02 July 2007

jury duty: coincidences, and semi-juries

Today I had jury duty.

During the lunch break I went to Reading Terminal Market, where I spent $9.05 for lunch (it was a large lunch, because I didn't want to be sitting around hungry); my pay for one day of jury duty was $9. (I was not chosen for a trial.) Coincidence? Probably, because I wasn't thinking "I'm spending my nine bucks on lunch" when I was walking around choosing where I would buy it.

Then I wandered down to a bookstore and found myself flipping through Steven Landsburg's book More Sex Is Safer Sex: The Unconventional Wisdom of Economics. . (I didn't buy it; it seemed interesting, but not $26 worth of interesting.) In particular, this book suggests that the jury system is broken (the link goes to the Freakonomics blogs, where he was interviewed a few weeks ago). The basic idea is that jurors have no incentive to do a good job. This is clearly true, although I'm not sure how to incentivize the jury system. When I got home I ran across an entry in the Freakonomics blog which mentioned that book. Coincidence? Maybe. Maybe not. (And I thin I saw the original Landsburg interview a few weeks ago and forgot about it, which may have primed me to be more likely to look at that specific book.)

Yet another coincidence: I went to high school with the judge's son. (I don't think this is how I got out of serving.) I also went to high school with the son of one of my panel-mates. It occurred to me as I was heading in this morning that if a few hundred people were called today, the chances I'd know one of them were not bad; there are probably about a million adults in Philadelphia, of whom I know a few hundred. I don't think anyone I know was there today (if so, I didn't see them) but as I said there were parents of people I knew. In some ways Philly is the largest small town in the country.

While waiting to be selected, it occurred to me that the voir dire procedure is set up so that no individual juror who is selected was biased. We were a panel of 50 for a sexual assault case, from which fourteen jurors were essentially selected; although I wasn't counting, I would guess that there were no more than twenty people who satisfied the following three conditions:

  • 1. doesn't possess a strong technical background (we had to write our occupations on the forms that were distributed; it seemed that all the people who were asked about their occupation by the judge were either people in technical fields or people who worked as lawyers, police officers, etc.);

  • 2. does not claim that jury duty would pose an extreme hardship;

  • 3. had not been sexually assaulted or had someone close to them sexually assaulted. It's often said that one in four people are sexually assaulted during their lifetime.

Still, would it have been such a horrible thing to have a sexual assault victim on the jury? A randomly selected panel of fourteen would probably have had at least one. I don't see why each individual juror has to be unbiased in order for the group as a whole to be unbiased.

Finally, some math. In Landsburg's book he suggests the following: break each jury up into two half-juries of six. If they come to the same conclusion, that's the verdict; he wasn't clear on what to do if they came to opposing conclusions. (Presumably it would be treated like current hung juries are.) In this study by Bruce Spencer it's suggested that juries are "right" about 88% of the time. This got me thinking -- how likely does this mean an individual juror is to be "right" about the verdict? If we assume that jurors make their decisions independently, that majority rules (which is a bit ingenuous because juries in criminal cases have to be unanimous), and throw out 6-6 results, it turns out each individual juror has to come to the right decision with probability 63.6% to recover this 88% probability. This is related to the post I made a couple weeks ago about the World Series; if one team is slightly better than another, they have a decent chance of winning a single game but not so good a chance of winning a whole series. The teams here are, of course, "guilty" and "not guilty".

So what does this say for the half-jury suggestion? Let's say that each juror, indepedently, has a 63.6% chance of being right, and there's a jury of six. Say the defendant is guilty. Then the probability that all six jurors will say this is (.636)6 = .066; the probability that five think he's guilty and one thinks he's innocent is 6(.636)5(.364) = .227; the probability of the 4-2 split is 15(.636)4(.364)2 = .325. So the probability of one of these three results is 0.618; the probability of having a 3-3 split is 20(.636)3(.364)3 = 0.248. So the probability of a half-jury finding the defendant guilty is (.618)/(1-.248) = .821. Not surprisingly, this is less than the chance of a full jury finding the defendant guilty.

But the chance that both half-juries find the defendant guilty is (.821)2 = .674; the chance that they both find him innocent (even though he did it!) is (1-.821)2 = .032. So the probability of finding the defendant guilty, given that there's a verdict at all, is .674/(.674+.032) = .955. In the end, this plan achieves much greater accuracy at the expense of increasing the number of hung juries. It seems worth considering, though. (Incidentally, you can't beat the hung jury problem by changing the sub-jury size; either at least one sub-jury is of even size or there's an even number of sub-juries, since 12 is even.)

I suspect, though, that this sort of thing would be rejected as being unnecessarily complicated. But the current voir dire process is byzantine enough that that hardly seems like a legitimate complaint.

edit (Tuesday, 9:16 AM): Landsburg has commented to this entry. In particular he points out that my assumption that juries would have the same accuracy in the arrangement with two half-juries as in the current system is incorrect; jurors would have more incentive to be accurate in his proposed system. This is true because in his proposed system the jurors are rewarded when both juries agree. But what I intended to show was that even without such a reward, his system still leads to a greater proportion of correct verdicts.

edit (Tuesday, 2:16 PM): Richard Dawkins suggested in 1997 that "Two juries of six members, or three juries of four members, would probably be an improvement over the present system". He also points out that jurors don't act independently, which is true; in my original analysis I was suggesting that even though jurors don't act independently, we'll assume that they act independently up until the moment they begin deliberation. This assumption is of course not true, but it was only a crude analysis.


Steven E. Landsburg said...

Your math quite misses the point,
by taking as given the probability
that any given juror gets the
verdict right.

The whole point of the "two
half-juries" is that the jurors are
rewarded when both juries agree;
this in incentive for them to work
harder at being accurate.

The right calculation would take
as given not the probability that
a juror is correct, but the
function that converts effort
levels to probabilities of
correctness. Then for a given
reward system, and taking as given
the behavior of 11 jurors, you
can work out the optimal effort
level of the twelfth; in
equilibrium, every juror must be
maximizing subject to the behavior
of the eleven other jurors.

Of course calculating the exact
equilibrium is impossible at this
level because we know very little
about the function from efforts
to probabilities. But we can make
a few reasonable assumptions about
that function and prove a few
theorems about the nature of
equilibrium (for example, comparing
the outcome with two half-juries
to that with three third-juries).
That's where the fun begins.

Patrick said...

It always seemed to me that the US tries a lot more cases by jury than Canada does.

I'm not sure that an external incentive would be useful for juries. People do a lot of things without any external incentive, like contributing to free software, or doing research (sort of, and after getting tenure). Adding incentives can sometimes lead to people adopting a "it's just a job" mentality.

The Probabilist said...


I'm not entirely sure that an external incentive would be useful either, because of the "it's just a job" mentality that you mention. This is why I didn't consider that as an option. Landsburg considers it, though.

One thing that comes to mind for me in an incentivized system is that it's not clear what the incentive should be -- assuming it's direct financial compensation -- because different people value their time differently.

Mary Pat said...

I've posted on my jury experience before (don't have the link handy), but I did a car-jacking case where the less educated (to put it politely) were removed during voir dire, and I stayed on even though I had a M.S. math, had been the victim of robbery before, and had had relatives who had been convicted of robbery. Also, there was an ex-cop on the jury. But that's probably because NYC had such a problem with so many people claiming exemptions that even active cops could be on a jury, and the sitting mayors have been on criminal juries.

And it's difficult to say what is a correct outcome for a case. In the case we tried, the guy being prosecuted had a good chance of being the perp, but the prosecutor did a crap job showing this. The jurors were definitely not independent in coming to a decision, in that some thought the guy was outright innocent, some (like me) thought probability was good that he did it, but there was a lot of missing information in the prosecution that was suspicious, and there was one woman convinced of guilt due to some of the stupidity from the defense and those of us in the second camp were the ones who convinced her to go with not guilty.

So, were we right or wrong? After we gave our verdict, the judge said he agreed with us (and for the reason I said - it was up to the prosecution to prove the case, and he didn't do it). If the guy had actually done the crime, then as a jury we were wrong, but given that we had to go with the evidence presented, there wasn't much we could do about it.