Monday, 25 May 2015

The De-struction of Game Theory: Nash Equilibrium and Nietzsche's Invariance



In my Weber-buch I applied to Max Weber's concept of "ideal types" the devastating refutation implicit in Nietzsche's critique of "scientific and logico-mathematical laws". In light of the unfortunate demise of John Nash just days ago, I am proposing to our friends possibly the most difficult piece ever published on this blog presenting what I have styled "Nietzsche's Invariance". My aim here is to expose the colossal stupidity of those economists who have turned Nash equilibrium, and thence game theory, into what they bathetically claim to be an important tool of economic analysis: and I achieve this by destroying - more than simply de-structing - the very rationale behind game theory - by proving that it can be applied only to mathematical "problems" and that it is utterly inapplicable to "practical" problems such as those tackled in economic analysis. If, after reading this, all those economist charlatans who waste their time on game theory do not hang their collective dunder-heads in shame - then they ought to have their heads "hanged" in the other sense! Cheers.
In game-theoretic situations or “problems", “means” and “ends” (or goals) are “mathematically linked” in axiomatic fashion. Yet, as Wittgenstein has established conclusively following Nietzsche’s own critique of logico-mathematics, such game-theoretic problematics categorically subvert the whole “nature and character” of the "game", of the “situation” or “problem”, that they intend to define or “resolve”! That is because the “solutions” (the “equilibria”, like “Nash equilibrium”) to the “game” are apodictically deducible from the “premises” and “axioms” in which the “game” is couched. But in that case, the “game” can no longer be defined as a “problem”! Because its apodictic formulation has turned it into an “inexorable fate”! And the “solution” is not a “solution” because there was never a “problem” in any meaningful sense! If we say, with Don Patinkin, that the “solution” to mathematical problems consists in “shortening the time of calculation [for finding the solution to the problem]”, then we would have to argue that the “truth” of a mathematical equation or “argument” depended on the “speed” of its computation or “application” to practical problems. But that is clear and sheer non-sense! Because the very definition of “truth” is “logical” and therefore not subject to temporal or practical considerations!  One of two things: either a mathematical equation, argument or axiom is an ab-solute identity, in which case it is a tautology (it says absolutely nothing: it is meaningless and “purposeless” and per-fectly useless); or else it is not an identity, in which case it may be “useful” but can never be “true” because “usefulness” and “truth” are categorically different things and the very “usefulness” of a mathematical identity or axiomatic discipline becomes a “value judgement”, contrary to the economist’s attempt to preserve the “logical objectivity” and “technical rationality” of such “axiomatic disciplines”!      
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            But In the construction of all “ideal types”, as in all mathematical operations, it is precisely “the logical per-fection”, the numerical “id-entities” involved in their equations, that deprive these operations of any possible content because either the identities are entirely “formal” and therefore “self-dissolving”, or else they have real contents, in which case the “entities” involved are not “identical”! Similarly, in “game theory” the “rational choices” are nothing more than “mathematical tautologies” in that the “axiomatic” rules of the game are de-fined tautologically from the outset. But such sets of rules vitiate the entire ontological “reality” of the situation of “choice” in that they transform their defined “problems” into “destinies”.

Wittgenstein's devastating illustration of this point is insuperable for its shattering simplicity: he takes the instance of the law and analyses the following statement: – “the law always catches the criminal”. Hence, this definition of “law” excludes the possibility of “evading the law” which is implicit as “transgression” in the very concept of “law”. Similarly, in physics, he looks at the notion of traction and concludes that – “we cannot walk on a perfectly smooth surface” – despite the fact that “walking” and “traction” are implicit in the concept of “perfectly smooth surface”: a "surface" is a space on which motion can take place, but a "perfectly smooth" surface can have no traction by definition, so that the phrase "perfectly smooth surface" is a contradictio in adjecto. In such cases it is no longer “meaningful” to speak of "game" or of “freedom”  or of “evaluations” or indeed of “science and rationality” in any sense at all – given that the “problem” is actually a “destiny” similar to that of Joseph K in Franz Kafka’s novels! 

Just as a logical “truth” cannot be “self-evident”, so a “law” –  juridical or scientific - has meaning only if those who are subject to it have the theoretical ability to evade it or to confute it successfully – otherwise it is no longer an observable and detectable entity “separate” from transgressive behaviour or from the “reality” it supposedly en-compasses. This is the kernel of Nietzsche’s invariance whereby “truth” cannot exist if it is postulated to be the real correspondence of “concept” and “reality” – and it cannot be “truth” if it is not so postulated. For “the Law” to be “a law” rather than a “fate”, it must admit or allow implicitly at least the possibility of successful transgression or evasion or  “grace”; for a “game” to be a “game” in game theory, it cannot postulate “common knowledge” or “flawless execution” or “perfect calculation” or “symmetric information” (unaffected by “adverse selection”, see axiom 5 in the insert below) – eventualities that game theory axiomatically excludes because these are “possibilities” that prevent its “games” from having “equilibrium solutions”! Similarly, a “surface” must allow implicitly of the possibility of “traction” or movement on it; but then it ceases to be “smooth”, defeating the original axiomatic definition. In all cases the axiomatic framework must either lose its “formal validity” as soon as it is “applied” to reality, or else it must be inapplicable to any reality and become a sterile tautology if it is to preserve its postulated “formal axiomatic validity”!
                                                ********************
A good illustration of the tautological nature of axiomatic models is given by the “conditions” or axiomatic assumptions that underlie Nash Equilibrium in game theory. If a game has a unique Nash equilibrium and is played among players under these conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are:
 1.The players all will do their utmost to maximize their expected payoff as described by the game.
 2.The players are flawless in execution.
 3.The players have sufficient intelligence to deduce the solution.
 4.The players know the planned equilibrium strategy of all of the other players.
 5.The players believe that a deviation in their own strategy will not cause deviations by any other players.
 6.There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on.    ************    
                                                                    
Philosophically, these axiomatic conditions (note especially the regressio ad infinitum in 6) turn the “game” into a “fate” or a “destiny”, depriving it thereby of the very essence of what constitutes a “game” or a "problem" - the problematic unpredictability of its outcome! The game then becomes meaningless and purposeless – and so does any “analysis” based on it! – except to the extent that it is used as a semeiotic device to regulate human behaviour, as in the process of Rationalisierung that we are enucleating here.

                                                *********************

The notion of axiomatic mathematical truth as “despotic” was not lost on the earliest theoreticians of the doctrine of the Ab-solutist State – the “statolatrists” – in Renaissance Europe. Yet again, it was Hannah Arendt who came closest to intuiting the complex problematic of logico-mathematical id-entities or “laws” and the theorization of ab-solute power in On Revolution:

There is perhaps nothing surprising in that the Age of Enlightenment should have become aware of the compelling nature of axiomatic or self-evident truth, whose paradigmatic example, since Plato, has been the kind of statements with which we are confronted in mathematics. Le Mercier de la Riviere was perfectly right when he wrote: 'Euclide est un veritable despote et les verites geometriques qu'il nous a transmises sont des lois veritablement despotiques. Leur despotisme legal et le despotisme personnel de ce Legislateur n'en font qu'un, celui de la force irresistible de l'evidence';26 and Grotius, more than a hundred years earlier, had already insisted that 'even God cannot cause that two times two should not make four'. (Whatever the theological and philosophic implications of Grotius's for-mula might be, its political intention was clearly to bind and
Foundation II:Novus Ordo Saeclorum 193
limit the sovereign will of an absolute prince who claimed to incarnate divine omnipotence on earth, by declaring that even God's power was not without limitations. This must have appeared of great theoretical and practical relevance to the political thinkers of the seventeenth century for the simple rea-son that divine power, being by definition the power of One, could appear on earth only as superhuman strength, that is, strength multiplied and made irresistible by the means of violence. In our context,contex~, it is important to note that only mathematical laws were thought to be sufficiently irresistible to check the power of despots.) The fallacy of this position was not only to equate this compelling evidence with right reason –the dictamen rationis or a veritable dictate of reason - but to believe that these mathematical 'laws' were of the same nature as the laws of a community, or that the former could somehow inspire the latter. Jefferson must have been dimly aware of this, for otherwise he would not have indulged in the somewhat incongruous phrase, 'We hold these truths to be self-evident', but would have said: These truths are self-evident, namely, they possess a power to compel which is as irresistible as despotic power, they are not held by us but we are held by them; they stand in no need of agreement. He knew very well that the statement 'All men are created equal' could not possibly possess the same power to compel as the statement that two times two make four, for the former is indeed a statement of reason and even a reasoned statement which stands in need of agreement, unless one assumes that human reason is divinely informed to recognize certain truths as self-evident; the latter, on the contrary, is rooted in the physical structure of the human brain, and therefore is 'irresistible'. (pp.192-3)

Arendt observes that “divine laws” and the “laws” of ethics and of States – in short, “all values” – differ from those of mathematics because the latter describe the constitution of the mind and therefore “cannot be resisted”, whereas the former, however “reasonable” they might seem, require “agreement” unless one appeals to a mystical “intuitus originarius”. Arendt, however, fails to comprehend the enormity of the problem she has dimly perceived, which is the reason why she is unable to enucleate it with the ruthless clairvoyance that Nietzsche applied to it. When Mercier calls Euclid a “despot” he is equiparating the “legislative” power of his geometrical axioms to the “ab-solute” power of despots in that both kinds of “power” effectually do not admit of “questioning” or “agreement”! Grotius, by contrast, is placing mathematical axioms above the power of Sovereigns and of God himself (!) – but in so doing he too is equi-parating the two powers in the sense that mathematical axioms in their “universality” offer a “guarantee” of “truth” and validity that even the power of Sovereigns and of God, in its “ab-soluteness”, cannot proffer. 

The significant feature that escapes Arendt is that both Mercier and Grotius interpret the “truth” of mathematical axioms as a “Value” – as an “ab-solute truth”, one that requires no de-monstration – that can stand as the ultimate, ab-solute guarantee of all human universal values, of that inter esse that is threatened by the arbitrariness implicit in the “ab-soluteness” (the “unanswerability”, the “unaccountability”, the “irresponsibility”) of any and all “political” or “divine” power! And because Arendt does not grasp the profound significance of this “equi-paration”, she is then unable to penetrate the next, the ultimate and most devastating conclusion – one that she eludes, or that eludes her, when she attributes the “self-evidence” of mathematical “truths” to “the physical structure of the human brain” (a “psychologism” already refuted by Wittgenstein and Husserl before him). 

Arendt seeks to keep separate and distinguish the “logical necessity” or “irresistibility” or “irrefutability” of logico-mathematics (as a “power of the human brain”) from the “political necessity” of human coercion. Yet, the devastating conclusion that Nietzsche was first to outline as the “con-clusion” or “com-pletion” or “ful-filment” (in the sense of “ex-haustion”, of “fully-ending”, Heidegger’s Voll-endung) of the Western metaphysical Ratio-Ordo is that it is precisely because human beings can conceive of logico-mathematical id-entities that we have ultimate proof of the complete value-lessness of life and the world! It is the very arbitrariness and con-ventionality of logico-mathematical id-entities that con-firms ineluctably the futility of all “Truths and Values”! 

Far from being “the ultimate and ab-solute guarantee” of the presence and reality of Reason and Order, of universality, in the human world logico-mathematical identities constitute the evidence of the ultimate instrumentality of human action, of its ultimate value-lessness, of the ultimate un-reality of “all values”, of all “Truth”! This is what Nietzsche meant by “the trans-valuation of all values”! 

Arendt completely fails to see that both logico-mathematical and juridical-ethical or practical “laws” are conventional (Nietzsche and Wittgenstein); that juridical-ethical and behavioural laws can also be given ab-solute logico-mathematical axiomatic form, as in game theory, and then they become a “fate” (in Wittgensteinian language games), which is the opposite of what “truth” is supposed to be! So, in fact, “self-evident truths” (Jefferson), whether logico-mathematical or practical, are not “truths” at all (thus, the Jeffersonian “we hold” can be applied to the former as well as the latter): – indeed, the required “ab-soluteness” of all ultimate values and truths demonstrates that there can be no such value or truth except for “truth as a value”.

Far from being the ultimate protection against political arbitrariness, it is the very human ability to erect con-ventionally the axiomatic rules of mathematics and logic to the status of ab-solute ultimate truth and value that reveals the utter “value-lessness” of life and the world and the im-possibility of “truth”!  It is for this precise and quite understandable reason that Mercier and Grotius both feel “tempted” to equiparate logico-mathematical necessity and political coercion – because “logico-mathematical necessity” is the ultimate instance of the ability of human beings to transmute a symbolic “con-vention” into “political coercion” and vice versa. This is the “secret” of the Rationalisierung!

(It will be recalled that in George Orwell’s 1984 the main character Winston Smith seeks refuge from the pervasiveness of Big Brother’s totalitarian power in the “truth” of the statement “two plus two makes four no matter what Big Brother says”. What Smith fails to perceive is that it is precisely the ability of human beings to devise logico-mathematical identities that exhibits the ultimate futility of “truth” as a “value” and that demonstrates instead its utter instrumentality, and therefore the possibility of Big Brother’s “ab-solute power”.)
 
Differently put, mathematical id-entities and logical axioms are borderline concepts (Schmitt, Politische Theologie); they de-monstrate (in the Wittgensteinian sense of “showing”, “pointing to” but never explaining meaningfully!) both the ultimate attempt and the ultimate inability of the human mind to con-ceive of “truth and value” as objective entities: they represent therefore not only the ultimate de-monstration of the “un-reality” of “truth and values” but also and most terrifying of all the possibility of turning human arbitrariness into a “science” and a “logic”. This is “the Will to Truth”. Arendt came frighteningly close to this terrifying conclusion when she wrote in On Revolution (already quoted above): -  

Whatever the theological and philosophic implications of Grotius's formula might be, its political intention was clearly to bind and
Foundation II:Novus Ordo Saeclorum 193
limit the sovereign will of an absolute prince who claimed to incarnate divine omnipotence on earth, by declaring that even God's power was not without limitations. This must have appeared of great theoretical and practical relevance to the political thinkers of the seventeenth century for the simple reason that divine power, being by definition the power of One, could appear on earth only as superhuman strength, that is, strength multiplied and made irresistible by the means of violence. In our context,contex~, it is important to note that only mathematical laws were thought to be sufficiently irresistible to check the power of despots.) The fallacy of this position was not only to equate this compelling evidence with right reason –the dictamen rationis or a veritable dictate of reason - but to believe that these mathematical 'laws' were of the same nature as the laws of a community, or that the former could somehow inspire the latter.

To echo Arendt by way of confutation, the fallacy of her position is failing to equate mathematical ‘laws’ with right reason and to believe that these mathematical ‘laws’ are not of the same nature as the laws of a community, or that the former cannot somehow inspire the enforcement of the latter! Even the most “irresistible laws” (Arendt), the “laws” of logico-mathematics, are just as “con-ventional” as “the laws of a community”: indeed, it is the ultimate “con-ventionality” of even logico-mathematical “laws” that demonstrates how all laws, including moral and juridical ones, are ultimately “con-ventional” and therefore political. This is what Mercier, more explicitly, and Grotius, implicitly, meant to say in the quotations that Arendt selected (which she reproposes in The Life of the Mind). It is the fact (understood in the Vichian and Nietzschean sense of verum ipsum factum, meaning that “the truth” is what human beings actually do, from the Latin facere, to do) that human beings can mis-take logico-mathematical con-ventions (agreements) for “irresistible truths” that evinces definitively the “con-ventionality” of all “truths” and all “laws” – their “legal” character, and therefore their lack of “legitimacy”, their dependency on some “authority” that is not and cannot be “ab-solute” (not requiring “further proof”). (On the antinomy of “legality and legitimacy” the reference is to Carl Schmitt’s homonymous work. We will examine the homology of Nietzsche’s Invariance and Schmitt’s notion of “decision auf Nichts gestellt” [made out of nothing] later.)
 
Here again what “betrays” Weber is his impossible aim to reconcile “freedom” and “rationality” (one could not imagine two greater “idols”, next to “the State” and “God”, for Nietzsche to desecrate and vilify [cf. his Twilight of the Idols]) – in circum-stances, in con-ditions – those of a “controlled experiment” - that make those very concepts utterly irrelevant to assessing – precisely! – the “political freedom” or less enjoyed by human beings in the “practical reality” that is under “scientific observation”. Not “scientific activity” per se is “ideological” (pace Heidegger and Marcuse and all other “late-romantics”) – because that would imply that human activities can be classified as either “ideological” or “non-ideological” (recall Nietzsche: “the real world has disappeared – and the apparent world…has disappeared with it!”) -, but rather its “abstract extension” as “empirical research” under the “real subsumption” of the capitalist wage relation to all spheres of human interest without regard to “consequences”. Because Weber (and Marcuse and Habermas even less) does not understand – nor does he ask – why and how human activity can be reduced effectually to instrumental action (the essence of the Rationalisierung), he gives proof of his failure to understand the “significance” of his own intellectual and political activity which in fact is often quite advanced in the direction of answering the “enigma” (see Freund discussed below) of the Rationalisierung. Weber looks for the “rationality” of “necessity” and forgets that he can find only the “in-variance”, either the ultima ratio of “dire necessity” or “the endless multiplicity of possible evaluations… [that] can be reduced to manageability only by reducing them to their ultimate axioms”. Finally, the “utopia” of the “ideal type” has degenerated into the logical purity, theaxiomatic tyranny of the Rationalisierung.

1 comment:

  1. One particular huge difficulty will be why these businesses usually are not also moral. These are allowing pupils gain a advantage just what exactly does you probably anticipate from them? game theory tutor

    ReplyDelete