Friday, March 16, 2012

Chess, intelligence and winning arguments

Arguments

We have all had arguments. Occasionally these reach an agreed upon conclusion but usually the parties involved either agree to disagree or end up thinking the other party hopelessly stupid, ignorant or irrationally stubborn. Very rarely do people consider the possibility that it is they who are ignorant, stupid, irrational or stubborn even when they have good reason to believe that the other party is at least as intelligent or educated as themselves.

Sometimes the argument was about something factual where the facts could be easily checked e.g. who won a certain football match in 1966.

Sometimes the facts aren’t so easily checked because they are difficult to understand but the problem is clear and objective. One famous example is the Monty Hall dilemma. In a certain game show a contestant is presented with three doors, behind one of which is a really desirable main prize and behind the other two are booby prizes. The contestant is asked to choose the door he thinks the main prize is behind. The game show host, who knows where the prize is, then opens a booby prize door. He can do that even if you picked the main prize door. The contestant is now offered the chance to change his choice to the other closed door. The question is – should he? The correct answer is that he should because it would double his chances of winning the main prize. This fact proved so hard to grasp that only 8% of the laymen and 35% of academics who tried it, got it right initially. Many of them damned Marilyn vos Savant for misleading the public with her answer in her Parade column Ask Marilyn. Among those academics getting it wrong, were a fair number of senior professional mathematicians and statisticians, including the great Paul Erdos. After a great deal of explanation, the proportion of people who accept the correct answer increased to 56% of laymen and 71% of academics. The figure is more like 97-100% for those who carried out an experiment or simulation, so in essence a large proportion of those approaching the problem through pure logic continue to fail to grasp it.

Sometimes the facts aren’t as mathematical or logical as the Monty Hall solution. Each party to the argument appeals to ‘facts’ which the other party disputes. The disputed facts could be anything from the validity of the theory underlying a phenomenon, or the empirical results supposedly shedding light on the topic. A good example would be the debate among economists about the causes of the most recent US recession and the most useful way out of it. On one side are those who think the solution is less government spending to reduce deficits, and simply leaving the economy to painfully sort out major structural mal-investments and imbalances. They believe intervention is likely to create worse problems later. The other side says aggregate demand is the problem and that the recession can and should be fixed via some kind of fiscal and monetary intervention. This side believes intervention will make things better, not worse, in both the long and short term. Both sides claim that the other side’s view has been thoroughly discredited by empirical events in the past, and will point to current events ‘obviously’ supporting their expectations, or when they haven’t will issue dire warnings that it will, soon. A similarly insoluble argument is being had around the ‘facts’ of global warming.

Sometimes the arguments boil down to differences in values. For example, what tastes better chocolate or vanilla ice cream, or who is prettier Jane or Mary? In these cases there isn’t really a correct answer – even when a large majority favors a particular alternative. Values also have a strong way of influencing what people accept as evidence or indeed what they perceive at all.

The unreasonableness of continuing disagreement

The interesting thing is that when the disagreement isn’t a pure values difference it should always be possible to reach agreement. Robert Aumann, a Nobel winning game theorist (who I’ve had the pleasure of chatting to) proved that under conditions of common knowledge it isn’t possible to agree to disagree – even when the parties start with completely different information, facts and theories. In a simplified format it goes like this. Party A says the answer is X. Party B, who considers the answer to be Y, hears this and, if rational, will think “A must either have access to evidence that I don’t or doesn’t have access to some evidence that I do.” Under the notion of evidence I include not only facts but also the reasoning process. B may say that “Notwithstanding the possibility that I may have missed evidence I am still very confident in the evidence that I have so I will say I think the answer is Y”. A now hears B’s answer and goes through a similar chain of reasoning. He thinks “Gosh B has evidence he is so confident about that exposure to the knowledge that I have different evidence hasn’t shaken him. He must think his evidence is particularly strong. I should take that into account when evaluating my own evidence.” At this point A could decide he isn’t all that confident in his own evidence, and concede the argument to B. Alternatively he could decide that in spite of B’s confidence he still considers his own evidence to be persuasive, and re-affirm that he thinks the answer is X.

The ball now passes back to B, who now faced with A’s continued confidence in his evidence, even after making allowances for B’s confidence in his own evidence, must upgrade his view of the strength of A’s evidence relative to his own. He must then decide whether he is still thinks his evidence is strong enough to carry the day. He can decide “No it isn’t”, and concede the argument to A, or “Yes it is” and say he still considers Y to be correct. The process goes on until one of the parties concedes. At any point either party’s actual evidence can, and probably will, be shared and explained. Some readers may recognize this as an iterative Bayesian process. Others have extended Aumann’s analysis, and have shown that the process won’t go to infinity and should come to a conclusion in a reasonable number of iterations. The upshot of this is that if an argument doesn’t result in an agreement, at least one of the parties involved is being irrational or dishonest.

The rest of the article makes the unrealistic assumption that people will be rational and honest when arguing.

IQ and relative correctness

Item response theory connects a latent trait e.g. intelligence or IQ, with the probability getting a particular item in a test correct. Typically they look like this.



The formula producing these lines is

p(X)=exp(a*IQ+b)/(1+exp(a*IQ+b)).

The “a” coefficient tells one how steeply the probability rises as IQ rises i.e. how much solving the item depends on ability as measured by IQ versus how much the solution depends on uncertainty and luck. The “b” coefficient tells one the difficulty level of the item.

Suppose we select two IQ levels and compare the probabilities of a correct answer for each IQ level. With a bit of arithmetic we can show that the ratio of

p(IQ2)/p(IQ1)=exp(a*( IQ2 - IQ1)).

Suppose two people with IQs at level 1 and 2 respectively disagree about the answer. In effect they argue about it. Then the probability of person with being right in the event of a disagreement is

p(2 is right) = p(2)*(1-p(1))/((p(2)*(1-p(1)) + p(1)*(1-p(2))). Similarly for p(1 is right).

More arithmetic gives

p(2 is right)/p(1 is right) = exp(a*( IQ2 - IQ1)) too.

Therefore, when two people argue over the correctness of something, the probability of who is right is determined by the difference between their respective abilities and the degree to which solving that problem actually depends on ability. The difficulty of the item is irrelevant.

A chess diversion

Chess’s ELO rating system uses a similar method to calculate the probability of a player winning, but use base 10 rather than base e. So according to the ELO rating system used by FIDA

p(player a)/p(player b)=10**(Ra – Rb)/400 (which obviously = exp(a*( IQ2 - IQ1)).)
- where Ra is the ELO rating of player a.

This means that if the player’s ratings differ by 200 points then the highest ranked player should take roughly 3 out of every 4 wins between them. A ratings difference of 400 points means 10 wins for the highest ranked player for every win for the lower ranked player. The median rating for members of the US Chess Federation was 657 and rating of 1000 is regarded as a bright beginner. International Grandmasters typically rate 2500+, and the very best players have ratings slightly over 2800. To give you an idea of the differences in skill, consider that if the very best were to play an average player the ratio of wins is likely to be 227772 to 1. The difference between a grandmaster and a good beginner would be 5623 to 1.

The identities above mean that we can easily convert an IQ difference into an equivalent ELO ratings difference, using the following formula

Ratings difference = 173.7178*a*(IQ difference)
- “a” is the coefficient telling us how much the item depends on IQ for its solution.

I looked at the distribution of FIDA ratings in order to convert them to an IQ metric. For example, about 2% of chess players have ratings as 2300 or higher and 0.02% have ratings over 2700. If IQs are normally distributed, with an SD of 16, then these ELO ratings would correspond to 132 and 157 on the IQ scale. Note this doesn’t mean that chess players with a rating of 2700 will actually have an average IQ of 157 – it’s just a different way of specifying the same thing e.g. like a change from the Imperial to the Metric system.

I did however find a study (1) that allowed me to map real IQs onto chess ratings in experienced players. The equation is

Chess rating = 18.75*IQ – 275.

It turns out that the expected real IQs are very close to the IQ metric I calculated from the ratings distribution. (Note that the equation I developed is quite different from the one hypothesized by Jonathan Levitt (3) i.e. Rating = 10*IQ + 1000.) One should also be aware that the equation gives an average IQ – the actual IQs vary quite a bit around the expected figure. For example, the authors show that threshold effects exist and that the minimum IQ needed to achieve a rating of 2000 is around 85-90. This is 30-35 IQ points lower than the expected IQ. Also from his peak rating Garry Kasparov’s expected IQ is 167 (and wild claims of 180+ have been made) but his actual IQ was measured at 135 (in a test sponsored by Stern magazine), some 32 IQ points lower.

I suppose one could derive a rule that the minimum IQ required for a peak rating is some 32 IQ points (or 2 SDs) below the expected IQ. Alternatively, it means that if you have a combination of memory and industry in line with elite professional chess players, your peak rating is likely to be 600 ELO points higher than it would be if you were like an average chess player in these respects. Your chances of winning could be as much as 31.6 fold higher than your IQ suggests – or that much lower. That says something about the relative value of focused application.

Assuming that the distribution of combined effort and memory is symmetrical, it also means that a 64 IQ point advantage can not be overcome - even if the brighter player is also among the very laziest with a bad memory, and the less intelligent player has a superb memory and is among the most dedicated.

Even after accounting for IQ and work the predicted ratings are still a little fuzzy so perhaps random factors play a role too.

IQ and ELO rating differences in other domains

Here I look at converting the effect of IQ gaps to ELO rating differences, across a variety of domains

Let’s get back to the IQ to ELO rating conversion. Recall that the equation is

Ratings difference = 173.7178*a*(IQ difference).

All that remains is to find “a” for everything we are interesting in.

Clear Objective problems

The obvious place to start is IQ test items. The “a” coefficient for more fuzzy IQ test items tend to be around 0.046, and around 0.086 for really efficient IQ test items. That means that for fuzzy items each additional IQ point is worth 8 ELO points, and it’s worth 15 ELO points for good IQ items. If these items are used in a weird tournament where players compete to solve puzzles instead of play games, and we set the bar at a 3 to 1 win ratio (a 200 point ELO rating difference), then fuzzy items will require a 25 point IQ gap, and efficient items a 13-14 point IQ gap.

Physics mastery

Using information from an article by Steve Hsu (2) I worked out that a 3 fold advantage at “winning” at a physics exam – where a ‘win’ is an A in the exam when your opponent failed to get an A – requires an IQ gap of 12 points. If however a win is defined as a 3.5 GPA (where your opponent fails to attain this), then a mere 6 IQ points will provide a 3 fold advantage.

Crime

We could view cops and killers as being involved in a grim contest. In the USA around 65% of all murders are solved. That converts to an average “murder” ELO rating difference between police and murderers of 108 ELO points. It is also known that the mean IQs of murderers and policemen are 87 and 102, respectively. So successfully solving murders is a puzzle then the “a” coefficient is 0.041, and each IQ point difference is worth 7.2 ELO points. A 3 fold advantage could be had with a 28 point gap between cops and killers. In other words some 31% of outstanding murders could be solved if the USA selected its policemen to have an average IQ of 125 i.e. to be as smart as an average lawyer. I’m not sure if that’s worth it but maybe some cost benefit analysis would help. Such an analysis would have to take into account the drop in murder rates (with a life currently being valued at $2 million) due to the greater odds of being caught, and the opportunity cost of taking professional level IQs out of the pool for other professions, where they might be even more productive.

Controversial issues

Finally we get back to real arguments – disagreements over controversial topics. According to my Smart Vote concept (see http://garthzietsman.blogspot.com/2011/10/smart-vote-concept.html), if proportionally more smart people systematically favor an alternative then that alternative is likely to be correct or better. Using that definition of “correct”, and information in the General Social Survey, I calculated the “a” coefficients and ELO to IQ ratios etc for a few controversial questions. Typically it would take an IQ difference of 30-50 points to gain a 3 fold advantage in a rational argument.

Tasks with a high level of uncertainty

For comparison I looked at an intellectual game that includes a large element of chance – backgammon. A rating equation gives

p(player a)/p(player b) = 10**Rating diff/2000

for individual games. The difference from chess is that it divides the rating difference by 2000 instead of 400. It would take the result of a string of 21-22 games to provide the same test of relative skill as does a single game of chess. Controversial questions are less fuzzy, less uncertain, or more tractable to intelligence, than backgammon. If people ‘played’ a series of arguments over controversial questions instead of a series of backgammon games, then it would take maybe 5-6 such arguments to provide the same test of relative skill/wisdom in arguments, as a single game does in chess.

A summary table



Some meaningful IQ or ELO rating differences

Some research shows that friends and spouses have an average IQ difference of 12 points, that for IQ differences less than 20 points a reciprocal intellectual relationship is the rule, for IQ differences between 20-30 points the intellectual relationship tends to be one way, and that IQ differences greater than 30 points tend to create real barriers to communication.

An IQ gap of 12 points implies a roughly 67-72% chance of winning an argument over a clear objective issue, like verbal or math problems, and close to a 57-61% chance of winning an argument over a controversial question. A 30 point IQ gap implies an 87-91% chance of winning a verbal/math item argument, and a 63-67% chance of winning an argument over controversial issues. It seems as though there is a very fine line between intimacy and incomprehension on controversial issues (a mere 6% difference) but a fair gap on more objective issues (a 20% difference).

Perhaps it isn’t so much the IQ gap that matters to people, as the proportion of differences of opinion they win or lose i.e. the ELO rating difference. That in turn depends on the balance of clear objective, uncertain, and controversial issues in their disagreements. In general however, it seems that people don’t like to lose more than 2 in 3 disagreements, and when they lose more than 3 in 4 of them they feel like they aren’t on the same planet anymore. Those proportions correspond to ELO ratings differences of 100 and 200 respectively. An ELO rating difference of less than 100 feels tolerable and reciprocal while a difference of more than 200 feels unfair or unbalanced. If that theory is right, when most of the issues are fuzzy or uncertain the larger IQ differences should occur between friends and spouses, but when they are mostly clear objective issues then those IQ differences will be smaller.

References

1. Individual differences in chess expertise: A psychometric investigation. Roland H Grabner, Elsbeth Stern & Aljoscha C Neubauer. Acta psychological (2006) but I got it from www.sciencedirect.com.
2. Non-linear psychometric Thresholds for Physics and Mathematics. Steven D.H. Hsu & James Shombert. http://arxiv.org/abs/1011.0663
3. Genius in Chess. Jonathan Levitt.

26 comments:

  1. Garth do you have references for Erdos being confused by the monty-hall problem? I've heard this before but simply can't bring myself to believe it. Once you try to think how you'd simulate it it really is not that hard a problem.

    Also arguments and chess both rely on things things that have (almost) nothing at all to do with IQ. Most notably specific knowledge. What is Kasparov's go rating?

    Finally alot of arguments I've had have ended in agreement on a position entirely different to either parties start position.

    ReplyDelete
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    1. The Monty Hall problem is often mis-stated when posed in conversation. The standard solution (switch doors, win 2/3 of the time) depends on Monty Hall's having pre-committed to open a door, but never to open the winning door.

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    2. Well yes. It also depends on Monty being equally likely to open either wrong door in the case where you got it right (as oppossed to say, you knowing he opens the door with the lowest number he can)

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  2. Jonathan: the Erdos thing apparently is sourced to https://en.wikipedia.org/wiki/The_Man_Who_Loved_Only_Numbers with some excerpts in http://archive.vector.org.uk/art10011640

    Garth:

    > Finally we get back to real arguments – disagreements over controversial topics. According to my Smart Vote concept (see http://garthzietsman.blogspot.com/2011/10/smart-vote-concept.html), if proportionally more smart people systematically favor an alternative then that alternative is likely to be correct or better. Using that definition of “correct”, and information in the General Social Survey, I calculated the “a” coefficients and ELO to IQ ratios etc for a few controversial questions. Typically it would take an IQ difference of 30-50 points to gain a 3 fold advantage in a rational argument.

    Problem is, you have inappropriately generalized from IQ test questions to 'rational argument'. Not only is this a stretch, but we have a lot of evidence that it is *not* true.

    There are tons of fallacies in inductive and deductive logic that intelligence helps little or not at all with! I've examined the sunk cost fallacy in some detail (http://www.gwern.net/Sunk%20cost) as one of the irrational arguments that intelligence does not save one from, and there are literally dozens of others: Stanovich's 2010 book _Rationality and the Reflective Mind_ cites them and discusses the issues at length as he develops a process-1/process-2 architecture. There was one fallacy, IIRC, where intelligence was even a slight penalty to correctness.

    (Very brief summary: IQ tests only measure an ability to abstractly 'simulate' problems and apply learned strategies, but half the battle in life is knowing when to use them in the first place and 'override' a quick superficial understanding. Hence, someone can have a high IQ and routinely commit fallacies, and they do just that.)

    On a side note:

    > Scored an IQ of 185 on the Mega27 and has a degree in psychology and statistics and 25 years experience in psychometrics and statistics. He is available to do any stats analysis you might profit from.

    This is a really awful capsule biography.

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    1. By 'rational argument' I meant the mythical case where people are fully open to evidence and admit when they are wrong.

      In cases like the sunk cost fallacy high IQ people usually make mistakes like everyone else. The point however is that on average they make them less often. Typically you see stuff like 70% of a smart group get it wrong but 90% of the dull group do. That trend, and not the % correct, is what points to the better answer.

      I have a big collection of common logical mistakes people make (gleaned from Tversky & Kahneman mostly, but also many others). On all those I've tried so far - including the sunk cost fallacy - the brighter group does quite a bit better, even though most still get them wrong. Contrary to what you are saying here the Smart Vote does particularly well on such fallacies.

      Thanks for the references. I will look them up.

      I would love to chat to you more about these fallacies.

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    2. Oh I should also note that when there is no systematic relationship between IQ and opinion then intelligence is of no help with that problem. I do not claim that high IQ can answer all questions.

      My claim is rather that when there is a systematic IQ related difference of opinion, that cannot be explained by bias, then there exists a better answer and it is the one (or those) favored more often by the brighter group - not necessarily by the majority of bright people.

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  3. "the probability of who is right is determined by the difference between their respective abilities and the degree to which solving that problem actually depends on ability. The difficulty of the item is irrelevant."

    This is misleading. In general, easy problems are less dependent on ability than hard problems.(until you get to problems that are so hard no one can answer them). everyone gets 1+1 right.

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    1. The probability of an individual solving the problem depends on the item difficulty as well as its correlation with ability but who's right when 2 people disagree depends only on the correlation.

      More accurately is depends on the gap between the IQ below which no one can solve it and the IQ above which everyone can. The size of this gap is not necessarily related to item difficulty or even correlated (once complexity is equated).

      Delete
  4. As a FIDE 2400+ player, I take issue with the statement that one can easily fact-check who won a certain football match in 1966. The entire Engineering School of my old university (Oxford) attempted that, and concluded that England failed to win that match by 6 centimeters, which is not even 2.5 bloody inches! What with the quality of the video in those days, you can't even really call that a fact-check, now can you?

    :-) aside, I may be able to test some of your assertions about differences in Elo...

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    1. I concede your point. I'll have to use a different example in future.

      I would appreciate some collaboration on ELO differences - especially by someone with a decent ELO rating.

      I once asked a former South African chess champion how it felt to play against people with ratings 100, 200, 300 or 400 points above and below him. Unfortunately I didn't get a helpful answer.

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    2. Sure---if I can. One problem I've already discovered is that it's hard to get a good sample of games between players 200 or more Elo points apart. International round-robin tournaments tend not to have that reat a rating difference. Open/Swiss tournaments have more variable conditions. I took a small sample but got strange results---and if you look at my papers, the error bars are already pretty sizable on my good-sized training sets.

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    3. Hi Kenneth

      Your papers are fascinating. Have you applied any of these intrinsic metrics to Magnus Carlson's recent games?

      When I asked about how it feels to play against someone a certain number of Elo points above or below yourself I had in mind some kind of psychological measure like
      8 totally overwhelmed
      6 thoroughly intimidated, feel utterly foolish and incompetent
      4 definitely feel outgunned but feel I can last a fair time
      2 feel opponents advantage but it is so slight that feel random events will play a big role
      0 matches could go any way
      and a similar scale for feelings while being the superior player.

      I wanted to be able to map ELO ratings differences to such a scale. The idea is to get some idea of what it feels like to have a difference of opinion across a known IQ gap on problems with specific g (general intelligence factor) loadings.

      By the way I figure discussions will be better by email and tried to email you using the email listed on your site. I got a "no such domain exists message". Perhaps you could email me at garth dot zietsman at g mail dot com

      Delete
  5. If you're so smart, why do your charts all look like crap? The ones about IQ and crime had horrible kerning, and this one has some horrible pixelly resize thing going on.

    ReplyDelete
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    1. No matter what you do with typography, it will take 20 hours to get right, the software will be broken within the year, and someone will tell you that you are doing it wrong anyway.

      Delete
    2. To tell the truth I'm fairly ignorant when it comes to computers. It doesn't work when I try to copy Excel or Word Tables into blogspot but since it does allow images I've tried to convert tables into images. I'm not happy with the results either and would appreciate advice.

      Anyway thanks for the feedback.

      Delete
    3. This comment has been removed by the author.

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    4. @Garth:

      From your comment and the appearance of your images, I'm guessing you're converting tables to images using commands built into Excel or Word. If the tables originally look better when viewed on-screen in Excel or Word (perhaps using Print View), you'll probably get much better results with a simple cropped screenshot of the on-screen table.

      There are numerous free screenshot utilities to accomplish this, but the one I prefer is MWSnap (h++p://www.mirekw.com/winfreeware/mwsnap.html). It hasn't been updated since 2003, has a few minor bugs, and doesn't have any annotation features, but it still works well with WinXP_Vista_7. For your purposes, snapping screenshots using the "Any rect. area" button on the MWSnap GUI will probably work best (after clicking the button, click on any corner of the desired-size rectangle and then click on the diagonally opposite corner).

      For tables, charts, and other graphics, you'll probably get the best results saving screenshot images as PNG files, instead of JPG or GIF files. (JPG files are generally better for the color gradations found in photos, and GIF files are often visually degraded.)

      Hope that is helpful. ;)

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    5. @Garth:

      ...Unless the poor text image quality is due to screenshots taken on an old CRT monitor instead of a higher resolution LCD monitor... ...if that's the case, I'm not sure what to suggest. ;)

      Delete
  6. Have you seen:

    Argumentative Theory of Reasoning
    http://edge.org/conversation/the-argumentative-theory

    about this (fairly recent) paper:

    Why Do Humans Reason? Arguments for an Argumentative Theory
    Hugo Mercier and Dan Sperber
    http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1698090

    In general for controversial issues, humans first decide what
    they want to be true and then find arguments to support their
    side. Intelligence can actually make things worse because smarter
    people are better at manipulating facts to support their viewpoint.

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    1. Indeed that's true. For that reason any INDIVIDUAL smart person's opinion is suspect. Also particularly suspect are views that have been formed in a group context because of the greater social incentive to win the contest rather than find the truth. Assuming that such biases are randomly distributed independent judgments, aggregated over a wide range of people should get past those problems.

      If that is true then a significant difference in the aggregated independent opinions of a very smart and a very dull group should boil down to a whatever defines the groups. One of these differences is the ability to find correct answers. Others are of course systematic differences in interests in the outcome. The Smart Vote is valid to the extent that these non-cognitive interests are accounted for.

      Thanks for the links.

      Delete
  7. Looking at round robin tournaments surely will get odd results. I played tournament chess in HS. The best player in our school could almost always beat everyone else. The 2, 3, and 4 players could almost always beat #5 and below, and the #5 could almost always beat anyone below him. Herewith the problem- Call 2, 3 and 4 A, B, and C, because we can't figure out who best among them. I was A. I could ALWAYS beat B, who could ALWAYS beat C, who could ALWAYS beat me. Different styles of playing, different thought processes.

    I've seen the idea of A beating B beating C beating A used to illustrate absurdity before. And anytime I see it used as such, I bring this up. I've seen it in other one-on-one activities, also, between people of almost equal skill. Tennis and racquetball. Fencing. Because I've seen and been part of such a triad, I don't find them unlikely at all. Ratings systems probably have a tough time with such results...

    The top 4 of us all started entering USCF tournaments. During the first tournament, I won the first round, against a 1600 rating, and drew the second game, against an over 2000 rating. He showed me how I could have forced a mate after accepting my offer of a draw. At the end of the first tournament, I had a provisional rating higher then that of the other 3. Didn't last long; the law of averages showed up, and I settled at at about 1700, where B and C also settled out. The #1? Currently rated well over 2000- he still plays tournament chess. 39 years later.

    The median rating is only 657? Wow.... Didn't realize until now I was actually a good player.

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  8. I discovered your blog yesterday from a reference to the SmartVote on lesswrong.com. Wow. Amazing stuff so far.

    One minor correction: on your table above you have an item you call: "reduce taxes rather than reduce social spending" I'm guessing that should be "increase taxes rather than reduce social spending"?

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  9. "The Monty Hall problem is often mis-stated when posed in conversation. The standard solution (switch doors, win 2/3 of the time) depends on Monty Hall's having pre-committed to open a door, but never to open the winning door."

    Garth, this, of course, is nonsense. Look at it this way. There you are. You chose one door, one door is open with the booby prize and one door you didn't choose is left closed and available to you. Will it benefit you to switch? Well that depends on whether Monty Hall knew there was a booby prize behind the door when he opened it. It does? How could what Monty Hall knew or didn't know affect what is behind a door? It can't, of course. The problem is, and this is common with math guys, that they assume that the truth of an individual event is determined by the truth of a large number of essentially identical events. It's not.

    A wonderful example of this difference is the St. Petersburg Paradox. If one evaluates the problem from the perspective of an essentially endless number of rounds, one ought to be willing to pay just about any amount of money for the right to play a round. The return, mathematically, is infinite. Yet in reality, when give ONE opportunity to play, people rarely are willing to pay more than $1.

    There is another way of looking at this. That is the mathematicians are wrong in evaluating the rational choice based upon the set of all games played if we assume that Monte Hall does not know what is behind the door. If we assume that, there is a probability distribution of what will be behind the door that Monty opens. But that probability distribution has collapsed in this case. We should only be evaluating the results of those cases where Monty chose a booby prize. This is not only correct, it resolves the rather odd conclusion that what Monty did or did not know affects the rational choice.

    As I think that the Monty Hall problem itself is well g-loaded and might be generally diagnosstic of about 130 D15IQ, understanding why the rational expectation is not affected by what Monty Hall did or did not know is diagnostic of something North of 160.

    What is your take?

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  10. Make that $7. Also, I see that the actual result is 8% which would be a D15IQ of 121. Perhaps then the more difficult understanding of the second point is north of 150.

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