Law and Economics

Another Probability Problem

My student Luc Attlan suggested this problem to me: 

You are told that 2 envelopes contain a positive amount of money, and that one envelope contains twice as much as the other. You can pick either envelope, so you pick envelope 1. Before opening it, you are asked if you would like to switch to envelope 2. Unlike the “3 doors” problem though, no new information is gained at this point. It would, therefore, seem impossible that switching would make any difference in the expected outcome. However, the following reasoning would suggest that in fact, the expected amount in the other envelope is higher: if you assume envelope 1 (“E1”) contains x dollars, you know that E2 contains either 2x or x/2, and that each possibility is 50 percent likely. Therefore, you could say that switching has an expected value of 0.5(2x) + 0.5(x/2) = 1.25x . This means that if you are risk neutral, you should always switch. Is this correct?

TSU

Supreme Court Term Limits

In the Friday, April 8, edition of The Wall Street Journal Northwestern University Law School professors Steven Calabresi and James Lindgren had a wonderfully interesting short opinion piece entitled "Supreme Gerontocracy."  The article is available here. 

Calabresi and Lindgren note that it has been almost 11 years since there was an opening on the U.S. Supreme Court and that this group of nine justices has served together longer than any other group in U.S. history.  This reflects, among other things, the fact that the average tenure of Supreme Court justices has increased significantly in the recent past. 

"From 1789 until 1970, justices served an average of 14.0 years.  Those who have stepped down since 1970, however, have served an average of 25.6 years.  ...  The reason for this is not hard to find, Recently, the average age at time of appointment has been 53, which is the same as the average age of appointment over the rest of American history.  The retirement age, however, has jumped from an average of 68 pre-1970 to 79 for justices retiring post-1970."   

And there have been costs to this increase.  Most significantly, some contend that "decrepitude has been a problem with the last 10 justices to retire." 

Calabresi and Lindgren think that an average tenure of 25.6 years is too long for this institution -- any governmental institution -- to operate effectively.  They note that none of the 50 states allows its supreme court justices to serve for life and that most developed countries have mandatory retirement ages for their supreme court justices. 

To correct the situation, Calabresi and Lindgren argue for "a constitutional amendment that would limit the justices to an 18-year-term with one seat opening up every two years." 

The authors are participating in a conference on "Reforming the Supreme Court" held at the Duke Law School on April 9.  You can read more about the conference here. 

TSU

Monty Hall 5

Bear with me through another variation of the Monty Hall problem, which I began a week or so ago. 

Recall that I contended that it was crucial (or so it seemed) that Monty knew where the prize had been placed.  But try this variation.  The rules are the same: there are three doors, one of which hides $60,000 in cash; the other two, goats.  You've been selected to play the game, and you choose Door 3.  Before anything else can happen, one of the doors at the back of the auditorium opens, a breeze rushes through the auditorium, and Door 1 is blown open to reveal a goat.  (It was chance, not the knowing agency of Monty that revealed the information regarding Door 1's contents to you.) 

Monty is unexpectedly faced with a situation not of his own devising.  Nonetheless, he offers you the same deal as before -- keep Door 3, switch to Door 2, or take $35,000 in cash (on the assumption that you are risk neutral).  What to do? 

Here's my analysis, but I'm not convinced that it's correct.  I think that you should still switch.  The information that has been revealed still fits the model developed before -- namely, that there was a two-thirds probability that the prize lay behind Doors 1 and 2.  (Remember that the prize has been placed behind some door and stays there, unlike the situation in the black velvet back variant.) 

But there's a subtle point here that bears further thought.  If we were to incorporate the door-opening breeze possibility into our scenarios, then things get more complicated.  One-third of the time the breeze would blow open the door hiding the cash because the breeze, unlike Monty, doesn't know where the prize has been placed.  And one-third of the time the breeze will blow open the door that you originally chose.  So, to be as clear as I can, in the door-opening breeze scenario, you should switch only if the breeze has blown open one of the two doors that you did not choose and that does not contain the cash prize.  (In only that circumstance the breeze mimics Monty's knowing opening of the door hiding a goat.) 

But I may be wrong.  Please correct me if I cam. 

I may do one last posting on Monty, incorporating the comments of my friend Charlie Petit.  Till then, look at this Google page for references on Monty.  And try it with others using three cards, placed face-down, one of them designated the winner. 

TSU

Levitt on NPR

Scott Simon of NPR interviewed Steve Levitt about his work this past Saturday, April 9.  It's a wonderful interview, available here.  Steve Levitt's and Stephen Dubner's Freakonomics appears on Tuesday, April 12. 

TSU 

Monty Hall 4

There's still more to say on this fascinating issue. 

Try this variation on the Berger library example yesterday.  There are 100 doors on the stage.  I've placed $60,000 behind one of them; it's yours if you select it.  You choose a door -- say, Door 37.  The probability that the prize is behind that door is 1/100.  The probability that it's behind one of the other 99 doors is 99/100.  I, knowing where the prize lies, start opening doors to reveal goats and asking you to switch.  After doing this for a while only Door 37 and Door 11 remain closed.  Would you switch from your original choice to 11? 

OK.  Here's a variant to illustrate how crucial it is that Monty knows where the prize has been placed.  Imagine a different but related game.  Monty has a large black velvet sack in which there are three tiles.  Each of the tiles has a number on it -- 1, 2, or 3.  Monty puts the three tiles into the bag and shakes it.  You can't see into the bag, nor can Monty.  You are to choose a number -- 1, 2, or 3.  Monty will have his assistant draw one tile from the bag and then another tile from the bag.  If the number that you chose is the number of the last tile drawn from the bag, you win $60,000.  Otherwise, you get nothing. 

Suppose that you select tile 1.  Monty shakes the bag containing the tiles.  His assistant draws out tile 2.  Tiles 1 and 3 remain in the bag. 

Now suppose that at this point Monty turns to you and says, "You chose tile 1.  That tile and tile 3 remain in the bag.  Let me give you three options.  (1) You stick with tile 1.  (2) You switch to tile 3.  (3) I give you $35,000 in cash."  Assuming that you are risk-neutral, what should you do? 

Now, the correct thing to do is to take the cash.  The expected value of each of the remaining tiles is $30,000 (1.2 x $60,000, because each tile is equally likely to be the last drawn).  Why is the likelihood of each tile's being a winner 1/2 here when it was 1/3 and 2/3 in the door example?  Because Monty doesn't know which tile will be the last chosen or have any way of influencing which tile that will be. 

I'll have only a couple more variants of this before I move on to other things. 

TSU

Monty Hall 3

Back to the task of persuading you that you should switch.  I tried putting in a table to indicate the virtues of switching, but it didn't look very good.  So, if you will click Download monty_hall_table.doc, you will get access to the table in Word format. 

Notice that the table contains all nine possibilities and that if you always switch from the door that you originally chose, you will win six times. 

Here's one more variation that is due to one of my students, Eric Berger.  Imagine that I have hidden a $10,000 bill in a book in the library of the University of Illinois College of Law.  I know exactly where the book containing the money is.  Assume that there are 10,000 books in the library.  I invite you to choose one of the books.  If it is the book containing the money, you get to keep the money. 

Your chances of choosing the right book are 1/10,000 -- not very good.  Now suppose that you have chosen a book.  We put it on the table, unopened.  It might contain the $10,000 bill; it might not. 

Imagine that I now take another book from the shelves and open it to reveal that it does not contain the bill, and we put that book aside.  I ask if you would like to switch from the book you originally chose to another unopened book in the library.  (There are now 9,998 of those.)  And you might well say, "Why should I?"

Suppose that I keep taking books off the shelves, showing you that each of them does not contain the $10,000, and then setting them aside.  After an interminable time there are only two books left -- the one you originally chose and one other.  Should  you switch?  Sure.  You might argue that after you made your original choice, the probability that the $10,000 is hidden in one of the other 9,999 books in the library is 9,999/10,000 (almost 1).  I have now opened 9,998 of those books.  All that remain are the book you originally chose and one other unopened book.  Isn't it fairly clear that all of the 9,999/10,000 probability that was distributed across the books that you did not originally choose now rests on that other book? 

One or more posts on this in the near future. 

TSU

Congratulations, Illini and UNC

Congratulations to the University of North Carolina Tar Heels, who defeated the University of Illinois Illini last night, 75 - 70, to win the 2005 NCAA National Basketball Championship.  This has been a magical year for the Illini, who finish the season with a record of 37 wins and 2 losses.  There will be a rally for our basketball team tonight in Memorial Stadium, the football venue, and I won't be at all surprised if there are 77,000 people present to express their love for this team.  Go Illini! 

TSU

Monty Hall 2

Several posts ago I posed the famous Monty Hall three-door problem.  Monty has placed $60,000 behind one of three closed doors on the stage.  Behind the other two are goats.  You choose a door -- say, Door 1.  Monty then has his assistant open Door 3 to reveal a goat.  He then offers to let you keep Door 1, to let you switch to Door 2, or to accept $35,000 (on the assumption that you are risk-neutral).  The question, recall, is which of those three options you, the contestant, should choose. 

You should switch to Door 2.  (More generally, you should switch to the unopened door that you did not originally pick.)  Trying to explain this convincingly is, however, a challenge.  I'm going to accept that challenge in the next several posts. 

Here's the gist of the matter.  If you switch, you will win two-thirds of the time.  When you chose Door 1, the probability that the cash prize lies behind any one of the three doors is one-third.  The probability that it lies behind either Door 2 or Door 3 is two-thirds.  Once Monty has shown you that there is a goat behind Door 3, all two-thirds of the probability that the prize lies behind Door 2 or Door 3 falls to Door 2.  (There's a crucial assumption that I will get to in a moment.)  So, the expected value of Door 1 is $20,000 (1/3 x $60,000), and the expected value of Door 2 (after Monty has Door 3 opened) is $40,000 (2/3 x $60,000).  Therefore, the expected value of Door 2 is greater than that of either Door 1 or the $35,000 cash.  Switch. 

Of course, no one -- on first hearing -- is convinced by that line of argument.  Most people think that the one-third probability that was initially with Door 3 but has now been revealed to be zero shifts in equal amounts to Doors 1 and 2.  The argument made is, "Each of the still-closed doors gets half of the one-third that used to lie with Door 3, so that 1/6 of the probability formerly with Door 3 goes to Door 1 and 1/6 to Door 2.  So, each of those doors now has a 1/2 chance of hiding the $60,000.  So, each of the closed doors has an expected value of $30,000.  As a result, I'll take the cash."  That's incorrect. 

I said above that there is a crucial assumption that bears on all this.  Here it is: Monty knows where the prize is and is simply having fun.  That is, he will always have his assistant open a door hides a goat.  Why is Monty's knowledge crucial?  Ah, well.  That's a poser.  But we'll get to that in future posts. 

One last thing.  I'm not saying that if you always switch, you will always win.  No.  You will win two-thirds of the time.  If you never switch, you will win one-third of the time. 

In a post tomorrow, I'll try to explain why the argument two paragraphs ago is incorrect and will give some additional arguments in favor of switching. 

TSU

The Law and Economics of Blogging

I mentioned a couple of posts ago that I had been inspired to return to blogging by a talk that my friend and colleague Larry Ribstein gave at an IPLE (Illinois Program in Law and Economics) workshop last week.  Larry has now written up his wonderful remarks.  They're available on his webpage, here.  Well worth reading and thinking about. 

TSU