% addencum to hayek piece dealing with jordan et al
The question of how to measure the communications cost
of economic equlibration has been addressed by some
neo-classical economist, see \cite{Jordan} for a review of
this literature. The conceptual basis by which they
measure this cost differs somewhat from our own. They
conceptualise the economy as being characterised by
a set of agents each of whom may emit one or more
messages. The receipt of these messages by other agents
causes them to adjust their activity in such a way
as to bring the system into equilibrium.
These messages are assumed to be real valued variables,
and, taken in aggregate the set of possible messages
sent by all of the agents forms a Euclidean vector space.
The measure of informational cost of the system is taken
to be proportional to the dimension of the vectors.
This method of definition is highly abstract, and one
encounters problems if one tries to concretise it.
From an information theoretic standpoint, to treat
messages as real valued variables is to accord them
an infinite information content. If each message requires
an infinite bit string, it then makes little sense
to compare costs in terms of the number of such infinite
strings required to achieve a task.
This however, is a relatively minor problem, since their
theoretical work can almost certainly be recast in terms
of messages defined over a finite subset of the
integers. A more serious problem is whether the dimension
of the message vector is the appropriate metric to use.
In the work of Mount, Reiter and Jordan, each agent has
a response function that takes as a parameter the message
vector in the current time-step in order to calculate
what the appropriate action in the next timestep should
be. This implies that each agent must sent messages
to each other agent, so that the total number of messages
sent must be proportional to the square of the number
of agents. If we consider the simple case where the
agents each emit a single integer scalar as their message,
then the number of messages sent will be proportional
to the square of the dimension of the message vector.
Thus, in using the dimension of the message vector,
rather than its square, as a metric, the authors seriously underestimate
the amount of information that would have to be
transmitted in their model of a decentralised economy.
Were this a realistic model, it would if anything
demonstrate the impossibility of any large scale competitive
economy because of the highly non-linear cost functions
associated with the number of agents.
This applies most obviously to the total number of letters, telexes, or
email messages that would have to be sent. In addition,
it implies that each individual agent must spend a number
of person hours proportional to the number of other agents
in the economy processing their incomming mail.
The lack of realism in their models stems from two factors,
\begin{enumerate}
\item The idea that information can somehow be broadcast to
all participants in a single operation.
\item The idea that each agent must process messages from
all others.
\end{enumerate}
We have attempted to be both more realistic, and more conservative
in our estimates of the informational costs of the market economy,
since we explicitly count all individual messages sent, and
only compell a firm to accept information from its suppliers
and customers. Given these assumptions, which are much more
favourable to the market economy than those of Jordan, the
number of messages we take into account is lower bound on
what must actually occur. In particular we explicitly omit
all messages associated with the payment and clearing of
cheques between bank accounts.
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Jordan J. S, The Informational Requirements of Local Stablility,
in "Information Incentives and Economic Mechanisms", edited
by Theodore Groves, University of Minessota Press, 1987, pp 183-211.