COMPARING FINITE MECHANISMS, WITH AN APPLICATION TO EXCHANGE ECONOMIES


                                L. Hurwicz and T. Marschak


					ABSTRACT


This paper obtains finite analogues to propositions that a previous
literature obtained about mechanisms whose possible messages form a continuum. 

Upon reaching an equilibrium message, to which all persons "agree", a
mechanism obtains an action appropriate to the organization's environment.
Each person's private knowledge of the environment enters her agreement rule.

Continuum mechanisms are unrealistic because transmitting every point of a
continuum is impossible. Messages have to be rounded off and the number of
possible messages is finite. Moreover, reaching a continuum mechanism's
equilibrium message typically requires infinite time. That difficulty is
absent if the number of possible messages is finite.

The question therefore arises whether results about continuum mechanisms
have finite counterparts. If we measure a continuum mechanism's
communication cost by its message-space dimension, then our corresponding
cost measure for a finite mechanism is the (finite) number of possible
messages. 

We find that if two continuum mechanisms yield the same action but the first
has higher message-space dimension, then a sufficiently fine finite
approximation to the first has larger error than an approximation to the
second, where both approximations use the same number of (equilibrium)
messages. "Error" is the largest distance between the continuum mechanism's
action and the finite mechanism's action. We obtain bounds on error.

We apply these results to exchange economies with quasi-linear utilities.
The continuum literature's findings about the Walrasian mechanism's
informational superiority reappear when we replace continuum
exchange-economy mechanisms by their finite approximations. In particular, a
finite approximation to the Walrasian mechanism has lower error for a given
number of messages than a finite approximation to a direct revelation
mechanism whose allocation is (like a Walrasian allocation) Pareto-optimal,
or is even Walrasian itself.

While the paper deals with information-processing costs and not incentives,
it is related to the incentive literature, since the Revelation Principle is
central to much of that literature and one of our main results is the
relatively high cost of finite direct revelation mechanisms.