No matter how surprising, outlandish, or even impossible it may seem, one of the next challenges of modern applied mathematics is the modeling of human behaviors. This has nothing to do, however, with the control of minds. Rather, thanks to its innate reductionism, mathematics is expected to help shed some light on those intricate decision-based mechanisms which lead people to produce, mostly unconsciously, complex collective trends out of relatively elementary individual interactions.
The flow of large crowds, the formation of opinions impacting on socio-economic and voting dynamics, the migration fluxes, and the spread of criminality in urban areas are examples which are quite different but have two basic characteristics in common: First, individuals operate almost always on the basis of a simple one-to-one relationship. For instance, they try to avoid collisions with one another in crowds, or they discuss with acquaintances or are exposed to the influence of media about some issues and can change or radicalize their opinions. Second, the result of such interactions is the spontaneous emergence of group effects visible at a larger scale. For instance, pedestrians walking in opposite directions on a crowded sidewalk tend to organize in lanes, or the population of a country changes its political inclination over time, sometimes rising suddenly against the regimes.
In all these cases, a mathematical model is a great tool for schematizing, simplifying, and finally showing how such a transfer from individual to collective behaviors takes place. Also, a mathematical model raises the knowledge of these phenomena, which is generally initiated mainly through qualitative observations and descriptions, to a quantitative level. As such, it allows one to go beyond the reproduction of known facts and face also situations which have not yet been empirically reported or which would be impossible to test in practice. In fact, one of the distinguishing features of human behaviors is that they are hardly reproducible at one’s beck and call, just because they pertain to living, not inert, “matter.”
As a matter of fact, historical applications of mathematics to more “classical” physics (think, for example, of fluid or gas dynamics) are also ultimately concerned with the quantitative description and simulation of real world systems, so what is new here? True, but what makes the story really challenging from the point of view of the mathematical research is the fact that to date we do not have a fully developed mathematical model for the description of human behaviors. The point is that the new kinds of systems mentioned above urge applied mathematicians to face some hard stuff, which classical applications have only marginally been concerned with. Just to mention a few key points:
- A nonstandard multiscale question. Large scale collective behaviors emerge spontaneously from interactions among few individuals at a small scale. This is the phenomenon known as self-organization. Each individual is normally not even aware of the group s/he belongs to and of the group behavior s/he is contributing to, because s/he acts only locally. Consequently, no individual has full access to group behaviors or can voluntarily produce and control them. Therefore, models are required to adopt nonstandard multiscale approaches, which may not simply consist in passing from individual-based to macroscopic descriptions by means of limit procedures. In fact, in many cases it is necessary to retain the proper amount of local individuality also within a collective description. Moreover, the number of individuals involved is generally not as large as that of the molecules of a fluid or gas, which can justify the aforesaid limits.
- Randomness of human behaviors. Individual interaction rules can be interpreted in a deterministic way only up to a certain extent, due to the ultimate unpredictability of human reactions. It is the so-called bounded rationality, which makes two individuals react possibly not the same, even if they face the same conditions. In opinion formation problems this issue is of paramount importance, for the volatility of human behaviors can play a major role in causing extreme events with massive impact known as Black Swans in the socio-economic sciences. Mathematical models should be able to incorporate, at the level of individual interactions, these stochastic effects, which in many cases may not be schematized as standard white noises.
- Lack of background field theories. Unlike inert matter, whose mathematical modeling can be often grounded on consolidated physical theories, living matter still lacks a precise treatment in terms of quantitative theories whence to identify the most appropriate mathematical formalizations. If, on the one hand, this is a handicap for the “industrial” production of ready-to-use models, on the other hand it offers mathematics the great opportunity to play a leading role in opening new ways of scientific investigation. Mathematical models can indeed fill the quantitative gap by acting themselves as paradigms for exploring and testing conjectures. They can also put in evidence facts not yet empirically observed, whereby scientists can be motivated to perform new specific experiments aiming at confirming or rejecting such conjectures. Finally, mathematics can also take advantage of these applications for developing new mathematical methods and theories. In fact, nonstandard applications typically generate challenging analytical problems, whereby the role of mathematical research as a preliminary necessary step for mastering new models also at an industrial level is enhanced.
Istituto per le Applicazioni del Calcolo “M. Picone”
Consiglio Nazionale delle Ricerche