Pragmatic Information, Computation, and the Efficient Market Hypothesis

Abstract

We address a long standing gap in standard information theory, namely the inability of that theory to assign a measure to the amount of meaning in a transmitted message. \cite{Weinberger24} argues for a particular quantitative measure of meaning that \cite{Weinberger24} calls pragmatic information, which has many of the properties expected of it. We then prove that the amount of pragmatic information of a given message that can be extracted by a given receiver depends on the computational capacities of the receiver, in particular, the receiver's ability to recognize symbol strings at various levels of complexity within the Chomsky hierarchy of formal languages. A string may appear essentially random to a given receiver, but not to a receiver at a higher level in the hierarchy; hence, this receiver may not be able to extract any pragmatic information from such a string, even though another receiver at the appropriate level in the hierarchy could. Also, the maximum processing rates at which messages of different levels of complexity serve as a kind of pragmatic channel capacity, leading to a tradeoff between the amount of pragmatic information extracted and the extraction time. We then propose a re-framing of the efficient market hypothesis of quantitative finance to that of a participant-specific ``computational efficiency'', {\it i.e.} the claim that participants lack the computational resources necessary to use available pragmatic information to ``beat the market''. Given that market participants vary widely in computational resources, it is therefore no surprise some participants will find a given market computationally efficient, even though others will find inefficiencies. We argue that this situation will persist even in the face of any conceivable increase in compute.