Donald G. Saari

The geometry associated with apportionment and voting methods enables us to “see" what difficulties will arise, why certain complexities are inevitable, how to create supporting examples, and identify similarities of the problems afflicting both areas. Although several new results become apparent, such as the harsh consequences for a method satisfying IIA and a replacement for the modified divisors that are used with monotone apportionment approaches, the emphasis is to use this picture book approach to explain why certain behaviors must happen.

Of importance when selecting a voting method is whether, on a regular basis, its outcomes accurately capture the intent of voters. A surprise is that very few procedures do this. Another desired feature is for a decision approach to assist groups in reaching a consensus (Sect. 5). As described, these goals are satisfied only with the Borda count. Addressing these objectives requires understanding what can go wrong, what causes voting difficulties, and how bad they can be. To avoid technicalities, all of this is illustrated with examples accompanied by references for readers wishing a complete analysis. As shown (Sects. 1–3), most problems reflect a loss of vital information. Understanding this feature assists in showing that the typical description of Arrow’s Theorem, “with three or more alternatives, no voting method is fair,” is not accurate (Sect. 2).

Of importance when selecting a voting method is whether, on a regular basis, its outcomes accurately capture the intent of voters. A surprise is that very few procedures do this. Another desired feature is for a decision approach to assist groups in reaching a consensus (Sect. 5). As described, these goals are satisfied only with the Borda Count (BC). Addressing these objectives requires understanding what can go wrong, what causes voting difficulties, and how bad they can be. To avoid technicalities, all of this is illustrated with examples accompanied by references for readers wishing a complete analysis. As shown (Sects. 1–3), most problems reflect a loss of vital information. Understanding this feature assists in showing that the typical description of Arrow’s Theorem, “with three or more alternatives, no voting method is fair,” is not accurate (Sect. 2). JEL Classification D71 D72

The derivation of a theory of systems engineering has long been complicated by the fact that there is little consensus within the systems engineering community regarding precisely what systems engineering is, what systems engineers do, and what might constitute reasonable systems engineering practices. To date, attempts at theories fail to accommodate even a sizable fraction of the current systems engineering community, and they fail to present a test of validity of systems theories, analytical methods, procedures, or practices. This article presents a more theoretical and more abstract approach to the derivation of a theory of systems engineering that has the potential to accommodate a broad segment of the systems engineering community and present a validity test. It is based on a simple preference statement: “I want the best system I can get.” From this statement, it is argued that a very rich theory can be obtained. However, most engineering disciplines are framed around a core set of widely accepted physical laws; to the authors’ knowledge, this is the first attempt to frame an engineering discipline around a preference.

The derivation of a theory of systems engineering has long been complicated by the fact that there is little consensus within the systems engineering community regarding precisely what systems engineering is, what systems engineers do, and what might con- stitute reasonable systems engineering practices. To date, attempts at theories fail to accommodate even a sizable

It is well known that decision methods based on pairwise rankings can suffer from a wide range of difficulties. These problems are addressed here by treating the methods as systems, where each pair is looked upon as a subsystem with an assigned task. In this manner, the source of several difficulties (including Arrow’s Theorem) is equated with the standard concern that the “whole need not be the sum of its parts.” These problems arise because the objectives assigned to subsystems need not be compatible with that of the system. Knowing what causes the difficulties leads to resolutions.

Altruistic preferences or the desire to improve the well-being of others even at one's own expense can be difficult to incorporate into traditional value and utility models. It is straightforward to construct a multiattribute preference structure for one decision maker that includes the outcomes experienced by others. However, when multiple individuals incorporate one another's well-being into their decision making, this creates complex interdependencies that must be resolved before the preference models can be applied. We provide representation theorems for additive altruistic value functions for two-person, n-person, and group outcomes in which multiple individuals are altruistic. We find that in most cases it is possible to resolve the preference interdependencies and that modeling the preferences of altruistic individuals and groups is tractable.

Lower one- or two-dimensional coordination, or potential games, are popularly used to model interactive behavior, such as innovation diffusion and cultural evolution. Typically, this involves determining the “better” of competing solutions. However, examples have demonstrated that different measures of a “good” choice can lead to conflicting conclusions; a fact that reflects the history of game theory in equilibrium selection. This behavior is totally explained while extending the analysis to the full seven-dimensional class of potential games, which includes coordination games.