An Elaboration on the Monty Hall Problem

Many, many years ago I breezed into a Mensa meeting and saw then Chancellor of the Triple Nine Society, Cyd Bergdorf, and Ron Hoeflin working busily on some kind of problem.  I knew Cyd personally, but I knew Ron only by reputation.  Being the gadfly that I was, I walked over, sat down and asked what they were working on.  It was the now famous 'Monty Hall Problem.'  They explained it to me.  It goes like this:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I immediately said, 'You switch, of course.  And, by the way, the qualification that the host knows what is behind the door is unnecessary.  I'm going to go get a drink.'  This remark that the qualification is not needed has led to a never ending dispute with other high IQ people, most recently Garth Zietsman and Rick Rosner.  So, I am going to put the argument here where it may be able to garner a larger audience.

To make it very clear, I am stating the the reservation is unnecessary precisely in the context of the problem as stated.  Thee are a whole lot of Bayesian assumptions that, I argue, do not apply because there is nothing in the problem that states that it should be taken as one is a series of similar games.

First, I will reproduce verbatim what I found to be the most clearly stated positions of Rick Rosner and Garth Zeitsman.  If you want to read the full dialogue, it is available here. 

Rick Rosner
Three doors, you pick one, then Monty randomly opens one. 1/3 chance the game is wrecked, because Monty prematurely revealed a car - he's never supposed to reveal the car until you make your final choice of door. 2/3 chance the game isn't wrecked. Two equal possibilities among the 2/3 of games that aren't wrecked - car is behind your first choice, or car is behind the door Monty didn't open. Random door-opening seems to either wreck games or leave an equal probability among the remaining unopened doors.

Garth Zeitsman
There is an urn with one blue and two green balls. A selects a ball followed by B (randomly) who notes his ball is green. If A had selected a blue ball B would have had two ways to select a green ball. If A had selected a green ball B would have had one way in which to select a green ball. However A had two ways in which to select a green and one way to select a blue ball. So the chance of A
having selected a blue given that B's was green is 1*2/(1*2 + 2*1) = 1/2.

When B can look into the urn and deliberately pick out a green then his probability of selecting green is always 1 and the probability of A selecting blue given that B has green is simply the initial probability of his selecting blue.

Michael Ferguson
Now both Rick and Garth are correct if we assume that the game described is one element in a string of games where n>>1.  However, that is not stated in the problem.  We have no reason to assume that this game has ever been played before or ever will be played again.  If that was the problem, they would be correct.  We could expect to improve our odds from one third to two thirds if Monty knows what is behind the doors and chooses to always open a door with a goat behind it.  We would expect that our odds would be the same by either staying or switching if Monty doesn't know what is behind the doors.  However, that is not the problem as stated.

When we flip a coin, the natural odds of a fair toss coming up heads is 1/2.  If we choose one door out of three the natural odds of choosing a car is 1/3.  One might ask how this natural odds of 1/3 could somehow change to 1/2.  The answer is that it does not.  Something very different is the cause.

Suppose we play 12 games.  We would expect that we will choose a door car 4 times.  That gives us the 4/12=1/3 natural odds.  If Monty always opens a goat door we didn't choose, we will have 8 unopened car doors and the odds if we switch is 8/12 or 2/3.  However, if Monty doesn't know, we will have 4 games where we chose a car door, 4 games where the unchosen door has a car and we will have four games where Monty, as Rick puts it, wrecks the game.  So, now, switch or not, we should expect to win 4 cars.

So the odds to not change from the natural 1/3 to 1/2 actually.  Rather, we have created a subset that doesn't include four unopened car doors.  This artificially decreases the probability that the unchosen door has a car.  In other words, and this is absolutely the crux of the issue, the lowering of the odds does not reside in the individual games but rather is a characteristic of the subset that is created.  The changing odds is directly a result of opening car doors.

In other words, by resorting to Bayesian reasoning, Garth and Rick are calling into existence games where car doors are opened when, in fact, there is no indication in the language of the problem that there ever will be played any other games, with or without Monty knowing what is behind the door.  In the only case of the game that we know to exist, Monty revealed a goat.

This game naturally belongs to the set of games string of n length in which no car doors are opened.  By naturally, I mean that there is no way to take this game out of that set.  If I could state that this game belongs to a set of n games in which Monty Hall knows what is behind the doors.  I could take it out of that set by stating that Monty Hall doesn't know what is behind the doors.  But it irrevocably belongs to this set.

I then state that a common factor of all game strings, regardless of the value of n, is that we expect that there are twice as many cars behind the unchosen door than there are between the chosen door.  To illustrate the significance I use the following thought problem.

Suppose we play twelve simultaneous games. We choose one door in each of twelve sets of three doors. Our expectation is that we will choose a car four times and will choose a goat the other eight times. Then Monty opens one door in each set of three doors and reveals a goat. Now, we have chosen a car door four times and the twelve unchosen doors contain the other eight cars. Clearly, we should switch doors when asked.

Now after all this transpires, Monty tells us that he didn't know what was behind all the doors. We are surprised because the odds against him choosing only goats in twelve straight games is (2/3)^12 or approximately 130:1. Yet, it does not change the proper strategy; there are still only four cars behind the chosen doors and eight behind the unchosen doors.

Now suppose that we play three simultaneous games. The logic is the same. Our chosen door has only one car and the unchosen door has two. We should switch. We are not so surprised when Monty tells us that he didn't know what was behind the doors because the odds against have fallen to (2/3)^3 or just a little over 3:1. In fact, for n games in which no car doors are opened, no matter the value of n, switching doubles our chances.

This happens because there is nothing magical about Monty knowing or not. The advantage falls from 2:1 to 1:1 by the process of games with opened car doors being eliminated from the pool. In other words, the odds do not change one iota UNTIL a car door is opened. When we are presented one game in which a goat was revealed and n=1, there are no car door openings that can change the odds of having chosen a car door from 1/3 to 1/2. Consequently, in one game in which a goat was revealed, my initial probability of 1/3 of having chosen a car door remains. The unchosen door still contains 2/3 of the probability.

For some reason, even people at the highest IQ level can have difficulty in grasping this.   I want to make this clear that I am not arguing against Bayesian probability.  I am only arguing that it is applicable only in those cases where multiple 'runs' of the game are taking place.
 

Suddenly, Young Ladies Everywhere Have Decided That They Are Bi

In 2008 Katy Perry 'Kissed a Girl' and she liked it.  Four years later, young women everywhere, in unprecedented numbers, are kissing girls and liking it.  According to a 2010 Psychology Today article, the number of women who self-identify as bi or lesbian has increased from its traditional 1% to 2% to 15%.  Other studies have found that 50% of women report that they have been attracted to another woman.  Another longitudinal study found that, rather than being a 'phase', for a significant percentage of women bisexuality is a persistent pattern of behavior.  It is not readily apparent, but this is a matter of very great cultural, social and political significance.

In the mid-nineties I developed what I called PaleoSociology and it will be included in my book, 'The Cultures of Affluence.'  To put it very briefly, I speculated that all of human social impulses must have emerged because they were adaptive within the social environment in which they evolved.  In other words, if one could just properly describe the PaleoSociology, all of the perplexing human behavior would suddenly make sense.  I tried; I succeeded and then, because I recognized that society was not ready to be receptive to many of the conclusions,  I moved on to the next project.

A lot has changed in the subsequent twenty years and, as a result, I will soon describe it as a portion of the aforementioned book.  However, for now, this newly discovered female bisexuality was anticipated by my PaleoSociology.  It is not a fad; it is the discovery of a fundamental element of female sexuality.  As soon as we encouraged our young people to explore their sexuality without guilt, it was inevitable that a rise in female bisexuality would take place.  It wasn't a matter of if, it was only a matter of when.  So, historically high levels of female bisexuality will, as the research suggests, persist and it will likely rock our powerfully pair-bond oriented Western civilization to its very foundations. 

There is some disagreement whether during our evolution breeding populations suffered more regularly from female or male shortages.  However, there is no doubt that the size of the next generation's population is determined by the number of women, not the number of men.  However, if there are more women than men, at least some women are going to need to share.  In the primitive world where survival of breeding populations was far from a foregone conclusion the option of placing monogamy above reproductive maximization was simply not practical.  We are not descended from genetic pools that made that decision because they became extinct because of it.  Consequently, by one mechanism or another, when there were more women than men, some men got an extra wife, or two, or more.

Given that, then, the Paleosociology had the difficult problem of keeping every woman making babies without fomenting disruptive behavior among the men as they tried to become the one who got the extra girl.  That is why we generally find that in primitive societies the decision making process is very rigid.  In some cases the widow went to a brother of her deceased husband.  In other societies, the tribal chief got all the extra women.  This ostensibly had the added benefit of giving the best genes to the extra women.  That is true, assuming that the king selecting process was efficient.  At any rate, in nearly every case the community was not disrupted by competition for extra females because the selection was already made by rule. 

For several reasons that I will explain in 'The Cultures of Affluence', humans needed to be one of the few animals that pair bond.  There is a piece of doggerel that explains the dynamic that maintains the need for opportunistic polygamy within the context of a primarily monogamous species.  It goes
Higamous Hogamous
Women are monogamous
Hogamous Higamous
Men are polygamous.

Much is made of how men have little investment in a pregnancy and therefore casual encounters have upside without much downside.  Women end up investing a great deal of energy in the results of a pregnancy and consequently have loads of downside.  They have a strong incentive to hold out on sex until the man makes the commitment to assist in child rearing.

As attractive as that logic may be, it isn't right.  The real reason is the need to create a dynamic where men are amenable to multiple partners when there is a man shortage, but are willing to allow women to force the situation back to pair bonding as soon as the shortage is over.

Now back to the bisexuality.  We are almost home.  As just about everyone would agree, it is very difficult to get two heterosexual wives to create a stable and collegial polygamous home.  The likely competition for attention, support, etc. is obvious.  Sufficient socialization can get it done, but it is not easy.  However, if the two wives are also lovers the dynamic changes dramatically and generally for the better.  Rather than competing for male attention they may view their relationship as one in which they share a man.  The difficulty, if any, rests with the a sense of marginalization that may be felt by the husband.

Fortunately, male sexuality comes prepared to accept, in fact relish, it.  The 'lesbian problem' has been one of the greatest in the study of human sexuality.  In this interpretation the paradox does not exist.  Especially because dominant and ostensibly genetically well endowed males tend to get the extra women, the sexual dynamic of the FFM triad (FMF denotes a triad where the women are not coupled and is generally not stable) doesn't just improve the stability of the household but, through several mechanisms, improves  the stability of the community in general.

OK, that is a thumbnail explanation.  In the book I will go into great detail and, consequently, the logic will be much more compelling.  But, hopefully, most readers will understand that this evolutionary innovation means that some relatively significant percentage of the female population has an inherent proclivity toward bisexuality that can be triggered by environmental conditions.  The creation of stable FFM triads within populations that experienced chronic male shortages created a more stable community.  They basically made the man shortage go away.

Now, is there currently a chronic man shortage?  Clearly not.  The recent dramatic uptick in triggered female bisexuality comes from changing social norms.  Katy Perry's song did probably play a role.  In essence, as an icon for the young people, she put a stamp of approval on it.  However, it only did so because the conditions were right for it.

It will persist for an interesting set of reasons.  Primarily, the evolutionary forces that caused humans to pair bond originally don't exist any more.  We are seeing evidence of that in the dramatic increase in single mothers.  That is a temporary expression of this evolutionary event.  However, beyond a doubt, in the realm of marriage, what we are going to get coming out the other side of the Transformation is not going to be anything like what we had going in.  The explanation for this resides primarily within the logic of female reproduction.

The probability of success in rearing a batch of children in a more primitive environment was greatly enhanced with the full attention of the father rather than half of his attention.  That is no longer the case.  Children benefit from an active and intimate involvement of a strong male role model, but it is no longer a full time requirement for the father.  A man can provide that service to the children of more than just one woman, especially since the number of children per woman has fallen.  So, the reproductive logic for women has changed.  Acquiring the very best male genetic contribution becomes relatively more important than the level of support in parenting responsibilities he is likely to provide.

So, now a new kind of genetic deal can and will be struck.  The man gets to double his reproductive potential by having two wives.  The wives are able to attract a higher genetic endowment for their children than they could have accomplished alone.  In other words the man is willing to trade a bit of quality for quantity.  For bisexual women there is another very significant benefit.  Surprisingly, according to current research, bisexual women are more inclined to form stable long term relationships than women in general. However, every time they choose to accept a couple relationship, they are making a commitment to denying half of their sexuality or, conversely, to cheating.  The FFM triad structure allows them to fully experience their sexuality within the rules of the marriage.  This logic is far from complex and young bisexual women are going to get that pretty quickly.

When I first studied this back in the middle nineties, the few FFM triads that formed generally began with a 'bicurious' wife asking her husband to allow her to add a second woman to the relationship.  The dynamics are all wrong and this route to an FFM triad is almost surely doomed to end in disaster.  However, with this sudden explosion in young openly bisexual women, they will be able to form the FF bond first and then go find their genetically well endowed and mutually acceptable fellow.  The dynamics for this route to an FFM triad has the potential for great stability - even more than traditional heterosexual couples.

So, the jury is out, but it appears very likely that this sudden revelation by many young ladies that they like to kiss girls can lead to a growing number of stable FFM triads.  As we slowly try to adjust to the idea of gay marriage, we will then get hit with demands for marriage rights for triads.  Why not?  It is going to be very difficult to formulate an argument that explains why gays can get married but bisexuals are denied their preferred marital structure.  There will be an attempt at a Feminist argument, but it won't fly.  The FFM triad exists not because the husband has a lesbian fantasy or wants to subjugate women.  That is the FMF model.  Rather, FFM triads exist because they facilitate the wives' needs.  It is quintessentially a Feminist institution, even though it will still anger many Feminists. 

There is so so much great stuff I left out of this.  But it is all for the book.  Hopefully, this will stimulate an interest in the book and Paleosociology that I will be able to satisfy in the near future.  The most important take away from this article is the very real probability that demands for triad marriage are likely to come right on the heels of gay marriage.  Furthermore, there will be far more triad marriages than gay marriages.  One might anticipate that they will constitute up to 10% of all marriages and involve 20% of women.  So, when you consider the latter, do not do so without considering the former.  They are displaced temporally, but they are almost surely a package deal.

You are STUPID!! OH YEAH!! WELL SO AM I!!!

While enthusiastically granting that individual humans can, from time to time, commit the most inspiring acts of selflessness, generosity and kindness, these events are so inspiring partially because they are rare. Typically, humans are venal, self-centered and arrogant. I didn't start out as a misanthrope. But you all are talking me into it. This article is about arrogance, that is the arrogance of assuming that you are smart. You are not.

YOU ARE STUPID!!!! "I am not good at math." NO, you are STUPID!! "I don't test well.' Yes, that is because YOU ARE STUPID!! "I have emotional intelligence.' THERE IS NO SUCH THING!!! YOU ARE STUPID SO YOU RELATE WELL TO OTHER STUPID PEOPLE!! And here is my latest favorite, 'I rely more on intuition'. That is because YOU ARE STUPID!!!

The single most important thing that is interfering with your happiness is that YOU WILL NOT ADMIT THAT YOU ARE STUPID!! Republicans? STUPID!! Democrats? STUPID!! Hollywood personalities? INCREDIBLY STUPID. Can you take a partial derivative? IF NOT YOU ARE STUPID, STUPID, STUPID!!!. When someone asks you, who should be President, the correct answer is, 'I don't know. I AM STUPID!! I have no idea what the consequences are of one over the other.' Of course, you probably don't agree with any of this. But that is..... you got it.... BECAUSE YOU ARE STUPID!!

The world will be one of woe until and unless people, and the Professors, Pundits and Politicians need to be first in line, accept that THEY ARE STUPID and act accordingly. We need an Arrogance Anonymous and I think a twelve step plan. People stand up in front of their peers and say, 'I am Michael Ferguson and I AM STUPID!' You, of course, need to insert your name in there. I used my name, partially to not offend anyone else whose name I may have used. It would be true, but they may not like that I singled them out. However, mostly it was to communicate that this applies to me, too. Yes, I am vastly less stupid than you. In fact, I may be the least stupid person on the planet. More importantly, I am vastly less ignorant than you. BUT I AM STILL STUPID AND I KNOW IT!!

ARE YOU GETTING THIS PEOPLE!!??

So what should a twelve step plan look like? We mentioned the first step. Get up in front of the mirror, look at yourself and say, "I am STUPID!" Yes, the emphasis is important. You need to have a deep, visceral reaction to it. It is not good enough to have an intellectual reaction. You need to feel this deep down inside of you. YOU NEED TO FEEL INTELLECTUAL HUMILITY! Everyone, and I mean everyone, should be constantly imbued with a sense of intellectual humility and absolutely nobody is.

Next, of course, you need to go to the people you have called stupid in the past and tell them, 'I am stupid too.' This is important. You don't apologize for calling them stupid. They are. Your error was feeling, saying or acting as if YOU AREN'T STUPID TOO!! Yes, you may be less stupid than they, but this is an act of humility. It is not time for you to YET AGAIN WALLOW IN YOUR UNDESERVED INTELLECTUAL ARROGANCE!!

When someone says something with which you are disinclined to agree, GOOGLE IT YOU MORON!!! Who the HELL DO YOU THINK YOU ARE!? God did NOT give you an encyclopedia for a brain. I can't TELL you how many times I have witnessed arguments that would not have survived one trip to a search engine. Either educate yourself before you disagree or SHUT UP AND ADMIT THAT YOU ARE SATISFIED WITH BEING NOT ONLY STUPID BUT ALSO IGNORANT!! I know far more than you do. I am vastly less stupid than you. Still, every time I choose to reply to someone, first I GOOGLE IT!! Because I AM STUPID!!

Every time you disagree with them ALWAYS perform a reductio ad absurdum. There is a good chance you don't know what that means, BECAUSE YOU ARE STUPID!! Even if you do know what it means, it just indicates that you are less stupid than most. With a reductio ad absurdum, you assume that the other person is correct. Then you try to get it to lead to a logical inconsistency. It is unlikely that you will succeed, because YOU ARE STUPID! But in the process you might get some glimmering of the reason the other person believes it. It does not mean that they are correct. Rather, it will open your eyes to the possibility that their belief is no more stupid than yours.

Now, that is only four steps, but I can't think of another eight. That is because, I am sure, I AM STUPID!!

As an aside, to all of you in 'high IQ societies', I suggest you change the designation to 'low stupidity societies.' I said this a couple of decades ago, 'I am not sure that IQ tests are an accurate measure of intelligence. But when you see the questions a person must get wrong in order to score 100, you understand that they are a very effective measure of stupidity. In fact, it should be inverted and renamed Stupidity Quotient."

Someone suggested that there should be a 'National Stupidity Day.' I think that is a wonderful idea. You take the day off and use it to contemplate just how STUPID you are. For example, consider this. You limped through High School Algebra and then avoided Math. Now you blather on, with not a hint of equivocation, about the Federal Reserve or unemployment or the affairs of nations or tax policy. SHUT UP!! THESE ARE ALL MUCH HARDER THAN HIGH SCHOOL ALGEBRA. YOU ARE STUPID. AND YOU ARE IGNORANT. YOU DON'T KNOW WHAT YOU ARE TALKING ABOUT AND YOU ARE POLLUTING THE CIVIC DISCOURSE WITH YOUR INANITIES!!

Now, if you completed a couple of years of college level calculus, a year of statistics, and learned linear algebra, the above does not apply to you. But I assure you, you are still stupid and ignorant. I can say this because I have that much math and more. Yet, I am still stupid and ignorant. We will just need to consider something else on National Stupidity Day. Believe me, there will be no shortage of options.

In the evening the whole family will sit down to a feast of crow and relate for one another the stupid things they have thought, done or said in the prior year. If entered into with the proper attitude we can all get a great chuckle out if it. If we don't see the humor, well, it is time to visit Arrogance Anonymous.

I would like to suggest in the U.S. that National Stupidity day be the first Monday in November. If you don't understand why, it is a wonderful opportunity for you to reflect upon your stupidity.

Another person suggested that a single day is insufficient. I guess they imagine something like the extended Jewish holiday of Hanukkah. I suppose we could call it Moronukkah. While not endorsing the idea, I am not going to argue vociferously against it either. It is very doubtful that we, as a species, will ever be overly aware of our own stupidity. That is just not how we humans roll.

The point of this article is not really that you are stupid... or ignorant. It is actually that YOU ARE ARROGANT. YOU VASTLY OVERESTIMATE YOUR INTELLIGENCE AND YOUR KNOWLEDGE. BECAUSE OF THIS YOU DON'T SHUT UP, LEARN AND THINK IT THROUGH. RATHER YOU POP OFF AND MISTAKE IT FOR COGENT ANALYSIS.

Yes, I was taught to suffer fools gladly and believe me I tried. In the end, however there were just too many of them. I don't mind the stupidity, really. It is how we humans are. I don't even mind the ignorance when people 'fess up to it. But the problem is that people are too stupid to understand that they are stupid. They think they are smart because the other animals on this planet are even dumber. Humans - the least stupid animal on the planet. You should place this statement some place prominent lest you forget it.