We briefly discussed “the value of a human life,” to be used in cost-benefit calculations such as the one in the current problem set. We talked about the government’s figures, which are in the 4-7 million dollar range, depending on the agency. $5 million seems like a good compromise.

We then started a discussion on jury duty. If you were on a jury, what are the decisions and what are the states of nature. The states of nature are “actually innocent” (I) and “actually guilty” (G), and the actions are “acquit” (A) and “convict” (C). So there are four possible outcomes, as in the chart. We need to assign losses to each of the four. AI and CG are both correct decisions, and we decided that they are equally good, so we assign them a loss of 0. Of the other two decisions, CI and AG, we decided that CI is the worst, and AG is in the middle. So we assigned a loss of 1 to AG (the scale is arbitrary, all scales will give the same answers).

To determine a loss for CI, set up a tree like the one above, where the intermediate outcome, AG, is on the “certain” leg and the worst and best outcomes are on the probabilistic legs. We then have to choose a value of p that makes us indifferent as to which outcome we get. We decided that p=0.01 was about right, it corresponds to about a 1% error rate, a 1% chance of convicting an innocent person. This makes the loss for CI equal to 100.

To use these losses, we can now decide how certain we must be (that is, what our posterior probability of guilt must be) in order for us to vote to convict. The appropriate decision tree is shown below:

We choose the decision that minimizes the expected loss. This means that we will acquit of p<100(1-p), or p<100/101, or about 0.99:

Next time we’ll talk about death penalty decisions. In this case there are three actions, not two: Acquit, Convict and give a life sentence, Convict and impose the death penalty. I noted that even though Vermont does not have the death penalty, it is possible for a Vermont jury to be faced with this decision in a federal case, as in the Jacques (pronounced ‘Jakes’) case. So I asked everyone to think about this for next time.

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