Can this kind of intuition be theoretically formalized in a reasonable way? It seems like an interdisciplinary study might be necessary, since we’re dealing with an effect which is no less sociological than mathematical. Do game theorists do these kind of things successfully? I know the notion of common knowledge is developed and used in the game-theory literature (Aumann, Milgrom,…), but does the formalization describe reality in a satisfactory manner? My intuition is that in order to understand the behavior of the stock market, sociological tools must be applied.

About issues related to common knowledge, one reference I particularly like is Steven Pinker’s paper, The Logic of Indirect Speech. (Also see the blog (in Hebrew) מדע בזיוני).

For completeness, here’s an excerpt from the Pinker paper mentioned above, where he gives a brief description of what common knowledge means (page 5; references are available in Pinker’s paper):

”
[...] a concept that linguists, logicians, and economists have called common knowledge, mutual knowledge, and common ground (2,9,
25–30). In common knowledge, not only does A know x and B knows x, but A knows that B knows x,and B knows that A knows x, and A knows that B knows that A knows x, ad infinitum. As with other phenomena in linguistics in which a person is said to ‘‘know’’ an infinite number of things, the knowledge is not enumerated as an infinite list, of course, but is implicit in a finite recursive formula. In this case, it could be the formula y:
‘‘Everyone knows x, and everyone knows y’’(2). Moreover, common knowledge can be ascertained perceptually, by observing that x is perceptible or broadcasted in public circumstances.

The paradigm illustration of common knowledge is the story of the Emperor’s New Clothes. When the boy called out,
‘‘The emperor is naked!’’ he was not telling the onlookers anything they didn’t already know. Yet he was conveying knowledge nonetheless: Now everyone knew that everyone else knew, and that everyone else knew that they knew, and soon, and that common knowledge licensed the people to challenge the dominance relationship commanded by the emperor. The moral for the present theory is that language is an efficient way of generating common knowledge.
“

for Schur functions. We substitute and to get MacMahon’s generating function on the right. On the left we get pairs of weighted tableaux that can be merged into a plane partition. (This proof is due to Bender and Knuth.) It is not unreasonable to replace the right-hand side by

If we set and then we get MacMahon’s incorrect conjectured generating function for solid partitions. What about the left-hand side? If we expand in terms of Schur functions we get

where is the notoriously intractable “Kronecker coefficient,” i.e., the multiplicity of the irreducible character of in the tensor (or pointwise) product of and . Thus solid partitions are replaced by certain triples of tableaux weighted by . This is probably too subtle for most people’s taste, especially since we don’t end up with any kind of solid partition. However, it is a rather natural extension of the elegant proof of Bender and Knuth and suggests why solid partitions may remain forever intractable.

1) Algorithmic ideas that allow to do the same things quicker. (Especially notable are the “alpha and beta cuts” introduced in the 60s and described in the lecture.)

2) Better hardware (and possibly also better algorithms) which allow to evaluate more positions per second. Deep Blue was able to evaluate several millions positions per second. Junior is able to evaluate a few hundred thousands positions per second.

3) Methods to evaluate a position. Amir described only a fairly basic method and did not describe how modern position-evaluation takes place.

4) Heuristics in climbing the trees of possibilities. According to the lecture these heuristics enabled computer chess programs to become “tactical wizards”. Among the heuristics, regarding a move and a forced reply as a single move, adding the illegal possibility of not making a move as part of the analysis, and more.

Ban noted the emergent sense of “intelligence’ based on much computational power and a few successful heuristics.

5) Improvements in openings and in end games.

Among the questions asked: “Q: Does the program involve opponent learning?” “A:no” (Perhaps some form of opponent independent learning is taking place.) Q: What about ‘Go’ ‘Bridge’ Poker’ ‘Canasta’?

Game theoretically, ‘Go’ is most similar to chess being a deterministic game with complete information. Programs playing Go are apparently weaker than programs for chess.