Randomness and Complexity in Social Explanation: Evidence from Finance and Bankruptcy Law
| Publication year | 2010 |
Georgia State University Law Review
Article 10
3-21-2012
Randomness and Complexity in Social Explanation: Evidence from Finance and Bankruptcy Law
Bernard Trujillo
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Recommended Citation
Trujillo, Bernard (2007) "Rand omness and Complexity in Social Explanation: Evidence from Finance and Bankruptcy Law," Georgia State University Law Review: Vol. 24: Iss. 4, Article 10. Available at: http://digitalarchive.gsu.edu/gsulr/vol24/iss4/10
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RANDOMNESS AND COMPLEXITY IN SOCIAL EXPLANATION: EVIDENCE FROM FINANCE AND BANKRUPTCY LAW
Bernard Trujillo*
Quantitative models are useful tools for understanding and explaining both natural and social systems. Models often include a term representing a random or stochastic element. Random terms are commonly deployed in modeling social phenomena such as economic, financial, and legal systems. This article contrasts conventional random terms in quantitative models with alternative terms supplied by the mathematics and science of complexity. This article argues that complexity modeling can explain many of the social phenomena that interest researchers. This article concludes with preliminary applications of complexity modeling in finance and bankruptcy law.
Introduction: What Dynamics Explain Social Forms?
We want to understand the dynamics that generate the things we observe. What are the rules, equations, interactions, or forces that produce objects and events in the world? A meteorologist wants to understand the forces that yield a storm or a still night. A financial scientist wants to understand the influences behind the daily movement of stock prices. And a student of legal systems wants to understand the forces that explain the diffusion of doctrine across space, or the rise and fall of legal forms throughout time.
* Professor, Valparaiso University School of Law. A.B. Princeton University; J.D. Yale Law School. Thanks to Jay Conison, Marc Galanter, JoEllen Lind, Benoit Mandelbrot, Clint Sprott, and Victoria Trujillo. Copyright © 2008 Bernard Trujillo.
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Static
"A"
Linear (periodic attractors)
Deterministic
Dynamic
on-deterministic (Random)
"B"
Non-linear (Chaos) (aofiriodic attractors)
"Wild" (Mandelbrot)
"D"
Brownian
Figure 1: Taxonomy1
Figure 1 is a rough taxonomy of the sorts of dynamics that produce the things we see in the world.2 The initial division is between "static" systems, which do not change over time, and "dynamic" systems, which do.
The category of dynamic systems divides into "deterministic" systems and "non-deterministic" systems. Deterministic systems behave according to a specified set of rules or equations that determine the next state of the system based on the current state of the system. Suppose your rule is always to turn on your front-porch light only when both of your immediate neighbors have turned on their front-porch lights, and to turn your light off only when both of your neighbors have turned off theirs. If I know the rule and the
1. This is my own diagram, but leans on Strogatz and Sprott, both cited below.
2. The generation of this Figure relies on tables by Strogatz and Sprott. See Steven H. Strogatz, Nonlinear Dynamics and Chaos 10 (Westview 1994); Julien Clinton Sprott, Chaos and Time-Series Analysis 212 (Oxford 2003). The type of "wild" randomness denoted at level "C" is something of an intriguing wildcard, since it does not fit comfortably within the "point-to-point independence" definition of randomness set forth below. See infra Part I and note 34.
Trujillo: Randomness and Complexity in Social Explanation: Evidence from F 2008] RANDOMNESS AND COMPLEXITY IN SOCIAL EXPLANATION 915
current state of the lights on your street, I can predict the next state of your light.
Non-deterministic systems, on the other hand, exhibit state-to-state independence. Nothing in the arrangement of the system at time-one will determine the arrangement of the system at time-two. This sort of point-to-point independence is generally what we mean by "randomness."
Figure 1 lists two types of deterministic dynamics, along with the sorts of forms, or "attractors" that these dynamics produce. Linear deterministic systems ("A" in Figure 1) can be complicated systems of many parts, or they can be very simple systems with just a few parts. But every linear system is essentially modular - one can successfully analyze the system by breaking it down into parts and measuring each part separately. A linear system is no more or less than the sum of its parts.3 The out-product of linear systems is regular, or periodic.4
The other type of deterministic system listed in Figure 1 is "nonlinear."5 A nonlinear system ("B" in Figure 1) cannot be analyzed by breaking it into modules. Integral to the system is cooperation among, or competition between, variables making the nonlinear system always more (or less) than the sum of its parts.6 Characteristic of nonlinear systems is the emergence of new forms or behaviors that were not part of the initial system. Nonlinear systems are capable of generating "aperiodic" attractors, so-called because the trajectory of the attractor never repeats.
It is possible to predict the behavior of nonlinear systems in the very short term, but not much beyond that. Assuming we had perfect
3. Steven Strogatz, Sync: The Emerging Science of Spontaneous Order 50-51 (Hyperion 2003).
4. Strogatz notes that linear systems are incapable of rich behavior. Strogatz, supra note 3, at 51.
5. A chaotic system is a type of nonlinear deterministic system that exhibits sensitive dependence on initial conditions. Sprott, supra note 2, at 104 ("chaos is the aperiodic, long-term behavior of a bounded, deterministic system that exhibits sensitive dependence on initial conditions"). A common illustration of "chaos" is that a butterfly, flapping its wings in Brazil, can cause tornadoes in Texas. Chaotic systems are necessarily produced by nonlinear rules. It is also possible, however, for nonlinear rules to produce regular, periodic behavior (e.g. planetary motion).
6. Strogatz, supra note 3, at 50-51.
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knowledge of the system's governing equations and of all the variables in the system (heroic assumptions, indeed), we would be able, at time-one, to predict the state of the system at time-two. But even assuming heroic knowledge, we would probably be unable, at time-one, to predict the state of the system at time-three. And our ability to predict declines precipitously as the iterations of the system increase. Thus an entirely deterministic system can be (and often is) unpredictable as a practical matter.
Figure 1 also lists two possibilities for non-deterministic systems: "wild" (named as such by the mathematician Benoit Mandelbrot7 and denoted as "C" in Figure 1) and "Brownian" ("D" in Figure 1). I shall say more about these two types of randomness in Part I of this Article.
We can impose two further axes on Figure 1: predictability and capacity to generate complex structures or forms.
Predictability. The systems near the top of...
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