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TABLE OF CONTENTS - PROSEPascal's Wager and the Paradox of Kraitchik
- Albert Frank Pascal's Wager and the Paradox of Kraitchik - Albert Frank Recently, I was looking at the famous Pascal’s Wager. Original text: Examinons donc ce point, et disons : Dieu est ou il n'est pas ; mais
de quel côté pencherons-nous ? La raison n'y peut rien déterminer.
Il y a un chaos infini qui nous sépare. Il se joue un jeu à
l'extrémité de cette distance infinie, où il arrivera
croix ou pile. Que gagerez-vous ? Par raison, vous ne pouvez faire ni
l'un ni l'autre ; par raison, vous ne pouvez défendre nul des deux. Mais je gage peut-être trop. Voyons : puis qu'il y a pareil hasard
de gain et de perte, quand vous n'auriez que deux vies à gagner
pour une, vous pourriez encore gager. Et s'il y en avait dix à
gagner, vous seriez bien imprudent de ne pas hasarder votre vie pour en
gagner dix à un jeu où il y a pareil hasard de perte et
de gain. Translation : "God is, or He is not." But to which side shall we incline?
Reason can decide nothing here. There is an infinite chaos which separated
us. A game is being played at the extremity of this infinite distance
where heads or tails will turn up... Which will you choose then? Let us
see. Since you must choose, let us see which interests you least. You
have two things to lose, the true and the good; and two things to stake,
your reason and your will, you knowledge and your happiness; and your
nature has two things to shun, error and misery. Your reason is no more
shocked in choosing one rather than the other, since you must of necessity
choose... But your happiness? Let us weigh the gain and the loss in wagering
that God is... If you gain, you gain all; if you lose, you lose nothing.
Wager, then, without hesitation that He is. Here, I have to present Kraïtchik’s Paradox: More than twenty years ago, I read the book “La mathématique des jeux” of Maurice Kraitchik. ( First edition: Imprimerie Stevens, Bruxelles, 1930; Second edition - which I have -: Editions techniques et scientifiques, Bruxelles, 1953). It’s a fascinating book, with a lot of mathematical puzzles, considerations on magic squares, geometrical curiosities, … Who was Maurice Kraitchik (1882 – 1957)? He was a Belgian mathematician (born in Russia) whose primary interests were the theory of numbers and recreational mathematics, on both subjects of which he published a lot. He wrote several books on number theory (1922-1930, and after the war), and was the editor of the periodical Sphinx (1931-1939), which was devoted to recreational mathematics. During World War II, Kraïtchik emigrated to the United States, where he taught a course at the New School for Social Research in New York City on the general topic of "mathematical recreations." Kraïtchik was « agrégé » of the free University of Brussels, engineer at the “Société Financière de Transports et d'Entreprises Industrielles (Sofina)”, and director of the “Institut des Hautes Etudes de Belgique”. Among his books, let’s mention: Kraïtchik, M. Théorie des Nombres. Paris: Gauthier-Villars,
1922. In “La mathématique des jeux”, I considered during years one of the paradox he presents (page 133): “Deux personnes, également « riches » conviennent de comparer les contenus de leurs porte-monnaies. Chacun ignore les contenus des deux porte-monnaies. Le jeu consiste en ceci : Celui qui a le moins d’argent reçoit le contenu du porte-monnaie de l’autre. (au cas où les montants sont égaux, il ne se passe rien). Un des deux hommes peut penser : « Admettons que j’ai un montant de A$ dans mon porte-monnaie. C’est le maximum que je peux perdre. Si je gagne (probabilité 0.5), le montant final en ma possession sera supérieur à 2A. Donc le jeu m’est favorable…l’autre homme fait exactement le même raisonnement. Bien entendu, vu la symétrie, le jeu est équilibré. Où est la faute dans le raisonnement de chaque homme ? » Two people, equally "rich" put their wallets on the table. Both don’t know the amounts of money of each wallet. The game is : "the man who has the less money receives the money from the other" (if they have the same amount, nothing happens). One of the men may think : "I know I have an amount of A$ in my wallet. That's the maximum I can lose. If I win (probability 0.5), my final amount of money will be greater than 2A. So the game is in my favor"...the other man thinks exactly the same. Of course, because of symmetry, the game is equilibrated. What is wrong with the reasoning of the two men? I noticed that Martin Gardner, in “La magie des paradoxes” (Bibliothèque POUR LA SCIENCE - Diffusion Belin, extracts of Scientific American, 1975), page 114, gives the same problem, asking for an answer (« I was not able to solve it »). Martin Gardner (born in 1914) was the Mathematical Games columnist for
Scientific American. He originated the column in 1956, and his columns
appeared until his retirement from the magazine in 1986. He graduated
Phi Beta Kappa from the University of Chicago in 1936. Some of his mathematical titles (published by several editors): In the first months of 2000, I put this paradox (which, afterwards, was called “Kraitchik’s paradox”, a “name “ never used by Maurice Kraitchik!) in several magazines and on several lists. Marc Heremans did the same. As a result, we got more than 50 answers! Most of them did not answer to anything, or were very poor. Finally, two articles came, giving finally what I consider to be “The Solution”: One from Marc Heremans, and one from Erik Goolaerts. Also, Chris Langan wrote an interesting solution on: http://www.megafoundation.org/Ubiquity/Paradox.html Here is the solution founded by Marc Heremans: Paradox, antinomy or sophism, I don’t know which term best describes this statement. Still, it generates the simultaneous feeling of admiration and incredulity, close to the one that one feels when a devious lawyer misleads his public while pleading brilliantly an already lost cause. We have the conviction of having been fooled, certainly, but the tracks are covered so finely that it is difficult for us to unmask the deception. The attempts to resolve the paradox, which call on the general laws of logic and simple “common sense”, are shown to be useless because they confirm a logical impossibility of which we are perfectly conscious but do not tell us where the error lies. Let’s try to understand why the reasoning is not correct. A visual representation in the form of a matrix will help. ? The amounts of player A (a1, a2,….,an ) can be seen in the left
column and the amounts of player B (b1, b2,…,bn) are shown in the
top row 0 1 2 3 4 Total Remarks about the matrix: Notice (this will become important later) that the matrix is either symmetric or not, depending on the way you look at it. If one considers the amounts shown on either side of the diagonal made up of the zero wins/losses, one can note that the matrix is perfectly symmetrical, each positive value being matched with an equivalent negative value. All the mathematical expectations are complementary and cancel each other out. In half of the cases, the “game” is favourable for A; in the other half, it is favourable for B. The number of winning positions is the same as the number of losing ones and the losing amounts are equal to the winning ones. A second approach consists of no longer looking at all the possible occurrences, as above, but to regroup the data, taking into consideration the regression of the “ai” on the bi For each occurrence of “a”, we associate its average expectation of gain. Therefore, we are interested, in order of priority, in the first (a1,…,an) and the last column (E1,...,En). Seen from this angle, the matrix is no longer symmetrical. The expected values vary greatly from one amount ai to another. The number of winning positions is even superior to the number of losing ones! On the other hand, the amounts of the losses are greater than the amounts of the gains. We can note that the mathematical expectation of A is clearly positive when he holds an “average” amount! The reasoning proposed by A (refer to Albert Frank's previous article): “if I win – probability 0.5 – the final amount in my possession will be greater than 2A,” shows itself to be correct for the specific case of an average amount, but can not be generalised. Actually, the fact of winning on average when you hold an average amount does not at all mean to win “on average” in all possible cases. That would be to ignore the totally asymmetrical shape of the distribution of wins and losses around the average amount. In extreme situations, the wins and losses are not balanced. A loses much more when he is in possession of a high amount than he would win when he owns a small amount. Conclusion A’s error consists of reasoning that does not take into account alternative groupings of the data he uses. He goes from one grouping representation to another, transposing surreptitiously conclusions that could be drawn from the other. By itself, no grouping is “better” than any other, but if one accepts a reading of the matrix based on the second approach (asymmetrical), it is necessary to take into consideration the unequal values of the expectations resulting from the grouping of the amounts that were used in that line of reasoning. And it is quite easy to see that Pascal’s wager and Kraïtchik’s
paradox are nearly the same, with a totally similar structure. This is a *very* good example of how a problem can be sometimes solved because there is an isomorphism with another problem that has been solved before." In the same way that it has been demonstrated, in the Kraïtchik's
paradox, that when one of the players says "it is in my favour",
he is wrong (and the bet is equilibrated), we can say that Pascal was
wrong when he said " you must wager for the existence of god"
- He made the same mistake when looking at the expectation - and the bet
is equilibrated (you don't loose(or win) more - if you loose - betting
against his wager – To Grope or to Reason - Albert Frank The approach of a lot of problems can be made of several manners: let's keep here some possible approaches: - The groping. It is a frequent method, that sometimes gives results (if one has luck). - The use of knowledge (theorems, known previous analogous results). - The" brutal strength": if the number of possibilities is limited, to examine, with the help of a computer, all cases. - The approach by reasoning. For every problem, to examine WHAT can BE USEFUL TO ITS RESOLUTION . Let's give here two examples of problems that can be solved by groping, by" brutal strength ", or - and in a pure and aesthetic manner - by reasoning. Problem 1: . Four miners are at the same place, inside a mine. The mine is going to explode in precisely one hour. They don't move to the same speed, and are situated respectively to 5, 10, 20 and 25 minutes of the unique exit. They cannot move without light and they have only one lamp. Besides, they can move maximum simultaneously by two at a time (then the speed is the one of the slowest) . How are they going all to run away? The solution can be found by groping, or by examining all cases (they are not very numerous). It is a lot simpler to make the following reasoning: Let's call the miners 5, 10, 20 and 25. To get out of the mine, it is necessary that the two slower travel together - but they cannot leave together the first, it would be too long. Therefore :1°. 5 and 10 leave together, and 5 (or 10) brings back the lamp. 2°. 20 and 25 leave. 3°. 10 (or 5) returns to look for 5 (or 10). Total time: 15 + 25 + 20 min. (or 20 + 25 + 15 min.) = 1 hour. Problem 2: With matches, form the following motive:
The solution can be found by groping. It generally takes a long time. What reasoning can one make to manage immediately to solve the problem? Let's COUNT the matches: There are sixteen of them. As we have to get four squares, they will be therefore necessarily disconnected (no side common to several squares). It gives us the solution:
- _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ - _ The Truth is out There. The Truth is in Here. At times of course the Truth is merely Asinine. Interlocutor: Mr Nachbar, could you please explain exactly what you mean by these remarks? I'm not really clear here and we would like to know whether you meant..or you meant.Or whether you were referring to..Or quoting..Or was it something that happened to you?We would like to know.I mean your feelings and experiences here.. Oh dont' tell us that.But? Or did we do something? Or mmm .We would like to know..You're being...and you're being..hmm.I don't get it..well..Very good but really vague..and we would like to know..damn...like what do you mean by Here and There and does asinine really mean what it sounds like and are you calling me an ass?? I don't think you would, but it's not nice if you were but..Don't get angry but.. (pats head of interlocutor and smiles somewhat sadly and nervously yet with vague paternail condescension).Not angry...Well, one day, my boy, all of THIS will be yours..if you can handle it..Gotta go, lad..business appointment, yknow.Duty calls? Ah, like the army.lol.wish things were different..but..see you later alligator.. Paul It is everywhere - What is? That magical stuff - Sorry, don't know what you mean - and please get out of my head! Why must you be so hostile? I must be insane! Who are you? A visitor. Just think of me as a visitor - Please leave, now - visitor. You should share the wonder - What wonder? The magical stuff - What are you talking about? It is all around you ... and in you - Where did you come from; why do you speak in riddles? Ahhh, it is truly that, now, isn't it - a riddle. The source? Yes. It must be this world - Closer? Always were a few molecules here and there, a strange curiosity. Then we found your solar system. Thought the source was...you call it mars; quite a bit there. But nothing like here. There's so much here. This must be the source. Doesn't matter if it is the source or not, though. So much...and it binds all together. Binds all together? Yes. We are not bound, as you are. It is a wonder to see through your eyes. To hear, to taste, to smell and touch. It must be a strange gift to be locked inside with all this power around you. Power? What power? The magical stuff - colorless, flowing, binder of all you know of life. Laced with the power of the universe. Power... Yes, power. You call it - Water. Luftmensch Liberation Front - Paul Nachbar god.. i'm so desponding.. i never have any fun anymore... i'm all wrapped up in my existential haze... and i can't seem to escape it... and its' beginning to smother me.. smother every bit of personality, of spontaneity, of carelessness out of me... every bit humanity... a humanity which is now being supplanted by disposition of a luftmensch... just a airly contemplative fool... who can't experience things for what they are... but has to discern into the deep meaning of everything around him.. abstracting reality into lofty philosophical notions... "(private confessions of a luftmensch (anonymous--and not me,damnit!) It has come to our attention that there exists within us a large minority which has received much in the form of institutional, social, political and media discrimination, direct and indirect: the Luftmenschen-- crude translations in English: nerds, airheads, space cadets,eggheads, artistes and geeks etc. Folks not exactly "grounded" or "down to earth"-Luftmensch literally meaning "man of the air" Luftmenschen appear in all religious, gender, racial, ethnic, political, philosophical, income,professional,class, weight height, age, sociological, gender and EQ,IQ and personality categories.Such individuals, who have not yet realized their individual or common Luftmenschness, often repress or sublimate their true identities, go into states of denial or self-hatred,and attempt to negate these unpleasant though apparant "facts" about themselves through overcompensation and overachievement.They also quite frequently lead lives in mainstream society, which are filled with a type of quiet despair, often hidden behind apparant "success"We at the Luftmensch Liberation Front, propose to change such deleterious conditions both on the plateau of individual lives and of society as a whole:the traditional misrepresentation and persecution of the Luftmensch. Are you embroiled in the labyrinth of contemporary institutions of therapy and self-help doctrines or other Ideas and Ideologies with no clear solution in sight? Are you lonely, filled with contradictory feelings about self-worth and the meaning and purpose or value of everything around you? Do you flounder or bounce from one thing to the next--or into walls- or become lost in thought? Do you specialize in self-deprecating humor or irony?Do you beat yourself up into functioning in a world where you feel you do not quite..belong? Have you diagnosed yourself with a dozen different mental disorders, none of which quite fit your true self? Do you find yourself incapable of solving the "normal" life problems, including obtaining a "reasonable degree' of happiness that others your age --though possibly more 'superficial' or 'less intelligent' or 'less sensitive' or more 'limited' seem to solve with almost instinctual readiness? Do you seem to march to your own drummer but find that very few people can follow you or understand what you really are saying unless you play "their" roles, "compromise" and "conform"? No worries here: these are probably the signs of Luftmenschness and not
social or psychological pathology or deviations from any tradition, set
of values or some correct life path.Hey, those who really had those traditions,
values and correct life path ideas probably got there without much commentary
or analysis, didn't they? That is, you are not a deviant but a VARIANT
upon human nature.And in most cases, a variant with extremely high levels
of human potential, if treated with the proper respect care and support.So
BE a Luftmensch, friend, be YOURSELF and be PROUD of it. We are free,
we are many.. Palu NACKBAR,chiarman Lessons for Luftmenschen - Paul Nachbar There may be nothing new under the sun, but there are always plenty of old things to complain about. Genius may be one percent inspiration and ninety-nine percent perspiration but who really cares? Early to bed, early to rise, makes a man healthy, wealthy and dull. Seek and ye shall find--it's probably under the couch. She told me I was digging my own grave, but I said, with appropriate deference, that I had actually had a lot of help and did not want to hog all the credit. Humor is often the last preventitive measure against jumping out the window. Yes, I know, why should such things always be prevented? The sad truth to life is that most things are not good or bad but merely overrated in terms of their goodness or badness. Nietzsche once said that thoughts of suicide kept him warm on cold winter
nights. If we knew what we were doing, almost none of us would be doing it. This probably includes the people who tell us what to do. Love is a many-splendored thing, but also often vastly overrated. There is nothing like the warm and nostalgic memory of a wonderful and irreplaceable dog one once had long ago. It brings tears to one's eyes, especially since one no longer has to feed it, groom it, take it to the vet or clean up after it. I would do truly evil and horrible things but I dont' have the energy. It may take fewer facial muscles to smile than it does to frown, but falling asleep on the job is easy. Secretly, I am very proud of having been half asleep during some of what the Chinese call "interesting times". All sorts of dramatic things happened to them, much to tell about, but they know that deep down I had the right idea. Life gets better after forty. I shall tell myself that until I reach fifty. Then I will think of some other comforting lie I can easily believe. Have you ever looked with wonder at the miraculous activity and organization in a beehive or ant-hill and then yawned and chuckled with the superior satisfaction that you, naturally, are not a mere bee or ant or other lowly creature? Or at least imagine you are not.. And then walked away, desperately trying to think of something else. I wouldn't mind if computers did nearly all the work. As long as they could be programmed to have hurt feelings when I yelled af them. Shall I compare thee to a summer's day? Well, yes I could, but I could
also compare Usually sour and bitter or negative or pessimistic thoughts mean absolutely
nothing.and just pop up whether there is or is not a "reason"
for them. Is all that humanity really aspired to during all these centuries basically found in any mall or supermarket? If so I shall remain on my couch and let somebody else do the shopping. One needs some optimists around. Most of them are either not very bright, brain dead or totally hypocritical but they keep the rest of us going. If somebody is or seems happy for apparantly good reasons, there must be something to it. Love is often a contract between two people who annoy each other in ways
that Merit and rewards: it takes far more intellect to write a good sonnet
than it does, Truth usually means, in practical terms, what an individual can live with. But here it is sort of like shoes sizes: one size does not fit all. Hey, there have been a lot of wars over this, no? Living well, my old therapist used to scold me, is the best revenge.Well sometimes living well means doing absolutely nothing. Which would of course put the therapists out of work...oh my. Everybody always has some noble intentions, despite some modern social theories. which say the contrary.Sometimes a bunch of people with noble intentions form a critical mass and something important happens, which often causes a lot of problems. The commonsense approach means squashing these intentions before anything bad happens then perhaps sobbing "what have I done? It was all an ACCIDENT?" I know there are a lot of crazy people but is there really such a thing as mental health? Oops, I'm sorry, that question was bad for business....please continue as you were. The weird truth about winning and losing is that losing is sometimes more exciting and advantageous and that winning is often just shooting fish in a barrel. Emphasis here of course on sometimes.. DId you ever look at a tank of fish or lobster in a restaurant and wonder if THEY were thinking that they were in the process of a job application? When in Rome live like the Romans. Duh. What else could one live like here, a pineapple? The amazing thing about cats is that although their IQs are estimated to be only around 4 points, they use all of those points SO WELL...Makes one think. I guess despite all my whining and moaning and dramatizing I am really a happy man. If I had to imagine myself coming back in another reincarnation, I would just imagine myself still laying on my couch, looking suspiciously at the world from beneath my blanket, wondering what was happening here Oh one change: it would be nice if my girlfriend would lose some weight in this reincarnation..though of course I never said that.Of course in her version of this reincarnation, she would be laying on the couch pondering and I would be doing the dishes. The main advantage of a big vocabulary is that everybody else knows one has a big vocabulary. The main disadvantage of having a big vocabulary is that everybody also knows it. What are television sets and radios REALLY for, anyway? Sometimes it seems that the only real law of marketing anything is that whatever or whoever makes the most noise makes the most sales. How shameful here, or am I just a luftmensch? Power corrupts and absolute power corrupts absolutely. Sounds good. This was either said by somebody who was deeply envious or completely manipulative. Intelligence usually consists of playing games with the obvious.For good or bad of course. Ah, necessity...it forces us into doing something at least. Well, sometimes it is the mother merely of SPAM. Being able to hold certain contradictory ideas in my mind at once--like Twinkle Twinkle Little Star and the fact that those stars are really huge thermonuclear bombs seems...somewhat difficult.. More Lessons for Luftmenschen - Paul Nachbar My boss Barbara informed me rather kindly, from her point of view, that I was a "valuable commodity". I smiled and kind of fumed to myself here about the use of the word commodity, thanked her and walked away. A genuine smile crossed my face, though, when I imagined her being gored by a wild mastadon charging through her office. She is a very nice woman, but after the mastadon hit her....nothing was left. Poor dear. He who believes honesty is the best policy has never worked for an agency nor can last very long working for one. I think an extremely restrictive rule I made in my own club has intimidated and silenced people's spontaneous participation. I shall endeavor to change here.which of course is very difficult for anybody to do. From now on, unsolicited assassination attempts made against me are perfectly acceptable. HEY, THERE, not HERE. A fool and his honey are also soon parted. People generally imagine that major cities like New York and Paris are morally bankrupt, wicked,corrupt, degenerate and awful places where society is hopelessly divided into disgusting wars of group against group and dog against dog...and where all genuine values and traditions are scoffed at and worth nothing. Which is why, I guess, everybody wants to go there. If they don't shoot artists or drive them completely insane, drive them
out of the country or keep them permanently locked up,. eventually they
usually I am, alas, an honest man. But if I had only wanted to help people, I would have become a doctor.Uh.. How many poets does it take to change a lightbulb? None. They let somebody else do it; why bother? What would Shakespeare think of the thousands of scholars who made a living off of works he wrote hundreds of years before in fits of passion? Probably nothing. He has been dead for a very long time. Over a hundred years ago they used to say "Go west, young man!" That is for pursuing one's fortune. I guess they say now "Go up?" Does anybody really need another product? If only somebody REALLY knew what the meaningful and meaningless parts of conversation really were.. Yes the road to hell is probably paved with good intentions and all of my intentions are good and thus probably pave a lot of roads to hell and therefore I will spend today ALSO on my couch. Whoever invented clocks should have been shot a long time ago. What?
Too late? There is actually plenty of inteligent life in the universe, but only a small portion of that manages to do anything with this inteligence which is vaguely interesting to anybody else. If we didnt' have all these rules and structures..then I guess we wouldn't...have
all these rules and structures. And THEN what would happen? Hell if I
know. Do you know? I sure don't.Who came up with them anyway? I don't
know. Do you? No... Sometimes genuine wealth means being able to take a nap when one wants to. Zzzzzzzz. Why is it, even with my fondest and most sincere and positive intentions, so difficult to write a decent poems about banks, the stock market, tax law or commodities markets? Really I try...I'm a Realist too. If God only knew this...well he probably does but it bores Him silly. |
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