Soviet mathematicians made remarkable contributions to Econ and OR. Kirill Ilinski is a post-Soviet mathematician who made quite an impression on 'quants' with his 'Physics of Finance' which came out more that twenty years ago. I see one of his first papers in that field is available here and it is certainly worth reading.
Before I give you a sample, I draw your attention to this call for papers, a couple of years ago, from Chichilnisky et al.
What is the key tenet that rationalizes why quantum formalism in social science and especially in social choice and decision-making can be used?
Social Choice and Decision Theory are epistemic, impredicative, feature mimetic effects, and don't have well defined 'extensions' (i.e. no well defined sets exist) or graphs of functions save by arbitrary stipulation. Thus 'intensional fallacies' vitiate the entire enterprise.
The key tenet of quantum formalism is that a quantum state is described by a normalisable wavefunction (i.e. they can be added to each other or multiplied by a complex number) and thus the set of all possible wavefunctions is an abstract vector space.
The problem is that a lot of things in the Social realm are 'ontologically dysphoric'. They are not at home in the world. Yes, there is 'utility' or 'ophemility' or 'hedonic value' but there is also 'regret' and 'sacrifice' and the wish to die to the world so as to be born to what passes all understanding.
Now it may be that some real smart guy who knows a lot about sex or QTM or horses can do very very well in the world of Finance by seeing analogies between that arcane field and fucking, or gauge theory, or the gee gees. If the guy makes money and provides a useful service, that is all the 'rationalisation' we need.
Chichilnisky takes a different view.
Based on the axioms of quantum theory, we identify a class of topological singularities that encode a fundamental difference between classic and quantum probability,
this is that the former is a density function while the latter is a wavefunction. This means quantum probability doesn't have the third classical axiom, viz. - for every collection of mutually exclusive events, the probability of their union is the sum of the individual probabilities. This means, the 'Wigner '(quasi-probability density) function can be negative for quantum systems. It has been shown that 'magic state distillation' (for fault tolerant quantum computers) arises where there are negative elements in the discrete Wigner function. This negativity is equivalent to 'contextuality'. But this is like saying if all entanglements are known then quantum probability converges to classical. There are no singularities. This is like saying the 'Tube Map' of London (which ignores hills, valleys, rivers and other such obstacles) is good enough for getting around London moving in what appear to be 'straight lines'. Even if a singularity- e.g. a wormhole- arises in Trafalgar Square, it would make no difference to the Tube traveller. My point is, there is no 'fundamental difference' between relevant axiom systems for any particular purpose.
.and explain quantum theory’s puzzles and phenomena in simple mathematical terms so they are no longer ‘quantum paradoxes’.
because when you explain something which is counter-intuitive, it is no longer so counter-intuitive unless it is because your explanation is shit.
The singularities provide also new experimental insights and predictions that are presented in this article and establish a surprising new connection between the physical and social sciences.
Societies exist in physical space. That's the connection. But it isn't exactly new.
The key is the topology of spaces of quantum events and of the frameworks postulated by these axioms. These are quite different from their counterparts in classic probability and explain mathematically the interference between quantum experiments and the existence of several frameworks or ‘violation of unicity’ that characterizes quantum physics. They also explain entanglement, the Heisenberg uncertainty principle, order dependence of observations, the conjunction fallacy and geometric phenomena such as Pancharatnam–Berry phases.
Pancharatnam is classical. What makes quantum events different is that we think the thing is an artefact which will disappear with further 'fine graining'. But we may be wrong.
Somewhat surprisingly, we find that the same topological singularities explain the impossibility of selecting a social preference among different individual preferences:
it is impossible to have a mathematical representation of individual preferences or social preferences or anything at all which is 'epistemic', impredicative or intensional.
which is Arrow’s social choice paradox:
the dictator is a Banach 'fixpoint'. However, fixed point theorems may not work in QMT because of 'tunneling'. They can't arise in Social Choice because there is no graph of the relevant function.
the foundations of social choice and of quantum theory are therefore mathematically equivalent.
No. QMT has a useful mathematical representation. Social Choice does not.
...In classic physics, events, when seen as sets in a sigma-algebra,
i.e. they occur in a measurable space
are described by one sample space with a single basis of coordinates (one framework) as in classical probability.
It can be a Clifford algebra (i.e. have a quadratic form) and thus framework independent.
The concept of ‘unicity’ essentially indicates that a single framework can describe all observed events.
In every type of physics, you are seeking unique representations. But nobody thinks there should be a unique representation of your search for any such thing. But decisions and choices are epistemic and can represent precisely this type of search. That's why they have no mathematical representation because there can be no unique pre-order.
However, in decision-making, there is no reason to believe there is only one framework, which can capture all events.
There is no reason to believe there is any mathematical representation. The thing may involve ontologically dysphoric elements- e.g. deciding to sacrifice material prosperity so as to get into Heaven.
Ilinski, unlike Chichilnisky, has made good money in Finance. His first paper gives
a brief introduction to the Gauge Theory of Arbitrage.
Chichilnisky had a paper showing limited arbitrage is a necessary and sufficient condition for the existence of a competitive equilibrium.
I suppose, if some degrees of freedom are redundant- e.g. people at the Stock Exchange ignore irrelevant information about the color of the President's cat- then the dynamics may be constrained- i.e. the Lagrangian is invariant- and you can have a useful gauge theory. In modelling financial markets, this may be a useful enough starting point but, equally, it could give rise to instruments which are 'weapons of financial mass destruction'. I believe Ilinski's own work was useful. But, maybe, this was because he wasn't off his head on coke nor was his math bullshit.
Treating a calculation of net present values (NPV) and currencies exchanges as
a parallel transport in some fibre bundle,
So, mathematical valuation is a local realization of a 'connection' or 'endomorphism' but of what? The problem here is that the 'globe' of what is financial contracts and expands for epistemic reasons. This means the connection or endomorphism has an impredicativity problem. One could assume 'Muth Rationality'- which begs the question whether there can be a true economic theory for the thing- or else 'Aumann agreement' or something of that sort. But, maybe there ought not to be any such agreement. Maybe, markets are supposed to raise volatility to drive 'creative destruction' or just maintain liquidity or else promote the circulation of elites or some such thing.
we give geometrical interpretation of the interest rate, exchange rates and prices of securities as a proper
connection components. This allows us to map the theory of capital market
onto the theory of quantized gauge field interacted with a money flow field.
Presumably, arbitrage is seen as an adiabatic (i.e. fast) process. The problem is impredicativity which could expand even faster.
The gauge transformations of the matter field correspond to a dilatation
(redefinition) of security units which effect is eliminated by a proper tune
of the connection. The curvature tensor for the connection consists of the
excess returns to the risk-free interest rate for the local arbitrage operation.
This is the crux of the problem. We don't know what is or isn't risk-free till after the event. Speaking generally, you can't have an invariant Lagrangean because financial markets have singularities or feature non-locality. Sad.
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