Tag Based Co-operation in Artificial Societies

Ph.D. Thesis - David Hales 2001

Error in chapter 9

It has been pointed out to me by serveral readers that that probablistic analysis given in chapter 9 is not correct. Specifically Eq. 9.6 and 9.7 are not correct. Hence a more detailed analysis seems to be required that takes account of the tag-group dynamics in some way. This is still an open issue.
Comments kindly supplied by Tamas Vinko, Rameez Rahman, Hiroto Yonenoh (see e-mail below)

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from hyonenoh@sk.tsukuba.ac.jp
to David Hales
date 9 October 2009 11:09
subject Re: About "Cooperation Without Memory or Spaace:Tags,Groups and the Prisoner's Dillemma"

Dear. Dr. David Hales,

I've read chapter 9 of your thesis I got at:


I've learned a lot from your thesis. Thank you!

I have a question.
Equation (9.6) in page 169 reads:

pti(n) = 2/(n-1) ... (9.6),

It seems Eequation (9.6) represents a sort of probablityprobability.
But, if n = 2, pti(n) = 2, which cannot be a probability.

Is there some restriction for the value of n?

Is pti(n) something other than probability?

equation (9.7) in page 169 reads:

pmoti(n,m) = pti(n)*pmo(n,m) ... (9.7)

pmo(n,m) is the probability that "more than one agent( n > 1 )" will
be mutated into a co-operator over a single generation.

On the other hand, pti(n) is the probability that "those two ( and only
two n = 2 ) agents" will interact producing a mutually co-operative

In other words, while pmo(n,m) requires "n > 1 ", pti(n) requires "n = 2
So, I'm afraid that equation (9.7), which represents that pmoti(n,m) is
computed by multiplying pmo(n,m) by pti(n), isn' t valid, if n does not
equal to 2.

These months, I have tried to derive the expected average number of
generations required before two cooperative agents perform a game

It would be grateful if you kindly give me your advice.

Yours sincerely.


Hiroto Yonenoh
Graduate School of Systems and Information Engineering, University of
Tennoudai 1-1-1, Tsukuba, Ibaraki 305-0006, Japan
e-mail: hyonenoh@sk.tsukuba.ac.jp

Last Updated: Sept 2010