The Source of Some Paradoxes from Social Choice and Probability

This paper offers a powerful, simple method for understanding many "paradoxes" in social choice and probability theory. The approach is a geometrical one; the underlying principle emerges from a wide variety of examples ranging from elections and agenda manipulation to gambling and conditional probabilities

Symmetry and Extensions of Arrow's Theorem

By emphasizing the symmetry of certain set theoretic conditions, shown to be associated with Arrow's Impossibility Theorem, a characterization of "kinds of axioms" is obtained. More precisely, if the defining properties of a model satisfies these conditions, then the model must have a conclusion much like that of Arrow's theorem. Because the conditions are described in set theoretic terms, the applicability of these results extends beyond the usual setting of complete, binary, transitive rankings to space of utility functions, probability distributions, etc. In this manner, not only can new extensions of Arrow's theorem be obtained, but it is shown how the same "kinds of axioms" applies to, say, problems about the aggregate excess demand function, the Hurwicz-Schmeidler dictatorial result about Pareto optimal, Nash equilibria, the Gibbard-Satterthwaite theorem about manipulability, etc

The Optimal Ranking Method in the Borda Count

This is a comprehensive investigation of the Borda count method, which is shown to come out fairly well when compared with other ranking methods. This research was carried out in collaboration with the System and Decision Sciences Program

The Case Against Bullet, Approval and Plurality Voting

The author pursues his inquiry into the paradoxes of voting systems and shows that bullet, approval and plurality voting can have most unlooked-for consequences. This research was carried out in collaboration with the System and Decision Sciences Program

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory

Strategic and behavioral decomposition of games

k1 x k2 x .... x kn games are uniquely decomposed into their strategic and behavioral parts. The strategic part contains all information needed to determine a specified strategic outcome (e.g. Nash, Quantal Response Equilibria (QRE); the behavioral portion is what motivates cooperative approaches such as tit-for-tat, side payments, etc. the decomposition reduces complexity; e.g., the space of 2x2 games is reduced from R^8 to points in a three-dimensional cube. Applications include characterizing all 2x2 games, finding which Nash (or QRE) structures accompany certain classes of k1 x ... x kn games, and how results concerning two and three player tit-for-tat strategies differ

The Lack of Consistency for Statistical Decision Procedures

Simpson's paradox exhibits seemingly deviant behavior where the data generated in independent experiments support a common decision, but the aggregated data support a different outcome. It is shown that this kind of inconsistent behavior occurs with many, if not most, statistical decision processes. Examples are given for the Kruskal-Wallis test and a Bayesian decision problem. A simple theory is given that permits one to determine whether a given decision process admits such inconsistencies, to construct examples, and to find data restrictions that avoid such outcomes

The Lack of Consistency for Statistical Decision Procedures

Simpson's paradox exhibits seemingly deviant behavior where the data generated in independent experiments support a common decision, but the aggregated data support a different outcome. It is shown that this kind of inconsistent behavior occurs with many, if not most, statistical decision processes. Examples are given for the Kruskal-Wallis test and a Bayesian decision problem. A simple theory is given that permits one to determine whether a given decision process admits such inconsistencies, to construct examples, and to find data restrictions that avoid such outcomes

Child Well-Being in Rich Countries: UNICEF’s Ranking Revisited, and New Symmetric Aggregating Operators Exemplified

In a report published in 2007 UNICEF measured six dimensions of child well-being for the majority of the economically advanced nations. No overall scores are given, but countries are listed in the order of their average rank on the dimensions, which are therefore implicitly assigned ‘equal importance’. In this study we take ‘equal importance’ to mean that the final aggregation is symmetrical in the scores and the ranks, i.e. permuting them leaves the aggregate unchanged. We rank the countries by aggregating the numerical information using a variety of techniques, geared to the measurement scales we distinguish (‘ordinal’, ‘interval’, ‘ratio’). The aggregators are symmetrical and mildly demanding, emphasizing good performance across the board. The rankings obtained deviate from the UNICEF ranking, but not over-dramatically. Our purpose is not only to study alternative approaches for the particular data at hand, but also to introduce and exemplify new and useful aggregation techniques: we propose ways to select weights for OWA-operators and weighted geometric means, and we suggest how to circumvent the choice of a power for the power means. In addition we extend the Borda method so that it values dominance as well

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the 1-dimensional Newtonian gravitational three-body system (three masses on a line with $-1/r$ pairwise potential). We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change (with negative total energy). For two mass choices we have calculated 18000 full orbits (with initial states on a $100\times 180$ lattice on the Poincar\'e section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincar\'e maps for $10\times 18$ starting points. Our results show that the Poincar\'e section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: 1) There is a region of fast scattering, with a minimum of pairwise collisions and smooth dependence on initial values. 2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincar\'e section. For both 1) and 2) the initial and final states consists of a binary + single particle. 3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is the periodic Schubart orbit, whose stability turns out to correlate strongly with the global behavior.Comment: 24 pages of text (REVTEX 3.0) + 21 pages of figures. Figures are only available in paper form, ask for a preprint from the author