Steven Landsburg

Viagra and Logic

From Steven Landsburg:

Ohio State Senator Nina Turner (along with several of her counterparts in other states) has introduced legislation requiring men to undergo a series of humiliating procedures before they can fill their Viagra prescriptions. Here I am confident that Senator Turner is following in the admirable footsteps of Rush Limbaugh, by proposing a policy she doesn’t actually support in order to highlight its symmetry with a policy she finds appalling, namely recent legislation requiring women to undergo a series of humiliating procedures before they can have an abortion….

But is Senator Turner’s analogy a good one? It depends, I think, on the intent of the Ohio abortion law.

There are two possible motivations for that law. Motivation One is paternalistic, proceeding from the assumption that women will make poor choices about abortion and that we do them a favor when we discourage them. If that’s indeed the motivation, then Senator Turner’s analogy is spot-on. If we’re going to assume (with no substantial evidence) that women make poor choices about abortion, why not assume that men make poor choices about erectile dysfunction drugs? If we’re going to arrogate the power to override women’s choices, why not do the same for men?

But Motivation Two is that the legislature believes abortion is ipso facto a bad thing and wants to discourage it in any way possible, without regard to what’s in the best interest of the pregnant woman. If that’s the motivation, then Senator Turner’s analogy becomes much weaker (unless you’re really prepared to argue that erections are ipso facto a bad thing). A perfectly consistent person might fervently oppose this legislation but still consider Senator Turner’s implicit argument a bad one….

… I have the strong impression that Motivation One has been bandied about quite a bit by the proponents of these laws. So I think Senator Turner has got this right, and I admire both her logic and her gumption.

Perhaps Landsburg is trying to atone for his fit of political incorrectness in l’affaire Fluke. In any event, Landsburg has the wrong end of the stick (so to speak).

Motivations One and Two are not, in this case, independent and mutually exclusive, as Landsburg treats them. Motivation Two precedes Motivation One.That is, the motivation for pre-abortion procedures, such as fetal sonograms, is the belief that abortion is ipso facto a bad thing. The intention of legislators who vote to require such procedures is to reduce the number of abortions. (For more about the distinction between motivation and intention, see this letter to the editor of the Orlando Sentinel and a longer discussion in an old post of mine.)

Moreover, requiring men to undergo “a series of … procedures before they can fill their Viagra prescriptions” is not analogous to requiring woment to undergo pre-abortion procedures. In the case of pre-abortion procedures, the intention is to discourage a life-taking event; in the case of pre-Viagra-prescription procedures, the obvious intention is to protest pre-abortion procedures. If you think that the latter is on a moral par with the former, you suffer from an advanced case of pseudo-feminist hysteria.

Nina Turner, call your analyst.

Related posts:
A Useful Precedent
Law, Liberty, and Abortion
Substantive Due Process and the Limits of Privacy
Crimes against Humanity
Abortion and Logic

Are the Natural Numbers Supernatural?

Steven Landsburg writes:

…It is not true that all complex things emerge by gradual degrees from simpler beginnings. In fact, the most complex thing I’m aware of is the system of natural numbers (0,1,2,3, and all the rest of them) together with the laws of arithmetic. That system did not emerge, by gradual degrees, from simpler beginnings….

…God is unnecessary not because complex things require simple antecedents but because they don’t. That allows the natural numbers to exist with no antecedents at all—and once they exist, all hell (or more precisely all existence) breaks loose: In The Big Questions I’ve explained why I believe the entire Universe is, in a sense, made of mathematics. (“There He Goes Again,” The Big Questions Blog, October 29, 2009)

*   *   *

The existence of the natural numbers explains the existence of everything else. Once you’ve got that degree of complexity, you’ve got structures within structures within structures, and one of those structures is our physical Universe. (If that sounds like gibberish, I hope it’s only because you’re not yet read The Big Questions, that you will rush out and buy a copy, and that all will then be clear.) (“Rock On,” The Big Questions Blog, February 8, 2012)

With regard to the first quotation, I said (on October 29, 2009) that

Landsburg’s assertion about natural numbers (and the laws of arithmetic) is true only if numbers exist independently of human thought, that is, if they are ideal Platonic forms. But where do ideal Platonic forms come from? And if some complex things don’t require antecedents, how does that rule out the existence of God … ?

I admit to having said that without the benefit of reading The Big Questions. I do not plan to buy or borrow the book because I doubt its soundness, given Landsburg’s penchant for wrongheadedness. But (as of today) a relevant portion of the book is available for viewing at Amazon.com. (Click here and scroll to chapter 1, “On What There Is.”) I quote from pages 4 and 6:

…I assume — at the risk of grave error — that the Universe is no mere accident. There must be some reason for it. And if it’s a compelling reason, it should explain not only why the Universe does exist, but why it must.

A good starting point, then, is to ask whether we know of anything — let alone the entire Universe — that not only does exist, but must exist. I think I know one clear answer: Numbers must exist. The laws of arithmetic must exist. Two plus two equals four in any possible universe, and two plus two would equal four even if there were no universe at all….

…Numbers exist, and they exist because they must. Admittedly, I’m being a little vague about what I mean by existence. Clearly numbers don’t exist in exactly the same sense that, say, my dining-room table exists; for one thing, my dining-room table is made of atoms, and numbers are surely not. But not everything that exists is made of atoms. I am quite sure that my hopes and dreams exist, but they’re not made of atoms. The color blue, the theory of relativity, and the idea of a unicorn exist, but none of them is made of atoms.

I am confident that mathematics exists for the same reason that I am confident my hopes and dreams exist: I experience it directly. I believe my dining-room table exists because I can feel it with my hands. I believe numbers, the laws of arithmetic, and (for that matter) the ideal triangles of Euclidean geometry exits because I can “feel” them with my thoughts.

Here is the essence of Landsburg’s case for the existence of numbers and mathematics as ideal forms:

  • Number are not made of atoms.
  • But numbers are real because Landsburg “feels” them with his thoughts.
  • Therefore, numbers are (supernatural) essences which transcend and precede the existence of the physical universe; they exist without God or in lieu of God.

It is unclear to me why Landsburg assumes that numbers do not exist because of God. Nor is it clear to my why his “feeling” about numbers is superior to other persons’ “feelings” about God.

In any event, Landsburg’s “logic,” though superficially plausible, is based on false premises. It is true, but irrelevant, that numbers are not made of atoms. Landsburg’s thoughts, however, are made of atoms. His thoughts are not disembodied essences but chemical excitations of certain neurons in his brain.

It is well known that thoughts do not have to represent external reality. Landsburg mentions unicorns, for example, though he inappropriately lumps them with things that do represent external reality: blue (a manifestation of light waves of a certain frequency range) and the theory of relativity (a construct based on observation of certain aspects of the physical universe). What Landsburg has shown, if he has shown anything, is that numbers and mathematics are — like unicorns — concoctions of the human mind, the workings of which are explicable physical processes.

Why have humans, widely separated in time and space, agreed about numbers and the manipulation of numbers (mathematics)? Specifically, with respect to the natural numbers, why is there agreement that something called “one” or “un” or “ein” (and so on) is followed by something called “two” or “deux” or “zwei,” and so on? And why is there agreement that those numbers, when added, equal something called “three” or “trois” or “drei,” and so on? Is that evidence for the transcendent timelessness of numbers and mathematics, or is it nothing more than descriptive necessity?

By descriptive necessity, I mean that numbering things is just another way of describing them. If there are some oranges on a table, I can say many things about them; for example, they are spheroids, they are orange-colored, they contain juice and (usually) seeds, and their skins are bitter-tasting.

Another thing that I can say about the oranges is that there are a certain number of them — let us say three, in this case. But I can say that only because, by convention, I can count them: one, two, three. And if someone adds an orange to the aggregation, I can count again: one, two, three, four. And, by convention, I can avoid counting a second time by simply adding one (the additional orange) to three (the number originally on the table). Arithmetic is simply a kind of counting, and other mathematical manipulations are, in one way or another, extensions of arithmetic. And they all have their roots in numbering and the manipulation of numbers, which are descriptive processes.

But my ability to count oranges and perform mathematical operations based on counting does not mean that numbers and mathematics are timeless and transcendent. It simply means that I have used some conventions — devised and perfected by other humans over the eons — which enable me to describe certain facets of physical reality. Numbers and mathematics are no more mysterious than other ways of describing things and manipulating information about them. But the information — color, hardness, temperature, number, etc. — simply arises from the nature of the things being described.

Numbers and mathematics — in the hands of persons who are skilled at working with them — can be used to “describe” things that have no known physical counterparts. But  that does not privilege numbers and mathematics any more than it does unicorns or God.

Related posts:
Atheism, Religion, and Science
The Limits of Science
Three Perspectives on Life: A Parable
Beware of Irrational Atheism
The Creation Model
The Thing about Science
Evolution and Religion
Words of Caution for Scientific Dogmatists
Science, Evolution, Religion, and Liberty
Science, Logic, and God
Is “Nothing” Possible?
Debunking “Scientific Objectivity”
Science’s Anti-Scientific Bent
Science, Axioms, and Economics
The Big Bang and Atheism
The Universe . . . Four Possibilities
Einstein, Science, and God
Atheism, Religion, and Science Redux
Pascal’s Wager, Morality, and the State
Evolution as God?
The Greatest Mystery
What Is Truth?
The Improbability of Us
A Digression about Probability and Existence
More about Probability and Existence
Existence and Creation
Probability, Existence, and Creation
The Atheism of the Gaps
Probability, Existence, and Creation: A Footnote
Scientism, Evolution, and the Meaning of Life

Wrong Again

Steven Landsburg, who is supposedly a “hardcore libertarian,” is on the verge of surpassing Bryan Caplan as the most wrong-headed libertarian economist on my RSS reading list. Landsburg’s upside-down view of the world has led me to issue the following posts:

Now comes Landsburg with yet another tour through the land of tortured logic: “Another Rationality Test.” There, Landsburg sets out the following problem:

Suppose you’ve somehow found yourself in a game of Russian Roulette. Russian roulette is not, perhaps, the most rational of games to be playing in the first place, so let’s suppose you’ve been forced to play.

Question 1: At the moment, there are two bullets in the six-shooter pointed at your head. How much would you pay to remove both bullets and play with an empty chamber?

Question 2: At the moment, there are four bullets in the six-shooter. How much would you pay to remove one of them and play with a half-full chamber?

In case it’s hard for you to come up with specific numbers, let’s ask a simpler question:

The Big Question: Which would you pay more for — the right to remove two bullets out of two, or the right to remove one bullet out of four?

The question is to be answered on the assumption that you have no heirs you care about, so money has no value to you after you’re dead.

If you think the right answer is to pay more for the right to remove two bullets out of two, you’re wrong, according to Landsburg. Why? Well, Landsburg — with a train of “logic” that reminds me of Lou Costello’s “proof” that 7 x 13 = 28 — “proves” that removing two of four bullets has the same value as removing two of two bullets. Here’s how Landsburg does it:

[T]hink about four questions:

Question A: You’re playing with a six-shooter that contains two bullets. How much would you pay to remove them both? (This is the same as Question 1.)

Question B: You’re playing with a three-shooter that contains one bullet. How much would you pay to remove that bullet?

Question C: There’s a 50% chance you’ll be summarily executed and a 50% chance you’ll be forced to play Russian roulette with a three-shooter containing one bullet. How much would you pay to remove that bullet?

Question D: You’re playing with a six-shooter that contains four bullets. How much would you pay to remove one of them? (This is the same as Question 2.)

Now here comes the argument:

  • In Questions A and B you are facing a 1/3 chance of death, and in each case you are offered the opportunity to escape that chance of death completely. Therefore they’re really the same question and they should have the same answer.
  • In Question C, half the time you’re dead anyway. The other half the time you’re right back in Question B. So surely questions C and B should have the same answer.
  • In Question D, there are three bullets that aren’t for sale. 50% of the time, one of those bullets will come up and you’re dead. The other 50% of the time, you’re playing Russian roulette with the three remaining chambers, one of which contains a bullet. Therefore Question D is exactly like Question C, and these questions should have the same answer.

Okay, then. If Questions A and B should have the same answer, and Questions B and C should have the same answer, and Questions C and D should have the same answer — then surely Questions A and D should have the same answer! But these, of course, are exactly the two questions we started with.

In case the trick isn’t obvious on first reading — and it wasn’t to me — here’s what Landsburg does.

He starts by positing a situation in which removing two of two bullets (Question A/Question 1) and removing one of one bullet (Question B) have the same value, and therefore should be worth the same amount to the hypothetical, involuntary player of Russian Roulette. Why should they have the same value? Because, given the player’s unstated but implicit objective — which is to survive Russian Roulette, and nothing else — he stands to improve his chance of surviving by 1/3 in both cases. Probabilistically, removing two of two bullets from a six-chamber gun is the same as removing one of one bullet from a three-chamber gun. In both instances, the chance of being shot goes from one-third to zero.

The point of the preceding exercise isn’t to get the reader to think about probabilities; it’s to get the reader (a) to assume that Landsburg’s “proof” is on the up-and-up, while (b) planting the idea that removing one of one bullets from three chambers is just as good as removing two of two bullets from six chambers. The reader is being set up to fall for a real whopper, to which I’ll come.

The next step (Question C) is to juxtapose the threat of dying by three-shooter with an irrelevant but equally probable threat. The 50% probability of summary execution is irrelevant, in the context of the problem at hand, because there’s nothing the player can do about it. What the player can do is to influence his probability of surviving Russian Roulette, which increases by one-third if he buys the single bullet that’s in the three-shooter.

The point of the preceding exercise is to reinforce further the reader’s confidence in Landsburg’s “proof,” while planting the idea that it’s legitimate to ignore a 50% threat of certain extinction. I use the word “confidence” because the setup of this “proof” is like the setup of a confidence game. The reader, like the victim of a con game, is now ripe for plucking.

Landsburg’s answer to Question D (Question 2) does the trick. It parallels Question C, but it doesn’t have the same import with respect to the player’s objective: surviving a game of Russian Roulette. Landsburg waves away 50% of the threat to the player’s existence (the three bullets that the player can’t buy), even though the three bullets (one of which may come up 50% of the time) are part of the game of Russian Roulette, unlike the summary execution of Question C.  Landsburg, having dismissed the threat posed by the three bullets as somehow irrelevant, then focuses on the remaining bullet. Because this bullet can be in one of three chambers (the three not occupied by the other bullets), it seems that the player can increase by one-third his chance of surviving Russian Roulette if he buys one bullet. Thus, by Landburg’s “logic,” buying one of four bullets has the same value to the player as buying two of two bullets.

If you swallow that one, I’d like to sell you a bridge.

The fact that the player can’t buy three of the four bullets posited in Question D/Question 2, has no bearing on the probability that he survives Russian Roulette, which is his objective. With four bullets in the gun, his chance of survival is one-third. If he pays to have one bullet removed, the gun still has three bullets in it. Thus his chance of survival increases by one-sixth (3/6 [1/2] – 2/6 [1/3] = 1/6). Clearly, the value of removing one of four bullets isn’t the same as the value of removing two of two bullets, which increases his chance of survival by 1/3.

If you’re not convinced, let’s return to the original problem and restate it in a way that doesn’t distort the essential question: If the player’s objective is to survive a game of Russian Roulette, would he pay more for a six-gun with no bullets in it (Question 1) or a six-gun with three bullets in it (Question 2)? If you say that he’d pay the same amount for either gun, your name is Steven Landsburg.

Take Landsburg’s Money

REVISED 01/01/11 and 01/02/11, with the addition of new material (clearly indicated).

Economist-mathematician Steven Landsburg recently offered a problem and later posted a purported solution to it. Landsburg is so confident that his solution is the correct one that he has offered to bet significant sums of money that it’s the correct one. And, as of this morning, Landsburg still insists that he has the right answer.

The problem is one (among many) that Google has posed to candidates for employment. Landsburg states it as follows:

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female? [The actual wording of the question, according to this source, is slightly different, but Landsburg's paraphrase is faithful to the meaning.]

He adds:

Well, of course, you can’t know for sure, because, by some extraordinary coincidence, the last 100,000 families in a row might have gotten boys on the first try. But in expectation, what fraction of the population is female? In other words, if there were many such countries, what fraction would you expect to observe on average?

I first heard this problem decades ago, and so, perhaps, did you. It comes up in job interviews at places like Google. The answer they expect is simple, definitive and wrong.

And no, it’s not wrong because of small discrepancies between the number of male and female births, or because of anything else that’s extraneous to the spirit of the problem. It’s just really wrong. The correct answer, unlike the expected one, is not simple.

According to Landsburg, the “obvious” — but wrong — answer is that one-half of the children are boys and one-half of the children are girls. Landsburg rejects the “obvious” answer, with this explanation:

I’ll start with the case where there’s just one couple. Here are some possible family configurations, with their probabilities:

From this we see that the expected number of boys is

which adds to 1. And the expected number of girls is

which also adds to 1. Sure enough, the expected number of girls is equal to the expected number of boys.

But the expected fraction of girls is

which adds to 1-log(2), or about 30.6%.

For a population of k families, a similar calculation gives an answer of approximately (but not exactly) (1/2) – (1/4k), which, when k is large, is approximately (but not exactly) 1/2.

Elsewhere, Landsburg offers to make bets with readers who disagree with his analysis, and to settle matters through the use of simulation. But (a) I’m not interested in betting and (b) I prefer to treat the problem as one of mathematical expectation, which Landsburg also (rightly) prefers. He suggests the use of simulation only as a way of convincing some skeptics of the correctness of his analysis.

Interestingly, Landsburg’s “solution” — an expected girl fraction of 0.3068 for one family and, presumably, not quite 0.5 for the entire country — is at odds another person’s solution (girl fraction ~0.61),to which Landsburg points favorably. This discrepancy suggests some confusion on Landsburg’s part, which is evident in his depiction of the possible configurations of a single family (the children in the family, actually). He takes a special case — which omits the possibility of a first-born girl. He then generalizes from that special case.

__________________________________________________________________________________________

This section added 01/01/11 and revised slightly for clarity on 01/02/11:

What I took for an omission on Landburg’s part (the possibility of a first-born girl), isn’t an omission — in Landsburg’s view. His estimate of the girl fraction for a family is for a completed family (his term, not mine). Thus the configurations B, GB, GGB, GGGB, etc.

But there’s no such thing (in the context of a single family) as a completed family. (In a large number of families, there may be a completed families, but every one of them will be matched by an equal number of uncompleted families. More about that below.)  No particular couple ever has a boy with an a priori probability = 1, which is what Landsburg implies (inadvertently, I’m sure) when he focuses on B, GB, GGB, GGGB, etc., where the B in each case signifies the end of a possible sequence of children.

On the contrary, if PB = 1/2 at all times (and not 1 at arbitrary times) every boy must be accompanied by a girl, with equal probability. (Alternatively, shades of Schrödinger’s cat, the probability wave collapses to PB =1 when Landsburg decides that  enough kids are enough.) Here’s a schematic depiction of what happens when Landsburg doesn’t play with the probabilities:

On the left side of the vertical line, the probable first-born boy, being only probable, is followed by a probable boy or girl, and so on. On the right side, the probable first-born girl is followed by a probable boy or girl, and so on. On both sides, the possible configurations take the form Child 1, Child 1 + Child 2, Child 1 + Child 2 + Child 3, etc.

Landburg, in effect, has restricted his view of the possible  configurations to the left side of the diagram.With the whole diagram in view, it’s obvious that the fraction of girls in each stage, and through each stage, is always 1/2.

End of section. Original post continues below the line.

________________________________________________________________________________________

To analyze the problem correctly it’s helpful to spell out all of its conditions and (sometimes implicit) assumptions:

  • The basic rule — couples have children until a boy is born, but not after that — means that no couple in country X will have more than 1 boy, but the number of girls is limited only to the number of children that a couple generates before having a boy.
  • A couple can generate children endlessly, if necessary. That is to say, there’s no limit on the number of children a couple may produce and no limit on the time in which they may produce them.
  • The probability that any given birth will result in a boy (B) or girl (G)  is 1/2 for each; that is PB = PG = 1/2. (The actual fractions, I gather, are about 0.52 boys and 0.48 girls per birth.) These probabilities never vary, and are always the same for every couple.
  • Given the open-ended nature of the problem, it’s possible that some couples will have an infinite number of children without producing a boy. But, given the preceding statement, the first-born of half the couples will be a boy; those couples will have no more children.
  • The situation begins at a finite time (t = 0) and only those children born to the couples in country X after t = 0 are counted.
  • Children are born at a uniform rate, so that all the first-borns are born at  t = 1; all the second-borns at  t = 2; and so on. (This assumption and the preceding one don’t affect the results, but they allow for a simple illustration of the problem.)
  • In computing the fraction of boys and girls in the population, it’s assumed that there are no abortions or miscarriages, and that no children die at birth or later.

Perhaps the solution to the problem will be easier to see if the problem is recast, so that B = blue ball, G = green ball, and “couple” becomes “player”:

  • There’s a large but finite number of urns. Each is full of colored balls. Half of them are blue (B); half of them are green (G).
  • Positioned at each urn is a player whose job it is to make a blind draw of one ball from his urn at a regular interval (say, 1 minute). Each ball is kept by the player who draws it.
  • As each ball is drawn, the keeper of the urn from which it is drawn replaces it with a new ball of the same color, and mixes the balls thoroughly to ensure the randomness of the next draw.
  • When a player draws a B, he keeps it but doesn’t draw any more balls.
  • When a player draws a G, he keeps it and draws another ball (from the replenished urn) a minute later. This continues until the player draws a B.
  • It’s possible that some players will never draw a B.

All of the other assumptions stated earlier apply in this case (e.g., PB = PG = 1/2, PB and PG never vary and are always the same for every player).

Consider the following illustration of the results of the first four rounds of play:

Illustration of the General Case ( with 10,000 urns)
B G Total G fraction
Minute 1 5,000 5,000 10,000 1/2
Total 5,000 5,000 10,000 1/2
Minute 2 2,500 2,500 5,000 1/2
Total 7,500 7,500 15,000 1/2
Minute 3 1,250 1,250 2,500 1/2
Total 8,750 8,750 17,500 1/2
Minute 4 625 625 1,250 1/2
Total 9,375 9,375 18,750 1/2

The 5,000 players who draw a B at minute 1 stop drawing, but they keep the Bs that they draw. The 5,000 players who draw a G at minute 1 keep their Gs and make another draw at minute 2. That draw results in the selection of 2,500 Bs and 2,500 Gs, which are kept by the players who draw them. The 2,500 players who draw a G at minute 2 make another draw at minute 3, which results in the selection of 1,250 Bs and 1,250 Gs, and so on.

At this point, it’s important to note that the stopping rule has no effect on the fractions of B and G drawn in any round of play. Given that PB and PG are always the same for each and every player, as stated in the list of assumptions, it doesn’t matter whether or when players drop out of the game or join the game, or under what conditions they drop out or join, as long as PB = PG = 1/2, always and for every player. Given those conditions — which are central (implicit) assumptions of the original Google problem — every round of play, from the first one onward, results in equal (expected) numbers of B and G.

_______________________________________________________________________________________

This section added and revised 01/02/11.

In the example of 10,000 players with 10,000 urns, the number of players at minute 5 would be an odd number, and the number of players at minute 6 and beyond wouldn’t be an integer. But the example is about expected values (as it should be), so the lack of an even, whole number of players after minute 4 wouldn’t affect the import of the example.

To show what happens in the “end game,” I turn to a slightly different example, which begins with a number of players such that there can be exactly two left in the penultimate round, after all others have drawn a B. What happens when one of the two players draws a B while the other draws a G? Good question. First of all, the draws to and including that round will have resulted in an equal number of B and G. What happens next is a matter of pure chance. The final player — the one whose penultimate draw is a G — has an equal chance of drawing a G or a B on his next and final draw. The chance of drawing a B doesn’t suddenly jump to 1, nor does the chance of drawing a G suddenly jump to 1. The game could end there, with equal numbers of B and G having been drawn and a final draw to be made with PB = PG = 1/2. But there’s no reason to expect that the final draw will be a B, to the exclusion of a G, or vice versa.

Schematically:

(The Gs drawn by players 8191 and 8192 in rounds 1-11 would have been matched, in each round, by other players who draw Bs in those rounds. Players 8191 and 8192, in this example, would be the only players left for round 12.)

End of section. Original post continues below the line.

_______________________________________________________________________________________

Mathematically, the expected numbers of B and G (EB and EG) drawn by a single player are as follows:

EB = (1)(1/2) + (1/2)(1/2) + (1/4)(1/2)  + (1/8)(1/2) + … = 1

EG = (1)(1/2) + (1/2)(1/2) + (1/4)(1/2)  + (1/8)(1/2) + … = 1

The expected fraction of G is:

EG/(EG + EB) = (1/2)(1/2) + (1/4)(1/2) + (1/8)(1/2) + … = 1/2,

which reduces to PG/(PG + PB) = 1/2

Given N players, the expected numbers of B and G are:

EB = (N)(1/2) + (N/2)(1/2) + (N/4)(1/2)  + (N/8)(1/2) + … = N

EG = (N)(1/2) + (N/2)(1/2) + (N/4)(1/2)  + (N/8)(1/2) + … = N

Again, given the “rules” of the game (i.e., of the original Google problem), the expected fraction of G is always 1/2, at every point in the game and in its expected (but never reached) outcome.

The stopping rule is a red herring, intended (I suspect) to draw attention from the essential fact that EB = EG, always and for everyone, no matter how many players (couples) there are or when and under what conditions they join or leave the game (start or stop having children).

*     *     *

To Prof. Landsburg, should he read this post:

If you don’t immediately spot a fatal flaw in my analysis, why not have some members of U of R’s stat department check it out? That seems to me to be the best way to settle the issue.

If you conclude that my analysis is correct (in the essentials, at least), you won’t owe me any money because we haven’t made a bet. (You couldn’t send me money, anyway, unless you are able to penetrate my anonymity.) Just acknowledge my contribution prominently in a post on your blog and add a link to my blog in your sidebar (perhaps under the heading “Unclassified Blogs”).

*     *     *

This isn’t the first time that Landsburg has attracted my attention:
Landsburg Is Half-Right
Rawls Meets Bentham
The Case of the Purblind Economist

The Case of the Purblind Economist

Purblind: lacking in insight or understanding; obtuse

Steven Landsburg just doesn’t get it. Uwe Reinhardt lectures him about the folly of “efficiency” (or “social welfare”), and Landsburg continues to act as if there were such a thing:

Suppose you live next door to Bill Gates. Bill likes to play loud music at night. You’re a light sleeper. Should he be forced to turn down the volume?

An efficiency analysis would begin, in principle (though it might not be so easy in practice) by asking how much Bill’s music is worth to him (let’s say we somehow know that the answer is $10,000) and how much your sleep is worth to you (let’s say $25). It is important to realize from the outset that no economist thinks those numbers in any way measure Bill’s subjective enjoyment of his music or your subjective annoyance. Only a crazy person would think such a thing, and I’ve never met anybody who’s that crazy in that particular way. Instead, these numbers primarily reflect the fact that Bill is a whole lot richer than you are. Nevertheless, the economist will surely declare it inefficient to take $10,000 worth of enjoyment from Bill in order to give you $25 worth of sleep. We call that a $9,975 deadweight loss.

The problem with this kind of thinking should be obvious to anyone with the sense God gave a goose. The value of Bill’s enjoyment of loud music and the value of “your” enjoyment of sleep, whatever they may be, are irrelevant because they are incommensurate. They are separate, variably subjective entities. Bill’s enjoyment (at a moment in time) is Bill’s enjoyment. “Your” enjoyment (at a moment in time) is your enjoyment. There is no way to add, subtract, divide, or multiply the value of those two separate, variably subjective things. Therefore, there is no such thing (in this context) as a deadweight loss because there is no such thing as “social welfare” — a summation of the state of individuals’ enjoyment (or utility, as some would have it).

Landsburg persists:

Take a more realistic example: Should we spend, say, a billion dollars a year to subsidize end-of-life health care for poor people? It would be, I think, a terrible mistake to settle this question without at least asking whether the recipients might prefer that we spend our billion dollars some other way — say by subsidizing their groceries or just giving them cash. If so, the difference in value between what they’re getting and what they could be getting (as measured by the recipients) is a deadweight loss. The bigger that deadweight loss, the more we should reconsider our spending priorities.

Who is “we,” Prof. Landsburg? Do you presume to speak for me, one of the taxpayers who would share in the cost of subsidizing end-of-life health care for poor people? The “recipients” have no right to prefer anything. It is my money you’re talking about, not some pot of “social welfare” that sits in the sky, waiting to be distributed by omniscient economists like you. The deadweight loss, as far as I’m concerned, is whatever you take from me to “give” to others, in your omniscience. I have better things to do with my money, thank you, and whether or not they’re “charitable” (they are, in part), is no business of yours. Who made you the accountant of my soul?

Related posts:
Greed, Cosmic Justice, and Social Welfare
Positive Rights and Cosmic Justice
Inventing “Liberalism”
Utilitarianism, “Liberalism,” and Omniscience
Utilitarianism vs. Liberty
Beware of Libertarian Paternalists
Landsburg Is Half-Right
Negative Rights, Social Norms, and the Constitution
Rights, Liberty, the Golden Rule, and the Legitimate State
The Mind of a Paternalist
Accountants of the Soul
Rawls Meets Bentham
Enough of “Social Welfare”

Rawls Meets Bentham

Steven Landsburg writes:

Paul Krugman is at it again, casting aspersions on everyone who opposes extended unemployment benefits while offering absolutely no positive argument for those benefits. Let me explain what would count, to an economist, as a positive argument.

There’s no question that extending benefits would be good for the currently unemployed, and no question that it would be bad for those who are called on to foot the bill. Economists usually deal with that kind of conflict is by asking what policy you’d prefer if you had amnesia, and and didn’t know your own employment status…. The amnesiac is an impartial judge who is forced to care about everyone, because he/she might be anyone.

I have no wish to defend the indefensible Paul Krugman, but Landsburg’s attack is equally indefensible, combining — as it does — John Rawls’s “veil of ignorance” and the utilitarianism of Jeremy Bentham and his philosophical progeny. The “veil of ignorance,” according to Wikipedia, requires you to

imagine that societal roles were completely re-fashioned and redistributed, and that from behind your veil of ignorance you do not know what role you will be reassigned. Only then can you truly consider the morality of an issue.

This is just another way of pretending to omniscience. Try as you might to imagine your “self” away, you cannot do it. Your position about a moral issue will be your position, not that of someone else. Moreover, it will not truly be your position unless you put it into practice. Talk — like happiness research — is cheap.

Pretended omniscience is the essence of utilitarianism, which is captured in the phrase “the greatest good for the greatest number” or, more precisely “the greatest amount of happiness altogether.” From this facile philosophy grew the patently ludicrous idea that it might be possible to quantify each person’s happiness, sum those values, and arrive at an aggregate measure of total happiness for everyone.

But there is no realistic worldview in which A’s greater happiness cancels B’s greater unhappiness; never the twain shall meet.  The only way to “know” that A’s happiness cancels B’s unhappiness is to put oneself in the place of an omniscient deity — to become, in other words, an accountant of the soul.

Landsburg, in the space of a single post, has put himself in company with “liberals” like Krugman, who arrogate to themselves the ability to judge the worthiness of others. A pox on both their houses.

Related posts:
On Liberty
Greed, Cosmic Justice, and Social Welfare
Positive Rights and Cosmic Justice
Inventing “Liberalism”
Utilitarianism, “Liberalism,” and Omniscience
Utilitarianism vs. Liberty
Beware of Libertarian Paternalists
Negative Rights, Social Norms, and the Constitution
Rights, Liberty, the Golden Rule, and the Legitimate State
The Mind of a Paternalist
Accountants of the Soul

Landsburg Is Half-Right

*     *     *

God does not play dice with the universe. — Albert Einstein

Einstein, stop telling God what to do. — Niels Bohr

*     *     *

In a post at The Big Questions blog, Steven Landsburg writes:

Richard Dawkins . . . [has] got this God thing all wrong. Here’s some of his latest, from the Wall Street Journal:

Where does [Darwinian evolution] leave God? The kindest thing to say is that it leaves him with nothing to do, and no achievements that might attract our praise, our worship or our fear. Evolution is God’s redundancy notice, his pink slip. But we have to go further. A complex creative intelligence with nothing to do is not just redundant. A divine designer is all but ruled out by the consideration that he must be at least as complex as the entities he was wheeled out to explain. God is not dead. He was never alive in the first place.

But Darwinian evolution can’t replace God, because Darwinian evolution (at best) explains life, and explaining life was never the hard part. The Big Question is not: Why is there life? The Big Question is: Why is there anything?

So far, so good. But Landsburg doesn’t quit when he’s ahead:

Ah, says, Dawkins, but there’s no role for God there either:

Making the universe is the one thing no intelligence, however superhuman, could do, because an intelligence is complex—statistically improbable —and therefore had to emerge, by gradual degrees, from simpler beginnings

That, however, is just wrong. It is not true that all complex things emerge by gradual degrees from simpler beginnings. In fact, the most complex thing I’m aware of is the system of natural numbers (0,1,2,3, and all the rest of them) together with the laws of arithmetic. That system did not emerge, by gradual degrees, from simpler beginnings. . . .

Now I happen to agree with Professor Dawkins that God is unnecessary, but I think he’s got the reason precisely backward. God is unnecessary not because complex things require simple antecedents but because they don’t. That allows the natural numbers to exist with no antecedents at all. . . .

What breathtaking displays of arrogance. Dawkins presumes that the only kind of intelligence that can exist is the kind that comes about through evolution. Landsburg wishes us to believe that complex things can exist on their own, without antecedents, which is why there is no God. (He fudges by saying “God is unnecessary” but we know what he really believes, don’t we?)

Landsburg’s “proof” of the non-existence of God is the existence of natural numbers, a “system [that] did not emerge, by gradual degrees, from simpler beginnings.” Landsburg’s assertion about natural numbers (and the laws of arithmetic) is true only if numbers exist independently of human thought, that is, if they are ideal Platonic forms. But where do ideal Platonic forms come from? And if some complex things don’t require antecedents, how does that rule out the existence of God — who, by definition, embodies all complexity?

Related posts:
Same Old Story, Same Old Song and Dance
Atheism, Religion, and Science
The Limits of Science
Beware of Irrational Atheism
The Creation Model
Evolution and Religion
Science, Evolution, Religion, and Liberty
Science, Logic, and God
The Universe . . . . Four Possibilities
Einstein, Science, and God
Atheism, Religion, and Science Redux
A Non-Believer Defends Religion
Evolution as God?
The Greatest Mystery