
In Game Theory, No Clear Path to Equilibrium - algui91
https://www.quantamagazine.org/in-game-theory-no-clear-path-to-equilibrium-20170718/
======
notthemessiah
This shouldn't be surprising to anyone who knows that Kakutani/Brouwer's
fixed-point theorem (the basis of Nash's formulation of game theory and
General Equilibrium Theory in neoclassical economics) was proved non-
constructively. It assumes that there'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'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]

[1] Lawvere - Conceptual Mathematics

[2] [https://en.wikipedia.org/wiki/Brouwer_fixed-
point_theorem#A_...](https://en.wikipedia.org/wiki/Brouwer_fixed-
point_theorem#A_proof_using_homology)

[3]
[https://en.wikipedia.org/wiki/Arrow%E2%80%93Debreu_model](https://en.wikipedia.org/wiki/Arrow%E2%80%93Debreu_model)

[4]
[http://coin.wne.uw.edu.pl/mbrzezinski/teaching/HE4/BlaugForm...](http://coin.wne.uw.edu.pl/mbrzezinski/teaching/HE4/BlaugFormalistRevolution.pdf)

[5] Weintraub. Stabilizing Dynamics: Constructing Economic Knowledge

------
ucaetano
A more precise title for article would be: _" In Game Theory, No Clear,
Universal, Fast Path to Equilibrium"_.

And I'm not an expert in either field, but the article seems go steer pretty
close to P=?NP. The article seems to acknowledge that "brute force"
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.”

On the other hand, a lot of games have "additional structure that greatly
reduces the amount of information each player must communicate", so you can
apply simplifications to solve them. In other words, use heuristics.

~~~
sjg007
Yes. People are looking at this.

[https://cstheory.stackexchange.com/questions/25148/why-is-
co...](https://cstheory.stackexchange.com/questions/25148/why-is-computing-
pure-nash-equilibria-np-complete)

~~~
ucaetano
Cool! Thanks for that, guess that my GT and complexity skills aren't that
rusty!

------
andrewprock
TL:DR, if you don't know the rules of the game, you cannot solve it.

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.

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's utility functions with respect to money are usually both non-linear,
and non-monotonic.

If you don'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.

~~~
ABCLAW
Player preferences aren't within the bounds of the rules of the game, though.

You can have 100% knowledge of the bounds of the game's rules and still have a
tremendous amount of difficulty in ascertaining player preference. The article
makes it clear that the process of iterative playing of computationally
complex decision games does not assure very significant approximation of
preference.

In other words, you need to do something more than just play in order to
reliably reach a place close to equilibrium.

~~~
andrewprock
On the contrary, the convergence properties of fictitious play
([https://en.wikipedia.org/wiki/Fictitious_play](https://en.wikipedia.org/wiki/Fictitious_play))
are well understood.

The key quote in the article is: "there’s no guaranteed method for players to
find even an approximate Nash equilibrium unless they tell each other
virtually everything about their respective preferences."

This is a layman's way of saying that "if you don't know the utility functions
of the players, it's hard to make computational inferences". The example in
the next paragraph with a game with 2^100 leaf nodes carries with it an
implicit assumption that the utility function for each of the leaf nodes is
arbitrary.

Compare this to the game of Go, a game which has in excess of 2^2000 leaf
nodes. The success of AlphaGo indicates that sheer size is not a barrier,
rather it is being able to parametrically express the utility function for
arbitrary nodes which is important.

~~~
ABCLAW
I don't think you've really understood, because your contradictory response
supports mine: Even when fictitious play is a good approximation for a real
system (and they aren't because firms do not adopt static strategies),
strategies do not always converge, which means iterative play does not
necessarily lead you to terminal equilibrium states.

But maybe we can salvage equilibria as a functional tool. Can these systems
arrive at 'almost equilibrium states?" Knowing this would still give us some
predictive oomph. This article states that even approximations might be out of
reach, but there are ways to speed the development of the metagame along.

On your other point, you've missed half of the article's content. It isn't
saying "if you don't know the utility functions of the players it's hard to
make computational inferences". It is saying that the amount of information
required to obtain the utility function of the players for even trivial games
is well beyond the scope of most systems we have, and accordingly the
assumption that even approximate equilibrium will be reached requires a good
independent rationale. This is a MUCH larger issue than the base computability
of the problem and jumps into the epistemology of economics and policy.

This is expanded upon because those assumptions regarding reaching equilibrium
are often used in justifying financial, policy and other decisions where
billions of dollars are at stake. See the Cournot and Bertrand competition
models and their successors for a view into how that faulty assumption can
lead you to terrible policy decisions.

------
hamilyon2
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'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.

------
lisa_henderson
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:

"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. "

[http://ieeexplore.ieee.org/document/7061317/?reload=true](http://ieeexplore.ieee.org/document/7061317/?reload=true)

~~~
alimw
Can you say more about the connection you make?

