side of everything"), but they have some very well-respected bloggers and their posts are usually non-technical and high quality.

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.

3 comments:

  1. I don't know anything about "payoff bimatrices", but it is astonishing that someone who is devout enough to not spell out God is also so irreverent. Economics is totally beside the point here. It's like criticizing the Bible for implying that pi equals 3. What's next, economists try to answer Hamlet's question "To be or not to be?" with a payoff matrix?

    The thing I take from this story is that Abraham is a jerk and should have learned his lesson in Genesis 12.

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  2. Payoff matrices show "payoffs" for a simultaneous-move game theory problem. The payoffs do not usually have units associated with them; sometimes they can be dollars, although in this case I believe they are "utils," a very arbitrary measure of "utility." The payoff matrix shows each player's payoffs for each possible outcome; the first number represents Avimelech's payoffs, while the second number represents God's payoffs.

    Yes, it is strange that someone would try to analyze the Bible using game theory. The whole exercise is a bit silly, but his analysis confused me so I was trying to figure it out.

    I didn't know you read the Bible. I was not aware of Genesis 12 until now, but yes, it would seem that Abraham should have learned not to keep saying his wife is his sister. (As a side note though: If Abraham was 75 in Genesis 12 and Sarai was extremely attractive at the time, surely Sarai was a lot younger -- wouldn't this tip off Pharaoh that Abraham and Sarai weren't brother and sister?)

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  3. I started to read the Bible out of curiosity and literary interest. Actually Abra(ha)m and Sara(h/i) _are_ brother and sister -- they have the same father) -- and are also married. (Thus Abraham wasn't technically lying.) Maybe this should have tipped Pharaoh off regardless. Then again, in Genesis the main characters often live 500-1000 years, so I'm not sure just what would seem unusual.

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