Israelis don't need lectures from American leaders

Seattle Times syndicated column "Israelis don't need lectures from American leaders" (August 19th, 2010) by George Will offers a curious observation. Specifically, that under threat of school buses exploding, Israeli parents would send their children on separate buses to lessen the likelihood of losing two children.

During the onslaught, which began 10 Septembers ago, Israeli parents sending two children to a school would put them on separate buses to decrease the chance that neither would return for dinner.

Corrections finds this rather peculiar. Either sending both children on the same bus, or sending them on different buses can be justified depending on the parent's risk aversion. However, Corrections would expect state-dependent utility to cause parents to send both their children on the same school bus. (Enough to be curious about Will's claim).

The idea is as follows. Parents cannot reduce the expected number of children killed by a terrorist device or gunfire by splitting them up or putting them together. All they can do is choose the spread of their losses. Let us imagine, assuming tractable, rather than realistic, numbers, that there is a 50% chance any given bus is blown up.

In this case, then parents are deciding between two options, given they send their children to school.  If they send both together, then they have a 50% chance of losing none, and a 50% chance of losing both.  There is a 0% chance of losing just one, as both will live or die together.  However, if they send both apart, then each has an independent 50% chance of being killed.  With independent chances (assuming negatively correlated shocks, or negative covariance between attacks, would only strengthen our point), their probabilities will now be 25% chance of losing zero children, 50% chance of losing one, and 25% chance of losing both.

The two options are displayed graphically below (click to enlarge).  The red line indicates the decision to send both children apart.  The blue line indicates the decision to send both children together.  As we could calculate, the expected number of children killed is the same: one child.  What is different is the spread of the children killed.

It is up to each individual to discover which choice they prefer.  However, Corrections would have conjectured that parents would choose to keep their children together, because they have state-dependent utility.  Far from dismissing or trivializing the pain a parent would feel, Corrections is conjecturing that, if human beings have some maximum level of pain, and the loss of a child raises one to that level, then individuals are at a "censoring" point for their "gambles" with their children's lives.

Our stylized conjecture about the pain of losing a number of children is displayed graphically below (click to enlarge).  The idea is as follows:  a parent hits a "maximum pain" point as soon as a single child dies.  The loss of another cannot raise their pain past the point at which it is already.  (We note that our general point also survives any two-children-lost pain-level adjustment that does not raise the pain loss of two children to greater than double the loss of one). If we then plot the expected pain of each decision, we see that sending both children on the same bus, because it is "upward censored", gives an expected pain level of .5.  However, sending both children on different buses  gives an expected pain level of .75. The two expected levels of pain are also displayed on our graph, blue denoting the pain of sending children apart, and red denoting the pain of sending children together.

As we can see, the expected pain of sending children together would appear to be less than the expected pain of sending them apart, due to the "censoring" of pain--the pain of the loss of two is less than double the pain of the loss of one.

However, Israeli parents, according to Will's article, are not behaving in the way Corrections would suggest.  Faced with theory and reported empirics not meeting, methodologically, several options are open to our understanding:

1. Parental utility does not fall into the broad class of relative utilities we conjecture.
2. Will's facts are wrong.
3. Will's facts are right, his reasons are wrong (reasons we miss).
4. Our broader framework of expected utility maximization is incorrect.

Corrections ascribes the failure of our model as most likely to be due to either Option 2 or Option 3.  Supporting Will's claim, a cursory examination of the ~170 bus casualties in Israel in the last 10 years yields  few people with the same last name killed in the same explosion or gunfire, and fewer same-age siblings.  Suggesting Option 2 is the likelihood that the observation came purely from anecdote.