Why 'common knowledge' is uncommon nonsense

Alexander Pope, in 'Essay on Man' asks us the question-

Is the great chain, that draws all to agree,

And drawn supports, upheld by God, or thee?

Pope, whom the Germans considered a philosopher rather than, as we do, a gifted versifier, was thought to be expressing the Newtonian- or, later, Kantian- orthodoxy re. the existence of 'naturality' (non arbitrariness) or 'categoricity (having a unique model) such that Nature and God were as but the same Book in which all might read and, save by reason of stupidity or a myopic type of self-love, arrive at the same coordinating or prescriptive conclusions.

This was not Pope's own view. God allots each creature a place in His plan. The 'untutored Indian' is condemned to no inferior fate (though 'Christian thirst for gold' may claim otherwise) if the humble Heaven granted him is one his dog is welcome to follow him into.

But, this is what happens when Yuddhishtra- the 'Just King' who must learn Statistical Game Theory to overcome 'vishada' (mental illness)- gets to Heaven but refuses to enter unless the dog 'Dhruva' too enters with him. 

Sadly, neither Hindus nor Germans nor lovers of English Literature have understood what Pope actually said. I have. But then, I was born in 'Benares on the Rhine' and had to study a bit of Statistical Game Theory when I was sixteen or seventeen (after which I devoted myself to drink). 

To be fair, Pope didn't say what I said he said. But that's because Pope was a genuine poet and thus remained 'untaught by the wisdom he had uncovered, the laws he had revealed.' The Hindu does not expect his faithful dog to accompany him to Human heaven. But if a dog, or any other creature, attaches itself to a man arduously climbing to that destination, it deserves an equal admittance to what lies beyond the gates even if the man himself, like Yuddhishtra, decides its comforts can't justify the horrible shit show that is life on Earth. 

Obviously, for Pope's readers there were two types of 'Indian'- some were noble and 'Red', some 'Black'. Notoriously, the 'Black' retained an ancient belief in metempsychosis. They had no use for the type of Christian Heaven which Christian 'greed for gold' could buy entry into. Thus, Pope's admonition-  "Go, like the Indian, in another life / Expect thy dog, thy bottle, and thy wife"- means the opposite of what Bolingbroke- the Alcibiades, or Petronius, of his age- would have received it as. The fact is, your wife- whom you oppressed in this life- will be your husband in the next life unless you are her puppy dog because your dog attained the state of God and decided this was a sweeter way for you to discharge your karmic debt. 

The bottle, however, it is my fervent hope, is the impossible, perhaps because wholly Eucharistic, 'fixed point' of all Creation's 'eigenform' such that Landau's problems forever remain unsolved- or rather, not properly posed. What I mean is that the 'merit' of prime gaps may themselves offer a superior encryption such that all revealed language remains its own Pascalian cypher. Obviously, when I say 'what I mean is' what is meant is that I'm drunk off my head. 

In Game Theory, Common knowledge is stuff everybody knows everybody knows everybody knows. Sadly, everybody also knows that nobody knows what they know or, at least, are not able to act as if they do.

This is because Man knows that Alexander Pope- not some unreadable shithead, like Marx- was right.

 Even if there are no 'accessibility' or 'computability' problems, strategic considerations, resource or endowment diversity, etc, etc there is a fundamental mathematical problem with 'Common knowledge' viz. if knowledge is finitary, and has a language, 'Common Knowledge' can't be a well defined formula in it because it is an infinite conjunction. The workaround is a 'fixed point' theorem. The problem is 'singularities'. If the universe is full of black holes and white holes, why should 'knowledge' be different? Following Chichilinsky (but also, I believe, D.G Saari) we can equate singularities with Arrowvian dictators or just departures from 'naturality'- i.e. arbitrariness. This is related to uncorrelated asymmetries which in turn are related to costly signals. In other words, 'Knowledge' is a type of information which is biologic or arbitrary & Popean. What it can't be is mathematical or represent indefeasible propositions from the STEM subjects. 

The problem with Nash's 1950 paper

There was a time when people thought that the universe must obey whatever stupid logic or mathematics cretinous pedants came up while sitting comfortably in their armchairs. Then, it turned out that denser objects don't fall faster and gravity bends light waves and, by the Madam Wu experiment, 'incongruent counterparts' don't exist, and the Bell inequality, not Von Neumann's theorem, is correct. 

In the field of Econ, Arrow's theorem showed that math is useless unless you use words with a degree of precision. A Dictator is a guy who kills or incarcerates his opponents. He isn't a guy who always happens to vote for the winning candidate. 

Social Choice, like individual choice, is something done by a Brouwerian 'creative subject' and thus only capturable by a choice sequence which may be wholly 'lawless'. The same thing can be said of the word 'strategy' in game theory. It isn't an option forced on you- like a gangster giving you the option to hand over your money or get a bullet in the head- but an approach you can take to getting what you want- which itself is something you can decide to alter as the situation evolves. 

If you have taken a lot of trouble to rig a game, you may be able to use mathematics to prove to yourself that the game is indeed rigged. This, is what is established by one of most cited Econ papers of all time- viz Nash's 'Equilibrium points in n-person games' from 1950. It relies on Kakutani, and therefore, Brouwer's fixed point theorems and applies only to endomorphic, compact, convex continuous functions. The problem here is that there is no way of telling whether something we believe to be a function is actually a function. This is because the 'intension' which the function is supposed to fulfil may not have a definite or decidable 'extension'. 

One may define a concept of an n-person game in which each player has a finite set of pure strategies and in which a definite set of payments

this is an 'intension'. Is the 'extension' knowable? By arbitrary stipulation- sure. There is some exogenous Game-master who fixes the pay-off matrix in advance. But, in that case, Nash is offering a proof of something which has been arbitrarily imposed. In other words, he is merely showing that, as Game-master, he can rig things a particular way. But that isn't very interesting. I have a proof that all cats are dogs. This is because I define 'dog' as cat. But this merely proves I can utter any arbitrary shite.  

Suppose we have to play a game rigged in advance by a Game-master. We may choose, either in a coordinated or uncoordinated manner, to give different 'psychic' or other pay-outs to different outcomes and thus frustrate the Game-master. Here, the 'intension' of 'pure strategy' has a different 'extension' from that which the Game-master stipulated, and thus his calculations are confounded. 

to the n players corresponds to each n-tuple of pure strategies, one strategy being taken for each player. For mixed strategies, which are probability distributions over the pure strategies, the pay-off functions are the expectations of the players,

which are formed according to a rule Nash imposes 

thus becoming polylinear forms in the probabilities with which the various players play their various pure strategies.

So, pure strategy just means 'arbitrarily imposed option'. It isn't a strategy at all.  

Any n-tuple of strategies, one for each player, may be regarded as a point in the product space obtained by multiplying the n strategy spaces of the players.

It is the matrix arbitrarily imposed by the game-master on the basis of options permitted to each player. 

One such n-tuple counters another if the strategy of each player in the countering n-tuple yields the highest obtainable expectation for its player against the n − 1 strategies of the other players in the countered n-tuple. A self-countering n-tuple is called an equilibrium point.

It is the outcome the game has been rigged to produce.  

The correspondence of each n-tuple with its set of countering n-tuples gives a one-to-many mapping of the product space into itself.

Thus making it endomorphic but if players can assign their own private values to outcomes, this ceases to be the case. 

From the definition of countering we see that the set of countering points of a point is convex.

Only if no player can introduce a private valuation to outcomes 

By using the continuity of the pay-off functions

which has been arbitrarily imposed 

we see that the graph of the mapping is closed...

by arbitrary stipulation.  

Since the graph is closed and since the image of each point under the mapping is convex, we infer from Kakutani’s theorem 1 that the mapping has a fixed point (i.e., point contained in its image). Hence there is an equilibrium point.

Why do people engage in zero-sum games they are bound to lose? The answer is that there can be a psychic or reputational benefit in taking part in the game. Otherwise the equilibrium point would be for only equally well matched players to ever indulge in that particular type of game.  

A superior solution concept to Nash is Aumann's correlated equilbria. Add in Brouwer type choice sequences as 'strategies' and you can still have fixed points though they may not be computable or accessible in any way. Still, they can drive dynamics. 

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Aikin & Talisse on deep disagreement

A disagreement is interesting where- rather than being the predictable  consequence of heterogeneous preferences or interests- it represents a departure from the conditions for Aumann agreement- i.e. a situation where two or more parties have Bayesian rationality and share common priors and posteriors. The question arises, why do they have different types of rationality? Can they do better by changing their heuristics? Alternatively, why are their priors or posteriors different? Knowing this may help expand the common information set.

Deep disagreements are particularly interesting. It may be that they represent  phenotypic diversity or confer survival value. This points to something else Aumann has looked at- viz. why 'regret minimization' militates against unanimity and its associated 'Bayesian rationality'. The existence of Knightian Uncertainty means that a 'deep disagreement' may be associated with an ambiguity in classification of states of the world such that something 'paradigm busting' is around the corner.

It is foolish to confuse mutual hatred, or the difference between sanity and lunacy, with deep disagreement. Predictably, this is what Aikin & Talisse do. They write in 3 Quarks-

Deep disagreements are disagreements where two sides agree on so little that there are no shared resources for reasoned resolution.

This is not a disagreement. There may be no contact at all between the two sides. If there is contact, there may be mutual hatred, not disagreement.

In some cases, argument itself is impossible. The fewer shared facts or means for identifying them, the deeper the disagreement.

Only in the sense that cats and dogs have deep disagreements. 

Some hold that many disagreements are deep in this way. They contend that reasoned argument has very little role to play in discussions of the things that divide us. Call these the deep disagreement pessimists – they claim that many of the disputes we face cannot be addressed by shared reasoning.

Shared reasoning is as ineffective with people as it is with cats snarling at barking dogs. By contrast, a change in the incentive system or conditions of life may be mutually beneficial. There are ways in which you can make your cat and your dog get along such that both are happier and your home is harmonious. This happy outcome is the result of a certain type of reasoning and application of mind on your part. But it isn't a process 'shared reasoning' in which Fido & Chairman Miaow have demonstrated their ratiocinative prowess. 

It is seldom the case that matters are resolved through 'shared reasoning' because uncorrelated asymmetries exist- i.e. it is in the interest of both parties to chose different protocols or principles. But, where utility is transferable, disagreement or disagreeableness can be mitigated by some rational, economic, process or mechanism. But this is transactional, not epistemic. It is foolish to think that what makes people get along is not mutually advantageous transactions but, rather, some sophomoric debate about values or categorical imperatives. Philosophy makes a fool of itself when it pretends Society works on the same principles as an idealized academic Symposium.

There are also deep disagreement optimists. Their view is that deep disagreements are intractable only for contingent reasons – perhaps we have not yet surveyed all the available evidence, or we are waiting on new evidence, or there is some background shared methodological principle yet to be uncovered. With deep disagreement, the optimist holds, it is hasty to give up on rational exchange, because something useful is likely available, and the costs of passing such rational resolution up are too high. Better to keep the critical conversation going.

But who will pay to keep it going? Talking has an opportunity cost. The potential pay-off from talking has to exceed that cost for rational people to indulge in it. 

Disputes among pessimists and optimists regularly turn on the practical question: Are there actual deep disagreements?

No- if some state of the world, one party affirms as possible, obtains.

The debates over abortion and affirmative action were initially taken to be exemplary of disagreements that are, indeed, deep.

They are not deep at all if the Sky turns as red and blood and the Angel Gabriel appears with a flaming sword saying 'I iz gonna fuck up all youse pro-Lifers! Also, Whitey, yore ass gonna fry sho nuff!'  

Later, secularist and theists outlooks on the norms of life were taken to instantiate a divide of the requisite depth. More recently, conspiracy theories have been posed as points of view at deep odds with mainstream thought.

If Q-Anon gets hold of cctv footage showing widespread pedophilia in Federal buildings then they win. There is no deep disagreement. 

This brings us to QAnon.

Here’s QAnon’s core doctrine in a nutshell: (a) there is a cabal of Satanic and child-sex-trafficking Hollywood and Washington elites who drink ‘adrenochrome’ from tortured victims to prolong their lives; (b) Donald Trump, along with select other patriots, is waging a secret war against this group; and (c) an insider with ‘Q-level’ clearance in either intelligence or military command is leaking information out to readers on internet message boards. So Q-believers think they have insider information about a monumental war of good versus evil that’s being fought right in front of everyone’s eyes, but is nevertheless largely unseen.

Till they get hold of all that damning cctv footage which is currently safely hidden up Trump's rectum.  

Conspiracy theories typically are based in accounts of secret dealings that not only run contrary to widely accepted views but are also evidentially sealed off from them.

No. Conspiracy theories have truth makers. Sometimes, they turn out to be true in some particular and, in the face of hard evidence, the 'mainstream' has to accept that the nutjobs were right. 

Because conspiracy theories are built around a contrast between what is widely believed and what is known only by those with special access to the truth, they can thrive only among a community of conspiracy believers.

Nonsense! A conspiracy theory doesn't start off 'widely believed'. It takes time for it to disseminate. Nobody has 'special access' to the truth, though there may be specific information silos which, however, some who have access to them, believe are of great and malign significance. A UFO conspiracy theory- or a theory about the Deep State- is likely to have some proponents who actually worked in some clerical capacity within a specific information silo. Similarly, conspiracy theories about supposed medical and technological evils generally have at least one credentialized 'expert' on side. Kary Mullins is a Nobel Prize winning scientist who endorses some truly crazy shit. But he's hella smart.

Accordingly, what outsiders present as evidence against the conspiracy theory gets explained away by those on the inside.

But this is true of any hypothesis- including the one that a [articular theory is 'conspiratorial'. 

Purported evidence against the conspiracy theory is often transformed into further evidence that those on the outside are deluded, duped, and gullible.

But this is also true of the anti-conspiracy theory. In some cases, suspicions about financial institutions or intelligence agencies have turned out to be correct. There really was a conspiracy. It was the 'insiders' who were deluded, duped and gullible. 

Was the Iraq War a conspiracy to unjustly enrich certain bad actors? Or was it necessary for the War against Terror? Clearly, if there was a conspiracy, not everybody was involved. Equally, if there was no conspiracy, some bad actors did exist. Still, it may be expedient for us to decide that the thing was a Conspiracy, not a Crusade.

In the end, conspiracy theories thrive partly because in adopting them, one adopts the view that all possible evidence confirms the theory.

No. Theism, of a particular mystical sort, may thrive for this reason. But so may a sufficiently sophisticated Structural Causal Model for a narrow enough class of evidence. That is why, for the research program to remain Scientific, the scope of observation must expand. 

We can say, then, that it is part of the nature of conspiracy theories to be epistemically sealed in this way.

No we can't. The nature of Scientific or Mathematical Theories is to be epistemically sealed in a protocol bound manner. But this is also true of legal arguments or opinions given by professional experts. Without protocols, nothing is epistemically sealed. Conspiracy theories are likely to be loosey goosey. We don't expect their exponents to be sticklers for protocol.  

So it is with QAnon. Observe some popular pairings of counter-evidence and correlate sealing strategies:

Ellen DeGeneres and Oprah Winfrey have been arrested for child pornography, but they are still allowed to tape their respective shows, so as not to alert the others in the cabal.

Either this is true or it is not true. By and by, this canard won't be repeated because both women will continue their successful careers. So, there is no real 'sealing strategy'. There is just a rumor of a silly sort. 

Wayfair trunks are means for ordering and delivering abducted children to their abusers, but only select users may order them.

An urban legend is something less than a 'conspiracy theory'. It may be convenient to pretend one has fallen victim to a rumor of this sort rather than to have engaged wittingly in seditious conspiracy.  

Kamala Harris has a body double, but one cannot tell the difference on video.

This is true enough. I had to give up pant suits because I was afraid of being mistaken for her. 

Hillary Clinton was to be arrested for child sex trafficking in 2017, but she was allowed to continue her activities but under heavy surveillance.

I am not a child. Any sex trafficking which I have been subjected to since 2017 is of a reprehensible type. But it does not amount to pedophilia- though, no doubt, I wear diapers.  

These reports make no promise that there will be publicly available confirming evidence.

But, by lapse of time, they are rebutted. It's like the Mayan prophesy of the end of the World. When the world doesn't end, the thing is dead in the water. As I say, it may be convenient to pretend to believe in this sort of shit so that the quantum of punishment is reduced if your seditious conspiracy is prosecuted.  

Repeating a rumor is a type of 'cheap talk'. It is a fake type of 'common knowledge'. But it provides a sort of alibi. You can pretend you were just going with the flow- wishing to belong to a 'pooling equilibrium'. You weren't doing anything naughty or premeditated. The difficulty arises with 'costly signals' which establish a 'separating equilibrium'- in this case, one of greater culpability. An actual conspiracy may well feature a full fledged conspiracy theory. But, it would be wise to disguise it so as not to give away the conspiracy's plan of action. 

A prediction of a cataclysmic even on a given date in the near future has two effects. On the one hand it acts like a 'Kavka toxin' increasing current levels of belief. On the other hand, it will lead to 'cognitive dissonance' after the prediction fails to materialize. Some will fall away to reduce this dissonance. But those who remain will be far more deeply committed. Something like a Church is created whose members will be prepared to 'tithe' and to provide martyrs from time to time. This could be very valuable.

Aikin & Talisse don't get that a failed prediction can be a good thing. It creates a 'separating equilibrium' on the basis of cognitively costly signals. Cognitive dissonance now operates to create greater loyalty and a greater desire to win reputational advantage by a greater show of commitment to the cause. 

Thus Aikin & Talisse feel Q Anon, and- perhaps Trumpism- have been thoroughly vanquished.  

The fact that Inauguration Day has come and gone without incident is evidence that what was taken to be a deep disagreement with no shared intellectual resources for rational disputation was, in fact, a disagreement waiting for crucial testing. That test has come. Though the QAnon conspiracy theorists may elect to incorporate yet another epicycle into their worldview, that they recognize the need to regroup and revise is a minor win for reason and argument. 
It also suggests that the among Q-believers and those who reject the view is not absolutely deep after all.

The above suggests that Aikin & Talisse are as stupid as shit. There can be little 'deep disagreement' about that. Meanwhile the people who got into Q-Anon will find some other type of clickbait scam to engross their time.

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Aumann agreement and Arjuna's Vishada

The Bhagvad Gita tackles Arjuna's Vishada (Depression) which arises when he realizes that by doing his duty as a warrior in the Kurukshetra War he will have ended up bringing about the worst possible outcome for all sides- including his own.
Previously, a Gandharva (demi-god) had granted Arjuna the gift of' 'chaksushi vidya' such that Arjuna could always visualize whatever he wanted to know, that too in the manner he wished to visualize it. Arjuna did not accept it but the Gandharva didn't take it back either. This meant that it could vest in Arjuna the moment he ceased to be master of himself (which is what happens when you suffer 'Vishada'). The dramatic element in the Gita has to do with whether Lord Krishna will be able to prevent Arjuna from finding out, using the Gandharva's boon, that Karna is his true eldest brother because then the war can't go ahead as Karna wants. 

"The Gandharva said 'O Arjuna, I would like to impart to thee the power of (producing) illusions which Gandharvas alone have... This science is called Chakshushi. It was communicated by Manu unto Soma and by Soma unto Viswavasu, and lastly by Viswavasu unto me. Communicated by my preceptor, that science, having come unto me who am without energy, is gradually becoming fruitless. I have spoken to thee about its origin and transmission. Listen now to its power!rly One may see (by its aid) whatever one wisheth to see, and in whatever way he liketh (generally or particularly). One can acquire this science only after standing on one leg for six months. I shall however, communicate to thee this science without thyself being obliged to observe any rigid vow. O king, it is for this knowledge that we are superior to men. And as we are capable of seeing everything by spiritual sight, we are equal to the gods.'

Unlike Yuddhishtra, whose Vishada was only dispelled by having to learn Statistics and mathematical Game theory, Arjuna just gets, by 'moral luck', the equivalent of a perfect information, Bayesian predictor for everything. What's more, it has a neat little g.u.i such that it can show its results as a perfect simulation of a real world video, or even a Supernatural video, where Gods or Abstractions or  'Emergents' do the Narration, which is super-cool.

What has this to do with Aumann's agreement theorem? (which states that two people acting rationally (in a certain precise sense) and with common knowledge of each other's beliefs cannot agree to disagree. More specifically, if two people are genuine Bayesian rationalists with common priors, and if they each have common knowledge of their individual posteriors, then their posteriors must be equal.<sup>[1] Ziv Hellman has recently extended this result.)

Well, the Gita looks forward to a piquant situation where Arjuna fights his eldest brother, Karna, and slays him under the influence of 'Manyu' (dark anger).  Karna knows that if he reveals his birth, then Arjuna won't fight him and there will be no War. But Karna wants the War to go ahead- the condition for it is that 'common knowledge' is blocked by Karna's desire that his paternity remain unknown. It may be that Arjuna, who wants to obey his eldest brother, is constrained in some magical way by this wish of Karna. However, Arjuna- as is quite natural- does want to know the outcome of the War and immediately gets a vision of the horror and destruction and futility of it all. He still doesn't see the worst aspect of it for him personally- which is that he kills Karna in a mood of dark fury, exulting in revenge, and thus violates his own Dharma by committing the moral equivalent of parricide- but, perhaps, he doesn't want to see this because what he wants is already determined by Karna's desire that his paternity remain a secret and the War proceed as a great sacrifice by which warriors gain Heaven.

Interestingly, the Gita would still have the outcome if Arjuna had no special feeling of filial piety.Suppose Arjuna and Karna were not younger and older brother but two independent scholars of equal stature. Suppose both want 'common knowledge' except if at some later point they might regret it. Suppose, further that they each receive Arjuna's chaksushi vidya so they have zero computational or informational constraints. In that case, 'regret minimization' or 'Hannan consistency' militates for not Aumann agreement but disagreement iff the multiplicative weights index algorithm is efficient. Another way of saying the same thing is- Life is what makes the price of anarchy negative- or, as Heraclitus put it, Gods and Men die each others lives, live each others deaths.</sup>

Reading this interview with Hannan, what astonished me was how far ahead the Indians (mainly Bengalis and Maharashtrians) were in Statistics some seventy years ago. But, truth be told, there was nothing surprising at all about this.
The Indian clerisy- Revenue officials in the main- were doing boring Statistical Decision theory most of the time though, no doubt, with a bit of ars dictaminis Sycophancy as the cherry on top.

Thus, the traditional Indian reading of the Gita was always, au fond, Decision Theory based. It's basic finding- viz. Prescriptivity is vector not scalar- agree to disagree and then try to kill each other but stop short of complete genocide so some diversity at the margin is retained- is pretty much what Nature- the chrematistic aspect of Life- keeps telling us.

Since Hannan's result was known to the Indian Math/Stats community some sixty or seventy years ago and since, furthermore, Game theory was going from strength to strength over the same period, how come Maharashtrian Statisticians, like Kosambi, and Bengali Academics, like Amartya Sen, write utter shite about the Gita?

Well, clearly, it's because they are smarter than you and me- 'drunkards, fools and fishermen' that we are- because, though the Mahabharata was written for us, only the savants are blinded by it and proceed to enact their genocidal illusory wars coz Aumann agreement actually means not that you can't agree to disagree but that everybody has to just go on shitting all over everybody else lest, by some mischance, whatever strain of bacteria it is that makes our own farts pleasant to only our own nostrils go entirely extinct and it is to that great end that Poetry as Socioproctology too is a true emanation of the Blessed Gita.