Abstract
Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the so-called Random Field Ising model (rfim) provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilizing self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and that account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of rfim-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith’s invisible hand can fail badly at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria from being reached. As a theoretical challenge, the study of so-called “detailed-balance” violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.
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Notes
As a recent anecdotal, but pittoresque piece of evidence: the explosion of “love-locks” on the Pont des Arts in Paris since 2008.
The converse is in fact also true: discontinuous behaviour at the agent level may end up being smoothed out at the macro level!
See also the earlier insightful paper by Becker [29].
The paper Discrete choice with social interactions by Brock and Durlauf [28] has 1034 Google scholar citations at the time of writing.
Think for example of the tax-evasion culture in some countries, that has recently become an acute issue.
Any other i-dependent value could have been chosen, since this simply amounts to shifting the value of idiosyncratic field f i .
The distinction between the two interpretations will be discussed again in Sect. 6.1.
It is interesting to notice that in Kirman’s ant recruitment model [24], one rather has P(N +→N ++1)=P(N +→N +−1)=μϕ(1−ϕ) because change of opinions are supposed to happen during two-body “encounters” where one of the two convinces the second one to change his mind. In the present setting, encounters are not necessary.
Note that there our choice of J differs by a factor 2 from the usual convention.
In agreement with the above convention, f i large means that agent i is more prone to join, i.e. his threshold is lower.
These effects, strictly speaking, disappear in finite dimensional lattices because of nucleation. But the time scales associated with nucleation may be so large that these metastable states still have a real existence.
Peyton Young in [31] contrasts instrumental conformism, when it is beneficial to do what others do, to informational conformism, when we conform to what others do because it conveys information on best actions. The problem, we believe, is that often nobody really has any idea of what’s going on, so the “information” provided by the others actions is close to zero.
The rfim is actually well known to have a rather wide critical region, as emphasized in [41].
We emphasize again here that if J=0, or if the variations were due to different speeds v in different countries, one should observe κ=1. The way h and w are extracted from data is detailed in [48].
I personally know people who cashed their money just after Lehman’s default, and left their bank with a plastic bag full of bank notes.
In the following expression, 〈⋯〉 denotes an averaging over all positions \(\vec{R}\).
Note that here we neglect the immediate influence of others in the actual decision to vote, i.e. the JS i S j interaction term. See the discussion of this point in [101].
Taking the relevant inter-town distance to be ∼10 km and the time for opinions to get closer to be a few months, one can estimate D to be of the order of magnitude of a few hundreds km2 per year.
Other arguments, inspired by statistical physics, could go as follows [111]: imagine a choice consisting in two sub-choices concerning issues in disjoint sets \(\mathcal{A},\mathcal{B}\). These two sub-choices are furthermore assumed to be independent, i.e. the choice of an alternative α in \(\mathcal{A}\) does not impact the utility of any of the alternatives b in \(\mathcal{B}\), and vice-versa. The utilities are therefore additive in that case: U α⊕b =U α +U b . Since the sub-choices are independent, one should also have P α⊕b =P α ×P b . Looking for probabilities that depend on the total utility of the choice therefore selects the exponential form. One could also try to replicate the usual canonical construction of the Boltzmann weight, by arguing that a choice is never in isolation but interacts with many other choices, while the agent is only interested in the total utility. Sub-optimal choices (corresponding to a finite β) are allowed because they only have a small contribution to the total utility. However, these arguments are somewhat ad-hoc.
One should of course keep in mind that the time needed to reach full equilibrium might be very large, or even infinite if there is a phase transition and ergodicity breaking—something that requires the number of states to be infinite.
More rigorous work is needed on this whole issue; in particular on whether the existence of a transition in the Ising model is sufficient to ensure loyalty formation.
Interesting situations could occur when these two effects are in conflict, for example when overcrowding or saturation effects prevent full condensation.
This is one of the ambitions of the CRISIS project, see: http://www.crisis-economics.eu/home.
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Acknowledgements
I thank J. Batista, S. Battiston, E. Beinhocker, E. Bertin, G. Biroli, Ch. Borghesi, M. Buchanan, D. Challet, R. Chicheportiche, R. Cont, P. Contucci, D. Farmer, P. Jensen, S. Grauwin, C. Hommes, A. Kirman, J. Kurchan, F. Lillo, M. Marsili, M. Mézard, Q. Michard, J.P. Nadal, A. Orléan, L. Papaxanthos, M. Potters, J. Sethna, M. Tarzia, S. Wolf, M. Wyart, F. Zamponi and Y.C. Zhang for very useful conversations on these subjects over the years, and S. Fortunato and S. Redner for giving me the opportunity to gather my thoughts on the subject. I thank in particular J. Batista, G. Biroli, D. Challet, A. Kirman, M. Marsili, D. Sornette, S. Wolf and the referees for carefully reading the manuscript and giving useful advice to improve it. This work is part of the European project CRISIS. I would like to dedicate this paper to Alan Kirman, whose work has been extremely inspiring to me.
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Bouchaud, JP. Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges. J Stat Phys 151, 567–606 (2013). https://doi.org/10.1007/s10955-012-0687-3
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DOI: https://doi.org/10.1007/s10955-012-0687-3