Mathematics for Economics: A Statistical Mechanics Perspective

by Pierluigi Contucci and Francesca Romiti

Will the noble (and sometimes snobbish) queen of sciences mathematics have a role in the future study of economics? Will the role it plays (if any) be as crucial as in the physical sciences? We argue that mathematics will very likely have a pivotal beneficial mutual exchange with economics, particularly through the study of statistical mechanical models of complex systems.

In recent times, we have witnessed a large-scale economic turmoil whose future is hard to predict. The crisis has so deeply involved the world's population that it is mentioned constantly in the media and is having a daily influence on government agendas. People's reactions and opinions are as diversified as their experience of these difficult matters.

How can mathematics be of help in such a situation? The spectacular success of mathematics in the physical sciences is based on a long interaction between theory and experiment, with trial and error procedures and multiple cycles of feedback. Reality is quantitatively 'understood' when a theory (whose language is mathematics) is able to deduce observed phenomena from a small number of simple principles and predict the output of new experiments. When even a single experiment contradicts the theoretical predictions, the whole machinery must be modified at the cost of replacing some of the principles with new ones.

Economics, however, has followed an apparently different path. On the one hand, the large amount of available data has only begun to come under serious consideration in the last century. The discovery that the tails of the probability distributions of price changes are generally non-Gaussian is quite a recent achievement. On the other hand, the axiomatic method of deductive science has been applied without performing proper feedback checks with observations: the principles of rationality of economic agents, market efficiency etc, have prospered within some schools of economics more like religious precepts than scientific hypotheses.

Yet testable and predictive theories have appeared in economics. The study by D. McFadden (2000 Nobel Laureate in Economics) on the use of the San Francisco BART (Bay Area Rapid Transit) transportation system is a celebrated example. It is interesting to notice that from a mathematical point of view, this work is equivalent to the Langevin theory of a small number of types of independent particles. However, when applied to cases in which the particles or agents interact and the resulting peer-to-peer effects play a more substantial role, that theory turns out to be inefficient.

There are several factors responsible for the delay in the advent of the scientific method within the economical sciences. The intrinsic difficulty of its topics and the gap between these and the available mathematical techniques is one. Until a few decades ago, mathematics only treated simple models with translation or permutation invariance. From the point of view of statistical mechanics, only uniform interactions were understood. However, as the physicist Giorgio Parisi likes to phrase it, science has become more robust and the theory of complex systems has made enormous progress. Among the things that have been learned is how to treat systems in which imitative and counter-imitative interactions occur, and where interactions themselves are generally random variables and are related to novel topological properties.

The challenge we now face is to fill the gap between phenomenological and theoretical approaches. Data analysis must increase in depth and in particular must follow a theoretical guide. Performing an extensive search of data without having an idea of what to hunt for is an illusion no less dangerous than the search for principles without regard for experiments. In the same way, the refinement of the theoretical background of economics must work in parallel towards data searching and analysis. The group of the Strategic Research Project in Social and Economical Sciences at the University of Bologna is working on these themes (see Link 1). Among the approaches being explored is the use of statistical mechanics to extend the McFadden theory to include interacting systems. There are good indications that this approach could lead to interesting results. First, it has the potential to include sudden changes in aggregate quantities even for small changes of the external parameters, as happens in an economic crisis (Link 2). Second, it may eventually make use of the complex systems theory of spin glasses, whose applicability to economics is well understood (Link 3). Third, it has built into it the capability to include acquaintance topologies, especially those that have been observed in network theory like the small-world and scale-free (Link 4).

Mathematics can provide a substantial contribution in the crucial phase of checking that questions are well posed and then solving them. It is obvious that new mathematical instruments will be necessary and that the process of developing them will be lengthy. An early phase in which mathematics will be involved is the so-called 'inverse problem'. Unlike physics, where interactions between agents have generally been established by pre-existing theories, in the realm of economics effective interactions should be deduced from data, possibly at a non-aggregate level. From a mathematical point of view, the computation of interaction coefficients from real data is a statistical mechanics inverse problem, a research setting to which many fields of science are turning their attention. The inverse problem solution is structurally linked to the monotonic behaviour of observed quantities with respect to parameters (Link 5).

For the time being, the simple economics models considered in mathematics and derived from theoretical physics look like rough metaphors of reality. Nevertheless, they are able to describe some features of the observed phenomena and are in any case a necessary step towards a more refined approximation of reality.

Finally, the attempt to provide soluble and tractable mathematical models for economics will be an important opportunity to fertilize mathematics itself with new paradigms and to develop new parts of Galileo's 'language of nature'.

Links:
(1) http://www.dm.unibo.it/~contucci/srp.html
(2) http://arxiv.org/abs/0810.3029
(3) http://arxiv.org/abs/0904.0805
(4) http://arxiv.org/abs/0812.1435
(5) http://arxiv.org/abs/cond-mat/0612371

Please contact:
Pierluigi Contucci
Bologna University, Italy
Tel: +39 051 2094404
E-mail: contuccidm.unibo.it