Dyadic Belief Modelling

Let A be a person, and A(B) be a description of person B as given by A.

A(A) will be very close to A, because of A's familiarity with and privileged access to particulars of A's character. Often, deviations between A(A) and A seem like evolutionarily adaptive non-conscious self-deception, producing mistaken inferences which support systematically distorted conclusions of one's own competence or status, which mistaken beliefs make belief-aligned social impression management efforts more persuasive in the minds of others.

Differences between A(B) and (B) will be mostly due to A's unfamiliarity with B, and access to particulars of B's character being costly for A, but there are also distortion effects from evolutionarily-politically-motivated social perception of others,  and distortions of convenience from using resource-inexpensive mental models.

At two levels deep, A(B(A)) corresponds to this situation: we go to A and say, "We asked B for a description of you. What do you think they said?".  A's answer, A(B(A)), might be thought of as properly contained within A(B) - since B's beliefs about A are features of B, but it's also sometimes useful to maintain the distinction. The distortions of A(B(A)) from A(A) will stem from A's guesses of what particulars B knows, or does not know, or misbelieves about A, relative to A(A). Of course, we expect that A(B(A) will be a multiply distorted version of A, but there are also a few improbable opportunities for A(B(A)) to be a more accurate account of A than either A(A) or B(A): A could inaccurately fill in details from A(A) which are unfamiliar to B, and A could incorporate B's presumed-mistaken but actually-accurate beliefs about A, correcting for A's self-misperception.

A(B(B)) is A's account of B's self-misperceptions. It's also properly contained within A(B). Not much to see here.

A(A(A)) is A's account of A's self-misperceptions, properly contained within A(A). Differences between A(A(A)) and A(A) should probably all be due to failures of logical closure; a deductively productive mind can be mistaken about many of their attributes (so that A(A) differs from A), but their beliefs should probably not be mistaken about those beliefs themselves (so that belief in belief contracts to belief, and A(A(A)) becomes equivalent to A(A), and not merely contained within it). Likewise, A(A(B)) should contract to A(B).

At 3rd level, if A thinks that B can contract belief in belief, then A(B(B(A))) becomes A(B(A)), and A(B(B(B) becomes A(B(B)). Of A's eight possible 3rd-level representations of the two people, the only ones which are not trivially contracted are A(B(A(A))) and A(B(A(B))), both of which are properly contained within A(B).

 The first kind is useful for correcting self misperception: Alan muses to himself, "Bettie and Barb say that I under-estimate myself as a programmer, so maybe I'm better than I think," and then contracts his belief in belief with an update.

The second kind, A(B(A(B))) could be elicited in this way: we go to Alan and say, "We asked Bettie to describe herself as you would describe her. What did she say?". Alan thinks of Bettie, A(B), and how she thinks of herself, A(B(B)), and what she knows about Alan, or does not know, or misbelieves (relative to A(A)), which is A(B(A)), ... and then he segfaults and asks you to repeat the question.

"We asked Bettie to describe herself as you would describe her. What did she say?"

Alan starts describing Bettie as he would normally describe her, making small omissions where he thinks Bettie thinks he doesn't know about her, or small substitutions where he thinks Bettie misperceives herself and doesn't know what Alan thinks, and other substitutions where he thinks that Bettie thinks his beliefs are mistaken about her relative to her B(B), when in fact he and Bettie agree, and probably other things, I don't know. This is probably useful for impression management, but I don't want to think about it any more today.