Problematic Probabilities

For background on the Tirhas Habtegiris case, read Steven Landsburg’s column on the subject. I won’t say Tyler Cowen is wrong, but I will say his explanation is rather misleading. Here’s what Tyler has to say:

I do not accept that ex ante choices provide definitive information about ex post values. Taking another context, assume you think the chance of cancer is only five percent when in fact for you it is (for genetic reasons) fifty percent. You fail to buy insurance, but does that mean you don't value your life much? In this case I am willing to downgrade the economic values we might infer from market behavior; it was based on faulty information.

Now take the underinsured, now-dead woman (Tirhas Habtegiris). She thought, either implicitly or explicitly, that her chance of needing "plug-pulling insurance" was small. Maybe this seemed reasonable at the time. But in reality she needed that insurance with p = 1. She was wrong.

People do make mistakes, no question about that. They underestimate the likelihood of some bad events while overestimating the likelihood of others. For this and other reasons, they can make the wrong decision about buying insurance. Okay. But looking at the outcome ex post is emphatically not the correct way to find out the right decision to have made. The fact that Habtegiris did ultimately need a ventilator does not mean her chance of needing it was p = 1. Say I’m playing craps, and the shooter throws snake eyes, causing me to lose my bet. Does that mean the likelihood of snake eyes was p = 1? Of course not. It was p = 1/36, and that is the relevant probability (or one of the relevant probabilities) for deciding the wisdom of making craps bets. Likewise, the relevant probability for Habtegiris’s deciding (not) to get ventilator insurance was the best medical estimate of her likelihood of needing a ventilator at the time she could have bought the insurance, considering her genetics, medical history, and condition as it was known at the time. It’s possible that she underestimated this probability. But the mere fact that she did get sick doesn’t prove it.

Tyler gets at a different point: whether Habtegiris should have bought insurance or not, it’s nonetheless true that eventually she did need a ventilator with probability one. So we – as compassionate onlookers or hospital officials or legislators – can’t escape the decision on whether to fund her ventilator. As Tyler says, “You've just got to decide how much a particular life is worth.” For answering that question, ex post information (does she in fact need a ventilator now?) is relevant. But is that what the debate is really about? Habtegiris may have been an especially nice lady, but I don’t think anyone’s basing their argument on that. Those outraged by the decision to let her die are not making a specific claim about how Habtegiris should have been treated, but a general claim about how people in situations like hers ought to be treated. They’d like everyone in cases like hers to get care. If we adopted a policy like that, we would effectively be providing everyone with ventilator insurance, as we’d be committing in advance to fund ventilators for people who don’t yet need them but might one day. And the wisdom of providing insurance depends on ex ante probabilities, not ex post outcomes.