In my research for my earlier post (Re: Nothing happens in the
Universe of the Everett Interpretation) I found the following:
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
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
"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. I see this as the natural reaction to the
implications of the idea of incomplete knowledge. We see a nice
discussion of this here:
"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
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. 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
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
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.
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to firstname.lastname@example.org.
To unsubscribe from this group, send email to
For more options, visit this group at