Comparing finite mechanisms |
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Authors: | Leonid Hurwicz Thomas Marschak |
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Institution: | (1) Department of Economics, University of Minnesota, Minneapolis, MN 55455, USA (e-mail: hurwicz@tc.umn.edu) , US;(2) Walter A Haas School of Business, University of California, Berkeley, CA 94720, USA (e-mail: marschak@socrates.berkeley.edu) , US |
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Abstract: | Summary. This paper obtains finite analogues to propositions that a previous literature obtained about the informational efficiency
of 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 privately observed characteristic
(a part of the organization's environment) enters her agreement rule. An example is the Walrasian mechanism in an exchange
economy. There a message specifies a proposed trade vector for each trader as well as a price for each non-numeraire commodity.
A trader agrees if the price of each non-numeraire commodity equals her marginal utility for that commodity (at the proposed
trades) divided by her marginal utility for the numeraire. At an equilibrium message, the mechanism's action consists of the
trades specified in that message, and (for classic economies) those trades are Pareto-optimal and individually rational. Even
though the space of environments (characteristics) is a continuum, mechanisms with a continuum of possible messages are unrealistic,
since transmitting every point of a continuum is impossible. In reality, messages have to be rounded off and the number of
possible messages has to be finite. Moreover, reaching a continuum mechanism's equilibrium message typically requires infinite
time and 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 equilibrium 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 of the first has larger error than an approximation of the second if the cost of the first approximation
is no higher than the cost of the second approximation. An approximation's “error” is the largest distance between the continuum
mechanism's action and the approximation's action. We obtain bounds on error. We also study the performance of Direct Revelation
(DR) mechanisms relative to “indirect” mechanisms, both yielding the same action, when the environment set grows. We find
that as the environment-set dimension goes to infinity, so does the extra cost of the DR approximation, if the error of the
DR approximation is at least as small as the error of the indirect approximation. 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 informational inefficiency of finite Direct Revelation mechanisms.
Received: May 21, 2001; revised version: December 14, 2001
RID="*"
ID="*" Earlier versions of this paper were presented at the Decentralization Conference, Washington University, St Louis,
April 2000 and at the Eighth World Congress of the Econometric Society, August 2000. We are grateful for comments received
on those occasions. The second author gratefully acknowledges support from National Science Foundation grant #IIS9712131.
Correspondence to: T. Marschak |
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Keywords: | and Phrases: Mechanisms Mechanism design Information processing Exchange economies |
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