Dear Friends,

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-Fran├žois 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 <> 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. 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 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. 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 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.



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