hand... Does the theory attributed to A Smith have any mathematical justification, i.e. that a group of individuals making egoistic choices will lead to a solution beneficial for everyone? Can "beneficial" be understood as "optimal"? No trivial solutions like individuals with identical utility functions.
Handwaving and solutions appealing to complexity & great interconnectedness of things welcome! I'm thinking about it and arriving to opposite conclusions - no wonder, I don't even know what's a good place to start!

Try this: https://www.econlib.org/library/Essays/hykKnw.html

Thanks, but I'm looking for mathematical analysis of the such optimal solutions and proofs of their existence. I went through matching problems (stable marriage/Gale-Shapley, etc.) and optimal choice theory, but they don't refer directly to Smith, Hayek and faire tales.
Did Smith think about beneficial solution in Pareto sense, or a solution in which all individuals' needs are "saturated" (loss-less compression algorithms are the kind of optimisation I have in mind)? That's the kind of questions I have on my mind, and once clarified, how I want to approach answering them (mathematically or numerically).

Hey Kat,

It might be a bunch of basic stuff for you, but maybe some of the below might help:

- Economics: Arrow's Impossibility Theorem and this interesting paper https://ieeexplore.ieee.org/document/84 ... rs#authors
- Applied Ecology: good models survey https://www.nature.com/articles/s41559-020-01298-8#Bib1
- Complexity Theory: Porf Farmer's landing page http://www.doynefarmer.com/presentations

Rgds,
M

Sounds like you want something like the Fundamental theorems of welfare economics.

However, "Pareto optimality" can be far from a "solution beneficial for everyone" in your OP.
For example, Elon Musk can consume all the output and that can be Pareto optimal, as taking output from him and allocating it to others may lower his utility. However, Pareto optimality seems a useful notion when there are wasted resources, barriers to competition, unnecessary regulations, etc.

I am not an economist, so I'm sure this is only part of a good answer. 

Thanks, but I'm looking for mathematical analysis of the such optimal solutions and proofs of their existence. I went through matching problems (stable marriage/Gale-Shapley, etc.) and optimal choice theory, but they don't refer directly to Smith, Hayek and faire tales.
Did Smith think about beneficial solution in Pareto sense, or a solution in which all individuals' needs are "saturated" (loss-less compression algorithms are the kind of optimisation I have in mind)? That's the kind of questions I have on my mind, and once clarified, how I want to approach answering them (mathematically or numerically).

I doubt it can be cast as a mathematical problem.

Former Bank of England Governor Mervyn King describes (and repeatedly gently pokes fun at) the “grand auction” proof of the optimality of capitalist resource allocation (the Arrow, Debreu stuff) in his 2012 book The End of Alchemy. It is likely by some margin the least convincing economic "proof " you will ever encounter.
 
The problem of optimal resource allocation is solved by assumption in the proof by requiring perfect information and complete markets for every conceivable future contingency. This perfect endowment means that every future contingency can be planned and hedged in a one-time (T=0) grand auction market (which sounds very ironically exactly like the work of a lone social planner!) that is so perfect (bow before its majesty!) one trades at T=0 to cover all future contingencies, and therefore there is no need for credit and even money in this marvelous capitalist world.   

So, yes, there is a mathematical foundation, it is a proof determined by assuming all the issues away. Paul Samuelson's warning so long ago about the dangers of strong axioms was clearly not heeded.

Pareto Optimality obtains in a situation where an economic change can be effected that raises the welfare of at least one person without reducing that of anyone else. Because that is a very, very rare thing (externalities abound), it has been more broadly interpreted to mean that "winners" will "compensate "losers", otherwise the Pareto condition cannot obtain. In the real world, despite nominally progressive income taxation and the social safety net (that Hayek both favored, incidentally), winners generally do not generally compensate losers enough for Pareto Optimality to obtain. That last observation would be consistent with the empirical fact of increasing inequality of income distribution that of course is so richly deserved by we, the practitioners of financial alchemy!  

Thanks, that's very insightful. The lack of mathematical proofs in economics is a bit like weather - everybody complains about it, but no one seems to be doing anything to change it.
Anyway, I actually made this digression when trying to prove some result about the optimisation of neural network models at work. The NN optimisation problems pretty much resemble those in economics - at least to me, the naive. Since the NN theory becomes more and more advanced and sophisticated (maybe even will catch up with practice one day!), there may be some progress in economics one day.

If for fun you want to go the opposite direction from math and read a satirical poem that influenced the laissez-faire, check out The Fable of The Bees: or, Private Vices, Publick Benefits (1714), by Bernard Mandeville, first published anonymously in 1705. Who knew bees would cause such a stir?

Those bees should read Orwell!

I really think of a neural network trained on data as a social system with the problem of resource allocation: people are features/random variable which realise the data and the NN's expressive power are resources to allocate. It's undoubtedly a very bad idea - just as bad as all that physicists have done to economics and math-fin, only more obscure (the advantage of NN-based machine learning!).

I think the problem here is the same one you have in Game Theory... all of it kind of works only if you're sure everyone else involved is playing the same game as you.

You could describe their individual agenda with a flexible utility function, like that trained it in multiagent reinforcement learning models. (There the question is if they reproduce the game-theoretical strategies.) I played only with very simple models (because of computational limitations), but even a simple setting can result in a cornucopia of individual time-varying strategies as the players interact with each other. But they kind of play the same game - can work out comparative advantage, side against free-riders, etc. I just wondered how much overlap there can be between the outcome optimal for each individual (at the same time) and the group average, and what precisely is in the head operating the invisible hand

Is it fair to suggest that an exchange is a convenient and presumably efficient formalization of the invisible hand? If so, what is the "in the head" of an exchange? Facilitation of trade through price discovery, finding that mythical nirvana called equilibrium by equating supply and demand. 

Smith was confident but unclear why individual optimizing behaviour aggregates cleanly. But as mentioned earlier, externalities abound in life, and Adam Smith also had serious concerns about the incentive effects on the managers of joint stock companies, concern about people rolling the dice with other people's money.     

The exchange indeed seems like a convenient choice as a heuristic, but in reality I feel this question is more analogous to how you would model ecological systems, chemical reactions, metabolism or a computational collective intelligence... the work of Tadeusz Szuba or Doyne Farmer seem to be on the right track.