Analysis of Conflict

To all those who bleat consistently about the implausibility of the Nash equilibrium, here's some words from Roger Myerson (heavily edited):

"A solution concept can be viewed as a mapping that determines for every game a set of mathematical descriptions of how players should behave...The goal of game theoretic analysis is to generate a solution concept that has the following two properties:

(a) For each prediction in the solution set, there exist environments where this prediction would accurately prescribe how rational, intelligent players would behave.

(b) For any prediction not in the solution set, there is no environment where this prediction would be an accurate description of how rational, intelligent players would behave.

[Note: an environment consists of all those things not modeled in the game.]

Call any solution that satisfies (a) a lower solution, and that satisfies (b) an upper solution. Ideally we want a solution concept to be both an upper and lower solution, i.e. an exact solution.

This is hard to figure out in practice...a lower solution excludes all unreasonable predictions but may include some reasonable predictions, while an upper solution includes all reasonable predictions but may include some unreasonable predictions...it may be best to think of a Nash solution as an upper solution than an exact solution."

This is from Myerson's 1995 book "Game Theory: Analysis of Conflict". Better than any other text I've read by miles.

Incidentally, there's a curious symmetry between the idea of an exact solution as stated above (a) + (b) and the two "fundamental" welfare theorems. It would appear to suggest some deep connection between the "goal of game theoretic analysis" and the goal of resource allocation...I dunno....

Ant Gods

My friend JC, over at uglybutbearable.blogspot.com has a very interesting post on what God could represent. He takes the example of ants in ant farm will attempt to make their "God" in their image - thus the idea of a God these ants will carry may have nothing to do with their "real" "God" - the owner of the ant farm.

I'm going to try taking this in another direction; to help point out the issues I see with the belief in a God - or at least, the notion of a God as it is popularly believed.

Imagine a 2-d ant, in fact imagine a colony of 2-d ants. Consider the world they live in to be a perfect sphere. Now being 2-d, it is inconceivable for these ants to understand a sphere, since their understanding of the world is going to be limited by the physical characteristics of the world as they see it. Although they live on a sphere, for the 2-d ant a sphere will be nothing but a bunch of circles. They might discover that the circles get smaller as you move up or down - but there is no up or down in their world.

This will become a mystery then, probably a mystery they will be unable to solve. Why do their circles get smaller at some places and larger at others? The change in circle size will probably also affect physical characteristics of their world. Ants that are believers will put this down to a "God" - a mysterious supernatural force that affects their world in innumerable ways.

Now, through a series of scientific advances they may chance upon a mathematical proof that claims a reconciliation of all the phenomena they observe lies upon the existence of a 3rd dimension, in which the sphere exists.

Impossible, most ants will say. "That's ridiculous! A 3rd dimension!" Of course this doesn't mean that it isn't true!

It does not also make it any less revelatory or supernatural than a belief in a God.

There are 2 ways of interpreting the above statement - (a) God is an easy concept that people rely on to escape their non-understanding of the world therefore religion halts science; (b) it does not matter what you call it as long as you seek to understand it.

Being no expert in theology, I have no idea what the consensus among religions is regarding understanding the world but I'd be surprised if the teachings in all the Holy books tell you not to seek understanding. In this way I see religion and science co-existing. It is a narrow view on what religion and science represent that I guess underlies most of the debate between religion and science.

But then again, it's hard to reconcile this with the apparent statement in the Bible that the world is 5000 years old, for example. Or that there were flying saucers during the time Ram was king.

As far as I'm concerned however, as long as you maintain a curiosity about the world and a basic humility regarding the extent of our knowledge, there does not seem to exist any room for a debate. In this light, the problem that I see is not with religion/God per se, it is the interpretation of it - this is our Holy Book, and it is completely correct, so don't challenge it - that is the problem. Of course this is not to say that scientific beliefs are easily challenged either, but rather that they are allowed to be and that too in a systematic fashion, more or less.

The beauty of a logical proof demonstrating an aspect - physical or psychological or both - of the world and that helps generate insight - that beauty you can call God. A God, that works in a less mysterious way.

Of course there is the question - why do we find beauty in a mathematical proof? (Well some of us anyway). Is there is a "understanding the world" gene within us, a la Richard Dawkins' Selfish gene?

Old World

Sometimes you forget how good the good ol' stuff was:

The only place where you can dream, living here is not what it seems.
Ship of white light in the sky, nobody there to reason why.
Here I am, I'm not really there, smiling faces ever so rare.
A let's walk in deepest space, living here just isn't the place.

Stalks of light come from the ground, when I cry there isn't a sound.
All my feelings cannot be held, I'm happy in my new strange world.
Shades of green grasses twine, girls drinking plasma wine.
A look at love, a dream unfolds, living here, you'll never grow old.

Listen here

then listen to this

There's something very much in the spirit of the futuristic worlds conjured up by the best Science Fiction writers - philosophical debates of what it means to be human in a distant unfamiliar world is one way of summing it up...? There is pervading sense of loneliness through all of this work, and in the first song as well.

Which reminds me, I need to see Moon.