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[MUSIC]

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

Â dilemmas that have legal applications, the prisoners dilemma,

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

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

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

Â You may also, you may already have

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

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

Â 0:48

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

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

Â Imagine two prisoners are being interrogated in separate rooms about

Â a bank robbery.

Â Each has the choice of remaining silent or

Â confessing that the two of them committed the heist.

Â The choice is often referred to as cooperating or defecting.

Â Remaining silent is the cooperating strategy,

Â because if both prisoners cooperate and

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

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

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

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

Â 1:42

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

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

Â The striking result is that each prisoner does best defecting,

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

Â If you're prisoner A, and you believe prisoner

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

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

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

Â 2:42

Two points are worth mentioning about this game.

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

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

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

Â Second, I described the game with

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

Â But so long as binding contracts are credible threats of retaliation

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

Â 3:47

Finally, you should be on the lookout for

Â prisoner dilemma strategic interactions in other contexts.

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

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

Â quantity quotas creates a similar prisoners' dilemma dynamic.

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

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

Â owners to tragically overuse common land.

Â 4:26

The fishing industry would be better off

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

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

Â leading to overfishing.

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

Â decimated the available population to regenerate itself in later years.

Â 4:57

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

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

Â six points added onto your final paper grade.

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

Â 5:33

Our second perversity is the Condorcet voting paradox,

Â which gives rises to what is called Condorcet cycling.

Â Suppose three voters, 1, voter 2,

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

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

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

Â 6:40

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

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

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

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

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

Â These preferences give rise to Condercet cycling because if

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

Â It would keep cycling.

Â Condorcet cycling emphasizes the importance of procedure.

Â The person who sets the agenda can determine the outcome.

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

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

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

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

Â which proves that there is no satisfactory voting method, or

Â for that matter a nonvoting method, of aggregating preferences.

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

Â There was a time when he left a message

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

Â I still have that tape recording of his voice.

Â That's how much of a fanboy I am.

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

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

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

Â illusory disparities in improperly aggregating data

Â that disappear when the data are disaggregated, unquote.

Â For example, scholars analyzing 1973 admissions data from

Â the University of California at Berkeley uncovered quote,

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

Â Because the uncontrolled aggregate analysis showed that women applicants

Â have an lower overall acceptance rate than men applicants,

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

Â 9:26

A stylized version of this university example

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

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

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

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

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

Â 9:59

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

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

Â 500 applied to Math and 500 applied to English.

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

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

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

Â Specifically, imagine the departments admit men and

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

Â 10:35

Under these conditions, the overall admission rate of men applicants

Â at the university would be 50%, while the overall

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

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

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

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

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

Â Failing to control for department effects in a statistical analysis,

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

Â admits a lot fewer applicants than the math department.

Â Failing to control for department affects in a statistical analysis,

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

Â gender disparity disfavoring women, when in fact,

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

Â But this concern about the possibility of the Simpson paradox

Â ignores the important differences between two different kinds of discrimination

Â claims, disparate treatment claims and disparate impact claims.

Â 11:59

In a disparate impact case where intentional discrimination need not be

Â proven, defendant policies that produce unjustified racial or

Â gender disparities in the aggregate may give rise to liability,

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

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

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

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

Â and only 99% of white janitors with a diploma,

Â and 1% of blacks with no diploma, and

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

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

Â was not a business justification for

Â an employment decision with regard to hiring jobs like janitors.

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

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

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

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

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

Â even though within each category, there was a small,

Â a 1% advantage for black applicants.

Â If on has the appropriate appreciation of disparate impact,

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

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

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

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

Â applicants disproportionately apply to the English department.

Â In the disparate impact analysis, controlling for the tendency of

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

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

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

Â 14:29

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

Â if some uncontrolled characteristics, like the applicant department or

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

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

Â 15:08

The fourth and final perversity makes a great bar bet.

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

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

Â Here's a modified version of the setup.

Â Suppose you're on a game show, and

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

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

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

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

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

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

Â 16:32

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

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

Â Once door 3 is revealed to be a goat,

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

Â 16:53

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

Â from one-third to two-thirds.

Â This can be proven theoretically, and

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

Â the curious incident of the dog in the night time.

Â 17:15

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

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

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

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

Â have guessed the right card initially.

Â You'll quickly learn, though,

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

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

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

Â 17:53

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

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

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

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

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

Â 18:18

The Monty Hall problem is deeply related to law,

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

Â Most people wrongly discount the importance of the seemingly

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

Â