I What would the pdf look like for 'N' chess computers with the same rating

[iVenky]

If we have 'N' advanced computers (where N-> infinity) each with exactly the same rating to begin with and make them play with each other for an infinite number of games, what would the shape of the probability density function of the ratings eventually look like?

 

[Keith_McClary]

Are they reset to the same factory setting before each game?

 

[hutchphd]

I find this an interesting question partially because it is rather vaguely worded. For instance are the computers identical? If not how can their ratings be exactly the same? If not how does one quantify this? Would the result depend upon the tournament rules?
I guess that in the proposed infinite limit the ratings would end up the same and equal..

 

[iVenky]

Are they reset to the same factory setting before each game?

You mean the ratings?

 

[iVenky]

I find this an interesting question partially because it is rather vaguely worded. For instance are the computers identical? If not how can their ratings be exactly the same? If not how does one quantify this? Would the result depend upon the tournament rules?
I guess that in the proposed infinite limit the ratings would end up the same and equal..

Yes, it was a shower thought. The computers are exactly identical. So you mean it would be a uniform distribution?

 

[Keith_McClary]

You mean the ratings?

I mean the computers. Are they reset before each game? Or perhaps they are AIs that learn from experience?

 

[iVenky]

I mean the computers. Are they reset before each game? Or perhaps they are AIs that learn from experience?

Ah , I see, that's an interesting question that I didn't think of. Let's consider both.

1) If computers don't update their neural settings after every game
2) If computers learn after each game and update their settings.

 

[jedishrfu]

Given identical digital computers with identical software and barring any hardware glitches. They should by definition act in the same way.

The only variation would be when each started to process a task. If they used their clock to initialize any random seed generators then they could wildly vary in how they compute some probabilistic spread due solely to the random seed generator and anything that depended on the generator as well.

 

[pbuk]

By the central limit theorem, normal.

 

[BWV]

Deepmind, to my knowledge, has not published data on the millions of training games played by AlphaZero - that would be an interesting real life example. Do not need separate computers, can do these games within the same software.

But this would not be the same as the identical version of Stockfish playing itself, where I would guess the outcomes are set, as the program is not 'learning' as Deepmind does

 

[iVenky]

By the central limit theorem, normal.

That's what I am not sure of, if it would be normal or uniform.

 

[hutchphd]

I think by the central limit theorem the distribution of ratings would trend (as N gets very large) to an ever sharper spike at the initial exactly similar rating.
I realize I don't really know how the rating is assigned so this may be incorrect (but the central limit trend is true for some reasonable measure).
So perhaps this is not so interesting after all!

 

[pbuk]

That's what I am not sure of, if it would be normal or uniform.

Uniform within what range? How would you account for the discontinuities at the bounds of that range?

 

[BWV]

As ELO measures wins in zero sum games, ignoring draws and white vs black , the distribution of wins for identically skilled chess players would look like fair coin flips, so what is the distribution of ELOs for an infinite number of coin flippers?

 

[Office_Shredder]

As ELO measures wins in zero sum games, ignoring draws and white vs black , the distribution of wins for identically skilled chess players would look like fair coin flips, so what is the distribution of ELOs for an infinite number of coin flippers?

I think elo rewards the lower rated person more points for winning than it takes away when they lose, so it's a random walk that drifts back towards the starting point.

 

[Stephen Tashi]

So you mean it would be a uniform distribution?

What do mean by "it"? Define the random variable ( random vector?) that you are asking about.

Are you asking about a vector representing the ratings of each machine as a sequence of round robin tournaments proceeds? A uniform distribution on such vectors would imply that a vector of ratings that were "highly unequal" would have the same probability as a vector of ratings that were all equal.

 

[iVenky]

What do mean by "it"? Define the random variable ( random vector?) that you are asking about.

Are you asking about a vector representing the ratings of each machine as a sequence of round robin tournaments proceeds? A uniform distribution on such vectors would imply that a vector of ratings that were "highly unequal" would have the same probability as a vector of ratings that were all equal.

I mean the ratings of the computers