Less Wrong is a community blog devoted to refining the art of human rationality. Please visit our About page for more information.

Conservation of Expected Jury Probability

10 Post author: jkaufman 22 August 2014 03:25PM

The New York Times has a calculator to explain how getting on a jury works. They have a slider at the top indicating how likely each of the two lawyers think you are to side with them, and as you answer questions it moves around. For example, if you select that your occupation is "blue collar" then it says "more likely to side with plaintiff" while "white collar" gives "more likely to side with defendant". As you give it more information the pointer labeled "you" slides back and forth, representing the lawyers' ongoing revision of their estimates of you. Let's see what this looks like.

Initial
Selecting "Over 30"
Selecting "Under 30"

For several other questions, however, the options aren't matched. If your household income is under $50k then it will give you "more likely to side with plaintiff" while if it's over $50k then it will say "no effect on either lawyer". This is not how conservation of expected evidence works: if learning something pushes you in one direction, then learning its opposite has to push you in the other.

Let's try this with some numbers. Say people's leanings are:

income probability of siding with plaintiff probability of siding with defendant
>$50k 50% 50%
<$50k 70% 30%
Before asking you your income the lawyers' best guess is you're equally likely to be earning >$50k as <$50k because $50k's the median [1]. This means they'd guess you're 60% likely to side with the plaintiff: half the people in your position earn over >$50k and will be approximately evenly split while the other half of people who could be in your position earn under <$50k and would favor the plaintiff 70-30, and averaging these two cases gives us 60%.

So the lawyers best guess for you is that you're at 60%, and then they ask the question. If you say ">$50k" then they update their estimate for you down to 50%, if you say "<$50k" they update it up to 70%. "No effect on either lawyer" can't be an option here unless the question gives no information.


[1] Almost; the median income in the US in 2012 was $51k. (pdf)

Comments (5)

Comment author: Michaelos 22 August 2014 06:09:58PM *  8 points [-]

I think I have an idea of what they might be attempting to model, but I do see a few phrases that aren't clear on the site if that is the case.

There are three possibilities I think they are attempting to model:

A: Defense strikes you. (Because you seem to favor the plaintiff too much)

B: Plaintiff strikes you. (Because you seem to favor the defense too much)

C: Neither side strikes you. You remain on the jury.

What they might be trying to say is that income <$50k might increase the chance of A and income>=$50k might increase the chance of C.

So 'No effect on either Lawyer' might be better phrased as 'Given that answer you may be more likely to remain on the jury.'

Some answers would presumably have to indicate that because the two lawyers can't strike everyone.

Comment author: John_Maxwell_IV 23 August 2014 02:39:30AM 2 points [-]

Another way of making Jeff's point: in the same way plaintiff lawyers favor jurors with low incomes, perhaps defense lawyers should favor jurors with high incomes because they don't have the problem of being overly generous with damage rewards.

Comment author: pinyaka 22 August 2014 07:50:39PM *  3 points [-]

I think that you're assuming that the lawyers calculate something like expected utility before asking the question and then correct their expected value once they have information when it's more likely that they just start off with some generic starting point about how the juror is likely to vote and then use demographic information to pin down a better estimate.

Let's say that my employer pays me x amount per period and I think I may have gotten a raise to amount x+y. You're suggesting that I would go through the process of calculating my "expected" salary by taking (p-1)x+p(x+y) (where p is my estimation of the probability of getting the raise). Of course if I do this and then ask my boss whether I got the raise I will revise my expected salary upwards if I get an affirmative answer and downwards if I get a negative answer, but the calculating the "expected salary" first doesn't offer any benefit over just going and asking and then figuring out what my salary will be (as the two methods should provide the same final answer anyways) and so in this case I only revise my expectation upwards with a positive and downwards with a negative.

Comment author: pinyaka 22 August 2014 08:44:12PM *  0 points [-]

Also, to quote from the sequence article you've linked to:

Equivalently, the mere expectation of encountering evidence—before you've actually seen it—should not shift your prior beliefs.

So you should only move your expectation of how the juror will vote after you find out how much their household income is.

Comment author: owencb 22 August 2014 04:44:24PM 3 points [-]

The takeaway here could be that this is a poor model for how the lawyers work, or that the lawyers aren't doing a very good job. Do you have a view on which is more accurate?

Note that for some of the criteria, if one answer is seen as much more likely than the other then receiving that answer may lead to a very small rational update of expectations. This could reasonably be abbreviated as "no effect on either lawyer", even when the other answer would lead to a substantial effect. But your example of the income shows that this can't be the only thing happening, since that's so close to the median the updates would have to be a similar size either way.