[MUSIC]

In today's lecture, I'm going to speak about four paradoxes and

dilemmas that have legal applications, the prisoners dilemma,

the Condercet paradox, the Monty Hall problem, and the Simpsons paradox.

Knowing about these perverse possibilities is useful, because you might be able

to argue that they're at play in new legal context that you encounter.

You may also, you may already have

learned about many of these in college and can skip parts of this lecture.

But if you haven't heard of them, you're likely to hear about them in law school.

First, let's look at the Prisoners' Dilemma.

It's the most famous game in all of game theory.

Imagine two prisoners are being interrogated in separate rooms about

a bank robbery.

Each has the choice of remaining silent or

confessing that the two of them committed the heist.

The choice is often referred to as cooperating or defecting.

Remaining silent is the cooperating strategy,

because if both prisoners cooperate and

remain silent, it becomes harder to convict them of a serious crime.

The prosecutor who wants to disrupt their cooperation offers to let a prisoner go

free if he or she is the only one to defect and confess,

so as to secure a severe sentence for the other prisoner who remains silent.

The following table shows the payoffs, here meaning the time served to

each prisoner under the four possible permutations of cooperation and defection.

The striking result is that each prisoner does best defecting,

regardless of what she believes the other prisoner is going to do.

If you're prisoner A, and you believe prisoner

B is going to remain silent, then you go free if you defect.

But even if you believe that prisoner B is going to defect,

then you still minimize time served, two versus three years if you defect.

Two points are worth mentioning about this game.

First, and chillingly, both prisoners might confess, even if they're innocent.

If the prosecutor can credibly threaten these outcomes, even innocent prisoners

will find it rational to falsely confess to crimes they didn't commit.

Second, I described the game with

the prisoners being kept in separate rooms without the opportunity to communicate.

But so long as binding contracts are credible threats of retaliation

are not allowed, communication between the prisoners will not change the result.

Finally, you should be on the lookout for

prisoner dilemma strategic interactions in other contexts.

In antitrust, the choice of cartel industry members on whether to defect by

chiseling on the cartel price or cooperate by maintaining the price or

quantity quotas creates a similar prisoners' dilemma dynamic.

In property law, the prisoners' dilemma is closely related to

the tragedy of the commons, which gets it's name from the tendency of group

owners to tragically overuse common land.

The fishing industry would be better off

if it limited the number of fish taken in a particular year.

But individual boats have a prisoner dilemma like incentive to defect,

leading to overfishing.

An example is shown here, where the overharvesting of cod in the early 70s

decimated the available population to regenerate itself in later years.

Several professors have added a prisoners' dilemma temptation to their final exams.

The professor adds a question that says, select whether you want two points or

six points added onto your final paper grade.

But if more than 10% of the class selects six points, than no one gets any points.

Our second perversity is the Condorcet voting paradox,

which gives rises to what is called Condorcet cycling.

Suppose three voters, 1, voter 2,

voter 3, need to choose between three candidates, candidate A, candidate B,

or candidate C, and the voters have the following preferences.

Voters 1 and 2 each prefer candidate B to candidate C.

These preferences give rise to a Condorcet paradox, because A is preferred to B,

which is preferred to C, which is preferred back to A again.

No candidate is preferred to both of the other candidates by a majority.

The fact that a majority of the group prefers A to B and

a majority prefers B to C does not imply that a majority prefers A to C.

These preferences give rise to Condercet cycling because if

pairwise voting continued until a clear winner arose, the voting would never end.

It would keep cycling.

Condorcet cycling emphasizes the importance of procedure.

The person who sets the agenda can determine the outcome.

An agenda setter who wants candidate B to win would just need to have an initial

vote between A and C, and have the winner of that contest,

which would be C, then face off and lose to candidate B.

The Condorcet paradox is also related to the Arrow impossibility theorem,

which proves that there is no satisfactory voting method, or

for that matter a nonvoting method, of aggregating preferences.

Ken Arrow, by the way, is a real hero of mine in economics.

There was a time when he left a message

on an old time answering machine asking me to give a paper, and

I still have that tape recording of his voice.

That's how much of a fanboy I am.

The third perversity I'm gonna talk about is Simpson's Paradox,

which has direct application to questions about how to best test for discrimination.

The 11th Circuit wrote that the paradox raises the possibility of quote,

illusory disparities in improperly aggregating data

that disappear when the data are disaggregated, unquote.

For example, scholars analyzing 1973 admissions data from

the University of California at Berkeley uncovered quote,

a clear but misleading pattern of bias against female applicants, unquote.

Because the uncontrolled aggregate analysis showed that women applicants

have an lower overall acceptance rate than men applicants,

even though many of the departments admitted women at a higher rate than men.

A stylized version of this university example

can help us understand how the Simpson's paradox operates.

Imagine there's a university with just two graduate departments, math and English.

Of the 1,000 woman who apply for graduate admissions, imagine that 90%,

900 out of the 1,000, applied to the English department and

that only 10%, 100, applied to the math department.

In contrast, imagine that there are 1,000 men applicants, but they

are evenly divided in their applications between the two graduate departments.

500 applied to Math and 500 applied to English.

Finally, imagine that in each department, the admission rate for women is higher

than that for men, but that the admission rate in the English department for both

male and female applicants is markedly lower than in the math department.

Specifically, imagine the departments admit men and

women at the following rates, as shown in this graph.

Under these conditions, the overall admission rate of men applicants

at the university would be 50%, while the overall

admission rate of women at the university would be only 28%.

The paradox in this example is that even though women have a higher admissions rate

than men in each of the departments, 82 versus 80% and

22 versus 20%, they nonetheless have a lower admission rate for

the University as a whole, 28% versus 50%.

Failing to control for department effects in a statistical analysis,

and here the department effect that's most important is that the English department

admits a lot fewer applicants than the math department.

Failing to control for department affects in a statistical analysis,

such as a regression, would seem to give a false indication of

gender disparity disfavoring women, when in fact,

women have a statistical advantage of two percentage points in each department.

But this concern about the possibility of the Simpson paradox

ignores the important differences between two different kinds of discrimination

claims, disparate treatment claims and disparate impact claims.

In a disparate impact case where intentional discrimination need not be

proven, defendant policies that produce unjustified racial or

gender disparities in the aggregate may give rise to liability,

even if there is no disparity in subsets of the data.

Thus, in a stylized version of a famous civil rights case called Griggs

vs Duke Power, which was decided by the US Supreme Court, if an employer hiring

janitors had hired 100% of black janitors with a diploma,

and only 99% of white janitors with a diploma,

and 1% of blacks with no diploma, and

0% of whites with no diploma, there still might be a disparate impact problem.

And it's because the Supreme Court said that having a diploma

was not a business justification for

an employment decision with regard to hiring jobs like janitors.

The problem is that the policy of having such a higher employment of people

who had diplomas worked a disparate impact against African-Americans, who

are much less likely at this time period to have high school diplomas than whites.

And so that overall, only 28% of blacks were being hired.

This is a hypothetical that I constructed, versus 50% of whites being hired,

even though within each category, there was a small,

a 1% advantage for black applicants.

If on has the appropriate appreciation of disparate impact,

there still could be a concern that this raises a disparate impact problem.

Similarly, in the foregoing university example, the university's

policy of admitting a much higher proportion of math applicants than English

applicants has an aggregate disparate impact on women applicants, because women

applicants disproportionately apply to the English department.

In the disparate impact analysis, controlling for the tendency of

different departments to admit students at different rates would only be appropriate

if the university could establish a business justification for its much,

much lower acceptance rate in the department dominantly applied to by women.

At the end of the day, a Simpson's paradox discrimination reversal only can occur

if some uncontrolled characteristics, like the applicant department or

the applicant diploma status, is correlated with the plaintiff's

protected class and disfavored by the defendant's decision making.

The fourth and final perversity makes a great bar bet.

It's the Monty Hall problem, inspired by a game that the long time host of

Let's Make a Deal routinely played with audience members.

Here's a modified version of the setup.

Suppose you're on a game show, and

you're given the choice of three doors labeled 1, 2, and 3.

The host, Monty Hall, explains to you that behind one door is a fabulous prize,

that behind each of the other two doors is only a smelly goat.

You can pick any of the three doors, and after you pick, Monty says he will

always open up one of the unchosen doors and show you that it contains a goat.

This is possible whether or not you've chosen the right door or not.

Well, most people, unless they have been taught this game in school,

have the intuition that it doesn't matter whether you switch.

Once door 3 is revealed to be a goat,

there's a 50-50 chance that either of the other doors will contain the prize.

It turns out that switching doors doubles your chances of winning the prize,

from one-third to two-thirds.

This can be proven theoretically, and

indeed there's a cool proof of it by the autistic protagonist in the book,

the curious incident of the dog in the night time.

But you can also show it empirically with a deck of cards.

Play repeatedly with a friend, and every time, take the option of switching.

If you never switch, you get the prize only one-third of the time,

because there is only a one-third probability that you would

have guessed the right card initially.

You'll quickly learn, though,

that if you switch every time, you get the prize two-thirds of the time.

Cuz that's the only, if you only get the prize a third of the time if you don't

switch, then you have to get it two-thirds of the time if you switch.

You can turn this card game into a profitable bar bet.

I've played it where I promised to pay a $1.10 every time I lose

if the other bettor promises to pay me a dollar every time I win.

And we keep playing until one of us gets down and loudly pronounces

in front of the rest of the bar that the other is truth and righteousness itself.

The Monty Hall problem is deeply related to law,

because at heart it's a question about how we make evidentiary inferences.

Most people wrongly discount the importance of the seemingly

irrelevant piece of evidence, that there is a goat behind door number 3.