HCOL 196, March 18, 2011

I started out mentioning a podcast that I had just listened to. It is about 1/2 hour long but interesting. In particular, I mentioned that the book “Predictably Irrational” is largely about how we make suboptimal decisions because of things that are not relevant to the decision.

We  then discussed the death penalty problem. Here we have six outcomes, since there are two possible sentences, life without parole or death (in this example). The best outcomes are the correct ones: AI and CGL. Ranking them is difficult, some ranking them one way and some the other. For the sake of the example, since everyone thought they were pretty close, we assigned a loss of 0 to each. The next best outcome is AG, to which we assigned a loss of 1. It turns out that it doesn’t matter what the loss for CGD is so I just assigned a loss of  Z to it. We then assigned the remaining losses using test trees as we did last time.


To assign a loss for CIL (life but convicting an innocent person), the tree is similar to last time, except that since the penalty of life in prison is more severe than the 5-10 years of the last example, we are even more reluctant to make this mistake, so the probability of making the mistake is smaller. We agreed on p=0.001, which means a loss of 1000.

The worst outcome is CID, executing an innocent person. One student thought that p should be 0 in this case (I agree!), but it’s better just to  pick a non-zero probability that makes us comfortable. People proposed very small numbers, and we settled on p=0.0001, which gives a loss of 10,000,000.

(Click on image above to get full-size image in another window.)

Each juror would have to evaluate the losses for herself. But for these losses, the tree for making the decision is shown above. In this tree, p is now the posterior probability of guilt that each juror evaluates (we’ll discuss this next time). We see two things: (1) The death penalty cannot be the lowest expected loss decision, since it is always worse than life without parole. Basically, this result will be independent of your losses, as long as you think it is worse to execute an innocent person than to put him in prison for life. Just this one fact makes execution always much riskier than life in prison. The value of Z doesn’t affect this as long as Z≥0. Even if, perversely, you thought that Z should be negative (that is, a better outcome than acquitting an innocent person, which seems strange), this result would not change unless you thought that execution of a guilty person was just as hugely better than acquitting an innocent person as you thought that acquitting an innocent person was hugely better than executing an innocent person!

Then we evaluated the value of posterior probability of guilt that we need to have to convict and (as we learned) send the accused to prison for life. It turns out to be about p=0.999, which means you are very sure of guilt.


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