How to Effectively Solve Unsolvable Problems

In his essay “The Use of Knowledge in Our Society”, Hayek begins to describe a special type of problem only solvable by a special type of machine:

The peculiar character of the problem of a rational economic order is determined precisely by the fact that the knowledge of the circumstances of which we must make use never exists in concentrated or integrated form but solely as the dispersed bits of incomplete and frequently contradictory knowledge which all the separate individuals possess.

His “marvelous” machine is effectively and simply put a pricing mechanism created by the collective of the individual knowledge of mankind. This pricing system plays a significant role in the efficient evolution of our society:

…in a system in which the knowledge of the relevant facts is dispersed among many people, prices can act to coördinate the separate actions of different people in the same way as subjective values help the individual to coördinate the parts of his plan.
The whole acts as one market, not because any of its members survey the whole field, but because their limited individual fields of vision sufficiently overlap so that through many intermediaries the relevant information is communicated to all. The mere fact that there is one price for any commodity—or rather that local prices are connected in a manner determined by the cost of transport, etc.—brings about the solution which (it is just conceptually possible) might have been arrived at by one single mind possessing all the information which is in fact dispersed among all the people involved in the process.
The most significant fact about this system is the economy of knowledge with which it operates, or how little the individual participants need to know in order to be able to take the right action. In abbreviated form, by a kind of symbol, only the most essential information is passed on and passed on only to those concerned.

John Nash felt he had a significant solution conjecture that was also a sort of “effective solution” that he outlined in a series of letters to the NSA:

The significance of this general conjecture, assuming its truth, is easy to see. It means that it is quite feasible to design ciphers that are effectively unbreakable. As ciphers become more sophisticated the game of cipher breaking by skilled teams should become a thing of the past.

Bitcoin is another such “effective” solution in which malicious players are incentived to hold up the system rather than to defect and attack it.  Provided the players are paid more to hold the system up then to defect an equilibrium of stability is theorized to ensue. Satoshi, the writer of the bitcoin.pdf, effectively solved the byzantine generals’ problem with this paper.

Sometimes a solution requires a different point of view which is seemingly impossible.  We are often present a problem with the assumption of context, when truly it is the very context that is needed to change in order to solve it. John Nash and other great minds have spend much time trying to see outside their own boxes as well as those society imposes on us:

 I think there is a good analogy to mathematical theories like, for example, “class field theory”. In mathematics a set of axioms can be taken as a foundation and then an area for theoretical study is brought into being. For example, if one set of axioms is specified and accepted we have the theory of rings while if another set of axioms is the foundation we have the theory of Moufang loops.
So, from a critical point of view, the theory of macro-economics of the Keynesians is like the theory of plane geometry without the axiom of Euclid that was classically called the “parallel postulate”. (It is an interesting fact in the history of science that there was a time, before the nineteenth century, when mathematicians were speculating that this axiom or postulate was not necessary, that it should be derivable from the others.)
So I feel that the macroeconomics of the Keynesians is comparable to a scientific study of a mathematical area which is carried out with an insufficient set of axioms. And the result is analogous to the situation in plane geometry, the plane does not need to be really flat and the area within a circle can expand hyperbolically as a function of the radius rather than merely with the square of the radius. (This picture suggests the pattern of inflation that can result in a country, over extended time periods, when there is continually a certain amount of gradual inflation.)
The missing axiom is simply an accepted axiom that the money being put into circulation by the central authorities should be so handled as to maintain, over long terms of time, a stable value.

Effectively solving collusion is another possible solution in regard to Ideal Poker.

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