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Animal Understanding of Animal Understanding

February 21st, 2012

Colin Allen (in American Scientist) reviews a book by Robert Lurz which takes what I am inclined to call a typical philosopher’s miss-take on the issue.

Apparently Lurz notes that an animal’s responses to another animal’s looking at food could, in principle, be just an instinctive response to the looking behaviour rather than being based on having some mental model of the mental state of the competitor.  He calls this the “logical problem” and claims that economy of thought favours the version with no mental modelling. But what I see there is an illogical appeal to false economy.

The possibility of alternative explanations is not of itself a problem. Every set of results admits a variety of “logically” possible explanations, and it is only by something like the principle of economy that we can make a choice. But in making this choice, economy of thought, like entropy, has to be considered globally rather than just locally.

Lurz’s “logical problem” applies also to my attribution of thought to fellow humans. Logically they could all be just mindless robots, but having learned to recognize myself (which elephants and chimps among others can also do) and seeing that other people do in fact look like me, it would be reasonable (and economical) to assume that when their behaviour matches mine it does so for similar internal reasons. Without access to my own thoughts and my own apparent similarity to my peers, their consciousnesses might seem like an unnecessary extra assumption, but in the extended observational context which includes myself, their minds become necessary in order to avoid the unnecessary distinction between minded people and the mindless others. Since my own experienced existence does render the mind-concept necessary it would be wasteful not to use it also to help explain the behaviour of others – even though, in my absence it may not have been necessary and so might have properly been ruled out.

More importantly, a theory of mind enhances my ability to predict the behaviour of my peers in a wider variety of circumstances than merely reacting to certain sepecific behaviours would. What evolution favours in me it also favours in my chimpish cousin – and has been doing for thousands of generations.  So again economy of thought suggests that if it looks like a chimp thinking of a thinking chimp and acts like a chimp thinking of a thinking chimp then it may well be a chimp thinking of a thinking chimp even if it can’t quite talk like a chimp thinking of a thinking chimp.

If a chimp could think of a chimp-thinking chimp with his eye on a chink what might he think that the chimp-thinking chimp with his eye on the chink would think?

 

 

Silly Questions?

February 17th, 2012

Statistical pundit William M. Briggs has written a piece for ‘Significance’ on Why Do Statisticians Answer Silly Questions That No One Ever Asks?.

Briggs is right to object to instances where statisticians (or more often users of statistics) respond to silly questions with the answers to different (and often equally silly) ones without making it clear enough that they are not answering the original question. But he is wrong in his presumption that the questions asked usually make sense.

In fact it is common to talk of probability in quantitative terms in situations where it is by no means clear what a specific numerical probability would mean.  Such talk is what gives rise to many well known “paradoxes” which can only be resolved by clarifying the interpretation.

But although Briggs alludes to recent advances in Bayesian analysis, he doesn’t seem to understand them well enough himself  - at least not well enough to answer a simple question about what he means when he says  “a civilian needs little or no maths to understand what ‘the probability that A is better than B is 80%’ means”. 

Briggs response to the question of what that understanding might be is just “It means the evidence is such that the probability ‘A is better than B’ is 80%. Which is greater than 0% but less than 100%. Nothing more.

When challenged that this is like claiming to explain what “the hoy is gerflumptive” means by saying that it means “the evidence is such that the hoy is gerflumptive”, he responds with “I wasn’t being glib. Probability (see above) is a measure of truth, or closeness to truth. 80% is closer than 70% and less close than 90% to being true. What you do with this number is different than what the number is.

Well, I’m sorry, but giving “closeness to truth” as a definition of probability *is* glib.

(It’s also more than 75% wrong in that I can think of at least three measures of closeness to truth that are more common than anything to do with probability.)

He asks for examples and I say:

For example, in common language (as per my claim):

1. an approximate answer is often referred to as close to the truth

2. a false statement is sometimes referred to as close to the truth if its error arises from a fairly common misuse of terminology

3. a detective may be said to be getting close to the truth if he has a good idea of where to look for the deciding piece of evidence

etc.

Briggs responds to these with:

But two of these examples are non-probabilistic.

1. Given our background knowledge, an approximate answer is likely true
3. Ditto

2. You’ll have to clarify this. A falsity is not close to a truth; a mistake is still a mistake.

Cooper:

They were intended to be non-probabilistic as I was giving them as examples of why “closeness to truth” is not a good definition of probability.
1. The statement that the circumference of a circle is six times its radius has zero probability of being true but it is close to the truth.
3. Knowledge of the fact that the murdered duke wrote a deathbed note which will tell me whether it was Colonel Mustard or Professor Plum who poisoned him brings me closer to the truth without increasing the probability of either hypothesis.
2. Your attempt to define probability as “closeness to the truth” may be close to the truth but it has zero probability of actually providing a useful definition.

Briggs:

Alan,

I assume you meant your “2″ as a joke, but it has backfired on you. In a useful way, however. Let’s see.

1. A = “The circumference of a circle is six times is radius.” Now, there is no such thing as

    Pr(A).

But we can calculate:

    Pr(A | E) = 0

where E = “My knowledge of geometry as might be found in any high school or higher text”. Notice that this is completely different than B = “A is a good approximation”. We still cannot calculate

    Pr(B).

But we can calculate:

    Pr(B | E & F) = 1

where we have the same E plus information F = “A good approximation is being within plus or minus 20% of the radius” or some other F (different F might change the probability, of course).

3. A = “Duke says M or P killed him”. If B = “M killed the Duke” then

    Pr(B | A) = 1/2

and similarly for C = “P killed the Duke.” The probability

    Pr(Duke was murdered | evidence of dead body & foul play) = 1

which is the same as

    Pr(Duke was murdered by somebody | evidence of dead body & foul play) = 1.

But we cannot compute

    Pr(Duke was murdered by M | evidence of dead body & foul play) = unknown,

unless we condition on something more, namely a list of suspects.

2. I could write this out, but you’ll get the idea. The probability that I have provided you the true definition, given all this (and other information on the blog) is 1.

Cooper:

All I can say is that I think you must have missed my point – which was that there are common language senses of “closeness to truth” which have nothing to do with probability, and so that “closeness to truth” is not a good definition of probability.

This all started when I asked you what you would say ‘the probability that A is better than B is 80%’ means, and so far I haven’t seen anything not glib in response.

Briggs:

Alan,

I haven’t; you have failed to make yours. In order to disprove my thesis, you need to show an example that can’t be written in the forms (for example) that I’ve given.

Cooper:

Thanks for trying, but I don’t understand what you are saying. If you have given an intelligible answer to my question about the meaning of probability then I guess I’ll just have to accept that the subject is beyond me.

Seriously, am I nuts or is this guy cuckoo?

Social and Biological Construction(s) of Race

February 13th, 2012

This article by Razib Khan at Discover Magazine makes an important distinction which relates to my earlier post on the same topic (Mythical Myth #3). The fact that the strength of a concept can be widely misunderstood does not mean that it has no basis in fact, and to deny that it exists at all (when it so transparently does) can actually discredit and undermine efforts to prevent its abuse.

 

Contraceptive Coverage in US Health Care Plans

February 10th, 2012

The fact that, in the USA, having decent basic health insurance for all is dependent on some funny scheme involving employers is something that most of the “civilized” world finds hard to understand. But given that it is so dependent, thinking about the cost impact of paying for contraception as opposed to the alternative raises an interesting point about whose “freedom” is at stake when the catholic bishops insist on being able to exclude it for their employees.

Health insurance companies which are free to charge on a cost-plus basis have no incentive to require a cost-reducing preventive medicine if they are free to charge whatever the costs are without it. But If I was selling dependent coverage at a fixed rate independent of family size (as many group plans do), then I would probably be prepared to add birth control for free after negotiating the rate without it – unless of course the employer told me not to. So what may well be the case here is that the church was seeking to pay a premium for the right to *exclude* contraception from a plan which could have been cheaper with it.

If so, then the church has been making all this fuss because they want the freedom to pay extra themselves in order to deny their employees the freedom to get whatever birth control they need via the most economical route. If people can understand this (and if the media don’t suppress it) then it will be interesting to see where the mud finally sticks!

Will it “Take”?

February 8th, 2012

These messages are not new but maybe there’s  now at least a chance that they’ll rise in volume sufficiently to overhwelm the mainstream censors:

3quarksdaily links to Why economic inequality leads to collapse

and David Brin has some good words about the distinction between investment and rent seeking.

Time Series

February 5th, 2012

William M. Briggs, a climate change skeptic who has been in a recent running battle with the other side ever since getting a podium at the Wall Street Journal is having a go at clarifying his position on Time Series.

In Let’s Try This Time Series Thing Again: Part I, Briggs starts with the idea of such a series as representing something that is “measured without error” and adds the claim that “Something causes every observation to take the values it does”. Both of these reduce my expectation that he has anything useful to contribute so maybe I should look up a serious reference on the topic and see if he’s really as far off base as I think he is.

ACTA

February 3rd, 2012

This confession came to my attention via Michael Geist.

Coincidentally this came at the same time via 3QuarksDaily, and I was also pleased to see that Neil Young has joined those who see the excessive criminalization of media sharing (and especially of  private copying) as ill advised.

Personally I am not a big user of commercial media, but the arrogant presumption of assholes stealing my money to pay for copying I will never do, and threatening to lock media that I do buy into playability only on proprietary platforms, has put me firmly in the pro-pirate camp. And I expect to stay there until the media world either rots away or comes to its senses and adopts a more reasonable and respectful attitude.

Mythical Myths #17: Humans radiate proportionately more than the Sun

February 2nd, 2012

Sometimes a statement which is perfectly true is called a myth on the basis of a misstatement. A case in point is “Bad astronomer” Phil Plait’s treatment of the statement in the above title in the post at  Q&BA: Pound for pound, are humans hotter than the Sun? | Bad Astronomy | Discover Magazine.

The correct statement of the mythical “myth” is that pound-for-pound (or ml-for-ml) the human body radiates more energy per second than the sun. Of course it’s not “hotter” nor does it *have* “more energy”, but it *loses* energy relatively more quickly because it has a relatively much larger surface area compared to its volume. This is just a consequence of the relative inefficiency of large spheres as radiators.  The reason for this is because any old cc in the middle of the sun may be as hot as hell but they all absorb almost as much radiation from their neighbours as they emit themselves, and it is only those near the surface which contribute photons which actually escape. So the idea of pulling them out to compare with us defeats the whole point of the exercise. It’s all part of the same theme which explains why elephants have big ears and why mice each day have to eat a much greater proportion of their body weight than we do. If I was as fat as the sun I’d be pretty hot in the middle too and that’s precisely because I would then be getting rid of heat proportionately less rapidly than I was generating it.

Blasphemy Redux

January 30th, 2012

Kenan Malik has posted some thoughts about “blasphemy” that he addresses at the recent CFI conference.

In my opinion, the concept of blasphemy should rightly include ANY statement attributing any characteristic, act, or opinion to any god or gods.

But what offends ME more than any of that is the suggestion that communications which are merely offensive should be suppressed. Although I don’t agree with them, I hope the advocates of suppressing all offensive speech will take note of my offense and follow their own advice.

What is Wrong with SOPA and PIPA?

January 19th, 2012

 Clay Shirky says it pretty well. (And in Canada, Michael Geist and Howard Knopf give good coverage of the legal issues for us). These proposed laws reverse the burden of proof with regard to alleged copyright violation. This both takes away the right of due process and has the effect of adding prohibitive costs onto the shoulders of the providers of self-publishing services.

The Myth of Japan’s Failure

January 13th, 2012

This article at NYTimes.com makes some interesting claims which may raise questions about how we should measure national economic success. But I don’t know how to either judge its validity or figure out what are the lessons we should take from it. All in all, it’s somewhat intriguing but quite frustrating. So I hope that it generates some response and follow-up.

Rethinking the Growth Imperative

January 11th, 2012

 Kenneth Rogoff (formerly chief economist at the IMF and now at Harvard) has found the sense to ask: “does it really make sense to take growth as the main social objective in perpetuity, as economics textbooks implicitly assume?”

And he correctly identifies a major driving force behind the growth that does happen in the fact that “people are fundamentally social creatures. They evaluate their welfare based on what they see around them, not just on some absolute standard.”

But what this implies is that the major force driving growth is less its position as a “social objective” and more its status as almost everyone’s personal objective – not out of a wish to actually have more actual stuff but rather out of fear of losing status by falling behind in the race to get it.

And of course national policies are driven by the same competitive zeal.

Unfortunately neither of these attitudes is irrational since 4 billion years of experience tells us in our bones (well genes actually) that whoever isn’t winning is losing and losers don’t last.

Perhaps the growth factor in economists’ models should be seen less as a social goal than as an abstraction of the effects of competitive struggle. If so,  asking the economists to come up with a zero growth model won’t solve the problem. What is needed is to address the psychology that drives that growth. And given its deep roots that may be no small problem.

Meanwhile right wing blogger Will Wilkinson disputes Rogoff’s thesis on the social value of growth by foolishly identifying the correlation between wealth and happiness of members of  society at a fixed time  (and short term changes in the average happiness in societies that are advancing or retreating in wealth relative to other societies at the same time) with the completely different question of how global average wealth correlates with global average happiness.

Taking Offense

January 10th, 2012

When certain Muslims voiced their offense, various others took offense in turn.

Hmmm.

Actually I agree with all but the last line of the following excerpt from the  Ahmadiyya Muslim Students Association:

“Once a particular act is deemed to be offensive to another, it is only good manners to refrain from, at the very least, repeating that act. In this particular case, when at first the cartoon was uploaded, it could have been mistaken as unintentional offense. When certain Muslims voiced their offense over the issue, for any civil, well-mannered individual or group of individuals, it should then be a question as to the feelings of others and the cartoons should then have been removed.”

Generally I like “good manners”,  but they don’t trump other virtues such as “true compassion”, “honesty”, “genuine humility”,  or even  ”self defense”.  Nor is it usually  ”good manners” to correct the manners of others (or  at least of those others who have given every indication that they do not want to be corrected and in fact might even be “offended” by it).

But much more significantly, although displays of bad manners always offend me a bit (no matter who is their object, and even though, on another level, I may sometimes enjoy them), what offends me many times more strongly is the suggestion that anyone’s freedom to offend, in their own name and without direct physical harm, be curtailed by any means other than just ignoring them.    

But that might not be quite what’s happening here…

What complicates the situation is that the group posting the cartoon may be affiliated with a larger group that does not want to give offense, and (depending on the actual relationship) that larger group may be entitled to require the cartoon to be removed and is certainly entitled to disassociate itself from whatever opinion the cartoon represents. The larger group also has a right to apply whatever discipline is agreed to on becoming a member in order to enforce this disassociation –  almost certainly up to the point of expulsion of the “offending” members if they continue to associate their offense explicitly with the larger group.

It might even be argued by extension from the above that society as a whole has the right to criminalize the giving of offense to whomever it (society) is unwilling to offend (whether out of fear or just kind consideration),  but  I would dispute this on the grounds that membership in a society is not voluntary for the native born, and although we do of course restrict individual rights on various grounds the mere giving of offense is not and should not be among them.  (And in any case that is not is what is at issue in the UCLU-ASH case.)

In the absense of any offensive demand (as opposed to polite request) from outside the organisation that the cartoons be disallowed,  if it was any of my business then, depending on the context (which I don’t really know), I might (or might not) suggest that they be voluntarily removed from the normal view of those offended. But if the suggestion of real suppression were to be raised, then it would become imperative to me that they remain (whether or not I actually liked or despised them).

Anyone who feels impelled to avoid giving any offense at all should be aware that many such as I are offended by the excessive taking of offense (especially when this is used to excuse violent reaction), and so those who want to limit the giving of offense should encourage all with whom they associate to moderate their responses accordingly. Unfortunately, on all sides, it often seems that those most inclined to take offense are also those with the least scruples about giving it.

At first I was inclined to be offended by the AMSA treasurer’s statement, but on reading it carefully I note that it does acknowledge the freedom to insult as a part of the right to freedom of speech, does not actually claim offense to the author, and is careful to put the request for removal of the cartoon as a suggestion rather than a demand.  On the whole I find it quite reasonable in fact. But I do still think the last line of the part quoted above is wrong.  If it were right then the author should actually remove it because of the fact that, even though I am not offended by it, there undoubtedly are people who are offended. Of course the fact that it offends some people is not, in my opinion, any good reason to remove it (or even, I suppose, to edit it according to this implied suggestion). But if the author wishes to be able to claim self-consistency then leaving it unchanged is not an option.

Help Preserve the Canadian Public Domain

January 7th, 2012

Michael Geist calls on us to ‘ Speak Out on the Trans Pacific Partnership Negotiations‘.

My brief letter to the provided response address (consultations@ international.gc.ca) was as follows:

Re COPYRIGHT ISSUES
Please take note of my very strong opposition to any agreement which either extends the term of copyright protection for existing works,
or extends the criminalization of digital lock beaking for purposes in particular of maintaining access to legally acquired content via any medium of the user’s choice and more generally for any purpose that would not otherwise be criminal (including any use that does not violate a copyright).
Thank you.
Alan Cooper

(And by the way, the address on the gov’t website has a space which makes copying it not work – perhaps inserted deliberately to discourage responses?)

Actually I could live with “protection” of digital locks IF, and ONLY if, it only applied to media on the packaging for which the fact that the product works only on specifically identified devices was displayed in larger and clearer print than any other aspect of content description.

But the issue of copyright extension drives me really mad! This is just a gift of cash stolen from the public to whomever owns the rights half a century after the author’s death and obviously does not contribute in any way to the original authorship and funding decisions that were made more than half a century earlier. With sufficient data it might be possible to convince me that longer copyright terms for future works would help increase creative output, but there is no way that this applies to works already written and the benefit of an extension is often being given to someone who purchased rights from the author on the basis of a shorter term of protection. If copyright is extended, then authors (and estates) who sold rights with a shorter term should be entitled to sue for recovery of whatever value is provided by the extension.

A Darwinian Approach to Moral Philosophy

January 5th, 2012

Michael Ruse is someone whose name is often mentioned and he presents this ‘Darwinian Approach to Moral Philosophy’ at the ‘ Talking Philosophy’ blog as a summary of his apparently sometimes controversial views. So I think I’ll read it fairly carefully…

My first problem is in his second paragraph with

 I come away with the belief that ethics – meaning by this substantive or normative ethics (“What should I do?”) – is a product of natural selection (on individuals) to further reproductive success

Here my problem is with the idea that individual reproductive success is the driving selector behind the evolution of morality.  That may or may not be current gospel among biologists, but I suspect that it is wrong – and that the relevant selective force is increase of whatever gene complex causes the behaviour. Although this can be achieved via individual reproduction, it could also be achieved by giving the gene’s host a capacity to recognize and favour other carriers of the gene (with the understanding, here and henceforth, that by “gene” I really mean something more complex than the coding for a single protein).

I have yet to see if this difference will undermine my agreement with anything else he says. So let’s read on…

My next uneasy feeling comes with the “empirical claim” that

 ethical claims have the appearance and meaning of being objective claims, in the sense of not just subjective emotions but about external standards.

I certainly go with the “appearance” part but am not so sure about actual “meaning”. He’s right to say next that “If we thought it was all a matter of liking and disliking, ethics would break down rapidly” but our strong feeling that there’s something more objective doesn’t make it so – and the fact that that feeling is a requirement of the phenomenon should help us to recognize that it may be there for its own self-serving reasons. Moral behaviour may often conflict with what we would “like” to do (and we may be tempted avoid it if sure of not getting caught), but that doesn’t make it any less a “feeling”. Of course, at this point I may be misreading a bit. The “meaning” of  a “claim” may refer just to the intent of the claimant rather than anything more objective.

And just two lines further on I find that I was indeed misreading as Ruse says that “the belief about objectivity is erroneous” and “ ethics (meaning substantive ethics) is an illusion put in place by our genes to make us social cooperators“. So we are back pretty much in agreement on that score. Although  it’s not the ethical feelings or principles I’d call an illusion – just the sense that they have some context-independent source. But if I could read more than one line at a time I’d see that in the next sentence he says “But notice I am not saying that ethics as such is an illusion – I very much don’t think this – rather I am saying that the belief that ethics is objective is an illusion. ” (Is it unfair to complain when someone says “X. But notice I am not saying X”?)

Something similar happens in the next paragraph since after having admitted that the sense of objectivity is an illusion he asserts that “ “I like spinach” is subjective, but “Murder is wrong” is absolute or objective or binding or whatever.” I know what he means, but he seems bound by the philosopher’s typical faith in language and dichotomy. Maybe he’ll sort this out later in the piece, but it seems reasonable to have pointed out at this point that there’s a big space between “subjective” (meaning dependent on the point of view of an individual)  and “objective” (meaning in some sense completely independent of any body of  opinion). Moral precepts are non-subjective and universal in the sense of being common to all “moral” people (and considered binding on all people), but are still a function of the human context and so not objective in any absolute sense.

Actually, he does seem to get this even though he can’t quite bring himself to say it, and he does place morality as universal among humans though not necessarily beyond that. But his fear of moral relativism still makes him overly optimistic. The belief that “all humans share the same basic moral sense” may be true in a sense but is not necessarily true in the sense that he would like. What I mean here (as I elaborated earlier in my thinking about the failings of Sam Harris’ enterprise) is that even with the same basic tendencies there is no guarantee of the existence (let alone the discovery) of any universally correct answers to moral questions. We may be driven to strive for many different and potentially conflicting values with no guarantee of any relative priority between them and both our behaviour and our judgement of it may be inherently unstable. It is possible that even with the same basic drives different societies can settle on different and incompatible choices of what to optimize – with no commonly agreeable way of seeing which is better. Although I think Darwin’s use of “reared” in the alien bee-people example was really intended with the sense of “evolved” I do think it is possible that here on Earth identical twins reared in the hills of Afghanistan and those of San Fransisco may end up with equally internally consistent but mutually incompatible moral systems and no way of getting from one to the other by a process of successive improvement of self-judged value – and no real way of saying which is better.

But all-in-all, despite the excess of optimism, he seems pretty clear headed by comparison to others, and there is nothing up to this point that I can imagine making people angry.

However, in the last paragraph he alludes to a possible source of hostility from some, and it seems a bit disingenuous because it is essentially unrelated to the rest of the content. Without knowing more of the details  (eg who “struck” first, and about what,  etc) I  feel pulled into a family squabble that I want no part of.

 

Not Much Love Among Philosophers

January 4th, 2012

 Stephen Downes led me to this interesting dust up. Apparently Clark Glymour, a philosopher at Carnegie Mellon who can actually be “taken seriously” by computer (and other) scientists was offended that his dep’t got a low ranking in a recent academic ratings game and hit back with a polemicmanifesto” belittleing some other branches of his subject (though in my opinion perhaps not too unfairly – except perhaps for the associations with Hitler, Stalin, and  Pol Pot!).

Trust in Neurons

December 29th, 2011

This review in American Scientist claims that, despite its subtitle, Patricia Churchland’s book doesn’t tell us much about morality. And in a sense, if the description is true (and I’ll find out soon), then I am inclined to agree since I see a big difference between “morality” and “moral behaviour”.  But I still expect to find  lots of interest in the neuroscience of moral behaviour (trust and cooperation), and I  see that as foundational to a further study of the basis of actual morality itself (which I identify with the distrust and enforcement behaviours that are necessary in a population in order for moral behaviour patterns to survive and prosper).

What Are The Goals of the Atheist Movement?

December 23rd, 2011

What Are The Goals of the Atheist Movement? | Greta Christina’s Blog.

Or more to the point for me. What are my goals in wasting time on all this nonsense?

There must be some, so perhaps I’ll add more here when I think of them.
…more »

A Curious Definition of Freedom

December 14th, 2011

Afghanistan: Imprisoned Rape Victim Freed. – where in this context, apparently “freed” means transferred into the permanent custody of her rapist.

Taxing the 1%:

December 8th, 2011

According to this ( Taxing the 1%: Why the top tax rate could be over 80% | vox – Research-based policy analysis and commentary from leading economists), lowering the top tax rate just meant that the top 1%  had more reason to argue for taking a bigger share of the pie without actually causing any relatively better performance of the businesses and economies for which they were responsible.

…while standard economic models assume that pay reflects productivity, there are strong reasons to be sceptical, especially at the top of the income distribution where the actual economic contribution of managers working in complex organisations is particularly difficult to measure. In this scenario, top earners might be able to partly set their own pay by bargaining harder or influencing compensation committees. Naturally, the incentives for such ‘rent-seeking’ are much stronger when top tax rates are low. In this scenario, cuts in top tax rates can still increase top income shares – consistent with the observed trend in Figure 1 – but the increases in top 1% incomes now come at the expense of the remaining 99%. In other words, top rate cuts stimulate rent-seeking at the top but not overall economic growth…

 

The data seem to support this rent-seeking explanation  (as opposed  to actual increased productivity), since:

 

For example, countries that made large cuts in top tax rates such as the United Kingdom or the United States have not grown significantly faster than countries that did not, such as Germany or Denmark. Hence, a substantial fraction of the response of pre-tax top incomes to top tax rates documented in Figure 1 may be due to increased rent-seeking at the top rather than increased productive effort.