Stephen,
The article illustrates the dangers of a premature modellization of a
problem. The urge to have a mathematical model forces to narrow the thing
to be modellized and to isolate it artificially from a wider context that
is crucial for the understanding of the problem. The result is a fine
model, but a model that do not bring light on the outcome of the game, Just
gives a pretty mathematical notation for a infinite regression that may be
expressed in a single paragraph. The problem is that
this paralyzing infinite regression never appear in the real game with
humans.

I will show how social interactions can not be avoided if we want to to
explain the real outcomes of the  prisoner dilemma, and that the dilemma
can be explained by  social and moral or economical  or algorithmical
reasoning, rather than purely mathematical, because the problem involver so
many variables that mathematics are useless. The social and moral can be
reduced to economical and even algorithmical, but at the cost
of simplifications that make the  game unreal and uninteresting. I will
descend to the economical, for the shake of understanding of the social and
moral reasons above them. Still a single round of prisoner dilemma can not
be explained in a framework that obviates these levels.

the infinite regression of beliefs about beliefs that the (ideally
rational) players of the prisoner dilemma have to solve can be cut if we
consider that they are involved in many simultaneous game plays like the
prisoner dilemma. This set of game plays is what is called "life in
society" social life is a set of game plays where trust is the exchange
currency .  Each player have many bets in many games, he most of the time,
can choose who to play with. There is a simple way to know who is worth to
play with: There are the ones whose life depend on us and viceversa, The
more the other is with us and separated is from others, the more sure that
this one will collaborate.

 A way to ascertain this is to have a previous history of collaboration,
but if this still does not exist, an initial investment is necessary.
The bootstrap for a newcomer in a community of collaborators, must start by
leaving a certain quantity  of social currency in the basket of the
community in a way that can be seen by all. This is done as follows: The
newcomer must perform a sacrifice that imposes a significative cost for
himself. The higer the sacrifice, the more strong is the collaboration.
Sacrifices are not only to kill a cow of his own  or to study the Talmud,
or to accept membership forever or confront a fatwa,   but to insult and
fight the fans of a rival rock band, or to demote creationists in a public
weblog, or to wear weird clothes and piercings  This is the initial payment
and the periodic payments for being accepted in the group of collaborators.
The more doubts a persons raise, the higher the sacrifice it has to perform
to convince the collaborators.

 These are the payments to the group, that act as an insurance company.
That is the price for being sure that the other of the group will
collaborate with you. This explanation can not be reduced to math, or the
reduction is useless. Sitll, this explanation at the social, moral,
economical and even religious level is necessary for explaining a single
REAL outcome of the prisioner dilemma. And in the process of that, find
explanations for very weird phenomena for which maths are not, nor ever
will be, an appropriate modelization tool (except perhaps, less interesting
simplifications).

Why textual explanation are more efficient in complex contexts than
mathematics? because the human language make use of our intuitive
understanding, that has implicit and innate game theoretical reasoning.
this is because we are expert game players, we are paying everyday.  On the
other side, Math has to make explicit the complications of the game that
are occult in a phrase, that  otherwise, intuition would gasp inmediately.

 We can descend to math, but we must not remain on math if we would like to
understand the real world.


2012/11/23 Bruno Marchal <marc...@ulb.ac.be>

>
> On 22 Nov 2012, at 19:26, Stephen P. King wrote:
>
>  Dear Friends,
>
>     In my research for my earlier post (Re: Nothing happens in the
> Universe of the Everett Interpretation) I found the following:
>
> From http://en.wikipedia.org/wiki/Hierarchy_of_beliefs
>
> Hierarchy of beliefs
>
> "Construction by Jean-François Mertens and Zamir implementing with John
> Harsanyi's proposal to model games with incomplete information by supposing
> that each player is characterized by a privately known type that describes
> his feasible strategies and payoffs as well as a probability distribution
> over other players' types.
>
> Such probability distribution at the first level can be interpreted as a
> low level belief of a player. One level up the probability on the belief of
> other players is interpreted as beliefs on beliefs. A recursive universal
> construct is built—in which player have beliefs on their beliefs at
> different level—this construct is called the hierarchy of beliefs.
>
> The result is a universal space of types in which, subject to specified
> consistency conditions, each type corresponds to the infinite hierarchy of
> his probabilistic beliefs about others' probabilistic beliefs. They also
> showed that any subspace can be approximated arbitrarily closely by a
> finite subspace.
>
> Another popular example of the usage of the construction is the Prisoners
> and hats puzzle. And so is Robert Aumann's construction of Common knowledge
> (logic)."
>
>   I think that we can identify the concept of a "privately known type"
> with the concept of 1p as discussed in the UDA. My concept of a "reality"
> as *that which is incontrovertible for some finite collection of 1p* can
> also we seen as synonymous with Aumann's Common 
> Knowledge<http://en.wikipedia.org/wiki/Common_knowledge_%28logic%29>which was 
> mentioned above:
>
>     "Common knowledge is a special kind of knowledge for a group of
> agents. There is common knowledge of p in a group of agents G when all the
> agents in G know p, they all know that they know p, they all know that they
> all know that they know p, and so on ad infinitum."
>
>     Bruno seems to use a radically flattened version of this idea - as do
> most logicians - where the 'players' have complete knowledge (ala Bp&p) of
> each other.
>
>
> ?
> Bp & p is not complete knowledge. Dt is true, but BDt & Dt is false, for
> example.
>
>
>
>
> I see this as the natural reaction to the implications of the idea of
> incomplete knowledge. We see a nice discussion of this here:
> http://plato.stanford.edu/entries/nonwellfounded-set-theory/harsanyi-type-spaces.html
>
>     "Type spaces are mathematical structures used in theoretical parts of
> economics and game theory. The are used to model settings where agents are
> described by their *types*, and these types give us “beliefs about the
> world”, “beliefs about each other's beliefs about the world”, “beliefs
> about each other's beliefs about each other's beliefs about the world”,
> etc. That is, the formal concept of a type space is intended to capture in
> one structure an unfolding infinite hierarchy related to *interactive
> belief*.
>
> John C. Harsanyi (1967) makes a related point as follows:
>
> It seems to me that the basic reason why the theory of games with
> incomplete information has made so little progress so far lies in the fact
> that these games give rise, or at least appear to give rise, to an infinite
> regress in reciprocal expectations on the part of the players."
>
>     My claim is that we can avoid the problems of infinite regress by both
> taking the possibility of regress seriously and understanding that beliefs
> are the result of processes that involve the consumption of resources.
>
>
> You need a linear logic for this, but you cannot impose it, you have to
> get it from the material hypostases, so as to be able to distinguish the
> quanta and the qualia. if not you are just doing physics, and lost the
> relation with the mind body problem, as formulated by the UDA.
>
>
>
> No activity of a mind -thought- can be said to occur without the
> occurrence of work. Thought is an activity and does not just 'exist'.
> Existence per se is property neutral, therefore to speak of the content of
> thought we must not ignore the requirements of discovering what  it might
> be. The same solution to the homunculus 
> problem<https://groups.google.com/forum/?fromgroups=#%21searchin/everything-list/homunculus/everything-list/G8z8J0iYCGk/-TqrVQU8b7QJ>that
>  Roger and I have discussed previously applies: if there is a finite
> quantity of resources or, equivalently, physical work involved in the
> implementation of a homunculus then there can only be a finite number of
> them.
>     There is no problem in the infinite case that Bruno is considering
> because it can be collapsed with Kleene's theorems, but in so doing we make
> the possibility of distinguishing one mind from another (within a plurality
> of minds) vanish.
>
>
> ?
>
>
>
>
> The measure problem that Bruno complains about is the direct result of
> assuming perfect knowledge aka omniscient mind. This is a soluble problem
> but to find solutions we must give up the idea of accessible prefect
> knowledge.
>
>
> I do not complain on the measure problem. It is what all everythingers
> agree on. Then I show that comp makes the problem both:
> 1) equivalent with deriving physics from arithmetic/computer science,
> 2) completely formulable in arithmetical terms, by using the traditional
> definition of knowledge.
>
> Bruno
>
>
>  http://iridia.ulb.ac.be/~marchal/
>
>
>
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-- 
Alberto.

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