By Donald G. Saari
The following is a short reflection from the author of Mathematics Motivated by the Social and Behavioral Sciences, which was published by SIAM in 2018 as part of the CBMS-NSF Regional Conference Series in Applied Mathematics.
The title of my 2018 book, Mathematics Motivated by the Social and Behavioral Sciences, may seem like an oxymoron to many readers. But given that serious problems in the social and behavioral sciences confront us on a daily basis, the title actually reflects an important invitation for more mathematicians to get involved with this area of research. My book, which captures portions of my Conference Board of the Mathematical Sciences (CBMS) lectures on this general topic, identifies several relevant challenges. As I state in the preface:
“The mathematics needed to advance the social and behavioral sciences most surely differs from what has proved to be successful for the physical sciences. Remember, a strong portion of contemporary mathematics reflects a fruitful symbiotic relationship enjoyed by mathematics and the physical sciences over a couple of millennia: Advances in one area motivated advances in the other. As it must be expected, this intellectual relationship shaped some of our mathematics and influenced the way in which certain physical sciences are viewed. Centuries of experimentation in the physical sciences, for instance, led to precise measurements and predictions, which motivated the creation of mathematical approaches, such as differential equations, that allow precision predictions.
As fully recognized, it is unrealistic to expect “precise” predictions for many issues in the social and behavioral sciences. But researchers have access only to limited number of mathematical approaches, where favorite choices for theoretical models tend to involve methods designed for precision predictions — not much else is available. This comment underscores the need to develop appropriate mathematical tools that, rather than designed for exactness, reflect the current status for much of the social and behavioral sciences, which requires qualitative predictors.”
These sentiments lead to chapter one: “Evolutionary Game Theory.” The social and behavioral sciences are dominated by change. Everything changes: opinions, economics, politics, and preferences. Because the best way to model this phenomenon is unclear, researchers rarely examined “change” — until recently. Much like the story of a drunk who is searching around a streetlight for the keys he lost elsewhere because “the light is better here,” we tend to emphasize things that can be analyzed with currently available techniques. We seek results where there is sufficient “light,” such as attempting to find equilibria without any knowledge or exploration of the associated dynamics.
Multiple factors—including a lack of reliable information—hinder our understanding of how to model change. In many cases, that which is best known reflects behavior in specialized, local settings. The qualitative approach that I develop in the first chapter thus emphasizes the way in which one can connect local information with a global dynamic. Because we know so little about the dynamics, we should keep the emphasis on qualitative modeling — wherein refinements must come from the host area data.
Adam Smith’s invisible hand metaphor, which is a supply-and-demand aggregation process that combines the economic agents’ preferences and resources, serves as another example. The key word of “aggregation” is central across the social and behavioral sciences. Statistical methods, probabilistic predictions, migration, social movements, political processes, and so on all involve aggregations for which even reasonably correct assertions require sound methods. But to the best of my knowledge, there is no complete and general mathematical analysis that describes the potential pitfalls of aggregation rules and explores what can go right or wrong.
An obvious obstacle is the overwhelming number of dissimilar aggregation approaches that cloud the issue. One can handle this problem by embracing Occam’s razor, which in contemporary terms is the KISS philosophy (“Keep it simple, stupid”). My initial emphasis in chapter two was therefore to examine a particular aggregation class: voting methods. These methodologies are often linear aggregations, meaning that we can reasonably expect the successful transfer of any resulting lessons to more complicated settings. This is because a standard mathematical way of identifying a system’s characteristics involves the use of linear approximations of derivatives, tangent spaces, and so forth.
An examination of voting systems might seem mathematically trivial. After all, commonly used voting rules just sum ballots; what can go wrong? This question reflects my seriously mistaken initial attitude. A clue should have come from actual events with pundits wondering, “How did so-and-so win the election?” In fact, a telling measure of the intricacies of paradoxical outcomes is that one can use the complexity of chaotic dynamics to identify the characteristics, number, and types of these mysteries! The difficulties are mind boggling (a more detailed description is available in section 2.3 of my book); using a thousand of the fastest computers, it would be impossible to count (not even list) the plurality vote paradoxical outcomes that arise with only eight candidates — even if the counting had started at the Big Bang.
These troubling, unanticipated behaviors help identify unexpected properties of other aggregation tactics. Understanding paradoxical behavior in voting provides guidelines for the discovery of similar actions in aggregation methods, ranging from bizarre features of the aggregate excess demand function in Adam Smith’s supply-and-demand story to puzzling behavior in nonparametric statistics. Chapters three and four address a selection of these topics.
Now I’ll move to a different subject. When introducing vectors or eigenvectors to my students, I always confess that there are far too many vectors—even in just two dimensions—for one to intimately know them all. A convenient approach is to become acquainted with carefully selected choices, such as \(\textrm{i, j}\) or the eigenvectors, and then describe all other vectors in terms of their relationship to our newly acquired friends.
This commentary reflects the common mathematical methodology of dividing a construct into component parts to clarify the analysis. Although researchers apply aspects of this useful approach to differentiate features of observations in areas like psychology, it has not been generally adopted to address mathematical concerns in the social and behavioral sciences. The strong advantages of doing so are themes of chapters four and five. To ensure consistency in the described topics, my illustrating choices come from the first two chapters; I demonstrate how one can use symmetries to decompose voting rules and games (many of the game theory results involve joint work with Dan Jessie). Both decompositions significantly simplify the discovery of new conclusions.
The final chapter of my book addresses the customary reductionist approach; all readers are likely familiar with this whole-parts system analysis. This realistic approach handles a complex problem by dividing it into tractable parts, solving the questions that each part poses, and assembling the answers into a solution for the whole. Although it is widely used, the reductionist approach can suffer serious, unexpected problems. As outlined at the end of the concluding chapter, many of the complexities that I describe in my book reflect unanticipated consequences of this method. The positive angle is that understanding the source of the difficulties helps us identify the causes of many complexities that the social and behavioral sciences face. This description of “what can go wrong” extends to shed light even on problems from engineering and the physical sciences, such as the compelling dark matter mystery of astronomy. Comprehending the causes of problems focuses our attention in a search for resolutions.
My hope is that readers of this book will join me in exploring the mysteries of the social and behavioral sciences.
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Donald G. Saari is a distinguished research professor and director emeritus of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine. His research interests range from the Newtonian N-body problem to voting theory and evolutionary properties of the social and behavioral sciences.