order of quantifiers in a compuational model of the market

[José Manuel Rodríguez Caballero]

Dear FOM members,
  The Efficient Markets Hypothesis [1] was introduced by Louis Bachelier in
1900 in his doctoral thesis “The theory of speculation” (Henri Poincaré was
his supervisor). The strong form [2] of this hypothesis can be formulated
as: all available information that is relevant to investment decisions is
in the asset prices.

We can formalize the market in the framework of process calculus, e.g.,
pi-calculus [3]. Every investor is an agent, and the market is also an
agent. Investors send messages to the market specifying their trade, and
the market sends messages containing asset prices to investors.

Notice that the model of the market presented is deterministic, the
randomness that an investor perceives in the asset prices is the result of
its lack of knowledge of the internal structure of the other investors.
This is not the mainstream approach in mathematical finance, which is based
on stochastic calculus [4]. Now, let's consider the following assertion.

Definition 1. We say that the market will break if, eventually, all the
investors will stop trading (either because they don't have money or
because they are afraid to lose money).

Remark on Definition 1. The market will break means that the market, as a
deterministic algorithm, will halt, in the sense of the halting problem.

Claim A. Let n be a positive integer. There exists an investor Y such that,
given any market M and any set of n investors X_1, X_2, …, X_n, the market
will break.

Claim B. Let n be a positive integer. Given a market M and a set of n
investors X_1, X_2, ..., X_n, there exists another investor Y (depending on
the n previous investors and the market) such that the market will break.

My question is: are claims A and/or B true?

Personally, I am skeptical about claim A and enthusiastic about claim B.

I think this problem is about the theory of computability, and it can be
reformulated in a purely mathematical way, after eliminating all the
financial concepts and accepting some hypotheses. This looks like a problem
in information-theoretical cryptography.

Finally, we can imagine versions of claims A and B in which computational
complexity is added to the statements. Nevertheless, as a first approach, I
think it is better to ignore computational complexity for the moment and
focus on the information theoretical aspect of this toy model.

Kind regards,
Jose M.


References:
[1] Weatherall, James Owen. The physics of wall street: a brief history of
predicting the unpredictable. Houghton Mifflin Harcourt, 2013.
[2] Jordan, James S. "On the efficient markets hypothesis." Econometrica:
Journal of the Econometric Society (1983): 1325-1343.
[3] Milner, Robin (1999). Communicating and Mobile Systems: The π-calculus.
Cambridge, UK: Cambridge University Press. ISBN 0-521-65869-1.
[4] Shreve, Steven E. Stochastic calculus for finance II: Continuous-time
models. Vol. 11. New York: springer, 2004.
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