In Game Theory, No Clear Path to Equilibrium

This shouldn&#x27;t be surprising to anyone who knows that Kakutani&#x2F;Brouwer&#x27;s fixed-point theorem (the basis of Nash&#x27;s formulation of game theory and General Equilibrium Theory in neoclassical economics) was proved non-constructively. It assumes that there&#x27;s a continuous endomorphism in a convex space with no fixed points, and shows that you can use such a function to retract to the boundary of that space [1][2]. However, the boundary itself is full of fixed points, and this shows by contradiction that fixed points must exist for continuous endomorphisms. By presuming the non-existence of the fixed points in such a proof, you can&#x27;t use it to arrive at any fixed point. Nash uses a clever argument to show these fixed-point equilibrium strategies exist in games, and Debreu and Arrow use a similar argument in the space of preferences to find equilibrium prices. [3] Unfortunately, these theories have been used to promote the myth that markets are self-regulating despite all evidence to the contrary. [4][5]<p>[1] Lawvere - Conceptual Mathematics<p>[2] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Brouwer_fixed-point_theorem#A_proof_using_homology" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Brouwer_fixed-point_theorem#A_...</a><p>[3] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Arrow%E2%80%93Debreu_model" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Arrow%E2%80%93Debreu_model</a><p>[4] <a href="http:&#x2F;&#x2F;coin.wne.uw.edu.pl&#x2F;mbrzezinski&#x2F;teaching&#x2F;HE4&#x2F;BlaugFormalistRevolution.pdf" rel="nofollow">http:&#x2F;&#x2F;coin.wne.uw.edu.pl&#x2F;mbrzezinski&#x2F;teaching&#x2F;HE4&#x2F;BlaugForm...</a><p>[5] Weintraub. Stabilizing Dynamics: Constructing Economic Knowledge

A more precise title for article would be: <i>&quot;In Game Theory, No Clear, Universal, Fast Path to Equilibrium&quot;</i>.<p>And I&#x27;m not an expert in either field, but the article seems go steer pretty close to P=?NP. The article seems to acknowledge that &quot;brute force&quot; communications is a generic, universal way to solve every game, which could “take longer than the age of the universe”, thus being “completely useless, of course.”<p>On the other hand, a lot of games have &quot;additional structure that greatly reduces the amount of information each player must communicate&quot;, so you can apply simplifications to solve them. In other words, use heuristics.

TL:DR, if you don&#x27;t know the rules of the game, you cannot solve it.<p>A bit more detail: Most games make use of some form of utility function. A utility function generates numeric values for a given outcome. Utility functions are one of the sticky human aspects of games that are difficult to accurately model. This is often papered over by making assumptions that a players utility functions are linear, monotonic, or identical.<p>For example, where applicable, the monetary value of an outcome is often used in place of a players utility function. However, it is well documented that people&#x27;s utility functions with respect to money are usually both non-linear, and non-monotonic.<p>If you don&#x27;t know enough about the utility function of a player in a given game, there is very little you can infer about the structure of optimal strategies.

This is more of practical importance than it looks on yhe first glance. Take software development, for example. End users, marketing, sales, stakeholders and tech people are all playing a complicated game. Let&#x27;s assume everyone knows what he wants. Virtually no one is interested in withholding information, but communicating it somewhat efficiently is next to impossible. If a mediator-based equilibrium was more archievable in short timeframe (and you could prove that), it will benefit everyone.

On a similar note, recently a lot of research has looked at the extent to which utility functions fail to explore a space, and how the introduction of novelty is a crucial new strategy:<p>&quot;Novelty search is a recent algorithm geared toward exploring search spaces without regard to objectives. When the presence of constraints divides a search space into feasible space and infeasible space, interesting implications arise regarding how novelty search explores such spaces. &quot;<p><a href="http:&#x2F;&#x2F;ieeexplore.ieee.org&#x2F;document&#x2F;7061317&#x2F;?reload=true" rel="nofollow">http:&#x2F;&#x2F;ieeexplore.ieee.org&#x2F;document&#x2F;7061317&#x2F;?reload=true</a>