The overall purpose of this monograph is to integrate and critically evaluate the existing literature in the area of optimal joint savings population programs. The existing diverse presentations are all seen to be discussions within a unified framework. The central problem is to compare the desirability of alternative inter-temporal sequences of total savings and population sizes. Of critical importance is whether one regards persons as the fundamental moral entities or whether one takes Sidgwick's viewpoint that something good being the result of one's action is the baSic reason for dOing anything. The latter viewpoint is consistent with defining a complete social preference ordering over these alternative sequences. Since part of one's interest is to evaluate the consequences of various ethical beliefs a com parative study of several such orderings is presented; in particular the Mill-Wolfe average utilitarian, and Sidgwick-Meade classical utilitarian) formulations. A possible problem with the social preference ordering approach is that the ordering may indicate the desirability of increasing the population size, if this increases the total amount of good, even though people may receive less than the welfare subsistence level of consumption. However, there are other ways of evaluating actions and, if persons are the fundamental moral entities, then perhaps these actions should be evaluated by their implications for the rights of individuals i. e. people Who are currently alive, people who one can predict will exist in the future (e. g.

This collection of articles aspires to be a permanent record of ideas which are likely to become important determinants in the future of management sciences. These papers were initially presented at the first session on Multiple Criteria Decision Making QMCDM) organized under the auspices of The Institute of Management Sciences (TIMS). All works were prepared by leading spokesmen for three generations of OR/MS change agents. Special mention must be made of the dynamic role which Professor Martin K. Starr played in organizing the program of the TIMS XXII International Meeting. In May, 1973, Professor Starr, who was President of TIMS and Program Chairman of the Kyoto conference, requested me to chair the MCDM session. Throughout the long period of formative inter change, Dr. Starr demonstrated his full and continuing support of both the event and the MCDM field. On July 25, 1975, surrounded by the rocky gardens of the Kyoto International Conference Hall (KICH), located on the shore of Takaraga Ike, we engaged in a day-long discussion of MCDM. Our "talk together in Kyoto" was a professional experience of the highest intensity for participants, speakers and audience alike.

Fixed-point algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundary-value problems. Since Scarf's pioneering work [56,57] on obtaining approximate fixed points of continuous mappings, a great deal of research has been done in extending the applicability and improving the efficiency of fixed-point methods. Much of this work is available only in research papers, although Scarf's book [58] gives a remarkably clear exposition of the power of fixed-point methods. However, the algorithms described by Scarf have been super~eded by the more sophisticated restart and homotopy techniques of Merrill [~8,~9] and Eaves and Saigal [1~,16]. To understand the more efficient algorithms one must become familiar with the notions of triangulation and simplicial approxi- tion, whereas Scarf stresses the concept of primitive set. These notes are intended to introduce to a wider audience the most recent fixed-point methods and their applications. Our approach is therefore via triangu- tions. For this reason, Scarf is cited less in this manuscript than his contri- tions would otherwise warrant. We have also confined our treatment of applications to the computation of economic equilibria and the solution of optimization problems. Hansen and Koopmans [28] apply fixed-point methods to the computation of an invariant optimal capital stock in an economic growth model. Applications to game theory are discussed in Scarf [56,58], Shapley [59], and Garcia, Lemke and Luethi [24]. Allgower [1] and Jeppson [31] use fixed-point algorithms to find many solutions to boundary-value problems.