Archive for the ‘commentary’ Category

Lotteries as Cost-Saving Mechanisms

Thursday, 23 February 2017

In decision theory, it's useful to conceptualize all choices as amongst lotteries. Even a choice that has an absolutely certain outcome may be imagined as a sort of trivial lottery, where one outcome had the equivalent of a 100% probability and all other outcomes had the equivalent of a 0% probability. But most of the choices that people typically imagine may be made with certainty cannot, and things that actually can be chosen with certainty are not things to which people give much conscious thought. For example, in a restaurant. one cannot choose tea with certainty; one cannot even choose to order tea with certainty. One can chose to try to order tea; however, whether one's language-processing centers and apparatus of speech will do what one wants is somewhat in doubt. But most people don't recognize the vast majority of their choices as amongst lotteries because it isn't particularly useful for them to make the recognition.

That said, it's still interesting (to me at least!) to note how people respond to the things that nearly everyone does recognize as lotteries. If Group A wants the m members of Group B to do something for them, they can pay D to them each, for a total cost of m · D, or they can offer a prize P; and if m is a moderate-to-large number then almost always the least value of P that will motivate the group is rather less than the the value of m · D, even when there is no sense of supporting a worthy cause. In the clearest illustration, what Group A want of Group B is just money. Most people will give you a five-dollar bill for five one-dollar bills, but few will ordinarily give you that five-dollar bill for four one-dollar bills. However, perhaps a million people will give you five dollars for a one-in-a-million chance at four million dollars.

In the context of various social confusions, there are restrictions on selling chances at money in exchange for money.[1] But chances at money or at other prizes are fairly freely traded for information that is worth money. Think of how many times you are offered a chance at a large sum of money or at a valuable commodity (such as a vehicle) in exchange for taking a consumer survey or for providing contact information. You might refuse — I do — but a lottery is offered because information is provided by more people than could be motivated to do so for the same sum divided into simple payments.

It's often claimed that people are irrational to make bets in which the price of participation exceeds the probability of the payoff times the size of the payoff. I don't want to claim that; the issue is actually very nuanced. (There have been studies that attempt to estimate the extent to which a systemic misappraisal of probabilities affect behavior, but most or all of these studies are hopelessly tainted by the active desire to find irrational behavior and by some questionable presumptions concerning how uncertainty ought to be handled.) But, in any case, it's interesting that a group can conserve its resources by using a lottery to motivate behavior. And, returning to the point that in reality almost everything is a lottery, one has to wonder to what extent the world more generally is getting us to do things on the cheap.


[1] The inescapability of lotteries is fatal to ordinary attempts to condemn gambling as immoral. That something were immoral or unwise would not ipso facto be sufficient to justify outlawing it. And outlawing payment in money while allowing payment in commodities is absurd.

A Minor Note on the Myth of admin

Sunday, 12 February 2017

This evening, I was looking at a record of recent failed attempts to log into this 'blog. I found that relatively few attempts tried to do so with the popular username of admin, whereäs by far the majority were with the username oeconomist (that it to say with the second-level domain name). There is not and never has been an account here with username oeconomist; the would-be intruder was guessing mistakenly — but not unreasonably. If my logs are representative, then having an account name match a second-level domain name is less secure than having it be admin. With people avoiding admin, it is natural for crackers to try other likely candidates, including candidates whose probabilities are conditional upon the domain names.

Mind you that the reasoning of my earlier explanation of why the avoidance of admin doesn't add a discernible amount of security if passcodes are properly selected can be applied to avoiding a username that matches a domain name. An account with a known username and a well-chosen password of m+n characters is more secure than an account with a secret m-character username and an n-character password.

Choose a username that pleases you. Choose a password that is long and that looks like chaos, and make occasional changes to it.

No Need for Doors

Thursday, 9 February 2017

84 Lumber bought airtime within the broadcast of the 2017 Superbowl, and presented a video of a Latina mother and daughter travelling through what seems to be Mexico, plainly in hopes of entering America. Inter-spliced with the scenes of their travel are scenes of Americans, clearly constructing something. When the mother and daughter reach the border, they are confronted by a grey and terrible wall. But, as they seek for some hope, they find it — sunlight somehow shining through a section of that wall. Running to it, they find a door. In the awful wall, it was a great door that the Americans in the other scenes were building.

84 Lumber is being attacked for their video, on a theory that its purpose were to defend illegal immigration. Naturally, 84 Lumber denies that their message were any such defense; they now claim that the door were a metaphor for the institutions of legal entry.

I don't encounter a lot of people who will honestly speak in favor of illegal immigration. They ought to do so. There is nothing wrong with illegal immigration. Nothing.

The vast majority of people who oppose illegal immigration or want greater legal restriction on immigration do not do so from racism, and I am very sorry that they have been slandered and libelled; but recognizing the inappropriateness of that accusation doesn't serve to support a case for denying people entry.

Indeed, immigrants might come to our nation and do a variety of things that are violations of the rights of the people who are here now, or that are otherwise undesirable; but every genuine right that might violated by an immigrant could also be violated by someone born and raised here; more generally, every socially corrosive act that might be perpetrated by an immigrant could also be perpetrated by a native. A man or woman who was born here can violate the property and person of someone else; a man or woman who was born here can demand that his or her religion or language be give a privileged legal status; a man or woman who was born here can live at the expense of the taxpayers. None of these behaviors is made better or worse by virtue of where the person were born, nor by whether he or she were allowed to immigrate by the law. I will grant that groups coming from some foreign cultures have a greater share of members likely to do undesirable things of some sorts; but some groups native to America have a greater share of members likely to do undesirable things of some sorts.

There is a dire confusion of the legal with the moral, both on the part of those who insist that illegal immigrants are already in the wrong by virtue of having broken laws that are ostensibly ours in coming here, and by those who insist that there is no such thing as an illegal immigrant. Law can be wrong, and when it is wrong then it may be ignored without doing wrong. Those immigrants here in violation of law are neither wrong simply for being illegal, nor legal because they are not wrong to be here.

America is not a club nor a corporation. The persons and properties within the area occupied by America are not ipso facto in any way the property of all Americans. The right to trade, the right to give without condition, and the right to take that which is freely offered are not rights that in any way reflect nationality. Those who would do business with newcomers are within their rights; newcomers who would do business with those Americans are within their rights.

It's offensively absurd to claim an entitlement to exclude people by pointing to state-managed infrastructure and programmes. They weren't brought into existence through some sort of social contract; ultimately, they were effected through threats of violence; and generally they crowded-out alternative institutions that would have been created by free people. Of course, the welfare state cannot survive in a world of such freedom; it could not survive even if the progressives were allowed to pursue their wildest dreams of taxation, nationally or globally. But so much the worse for the false generosity and false security of the welfare state, which cannot avoid bankruptcy in this century, regardless of whether it keeps all of us trapped on one side or another of its jurisdictional boundaries.

Many people who are going or went through the process of legal immigration may feel that it is unfair for others now to jump the queue; but the queue should never have existed in the first place, and one only compounds the injustice by imposing it upon others.

There should be no queue, no wall, no need for doors.

Fired-up

Thursday, 2 February 2017

I returned home on foot this evening, carrying various things. As I got back to the apartment complex and was going to enter by way of the vehicular gate, I saw and smelled what appeared to be a fire outside of the central front pedestrian gate, so I investigated.

Yup, there was a small fire inside of what appeared to be a pylon or one of those tall butt receptacles, which was within an inch or so of the building, if not up against it.

So I first got my phone to call emergency services. The first dispatcher switched me to a fire department dispatcher, who was a fool following a flowchart. I started to tell him There's a small fire outside of 4050— at which point he interrupted me to tell me that he needed my location. So I told him my location exactly as I'd begun doing when he interrupted me — I didn't note to him that I'd been doing just that — and I told him what was on fire. At this point, I wanted to put down what I was carrying, and go get a fire extinguisher, which meant getting off the phone. Of course, the fire was worsening and the burning object was collapsing in a way that could further fuel the flames. But the dispatcher was demanding my phone number in case we get cut off. I shouted at him that I'd told him what he needed to know, and wasn't going to stay on the phone with him. My phone set resisted my attempt to hang-up, so I turned it off. I got-out my keys, got through the gate, was interrupted by someone who told me that there were a fire, put my packages down, went (barking about stupidity) to a case near the elevator, retrieved a fire extinguisher, then returned to put-out the damn'd fire.

I thought that I heard a fire truck, so I waited, and one indeed arrived. They decided that the fire were extinguished, and so went on their way. The complex manager showed-up, so I explained the situation to her. Then a cop showed-up so I explained the situation to him. Satisfied, he too went on his way. The manager stayed to deal with the clean-up. I grabbed my things, went on to my apartment, and grumbled sub-vocally about inhaling things that I didn't want to inhale.

When I restarted my phone, I found that I had a message waiting. The dispatcher said that they needed me to call back to tell them what were on fire.

The Endurance of Love

Tuesday, 31 January 2017

Many months ago, sitting in the bistro that I frequent, I overheard a woman declaring Love never dies! Now, I really don't know the context, and she may have been elliptically expressing some thought with which I would agree, but certainly love does sometimes die. I thought about how and why it dies.

There's something often mistaken for love, that virtually always dies. That's the romantic bliss that many people feel each in response to some other person. Given human variability, there is probably some small number of people who feel that bliss about some person, without cessation, for many decades and until these blissful people die. But it rarely persists for more than a very few years, if even that. People who mistake it for love involve themselves in ways that they should not. People who mistake it for love sometimes leave when they should not; the bliss evaporates and they think themselves no longer in love, or they do not feel the bliss in the first place and think that they never loved, because they do not recognize the love that they feel for the other person. As for me, I've never felt that bliss at all; I only know of it from reports; but I certainly have personal experience of loving someone.[1]

Real love is the distinctive, fundamental emotional response to finding a person whom one believes exemplifies one's values. And when one believes that the other person complements one's self, the love has a romantic aspect, a desire to build a shared life with that person. Those beliefs are usually unconscious. The typical person, when not in love, can make a list of what he or she would want in another person, only to find him- or herself later in love with someone who seems quite different from what was imagined. But, in all things, our actual values come-out in the course of felt desire, of choice made at cost, of action. In any case, that emotional response lasts for as long as one holds those values, believes that the other person manifests those values, and sees that person as one's complement.

It takes rather a lot for that response to die, because there is so much to be undone. Some highly personal values of the lover must change, or the person loved must come to appear to be very different, in a negative way, from what was believed. I'm inclined to say that love doesn't die unless it is killed. Sometimes it staggers along for rather a long while even when obviously mortally wounded. If no one kills love, then it lasts for the lifetime of the one who loves, and thus can abide beyond the lifetime of the one who was loved.

I think that many or most of us have seen love killed. I've more than once seen one person who felt but did not recognize love kill the love that the other person felt for him or for her, and then struggle to live with an unrequited love, perhaps never seen for what it were.

I certainly won't claim that it is better to have loved and then experienced the killing of that love than never to have loved at all. But there is a self-awareness that can be salvaged from the wreckage. One doesn't know whom one will love before one loves; but, after all, one emerges from a failed love with the experience of having loved, and thus of having one's actual values expressed. Perhaps the other person wasn't whom one thought, but it should be possible consciously to identify some of the attributes that one imputed to that other person which caused one to love him or her; a contrast with the discovered makes the imagined easier to see. Thus, one may have a more clear idea of what one may call one's personal destiny, though this is a destiny that may not be reälized and might even be absurd.

Love that doesn't die but that is unrequited or effectively unrequited is a different matter. One might still clarify one's values, even without a contrast between the one's earlier beliefs about a person and what one discovers about him or about her. But there may be no application of this knowledge; so long as one is in love, there is no next person to seek.


[1] Early in my relationship with my most recent girl-friend, she was deeply hurt to learn that I wasn't joyful. She didn't explain why she was hurt, and I did not understand during the course of that relationship why she had been hurt (and perhaps remained hurt). Now I infer that she mistakenly felt unloved.

λέγει αὐτῷ ὁ Πιλᾶτος τί ἐστιν ἀλήθεια;

Wednesday, 25 January 2017

Years ago, National Lampoon had a monthly column that they entitled True Facts. The title was a joke, not because the contents weren't true (they were an assembly of extraödinary news reports), but because facts cannot be untrue; something untrue is not a fact. Yet many people in various contexts were using terms such as actual fact, real fact, and true fact, almost as if it were possible for some facts to be false, imaginary, unreal. People still do, perhaps even more often. One can find lots of instances of people using imaginary fact; sometimes they do so ironically, but more often they are quite serious. By imaginary fact they mean a proposition that may be untrue, is likely to be untrue, or simply is untrue. In this retasking of the word fact, they've lost the use of the word to talk about facts, unless they add a word such as true. But, with that change in meaning, it not only becomes possible to use a term such as alternative fact to refer to a rival claim, but it becomes harder to see that untrue rival claims don't have equal standing with true rival claims, as they are all supposedly facts.

We aren't at all helped here that a great many people don't understand the words true and truth. That's not simply a problem of vocabulary. Truth is a hard concept, because it entails a meta-propositional act of mapping from a proposition back to itself. That is to say that, in most cases when we apply the word true or equivalent and certainly in the case of true facts, we are explicitly or implicitly making a proposition about a proposition. When we say It's true that I went to the store, that actual referent of the grammatic subject is not I, but the proposition that I went to the store, yet the upshot of this sentence is merely what would be conveyed in saying I went to the store. We perhaps don't need this device of recasting a proposition (I went to the store) as a meta-proposition (It is true that I went to the store), but it is useful because we are not omniscient, and must entertain propositions that are uncertain or discovered to be false; the concept of truth complements the conditions of falsehood and of uncertainty. Yet it is very hard to see that function, exactly because we use the concept to discuss itself. Truth is more easily named than described, if indeed a description is possible.

The difficulty in understanding the nature of truth makes it psychologically easier to embrace such notions as that all aspects of of past, present, and future are simply artefacts of individual belief or of group belief (expressed with formulæ such as truth is a social construct) or that what one wants or ought to want is to be treated a true. The word fact may then be used for components of narratives; embracing one narrative is seen as licensing one to accept propositions as fact that are alternative to components of rival narratives, and to reject propositions for no better reason than that they participate in rival narratives. Evolution of narratives is seen as licensing one to change the status of a proposition from fact to falsehood, or vice versa, even when discussing history. And we may even observe those socially identified as fact-checkers testing claims against narratives which are themselves never fact-checked, because the checkers implictly treat their favored narratives as the ultimate determinant of fact.

When Pilate asked What is truth?, perhaps he was truly curious as to the nature of truth, but he may merely have been asking why he should give a damn about it. Our political leaders have become ever more disdainful of truth. They have long offered us alternative facts, and their followers in each of our major political tribes and in most of the smaller groups as well have decided that, for them, these are the facts. Now we have an Administration that does so more baldly and less artfully. One might hope that this practice will explode on them; but, even if that explosion should happen, their opponents are likely to see an expansion of the envelope within which they may disregard the facts.

Gotta Wear Shades

Sunday, 22 January 2017

Thinking about there already being an effort to persuade voters that they must support whomever the Democratic Party nominates in 2018 and in 2020, I was naturally led to wonder what sort of candidates those would be; and thence, more generally, what politics would be like in about three years.

What politics will be is something within the context of what they can be, so I try to imagine the plausible scenarii. To me, these look very grim.

I expect the Republican Party to continue to reshape itself around Donald Joseph Trump, to be more informed by his poison. I don't expect the dissenters within the Republican Party to enjoy much success, especially as I don't expect them to find any help either from main-stream of the media or from the alternative led by Fox. Some of the better people in that tribe whom we call the political right will leave it, and even some people who were enthusiastic about Trump will see or sense where things are leading, and turn away; however, it will continue to recruit people who were not previously its members but are attracted to Trump or to Trumpism. And a very large share of those who were already members of that tribe will reshape their ostensible beliefs to fit the pronouncements and actions of Donald Trump.

(If Trump should leave office early, then the Party will presumably work to reshape itself around Michael Richard Pence, who will attempt to segue from whatever Trumpism then is, to a frightening social conservatism. But I don't expect Trump to be removed.)

Meanwhile, the aforementioned Democrats are in a dark political death-pit. Their most recent nomination was decided as a struggle between what progressivism has become (and was always destined to be) and crony capitalism posturing opportunistically as that progressivism. The leadership ensured that the crony capitalist got the nomination, but the subsequent result was, if anything, to intensify the need of the leadership to pander to the progressive urges of the Party's base. And the response of the base to Trump's election has been to move still further left and to become more overtly violent.

That base is about 25% of the nation and of the voters. It likes to imagine itself as a plurality, as a majority, or sometimes even as the other 98% or as the 99%. But the political left has inflated its apparent size by taking disproportionate control of the commanding heights of our social institutions, and by implicitly coöperating with the political right (also only about a quarter of the population) to marginalize those with views that do not conform to the left-right framework; remarkably, the political right has kept the political left viable, and vice versa. (These days, about a third of the American population wants the state to be less involved both in matters of sex and of religion and in matters ordinarily classified as economic. That makes them a larger group than either the left or the right, but not larger than the two combined.) Given how small they actually are, the left (like the right) cannot take political power without getting votes from those outside their tribe. If the Democratic Party cannot free itself from the influence of those who continue down the path of progressivism, then it cannot return to power based upon a positive appeal; its hopes would lie only in Republican over-reach.

Which over-reach would certainly have recent precedence. In 2006 and 2008, the Republican Party was in its own death-pit. They'd stumbled into it under the influence of neo-conservatism. The neo-conservatives pursued two wars, guided by a theory that the people in Afghanistan and in Iraq were aching for liberal democracy and prepared to fight for it. And they pursued fiscal and regulatory policies that triggered a major financial crisis. The Republicans lost control of both chambers of Congress, and then of the White House.

What pulled them from that death-pit was the Tea Party, which began coälescing with rallies, and then organized itself well enough to displace candidates of the Republican establishment in primary races. Those in the Republican establishment who held onto their positions fought vigorously with the Tea Party, which served to burnish the credentials of the Tea Party candidates and office-holders. Attempts by the left to treat the Tea Party Republicans as responsible for the prior actions of the neo-conservative Republicans were unsuccessful in persuading those outside the Democratic base. Ultimately, it seems that the Tea Party failed in its objectives; they did not succeed in capturing the leadership of the Republican Party, and some office-holders who had cultivated the support of the Tea Party betrayed their trust. The Tea Party appears now itself displaced by an uglier and angrier populism. But the Tea Party had the effect of allowing the Republican Party largely to free itself from association with the awful policies of the Administration of George Walker Bush.

That brings me to the Women's March on Washington, and to its sister Marches in other cities. It is not impossible that these rallies could be the beginning of a process that could reälign the Democratic Party, pulling it from its death-pit as the Tea Party pulled the Republican Party from its own. The odds, however, are very much against that reälignment.

First, though a great many of the women who participated wanted to be heard, rather that to be told what to say by celebrities and by iconsdefinitely shouting No! to goddamn'd Donald Joseph Trump, but not all saying Yes! to what the Democratic Party or its progressive base have been offering — the objective of most of those who called and led these marches in the first place was not to reälign the Party, but to reälign women. A new leadership of reformers would have to be found.

Second, while the Tea Party did not much have to fight the Republican base, that is exactly what anyone seeking fundamentally to reälign the Democratic Party would have to do.

Those without much hope for the Republican or Democratic Parties as instruments of positive change sometimes turn to third parties. Some exist, others could be creäted. We've already seen a collapse of the institutionalized myth that third parties must always be formed either around one issue or around one candidate. The left and the right will join together, through the media that they control and through bi-partisan commissions, to fight the emergence of of a real alternative; but, if the left continues to get ever more repulsive to those voters outside of the left and right, then the right will have a greatly weakened partner in their unholy alliance to exclude rival views. I don't imagine that I myself will find a party or Presidential candidate whom I can, in good conscience, support; but perhaps 2020 will be made less awful by the emergence of a better major party.

A Blame Game

Sunday, 22 January 2017

I have previously explored the logical absurdity of insisting that those who don't vote for one of the two foremost candidates in an election are effectively voting for the other of these two candidates. That analysis could easily be generalized to include ballot measures; where abstaining from voting has sometimes been claimed to be the same as opposing a measure, and sometimes been claimed to be the same as supporting the measure.

The primary purpose of claims of these forms is to pressure someone into voting for a candidate whom — or measure which — the potential voter finds unappealing. We especially see these claims when there are actual candidates, or formulated ballot measures. But we also see that purpose at play when an election has been held and candidates and proposal for the next election are not yet identified. Some of those declaring You are now to blame for X because you did not vote for Y! are hoping to motivate the audience to tolerate whatever is demanded by the claimant's faction in that next election. An alternative would be to promise to offer something better than Y in that next election; but they are engaged in brinksmanship, threatening to take the community off a cliff if a plurality don't agree to their demands.

Other motivations for such claims in the wake of an election are simple ventilation and blame-shifting. It is frustrating to lose an election, and a blow to the ego to acknowledge that one's own faction may be largely responsible for that loss.[1]

There's another, unrecognized motivation for these claims. Although there is a very great logical distance between refusing to support one of the two major political tribes and thinking as do members of the other major tribe, it is easy enough for tribal members to disregard that distinction almost perfectly. Thus, these absurd claims that refusal to support Y is operationally the same thing as supporting X implicitly become part of a more general psychological device of treating politics as all falling along a left-right spectrum, and thereby avoiding any challenge to one's thought or behavior that cannot be dismissed from the left by pointing to the right or from the right by pointing to the left, and saying You guys are worse!, even if the challenger is not one of those guys. The challenge may even be spuriously taken as proof that, after all, the challenger were really a member of the other tribe, as he or she is not challenging them, and them alone.


[1] For example, James Henry (Jim) Webb could have beaten any of the Republican candidates for President in 2016, and it is at least plausible that Bernard (Bernie) Sanders could have beaten Donald J[oseph] Trump; but the Democratic Party chose its weakest or second-weakest candidate. (Martin Joseph O'Malley might have been a still worse choice, as his mythology of Baltimore collapsed in the face of the police murder of Freddie Carlos Gray jr.)

Recognizing the Difference

Tuesday, 17 January 2017

The word problem is used for multiple concepts:

  • a challenge, especially a mental challenge
  • an unhappy state of affairs that exercises the mind
  • a seemingly insurmountable difficulty

These may actually be very different things, because some states-of-affairs may be changed, and some may not; but it is often hard to distinguish the former from the latter. Thence we use the same word for all three concepts, and thence we get the famous Serenity Prayer of Reinhold Niebuhr:

God, grant me the serenity to accept the things I cannot change, courage to change the things I can, and wisdom to know the difference.

A good economist is going to note that sometimes the costs of changing what we can would be higher than the benefit. Properly understood, that greater costs would mean that we were trading some greater problem for a solution to the problem under immediate consideration. The Serenity Prayer might lead one to mistake sensible calculation for cowardice, though poetry might have to be sacrificed to provide better guidance.

Still, the distinction between that which we can change and that which we cannot is most fundamental, and confusion between the two is a source of frustration and of anguish. That which displeases us and which we can change is a challenge about which we can usefully exercise the mind, until such time as either we develop a reasonable method of changing things or we recognize that it's not worth the cost. That which displeases us and which we cannot change is an unhappy condition of existence; and — ideally — we ought to come to terms with that condition, to accept it, so that it no longer exercises our minds. (And we ought, similarly, to accept those states the changing of which, though possible, truly would be too costly.)

One seldom-noted root of some of the confusion is a failure to distinguish between what the person in question may change and what someone else may change. The state-of-affairs might be a problem for a person who could change it, or it might not be a problem for that person; but, either way, by someone who cannot change it, it is best treated as a condition of existence.

It may be especially hard to see the distinction when the other person might indeed act to change things and, if that other person did, the first person would want to undertake quickly some course of action. But, likewise, there might be some potential event which would have to be impersonally caused (if at all) and which would call for a prompt response. One should not try to cause that which no person can cause, and one should not try to cause that which only another person can cause, even if there would be things to do should it happen.

It may be too much to hope for serenity from such realizations. It is certainly too much to demand that others be serene in the face of indifference, indecision, or madness on the part of those around them. Sometimes grim acceptance is the best that one can manage.

Deal-Breakers

Saturday, 7 January 2017

Elsewhere, Pierre Lemieux asked In two sentences, what do you think of the Monty Hall paradox? Unless I construct sentences loaded with conjunctions (which would seem to violate the spirit of the request), an answer in just two sentences will be unsatisfactory (though I provided one). Here in my 'blog, I'll write at greater length.


The first appearance in print of what's called the Monty Hall Problem seems to have been in a letter by Steve Selvin to The American Statistician v29 (1975) #1. The problem resembles those with which Monty Hall used to present contestants on Let's Make a Deal, though Hall has asserted that no problem quite like it were presented on that show. The most popular statement of the Monty Hall Problem came in a letter by Craig Whitaker to the Ask Marilyn column of Parade:

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?

(Before we continue, take car and goat to stand, respectively, for something that you want and something that you don't want, regardless of your actual feelings about cars and about goats.)

There has been considerable controversy about the proper answer, but the text-book answer is that, indeed, one should switch choices. The argument is that, initially, one has a 1/3 probability that the chosen Door has the car, and a 2/3 probability that the car is behind one of the other two Doors. When the host opens one of the other two Doors, the probability remains that the car is behind one of the unchosen Doors, but has gone to 0 for the opened Door, which is to say that the probability is now 2/3 that the car is behind the unchosen, unopened Door.


My first issue with the text-book answer is with its assignment of initial, quantified probabilities. I cannot even see a basis for qualitative probabilities here; which is to say that I don't see a proper reason for thinking either that the probability of the car being behind a given Door is equal to that for any other Door or that the probability of the car being behind some one Door is greater than that of any other Door. As far as I'm concerned, there is no ordering at all.

The belief that there must be an ordering usually follows upon the even bolder presumption that there must be a quantification. Because quantification has proven to be extremely successful in a great many applications, some people make the inference that it can be successfully applied to any and every question. Others, a bit less rash, take the position that it can be applied everywhere except where it is clearly shown not to be applicable. But even the less rash dogma violates Ockham's razor. Some believe that they have a direct apprehension of such quantification. However, for most of human history, if people thought that they had such literal intuitions then they were silent about it; a quantified notion of probability did not begin to take hold until the second half of the Seventeenth Century. And appeals to the authority of one's intuition should carry little if any weight.

Various thinkers have adopted what is sometimes called the principle of indifference or the principle of insufficient reason to argue that, in the absence of any evidence to the contrary, each of n collectively exhaustive and mutually exclusive possibilities must be assigned equal likelihood. But our division of possibilities into n cases, rather than some other number of cases, is an artefact of taxonomy. Perhaps one or more of the Doors is red and the remainder blue; our first division could then be between two possibilities, so that (under the principle of indifference) one Door would have an initial probability of 1/2 and each of the other two would have a probability of 1/4.

Other persons will propose that we have watched the game played many times, and observed that a car has with very nearly equal frequency appeared behind each of the three Doors. But, while that information might be helpful were we to play many times, I'm not aware of any real justification for treating frequencies as decision-theoretic weights in application to isolated events. You won't be on Monty's show to-morrow.

Indeed, if a guest player truly thought that the Doors initially represented equal expectations, then that player would be unable to choose amongst them, or even to delegate the choice (as the delegation has an expectation equal to that of each Door); indifference is a strange, limiting case. However, indecision — the aforementioned lack of ordering — allows the guest player to delegate the decision. So, either the Door was picked for the guest player (rather than by the guest player), or the guest player associated the chosen Door with a greater probability than either unchosen Door. That point might seem a mere quibble, but declaring that the guest player picked the Door is part of a rhetorical structure that surreptitiously and fallaciously commits the guest player to a positive judgment of prior probability. If there is no case for such commitment, then the paradox collapses.


Well, okay now, let's just beg the question, and say not only that you were assigned Door Number 1, but that for some mysterious reason you know that there is an equal probability of the car being behind each of the Doors. The host then opens Door Number 3, and there's a goat. The problem as stated does not explain why the host opened Door Number 3. The classical statement of the problem does not tell the reader what rule is being used by the host; the presentation tells us that the host knows what's behind the doors, but says nothing about whether or how he uses that knowledge. Hypothetically, he might always open a Door with a goat, or he might use some other rule, so that there were a possibility that he would open the Door with a car, leaving the guest player to select between two concealed goats.

Nowhere in the statement of the problem are we told that you are the sole guest player. Something seems to go very wrong with the text-book answer if you are not. Imagine that there are many guest players, and that outcomes are duplicated in cases in which more than one guest player selects or is otherwise assigned the same Door. The host opens Door Number 3, and each of the guest players who were assigned that Door trudges away with a goat. As with the scenario in which only one guest player is imagined, more than one rule may govern this choice made by the host. Now, each guest player who was assigned Door Number 1 is permitted to change his or her assignment to Door Number 2, and each guest player who was assigned Door Number 2 is allowed to change his or her assignment to Door Number 1. (Some of you might recall that I proposed a scenario essentially of this sort in a 'blog entry for 1 April 2009.) Their situations appear to be symmetrical, such that if one set of guest players should switch then so should the other; yet if one Door is the better choice for one group then it seems that it ought also to be the better for the other group.

The resolution is in understanding that the text-book solution silently assumed that the host were following a particular rule of selection, and that this rule were known to the guest player, whose up-dating of probabilities thus could be informed by that knowledge. But, in order for the text-book solution to be correct, all players must be targeted in the same manner by the response of the host. When there is only one guest player, it is possible for the host to observe rules that respond to all guest players in ways that are not not possible when there are multiple guest players, unless they are somehow all assigned the same Door. It isn't even possible to do this for two sets of players each assigned different Doors.


Given the typical presentation of the problem, the typical statement of ostensible solution is wrong; it doesn't solve the problem that was given, and doesn't identify the problem that was actually solved.


[No goats were harmed in the writing of this entry.]