Thomas Marschak

Restabilizing Responses, Inertia Supergames, and Oligopolistic Equilibria

I. Introduction, 71. — II. Convolutions: Response functions that preserve rationality, 73. — III. Inertia supergames and convolutions, 80. — IV. Conclusion, 89.

Communication Requirements for Individual Agents in Networks and Hierarchies

Stanley Reiter has been a pioneer in studying the informational requirements of resource allocation mechanisms. Reiter and Hurwicz were the first to show us how to work with one important measure of a mechanism’s informational cost, namely the number of variables communicated, or— more generally—the size of the mechanism’s message space (Hurwicz, 1972, 1977; Mount and Reiter, 1974). They developed techniques for determining the minimal message-space size if the mechanism’s outcome is to meet a specified standard. The techniques remain fundamental. The present paper, for example, is strongly influenced by them.1

On economies of scope in communication

A classic puzzle in the economic theory of the firm concerns the fundamental cause of decreasing returns to scale. If a plant producing product quantityX at costC can be replicated as often as desired, then the quantityrX need never cost more thanrC. Traditionally the firm is imagined to take its identity from a fixednon-replicable input, namely a ‘top manager’; as more plants or divisions are added, the communication and computation burden imposed on the top manager (who has information not possessed by the divisions) grows more than proportionately. Decreasing returns are experienced as the top manager hires more variable inputs to cope with the rising burden. Suppose it turns out, however, that when the divisions are assembled, and are given exactly the same totally independent tasks that they fulfilled when they were autonomous, then asaving can be achieved if they adopt a joint procedure for performing those tasks rather than replicating their previous separate procedures. Then the top managers rising burden must be shown to be particularly onerous—otherwise there may actually beincreasing returns.We show that for a certain model of the information-processing procedure used by the separate divisions and by the firm, there may indeed be such an odd unexpected saving. The saving occurs with respect to the size of the language in which members of each division, or of the firm, communicate with one another, provided that language is finite. If instead the language is a continuum then the saving cannot occur, provided that the procedures used obey suitable ‘smoothness’ conditions. We show that the saving for the finite case can be ruled out in two ways: by requiring the procedures used to obey a regularity condition that is a crude analogue of the smoothness conditions we impose on the continuum procedures, or by insisting that the procedure used be a ‘deterministic’ protocol. Such a protocol prescribes a conversation among the participants, in which a participant has only one choice, whenever that participant has to make an announcement to the others.The results suggest that a variety of information-processing models will have to be studied before the traditional explanation for decreasing returns to scale is understood in a rigorous way.

Mechanisms that efficiently verify the optimality of a proposed action

The best known achievement of the literature on resource-allocating mechanisms and their message spaces is the first rigorous proof of the competitive mechanisms informational efficiency. In an exchange economy withN persons andK+1 commodities (including a numeraire), that mechanism announcesK prices as well as aK-compenent trade vector for each ofN−1 persons, making a total ofNK message variables. Trial messages are successively announced and after each announcement each personprivately determines, usingprivate information, whether she finds the proposed trades acceptable at the announced prices. When a message is reached with which all are content, then the trades specified in that message take place, and they satisfy Pareto optimality and individual rationality. The literature shows that no (suitably regular) mechanism can achieve the same thing with fewer thanNK message variables. In the classic proof, all the candidate mechanisms have the privacy property, and the proof uses that property in a crucial way.‘Non-private’ mechanisms are, however, well-defined. We present a proof that forN>K,NK remains a lower bound even when we permit ‘non-private’ mechanisms. Our new proof does not use privacy at all. But in a non-private mechanism, minimality of the number of message variables can hardly be defended as the hallmark of informational efficiency, since a non-private mechanism requires some persons to know something about the private information of othersin addition to the information contained in the messages. The new proof of the lower boundNK invites a new interpretation of the competitive mechanisms informational efficiency. We provide a new concept of efficiency which the competitive mechanism exhibits and which does rest on privacy even whenN>K. To do so, we first define a class ofprojection mechanisms, wherein some of the message variables are proposed values of the action to be taken, and the rest are auxiliary variables. The competitive mechanism has the projection property, with a trade vector as its action and prices as the auxiliary variables. A projection mechanism proposes an action; for each proposal, the agents then use the auxiliary variables, together with their private information, to verify that the proposed action meets the mechanisms goal (Pareto optimality and individual rationality for the competitive mechanism) if, indeed, it does meet that goal. For a given goal, we seek projection mechanisms for which theverification effort (suitably measured) is not greater than that of any other projection mechanism that achieves the goal. We show the competitive mechanism to be verification-minimal within the class of private projection mechanisms that achieve Pareto optimality and individual rationality; that proofdoes use the privacy of the candidate mechanisms. We also show, under certain conditions, that a verification-minimal projection mechanism achieving a given goal has smallest ‘total communication effort’ (which is locally equivalent to the classic ‘message-space size’) among all private mechanisms that achieve the goal, whether or not they have the projection property.

Approximating a function by choosing a covering of its domain and k points from its range

Suppose that we would like to compute the value of a real-valued function cp for every point 8 that might be selected from some given set E, but that this is infeasible and that we have to settle for approximation in the following manner: a finite covering’ of E, denoted C and containing j sets-to be called cells-is chosen. To each cell of C we assign a real number, called an outcome, from a k-element set A C R of possible outcomes. Given the 6 that is currently of interest we find a cell containing that 8, and we take as our estimate of q(B) the outcome assigned to that cell. We define the error of this procedure as the supremum, over all 8 in E, of the possible distances between the outcome so obtained and the true value q(B). Our objective is to minimize error by a suitable choice of C, of A, and of the assignment rule. Now given C and A, the choice of assignment rule is a simple matter, It

Chapter 27 Organization design

Publisher Summary This chapter discusses an assortment of recent economic studies in the same way: as steps toward characterizing those organization designs that do well, according to some measure of gross performance, with the informational and administrative resources they require. The steps turn out to be diverse and modest, but the problem is difficult. Piecing the assorted contributions together, one is still far indeed from a unified theory of efficient organization design. The main stumbling block remains the modeling of technology and cost. Some elements of cost have been studied intensively and even elegantly. The chapter examines the Shannon theory in connection with transmission, and the theory of finite-state machines in connection with the assignment of output/state pairs to input/state pairs in one-step designs with memory. Techniques have been developed for the study of a designs gross performance, for example, the computation of expected payoff for a given team information structure, and these remain useful in efficiency studies when good cost models become available. The theory, even in its present form, has already been useful in revealing how difficult it is (1) to define certain widely current terms sharply and agreeably to most usages and (2) to verify certain widely held conjectures.