Broken Symmetry

Symmetry and Entropy in Market Bubbles and Crashes

From Dion Harmon, Marcus A. M. de Aguiar, David D. Chinellato, Dan Braha, Irving R. Epstein, Yaneer Bar-Yam:

Predicting panic is of critical importance in many areas of human and animal behavior, notably in the context of economics. The recent financial crisis is a case in point. Panic may be due to a specific external threat, or self-generated nervousness. Here we show that the recent economic crisis and earlier large single-day panics were preceded by extended periods of high levels of market mimicry --- direct evidence of uncertainty and nervousness, and of the comparatively weak influence of external news. High levels of mimicry can be a quite general indicator of the potential for self-organized crises.

Via Physics arXiv blog.

What I like about this work is how at the conceptual level it fits with other theories of markets making phase transitions between high and low entropy states, with symmetries breaking as the transition begins and picks up steam.

There is a lot of talk about whether Quantitative Easing has done much. Folks like Scott Sumner seem to be arguing that it has not because market expectations were adjusted at the moment QE was announced (or quickly thereafer). I'm more skeptical about that claim because I imagine that it takes more than days or even weeks for organizations to propagate new information about prospective rates through their entire ranks, such that there remains an impedance mismatch between Fed signals and market reactions. But at an even more fundamental level, the market response to Fed signals does not appear to be linear within some regimes, including the one we live in now, which looks to me like a liquidity trap, albeit not homogeneously sticky for all industries and firms. (Consumer internet, in particular, seems to be attracting capital investment at an accelerated pace compared even with hardware and software solutions providers.)

Are Biological Systems Poised at Criticality?

From Thierry Mora and William Bialek:

The first question is why biological systems should be nearly critical. What benets does operation at this special point in parameters pace provide for these systems? For birds, we have seen that criticality confers high susceptibility to external perturbations, and this enhanced reactivity endows them with a better defense mechanim against predators. Similarly, in the auditory system, being close to a bifurcation point allows for arbitrarily high gains and accurate frequency selectivity in response weak sounds.

In neural populations, the naive idea underlying the theory of branching processes makes criticality seem almost inevitable, a middle point between death and epilepsy. However, the function of neural networks is not only to be reactive, but also to carry and process complex information in a collective manner through its patterns of activity. The observation and analysis of metastable states, both in retinal acitivity analyzed within the maximum entropy framework [20] and in the activity of cortical slices analyzed with the theory of branching processes [79], suggest that criticality may be coupled to the explosion of these states, allowing for a wider set of coding options.

Via the Physics arXiv blog.

Networks of Economic Market Interdependence and Systemic Risk

by Dion Harmon, Blake Stacey, Yavni Bar-Yam, and Yaneer Bar-Yam:

The dynamic network of relationships among corporations underlies cascading economic failures including the current economic crisis, and can be inferred from correlations in market value fluctuations. We analyze the time dependence of the network of correlations to reveal the changing relationships among the financial, technology, and basic materials sectors with rising and falling markets and resource constraints. The financial sector links otherwise weakly coupled economic sectors, particularly during economic declines. Such links increase economic risk and the extent of cascading failures. Our results suggest that firewalls between financial services for different sectors would reduce systemic risk without hampering economic growth.

Via Brandon Keim at Wired.

Distinguishing Exogenous from Endogenous Market Crashes

Simoleon Sense links to what could be a useful bit of anlaysis from Sornette et al.:

[W]e have provided an objective test of the hypothesis that the largest negative markets moves are special and form two distinct populations. The exogenous crashes can be attributed to extraordinary important external perturbations and/or news impacting the market. The endogenous crashes can be seen as the natural deaths of self-organized self-reinforcing speculative bubbles giving rise to specific precursory signatures in the form of log-periodic power laws accelerating super-exponentially.

Sornette applied a similar approach to analyzing YouTube video traffic here.

I'm slowly warming up to Sornette's idea of log-periodic scaling. The mathematical part of this model, at bottom, arises from a treatment of fractional dimensions as complex rather than real-valued quantities.

Renormalization Group in Macroeconomics

Should I be relieved that this was published in 1996?

A friend clued me in to the fact that macroeconomic theorists were applying the Ising Model a long time ago.

But how far have Prof. Aoki and his students and colleagues gotten with this?

Self-Organized Criticality in Neural Networks

From the New Scientist:

In 2003, John Beggs of Indiana University in Bloomington began investigating spontaneous electrical activity in thin slices of rat brain tissue. He found that these neural avalanches are scale invariant and that their size obeys a power law. Importantly, the ratio of large to small avalanches fit the predictions of the computational models that had first suggested that the brain might be in a state of self-organised criticality (The Journal of Neuroscience, vol 23, p 11167).

To investigate further, Beggs's team measured how many other neurons a single cell in a slice of rat brain activates, on average, when it fires. They followed this line of enquiry because another property of self-organised criticality is that each event, on average, triggers only one other. In forest fires, for example, each burning tree sets alight one other tree on average - that's why fires keep going, but also why whole forests don't catch fire all at once.

Via Omnivore.

Mario Rizzo's Reply to Robert Lucas

From a new favorite, the ThinkMarkets blog:

what Hayek makes clear is that it would not be too much to ask of a theory that it at least point in the general direction of a problem. Lucas, and others, continue to think of recent events as purely a bubble phenomenon — and so there is a degree of randomness involved in the phenonemon, in their view. However, Hayek’s cycle theory pointed to an interest rate effects on higher-order investments that generally or systematically produce over- and mal-investment. But it says little about timing. Lucas seems oblivious to both.

Hayek's vision of the macroeconomy is at least consistent with an explanation of bubbles as a self-organized critical (SOC) phenomenon. For an introduction for those with background in physical sciences or mathematics, see the short treatise by Jensen. What these phenomena have in common is a large system of N components (such as atoms, molecules, sand grains, trees in a forest) that are driven into more-ordered states by a process that acts slowly over time (such as a low-frequency magnetic force, temperature gradient, or growth of a forest) and into less-ordered states by a process that happens rapidly through a mechanism of dense interconnections between components (such as electron-electron interactions, or the spreading of a forest fire). Not surprisingly since we now know that no system is deterministic, SOC does not let you predict when the system will relax. What it does let you do is predict what statistics will apply to the dynamical processes, and how those statistics are affected by the microdynamical variables.

In the case of a macroeconomy, SOC theory would suggest that an increase in money supply that is slow in comparison to the rate of exchange among market actors would generally lead to an overall increase in wealth over some period of time followed by rapid relaxations (i.e., crashes) once the patterns of activity encouraged by cooperation and exchange within a period exhaust their social benefits.

Jamming transition in air transportation networks

From Lucas Lacasa, Miguel Cea, Massimiliano Zanin:

In this work we present a model of an air transportation traffic system from the complex network modelling viewpoint. In the network, every node corresponds to a given airport, and two nodes are connected by means of flight routes. Each node is weighted according to its load capacity, and links are weighted according to the Euclidean distance that separates each pair of nodes. Local rules describing the behavior of individual nodes in terms of the surrounding flow have been also modelled, and a random network topology has been chosen in a baseline approach. Numerical simulations describing the diffusion of a given number of agents (aircraft) in this network show the onset of a jamming transition that distinguishes an efficient regime with null amount of airport queues and high diffusivity (free phase) and a regime where bottlenecks suddenly take place, leading to a poor aircraft diffusion (congested phase). Fluctuations are maximal around the congestion threshold, suggesting that the transition is critical. We then proceed by exploring the robustness of our results in neutral random topologies by embedding the model in heterogeneous networks. Specifically, we make use of the European air transportation network formed by 858 airports and 11170 flight routes connecting them, which we show to be scale-free. The jamming transition is also observed in this case. These results and methodologies may introduce relevant decision making procedures in order to optimize the air transportation traffic.

The frequency spectrum of the "brownian" states in the middle obey a power law that has been measured in other systems exhibiting flow congestion. Consider the possibility that other behavior modeled as "brownian" is actually the result of self-organized criticality in flow diffusion. Although it's difficult to visualize, it's not entirely unreasonable to consider "brownian" market price signals as flow-limiting nodes in a network of consumption and production. In other words, market price signals are the rate-limiting step in synchronizing flows of cash and goods within an economy.

Very nice, clearly written paper.

Via physics arXiv blog

Robust dynamic classes revealed by measuring the response function of a social system

From Riley Crane and Didier Sornette:

We find that most videos’ dynamics (≈90%) either do not experience much activity or can be described statistically as a Poisson process (verified using a Chi-Squared test). This large set of data is consistent with the endo-subcritical classification. For the remaining 10% (≈500, 000 videos), we find nontrivial herding behavior that accurately obeys the three power-law relations described above. Characteristic examples of endogenous and exogenous dynamics are shown in Fig. 3.

Via Computational Legal Studies blog.

By what mechanism does traffic transition form Poisson to Power-law statistics? Click on the "syncronization" category at on the right to see some speculations. It seems that spatially dispersed clusters are syncing up. In the exogenous case, that's in response to a driving force -- new information. In the endogenous case, the mechanism is more subtle, and may be an example of self-organization. See the Zanette and Kuperman summary especially.

What is the Systems Theory of Economics?

Systems theory is an attempt to put Hayek, Coase, Williamson, North, and Ellickson on more quantitatively rigorous foundations. Systems theory is an umbrella term for physics models of networks and dynamical systems that are used to understand how interconnected flows can end up in stable or unstable local equilibria, spontaneously synchronize, and generally evolve over time in feedback loops.

Here are some of the basic hypotheses about human behavior that might be associated with systems theory:

• Individuals have preferences for goods that can be measured in units of flow (quantity consumed/produced per unit time).
• Individuals seek to optimize the fit between their flow preferences and available flow using available information about past and present available flow.

These together suggest that individuals can be modeled as a nexus of flows of cash and other goods. In turn, the interconnected streams of these flows lead to additional hypotheses about the behavior of the integrated system of flows.

• Feedback loops in flow among individuals can result in more spatially and temporally stable preferences by synchronization of flow.
• A nexus of synchronized flows may be a firm or a market depending on the structure of interconnections. (I.e., the ability to synchronize flow over time determines what structures will persist.)
• Firms that synchronize with higher-bandwidth can operate at lower cost, sell at higher margin, or both.
• Firms with scale-free architecture are the most cost-efficient, but also the most susceptible to catastrophic failure.
• (Perhaps most surprisingly) Under certain circumstances, individual preferences will spontaneously synchronize through market price signals.

There is an evolutionary mechanism at work here in that the nexus of flows that constitutes a firm is in competition with other nexuses of flows for additional flow. Over time, some network structures will survive better. And the success or failure of a particular network will, of course, have an impact on how the preferences of individuals within it will evolve over time. There are deep connections between systems theory and the evolutionary theory of psychologists like Mihaly Csikszentmihalyi and sociologists like Donald T. Campbell. But the kind of data and scale of the models used by systems theory is more traditionally associated with economics.

Much of systems theory has been anticipated by New Institutional Economics, and in some cases sociology and psychology. The work from Haidt and Ehrenreich on hive psychology and collective joy, for example, is consistent within broad contours of the physics models of self-organized synchronization. Robert H. Frank's work on positional goods can be understood in terms of the sustainability of certain integrated flow structures over long times. In general, Thomas Schelling seems to have had these sorts of models in mind in his writing, although he tended to focus on the potentially destructive outcomes after long periods of repeated interactions.

I'm going to try to turn each of these bullet points into a separate post over the next few weeks or months.

Large-scale structure of time evolving citation networks

By E. A. Leicht, Gavin Clarkson, Kerby Shedden, and M. E. J. Newman.

As the plot shows, there is a marked trend for the average age to increase in step with the passage of time. This is precisely the behavior one would expect if the top authorities in the network are remaining the same as time goes by. Every once in a while, however, the plot shows a sudden and precipitous drop in the average age, indicating that a much younger set of vertices have, in a short space of time, taken over as the new leaders in the authority score rankings. Thus the plot indicates a repeated pattern in the evolution of the network in which a certain set of vertices—certain cases considered by the Supreme Court—remain the top authorities for substantial periods of time before being swiftly replaced by a different set. One example of such a turnover can be seen in Fig. 8 around 1900 and a smaller one around 1940, dates that, as we have seen, correspond roughly to the beginning and end of the Lochner era. Another very large dip in the curve occurs around 1970.

Available here. The authors don't say so in their analysis, but the associated graph suggests a mechanism of self-organized criticality.

Sync and Self-Organized Criticality

From an extraordinary article by Barbra Drossel:

The results of this paper might have implications for excitable media (e.g. spreading of diseases, autocatalytic chemical reactions, propagation of electrical activity in neurons or heart muscles). These systems essentially have three states which are called quiescent, excited, and refractory. If there is an excitation, it spreads to the quiescent neighbors. After excitation, a region is refractory to further stimulation and needs some time to recover its quiescent state. The model discussed in this paper describes an excitable medium in the limit where excitation spreads fast on the scale of the refractory time, and where spontaneous excitation is rare. In cases where this limit can be realized experimentally, such a system should show either synchronization or SOC, depending on whether the transition from the refractory to the quiescent state and from the quiescent to the excited state are deterministic or stochastic.

In experiments, higher dimensional systems have given rise to spiral waves. One can't help wondering about the connections between Kolmogoroff's theory of turbulence and the critical structure of networks. Maybe the "angular momentum" of feedback loops within networks is analogous to vortices in a turbulent flow.

Self-organized Criticality and Commerce Clause Jurisprudence

Consider the description of the natural phenomenon of self-organized criticality (SOC) offered by Jensen:

"What kind of systems will evolve into a SOC dynamical state?  A separation of time scales is required.  The process connected with the external driving of the system needs to be much slower than the internal relaxation processes.  The prototypical example is an earthquake.  The stress in the earth's crust is built up on the scale of years owing to the motion of the tectonic plates.  The stress is subsequently released in a few seconds or minutes during an earthquake."

This called to mind the famous metaphor for markets offered by Judge Learned Hand in the Schecter Poultry case:

"In an industrial society bound together by means of transport and communication as rapid and certain as ours, it is idle to seek for any transaction, however apparently isolated, which may not have an effect elsewhere; such a society is an elastic medium which transmits all tremors throughout its territory; the only question is of their size."

Judge Hand here is describing SOC dynamics.  The metaphor may be more literal than he imagined.  For there is some evidence of SOC dynamics in markets.

The implication is that the firms interact with one-another in the market the same way that tectonic plates rub against one-another under the crust of the earth.  Most of the time, we're not even aware of the tension building.  But every once in a while, our whole economy gets a good shake.

Incidentally, the Schecter Poultry case was the last in a line of Supreme Court cases stretching back to Lochner v. New York (1905) in which the Supreme Court ruled unconstitutional various restraints of contract (such as minimum wage and maximum hour laws) enacted by Congress.  The end of the line came in West Coast Hotel Co. v. Parish (1937), when the Supreme Court upheld a minimum wage law in the shadow of threats by President Franklin Roosevelt to "pack the court" with additional justices who would be more favorable to his desire for a more powerful administrative state.  At the time, the President and Congress felt that expanded Executive and Legislative authority were necessary to avoid the market failures of the Great Depression.  Does this sound familiar?  We should attend to these details of history, since good intentions like these often lead to unanticipated consequences.

Fortunately, unanticipated consequences are both good and bad.  One uanticipated consequence of the expansion of Congressional authority under the Commerce Clause was the Civil Rights Act of 1964, which was probably the most important step toward reconciliation of racial conflicts in the United States since the end of the Civil War almost 100 years earlier.  But the opaque and labyrinthine bureaucracies of Federal administrative agency authority is net a more negative consequence.

Market prices synchronize the phase of supply and demand cycles.

Market price signals permit dispersed people to synchronize production and consumption activities.*

Broken Symmetry readers know that the aggregate supply and demand curves are cumulative distribution functions that can be derived from the frequency distributions that describe the rhythms of consumption or production for a given good or service.  Specifically, the frequency distributions show the patterns of consumption or production for a population of people within a window of time.  Within different time windows, the peaks of these frequency distributions may slide left or right, and narrow or fatten.  For example, we can observe the supply cycle sliding right to some maximum mean frequency and then sliding left again.  As another example, we can observe the demand cycle sliding right to a higher maximum mean frequency when the good consumed is certain kinds of information (such as books or music).

Often, the frequency distributions in supply and demand will take a poissonian shape because the decision by each person within the population to consume or produce will often be uncorrelated with the decisions of other members of the population.  Other distribution shapes may arise, especially within smaller populations and shorter time windows.

Both frequency distributions (i.e., both the frequency distribution for supply and the frequency distribution for demand) correspond to a set (or "ensemble") of time-varying functions.  These two ensembles of time-varying functions (in supply and demand, respectively) together provide the the information needed to calculate the time-varying function of price.  What follows is a plain-English explanation of the dynamics.

Static Theory and the Cobweb Model

To visualize what is happening in time, it helps to start with an easier case, which has already been solved.  What I will here call "static theory" obtains at the limit where, for a given population within selected successive time windows, the frequency distributions (and hence supply and demand curves) do not change.  More specifically, they vary less from time window to time window than would be required to increase or decrease equilibrium price more than the transactions costs for exchange within that market.  In other words, if the supply or demand curves don't shift enough from window to window to increase or decrease equilibrium price more than transactions costs, we're at a "static" limit.

From the cobweb model, we know that when the static limit obtains, price will evolve as a damped, driven oscillation around the equilibrium price.  In other words, regardless of where the aggregate quantity of supply or demand begins, the suppliers and producers will eventually (over a succession of time windows) end up exchanging goods at the equilibrium price.  This is equivalent to saying that regardless of how many people are producing or consuming within the first time window, price will eventually (after a succession of later time windows) reflect the level corresponding to the point at which the aggregate supply and demand.  (Technically, within a price range of that point proportional to the transactions costs.)

The period of time required to reach equilibrium is a function of the frequency of supply, the frequency of demand, and the transactions costs.  Higher frequencies and lower transactions costs means faster return to equilibrium.

Non-equilibrium Price as a Phase Synchronization Device

With the static limit picture of a damped, driven oscillation in price in mind, we have our insight into what is causing non-equilibrium dynamics in price.  Specifically, it must be a mismatch in phase between supply and demand because, by definition, at the static limit the frequency distributions of supply and demand are not varying in time.

At any given moment, there are some people within the population that have to decide whether to undertake another cycle of supply or demand.  To a good approximation, each person within the population decides whether or when to undertake another cycle of supply or demand by asking whether, at the current market price, an additional cycle of supply or demand would be increase wealth.  When the current market price is below equilibrium, suppliers will not undertake a new supply cycle.  When the current market price is above equilibrium, consumers will not undertake a new consumption cycle.**

In other words, market price is a mechanism for synchronizing the cycles of supply and demand.

At any given moment in time, market price is a combination of the (positive or negative) phase difference (caused by the asynchronicity in supply and demand) plus the equilibrium price.

Beyond Static Theory: Dynamic Market Price

Having seen how market price is a mechanism for synchronizing supply and demand cycles in the static limit, I now ask whether removing the static limiting conditions should have an effect on price.

Not on the synchronization mechanism.  Although the "equilibrium" component of price will no longer be fixed in time (because the supply and demand curves may slide left or right or fatten or narrow with time), the phase difference component is calculated the same way by looking at the aggregate lead phase difference between supply and demand cycles.

Conclusions

Price at a moment in time is a function of both the relative aggregate quantity of supply and demand (which are a function of the frequency of supply and demand cycles) and the aggregate difference in phase between supply and demand cycles.

Although the phase of supply and demand cycles may be randomly distributed at an earlier point in time, as time evolves, a population of people will tend to synchronize supply and demand cycles so as to eliminate the phase difference component of price.  In the equilibrium limit, this component vanishes as supply and demand become perfectly synchronized.

* In electrical engineering terms, I am toying around with the model of markets as voltage-controlled oscillators, with the voltage being the phase difference signal.  VCOs are very useful devices in control theory.  When used in conjunction with negative feedback, VCOs can be used to build phase-locked loops.

** Note that the ability to store and use inventory or money will have important effects on the phase synchronization of the supply and demand cycles.  I'm going to ignore these for now, although I will assert that these effects probably dominated the price dynamics within the subprime mortgage market.