Patterns in a Complex System: An Empirical Study of Valuation in Business Bankruptcy Cases, Essays (university) of Financial Management

Download Patterns in a Complex System: An Empirical Study of Valuation in Business Bankruptcy Cases and more Essays (university) Financial Management in PDF only on Docsity! PATTERNS IN A COMPLEX SYSTEM: AN EMPIRICAL STUDY OF VALUATION IN BUSINESS BANKRUPTCY CASES Bernard Trujillo This Article applies complex systems research methods to explore the charac- teristics of the bankruptcy legal system. It presents the results of an empirical study of twenty years of bankruptcy court valuation doctrine in business cramdown cases. The data provide solid descriptions of how courts exercise their discretion in valuing firms and assets. This Article has two objectives: First, using scientific methodology, it explains the content of bankruptcy valuation doctrine. Second, the Article uses doctrine as a vari- able to explore the system dynamics that govern the processes of change over time. Significant findings include: (1) Courts tend to split the difference in valuations much less frequently than expected; (2) while early data show debtors' and credi- tors' valuation positions were close together, later data show the parties' valuations moved further apart; (3) bankruptcy courts' valuation approach is substantially influenced by whether the valuation includes a calculation for the time value of money; (4) there seems to be some geographic distribution of courts' acceptance of valuation models, with courts in southern circuits more likely to accept soft valuation models, and nonsouthern circuit courts more likely to accept hard valuation models; and (5) the evidence offers preliminary support for the hypothesis that bankruptcy system content may self-organize according to some complex deterministic dynamics. IN TRO DU CTIO N ............................................................................................................. 358 I. STUDYING DOCTRINE EMPIRICALLY ....................................................................... 361 A . Explaining D octrine ....................................................................................... 362 B. Using Doctrine to Explain Legal Systems ...................................................... 364 C . D esign of This Study ...................................................................................... 367 II. STUDY RESULTS AND INTERPRETATIONS ............................................................... 368 A . Party Success .................................................................................................. 368 B . Party Proxim ity .............................................................................................. 371 C . V aluation Standards ....................................................................................... 375 * Assistant Professor, University of Wisconsin Law School. J.D., Yale Law School, 1992; A.B., Princeton University, 1988. Thanks to Peter Carstensen, Bill Clune, Ted Eisenberg, Howie Erlanger, Marc Galanter, Neil Komesar, Bert Kritzer, Bob Lawless, Lynn LoPucki, Stewart Macaulay, Robert Martin, Art McEvoy, Clint Sprott, Martin Shapiro, Bill Whitford, and two anonymous referees. Errors are my own. The empirical work in this Article was funded by grants from the National Conference of Bankruptcy Judges, the American Bankruptcy Institute, and the University of Wisconsin at Madison Graduate School. I gratefully acknowledge the research assistance of Mike Morgalla, the statistical assistance of David Merrill and Menesh Patel, and the coding assistance of Chris Rechliz and Jonathan Schuster. Special thanks to Victoria Trujillo. 358 53 UCLA LAW REVIEW 357 (2005) 1. Definitions and Descriptive Statistics .................................................... 375 2. M odeling C ourts' V aluation ................................................................... 379 3 . Fin d in gs .................................................................................................. 380 D . V aluation M odels ........................................................................................... 383 E. Self-O rgan ization ........................................................................................... 386 1. V aluation Standards ............................................................................... 386 2. Parties' V aluation M odels ....................................................................... 392 3. Sum m ary of Findings .............................................................................. 397 C O N C LU SIO N ................................................................................................................. 398 A PPENDIX: M ETHODOLOGY ........................................................................................... 400 A . R eliability ....................................................................................................... 400 B. O rdered Logit M odel ...................................................................................... 402 C. Access to Documents of Potential Interest .................................................... 404 INTRODUCTION Bankruptcy law is a complex, adaptive system.! Despite the presence of a complicated statute and a lengthy set of procedural rules, the bankruptcy system leaves much of its most important work up to the discretion of actors 1. By "complex," I do not mean that the system has many moving parts, although bankruptcy certainly does. My sense of the term "complex" is borrowed from research in physics and other natural sciences. A system is "complex," in this sense, when it operates far from equilibrium. Equilibrium systems are characterized by balance, such that any flow of matter through the system represents a disturbance-a sort of crisis that must be managed so that the system may return to its characteristic balance. Far-from-equilibrium systems, on the other hand, are characterized by imbalance; these systems are paradigmatically poised on the edge of substantial change. Mathematically, the relationships between variables in a complex system tend to be nonlinear, the arithmetic is discrete (rather than continuous), and the geometry is fractal (rather than Euclidian). Most significantly for this project, the dynamics of complex systems are frequently observed to be self-organizing rather than entropic. A complex system endogenously adapts to environmental changes and tends to reach higher forms of order over time. Complex systems in nature include the ecological, such as a river bed, and the biological, such as a cell. The literature on complex systems is extensive, and these ideas are spreading from the study of natural systems to the study of social systems. See, e.g., PER BAK, HOW NATURE WORKS: THE SCIENCE OF SELF-ORGANIZED CRITICALITY 183-98 (1996); BENOIT B. MANDELBROT & RICHARD L. HUDSON, THE (MIS)BEHAVIOR OF MARKETS: A FRACTAL VIEW OF RISK, RUIN, AND REWARD (2004); JULIEN CLINTON SPROTr, CHAOS AND TIME-SERIES ANALYSIS (2003); Steven N. Durlauf, What Should Policymakers Know About Economic Complexity?, 21 WASH. Q. 157 (1998); Arthur F. McEvoy, Working Environments: An Ecological Approach to Industrial Health and Safety, 36 TECH. & CULTURE S145, S153-55 (1995); J.C. Sprott, Competition With Evolution in Ecology and Finance, 325 PHYSICS LETTERS A 329 (2004), available at http://sprott.physics.wisc.edu/pubs/ paper285.pdf; M.E.J. Newman, Power Laws, Pareto Distributions and Zipf s Law (Jan. 9, 2005), http://arxiv.orgfPS--:cache/cond-mat/pdf/0412/0412004.pdf. Applications of complex systems science to legal phenomena are relatively recent and hold much promise. See Thomas A.C. Smith, The Web of Law (Spring 2005), available at http://ssm.com/abstract=642863 (showing power law properties in the citation of legal materials as evidence that legal systems may be characterized by self-organizing criticality); see also Bernard Trujillo, Self-Organizing Legal Systems: Precedent and Variation in Bankruptcy, 2004 UTAH L. REV. 483, 528-43 (arguing that bankruptcy law is a self-organizing system). Patterns in a Complex System 361 courts do when asked to value? We find that, despite the lack of exogenous ordering, stable and navigable patterns of valuation have emerged. Part I of this Article discusses the study of doctrine, considers some criteria for studying doctrine empirically, and lays out the design of this study. Part 1I presents the statistical results of this study and some interpretations of those results. Part III summarizes the major findings and concludes with some questions and directions for future research. I. STUDYING DOCTRINE EMPIRICALLY There are at least three reasons for studying doctrine:'" (1) We want to know what the doctrine is; (2) we want to learn about the behavior of per- sons or institutions, and we think doctrine offers some explanation of that behavior; and (3) we want to explain what a legal system is and how it works, and we use doctrine as a variable to test propositions regarding a legal system's structure and dynamics. In the first type of study (what is the doctrine?), researchers regard doc- trine as the dependent variable, or the thing-to-be-explained. Researchers consider independent variables in the hope of explaining, and perhaps pre- dicting or ultimately reforming, legal doctrine. These independent variables can be both formal (for example, jurisdiction, time, judge, type of case, and party structure) and realist (for example, politics, economic incentives, race, class, gender, and social norms"). In the second type of research (what is the behavior of persons and insti- tutions?), the thing-to-be-explained is behavior. Doctrine serves as one among many independent variables that may shed light on behavioral hypotheses. In the third type of doctrinal study (what are the dynamics of a legal system?), researchers again treat doctrine as an independent variable that may help explain a targeted thing-to-be-explained. That thing, however, is not some behavioral pattern, but rather the structure and dynamics of the legal system itself. 2 10. Doctrine is the body of formally defined rules of a legal system. These rules are officially promulgated by the authorized agents of the system, and include statutes, court opinions, and admin- istrative regulations. A common feature of doctrine is that it is written, published, and available to the public. As a simplification, I will be taking the published judicial opinion as the central exam- ple of doctrine for the purposes of this Article. 11. See Marc Galanter, The Portable Soc 2; or, What to Do Until the Doctrine Comes, in GENERAL EDUCATION IN THE SOCIAL SCIENCES: CENTENNIAL REFLECTIONS ON THE COLLEGE OF THE UNIVERSITY OF CHICAGO 246, 252 (John J. MacAloon ed., 1992) ("1 make no distinction between believers in the model of rules and instrumentalists; nor between formalist believers in autono- mous rule development and their realist critics. Thus, where some observers detect a radical break, I see a striking continuity."). 12. See infra Part II.E.3 (discussing randomness versus chaos in system dynamics). 362 53 UCLA LAW REVIEW 357 (2005) This project intends no contribution to the second type of doctrinal research. The data in this study-published bankruptcy court cases-cannot credibly falsify any behavioral hypotheses. We cannot, on the basis of this work, draw conclusions about what backroom deals are struck or what dis- putes are not brought to formal adjudication. This project does, however, hope to contribute to both the first and third types of doctrinal study. A. Explaining Doctrine One objective of this research is to understand more about the doctrine of bankruptcy valuation. Two tendencies mark my pursuit of this objective: First, this Article suspends discussion of any normative theory of value13 in favor of straight description. Much doctrinal research is at least as interested in what the doctrine ought to be as in what the doctrine is."4 By contrast, this Article will try to refrain from imposing, ex ante, any overarching theory upon the data. 5 13. For useful general theorizations of valuation, see, for example, Margaret Jane Radin, Compensation and Commensurability, 43 DUKE L.J. 56 (1993); Cass R. Sunstein, Incommensurability and Valuation in Law, 92 MICH. L. REV. 779 (1994); Cass R. Sunstein et al., Assessing Punitive Damages (With Notes on Cognition and Valuation in Law), 107 YALE L.J. 2071 (1998). For two of the more useful articles theorizing valuation in the bankruptcy area, see Carlson, supra note 9, and Lawless & Ferris, supra note 9. 14. This blurring of is with ought is characteristic of what John Griffiths calls an "ideological" rather than an "empirical" position. See John Griffiths, What Is Legal Pluralism?, 24 J. LEGAL PLURALISM 1, 3 (1986). "Ideology," according to Griffiths, is a "mixture of assertions about how the world ought to be and a priori assumptions about how it actually and even necessarily is." Id. "Empirical," descriptive, or scientific approaches to the study of law, on the other hand, take the phenomenon as the primary object of study, without entertaining questions about how the phenomenon ought to be or what it necessarily must be. Id.; see also Galanter, supra note 11, at 251-53 (listing eight propo- sitions of conventional legal studies. Galanter also noted that the listed propositions have a dual, composite character, fusing both descriptive and normative. They are thought to state what is normal and typical in legal systems-to reflect the inherent and proper shape of legal reality. This fusion of factual and normative assertion ... establishes them as ideo- logical statements-statements about what a legal system ... ought to be like. Id. 15. See Comm'r v. Marshall, 125 F.2d 943, 946 (2d Cir. 1942). In the opinion of the court, Judge Jerome Frank wrote: The fallacy.., stems largely from lack of recognition of the eely character of the word "value." It is a bewitching word which, for years, has disturbed mental peace and caused numerous useless debates. Perhaps it would be better for the peace of men's minds if the word were abolished. Reams of good paper and volumes of good ink have been wasted by those who have tried to give it a constant and precise meaning. Id.; see also Old Colony Bondholders v. N.Y., N.H. & H.R. Co., 161 F.2d 413, 450 (2d Cit. 1947) (Frank, J., dissenting) ("[Ilt would be desirable to abandon the word 'valuation'-since that word misleadingly connotes some moderately rational judgment-and to substitute some neutral term, devoid of misleading associations, such as 'aluation,' or, perhaps better still, 'woosh-woosh."'). Heeding the pleas of Judge Frank, this Article attempts description rather than theorization. Patterns in a Complex System 363 This Article revolves around the data, and the data show only what bank- ruptcy courts have memorialized about their valuation .e.in. Second, this Article attempts to apply scientific methodology to doc- trinal study by treating doctrine as a quantitative unit. This methodological approach works particularly well in studying bankruptcy, which is an almost ideal laboratory for the empirical study of legal systems. In part because of the existence of over 330 specialty bankruptcy courts, the U.S. bankruptcy system emits a tremendous amount of visible data relative to many other legal systems. A system that emits large amounts of easily detectible data can be studied scientifically and quantitatively, without resort to anecdote or idiosyncrasy, and without the biases of expectation or experience. 7 This Article thus tries to make transparent all choices regarding selection and interpretation of data,"s and uses statistics to interpret the sample data drawn from this complex system- in-motion. 9 16. Nothing in this study tells us how bankruptcy courts perform valuations in contexts beyond Chapter 11 cramdowns. We cannot, for example, generalize from these data to draw con- clusions about areas of consumer bankruptcy valuation. These data only show the variation of forms that exist in the doctrine of business cramdown valuations. 17. The methodological approach of this research follows the inspiration of the evidence-based medicine (EBM) approach to the practice of medicine. EBM promotes the scientific study of the efficacy of medical treatments. Such research allows medical professionals to prescribe treatment, not because of tradition or path dependence, but because of a well-informed belief that the treatment will be effective. See generally DAVID L. SACKETT ET AL., EVIDENCE-BASED MEDICINE: HOW TO PRACTICE AND TEACH EBM (1997). A frequently updated database of evidence is available for practitioners of EBM at http://www.clinicalevidence.com/ceweb/conditions/index.jsp. I believe that an evidence- based law approach to doctrine can move us past anecdote and unexamined path dependence, and perhaps toward a systematization and verification of knowledge about legal doctrine. 18. Often, doctrinal research bases its conclusions on a set of data that is both highly selective and rather small relative to the total amount of available data-for example, by limiting the data set to a few appellate court opinions-and typically does not explain its criteria for data selection and interpretation. By scientific standards, both the selection and review of such data are idiosyncratic, and the idiosyncrasies are not expressly revealed. As a consequence, such research cannot be rep- licated by other scholars. 19. There is an interesting similarity between the rise of statistical methods in the physical sciences and the roughly contemporaneous rise of impressionist methods in the visual arts. Both are ways of representing systems-in-motion, and of knowing those systems by sampling them. Henri Matisse wrote an essay entitled Exacdtude Is Not Tnrh to accompany four sketches of his own face in a mirror. Matisse makes the point that the four renditions are all very different, but unmistakably represent the same subject. They are, if you will, four samples of the same complex system. See Henri Matisse, Exactitude Is Not Truth, in THEORIES OF MODERN ART 137-39 (Herschel B. Chipp ed., 1968). Matisse explained: These drawings seem to me to sum up observations that I have been making for many years on the characteristics of a drawing, characteristics that do not depend on the exact copying of natural forms, nor on the patient assembling of exact details, but on the profound feeling of the artist before the objects which he has chosen .... ... [TIhe leaves of a tree-of a fig tree particularly-the great difference of form that exists among them does not keep them from being united by a common quality. Fig leaves, whatever fantastic shapes they assume, are always unmistakably fig leaves .... 366 53 UCLA LAW REVIEW 357 (2005) precisely because it is not an unvarnished report of an objective event-what was the found value of the asset?-but rather a subjective account of an objec- tive event-what did the judge say about finding the value of the asset?2" If doctrinal data can explain something about system dynamics, then the data of this study may illuminate at least two features of the bankruptcy legal system: (1) the effects of discretion; and (2) the tendency to self-organization. Bankruptcy law contains a high amount of judicial discretion.29 Scholars and policymakers often paint judicial discretion as a problem, in part because -the outcomes of a discretion-rich system are, at least in theory, uncontrolled and subject to caprice. Yet despite the theoretical possibility that anything can happen in a discretion-rich space, the data in this Article show that, in fact, just anything does not happen. Rather, very constrained and navigable pat- terns have emerged in the area of bankruptcy valuation.0 This Article hopes to make transparent how bankruptcy courts use their discretion in the area of valuation. As a more general aspiration, one may hope that by measuring the uses and effects of judicial discretion in a specific area, legal actors and com- mentators may begin to understand and become comfortable with discretion as an omnipresent and unremarkable feature of complex legal systems." Another characteristic of bankruptcy system dynamics is self-organization, or the emergence of patterns that are not attributable to exogenous, extrasys- temic events. The data in this Article, echoing other empirical bankruptcy work,32 show that the bankruptcy legal system produces stable and navigable patterns of behavior, despite the absence of rigorous hierarchical control by appellate courts and Congress. 28. Studying doctrine to understand system dynamics is a content-neutral use of doctrine, in the sense that we will learn about the system's dynamics regardless of the doctrine's content-regardless of whether a given court, say, adopted or rejected a particular discounted cash flow model. Such research is designed not necessarily to predict future system content, but rather to describe the processes that produce future system content. See, e.g., RAYMOND T. NIMMER, THE NATURE OF SYSTEM CHANGE: REFORM IMPACT IN THE CRIMINAL COuRTS 3 (1978) ("My premise is that the reform process is an inde- pendently significant phenomenon, typified by common patterns and delimiting characteristics. This phenomenon can and should be studied quite apart from the specific goals of particular reforms."). 29. See Trujillo, supra note 1, at 490-500, 509-12. 30. See infra Part lI.D. 31. See Edward L. Rubin, Discretion and Its Discontents, 72 CHI.-KENT L. REv. 1299, 1299-300 (1997) (suggesting that judicial discretion is presented as a problem, although it is actually a ubiquitous and unremarkable feature of modem law); Rubin & Feeley, supra note 23, at 2037 ("[Judicial creation of law] can be described, understood, and justified. It is one of the basic, quotidian elements of our legal system."). 32. See, e.g., Jean Braucher, Lawyers and Consumer Bankruptcy: One Code, Many Cultures, 67 AM. BANKR. L.J. 501, 532 (1993); Lynn M. LoPucki, The Demographics of Bankruptcy Practice, 63 AM. BANKR. L.J. 289 (1989); Lynn M. LoPucki, Legal Culture, Legal Strategy, and the Law in Lawyers' Heads, 90 Nw. U. L. REV. 1498, 1506-07 (1996); Teresa A. Sullivan et al., The Persistence of Local Legal Culture: Twenty Years of Evidence From the Federal Bankruptcy Courts, 17 HARV. J.L. & PUB. POL'Y 801, 804 (1994). C. Design of This Study The data for this Article are drawn from reported opinions of U.S. Bankruptcy Courts in Chapter 11 cramdown proceedings that contain a judi- cial valuation of at least one asset.33 The database consists of 180 observations drawn from 145 published opinions reported in the Westlaw computer data- base, and decided from 1979 through 1998." A law student, working closely with me, coded each observation according to a substantial coding instru- ment.35 Following the initial coding, a substantial portion of the observations was recoded by a different law student to test for reliability." Besides the fact that the data are drawn from published cases,37 other potential limitations of the database include the possibility that the computer search failed to identify relevant cases, that Westlaw misclassified some rele- vant cases, and that the coding process registered some false negatives (by failing to classify an item as an observation despite the presence of a reported judicial valuation). Thus, while these data support statistical conclusions about the population of all reported cases from 1979 through 1998, the present database is properly understood as only a sample of that population. Each observation was coded for several variables. The meanings of most of the variables (such as date of adjudication or federal circuit) are obvious, but five variables merit some initial explanation: (1) party success; (2) party prox- imity; (3) finance element; (4) valuation standard; and (5) valuation model. 33. In this cramdown database, judicial valuations typically occurred in one of two contexts. In some, a dissenting class of creditors (typically a single secured creditor) sought cramdown rights under § 1129(b)(2)(A) through a judicial valuation of the collateral securing its claim. In others, the debtor's plan of reorganization proposed to extinguish a creditor's lien in a piece of collateral in exchange for compensation, and the court valued the item to determine if the compensation was appropriate and the plan was confirmable. The parties to these valuation proceedings were corporations. 34. The Westlaw search term used to acquire the cases was "51K3563 51K3564 51K3565 & DA(AFT 1978 & BEF 1999)" in the library "fbkr-bct." This term yielded a list of 388 cases, which generated 180 observations, reported in 145 cases. Observations consist of a reported numerical valuation of an asset by a court. If a single case reported valuations of multiple assets, each asset was coded as a separate observation. The earliest observation is from December 5, 1980. The latest observation is from November 20, 1998. 35. See infra Appendix (describing how a reader can access a copy of the coding instrument). A coding instrument is a questionnaire that a coder uses to transform written information into quantitative units. See generally HERBERT JACOB, USING PUBLISHED DATA: ERRORS AND REMEDIES (Sage University Papers Series: Quantitative Applications in the Social Sciences No. 07-042, 1984) (discussing methodology generally applicable to this study); ROBERT PHILIP WEBER, BASIC CONTENT ANALYSIS (Sage University Papers Series: Quantitative Applications in the Social Sciences No. 07-049, 2d ed. 1990) (same). 36. See infra Appendix, tbl.I-A (presenting reliability statistics). 37. See supra Part I.B (discussing the relationship between data's publication and its proba- tive value). Patterns in a Complex System 367 368 53 UCLA LAW REVIEW 357 (2005) (1) Party success is a numerical representation of the degree to which the parties in the valuation proceeding won or lost.38 (2) Party proximity is a numerical measure of how far apart the debtor and the creditor were in the dollar valuations they sought from the court.39 (3) Finance element is a variable for whether the valuation contained a calculation or discussion of the time value of money. Of the 180 observations, 74 (41 percent), contained a finance element, and 106 (59 percent) did not." (4) Valuation standard is a variable that attempts to represent the internal criteria used by a particular valuer (court, debtor, or creditor) when it assigned value to an item-for example, whether the valuer based the valuation on the item's anticipated use, or resale, or some mix between use and resale.41 (5) Valuation model tracks the way that parties argue for a particular valuation, for example, by stating what an asset would be worth if it were liquidated (a liquidation model), or by constructing a prediction of what cash flows an asset would earn over time (a discounted cash flow model).42 I. STUDY RESULTS AND INTERPRETATIONS This part of the Article presents the conclusions of statistical analysis of the data, and offers some interpretations of the results. A. Party Success The database illuminates some patterns of party success in valuation proceedings. We define success quantitatively, by comparing the valuation proposed by the parties with the valuation found by the court. Using two 38. See infra Part IL.A for the mathematical construction of the success variable and analysis. 39. See infra Part II.B for the mathematical construction of the party proximity variable and analysis. 40. A case was coded as containing a finance element if the court, as part of the valuation, also determined an interest rate or discount rate. 41. See infra Part II.B for the mathematical construction of the valuation standards variable and analysis. 42. See infra Part II.C for further analysis of valuation models. Patterns in a Complex System- 371 FIGURE 2 WINNER-TAKE-ALL ADJUDICATIONS OVER TIME Year Percent1982 100 1983 100 80% 1984 100 1 1985 60 60% 1986 0 1987 0 40% 1988 50 1989 0 20% 1990 75 1991 75 1992 67 0% 1993 37.5 9; 195 41 q 19 R' P 199 75' 1995 25 Year 1997 33 1998 67 While these data are not as well behaved as the data in the W-curve, one can detect a slight downward trend over time. Thus, while courts continue to deploy the winner-take-all strategy frequently, there has been a slight trend toward splitting the difference in the litigants' numbers. B. Party Proximity The database also illuminates parties' valuation strategy. We construct a party proximity variable to measure the distance between dollar positions staked out by the parties in the valuation proceeding.5' Values close to "0" indicate that the parties' dollar positions were very close together. Values approaching "1" indicate that the parties' dollar positions were very far apart. The party proximity variable is useful because it permits a dollars-to-dollars comparison without that comparison being distorted by differences in the time, place, or circumstances of the valuation. In this way, we can focus on the relative valuation strategies of the parties. 51. Observations must have dollar positions for both the debtor and the creditor, and the positions must be comparable. Thus, we omit from the database valuation positions such as "3% over prime v. 11% over prime" because there is no constant way to compare these values. Where a party offered a range for its dollar position, we took the midpoint of that range; n = 80. The formula for party proximity is (debtor's dollar position - creditor's dollar position) I (debtor's dollar position + creditor's dollar position). For example, in an observation from 1989, the debtor argued for a valuation of $9.2 million, and the creditor asked for $14.5 million. Applying the formula yields a quantity of -0.223629. For purposes of determining proximity, we took the absolute value of each quantity. I thank David Merrill for help in developing this version of the party proximity variable. 53 UCLA LAW REVIEW 357 (2005) Figure 3 shows the frequency distribution of the party proximity variable. FIGURE 3 PARTY PROXIMITY: FREQUENCY 25 20 15 10 0.05 0.15 025 0.35 0.45 0.55 0.65 0.75 0.85 1 Categories of Proximity Proximity Categories Freq. 0.05 22 0.15 20 0.25 22 0.35 5 0.45 3 0.55 2 0.65 1 0.75 2 0.85 2 1 1 For example, there were twenty-two observations in which the party proximity was 0.05 or below (very close together). There was one instance in which the party proximity was above 0.85;52 for the most part, parties' dollar positions tended to be close together. Considering the data over time, however, we see a clear trend away from litigants' arguing for numbers that are close together. Figure 4 shows the mean values per year for party proximity. 52. This largest observation, from 1993, was 0.8930. In that case, the debtor sought a valuation of $23 million and the creditor argued for $1.3 million. 372 Patterns in a Complex System FIGURE 4 PARTY PROXIMITY OVER TIME: DESCRIPTIVE -P 4, Year Part Proximity 1982 0.1430262 1983 0.2234069 1984 0.1869919 1985 0.1179881 1986 0.0744851 1987 0.3387833 1988 0.1273576 1989 0.1450607 1990 0.2073515 1991 0.1528005 1992 0.1657538 1993 0.2826830 1994 0.3172302 1995 0.2131415 1996 0.2640636 1997 0.4122704 1998 0.3086420 As a matter of descriptive statistics, it appears that parties' valuation positions moved farther apart over time. A standard linear regression, shown in Figure 5, demonstrates that the observed increase over time in the party proximity variable is statistically significant. 3 53. The slope estimate was 0.0100108 with a standard error of 0.0018757 and a probability of 0.000 (t-value of 5.34). This means that for each additional year, the value of the party proximity variable increased by approximately 0.01. A t-test is a simple statistical test that allows us to state the likelihood that an observed correlation between two variables could have been observed merely by chance. 376 53 UCLA LAW REVIEW 357 (2005) A valuer is using the common value when her valuation of an item is a best guess of how others will value the item in some common market.6" Common value models assume that the item has "exactly the same value to each bidder,"6 and that the variation in bids is explained by bidders' incomplete information regarding that value. Take, for example, an art dealer who bids for a painting she intends to resell. Her valuation is based on her informa- tion about what the painting will fetch in the resale market. Her valuation of the item will almost certainly change as she learns more about the valua- tions of others. Using these categories of valuation, we assigned a code to each instance of valuation in the database. The possible codes ranged on a continuum from extreme common value to extreme independent value. Initial coding of the data resulted in assigning a number from "1" to "8" (with "1" representing extreme common value and "8" representing extreme independent value) for each valuation by courts, debtors, and creditors. Subsequent statistical analysis62 showed that the valuation standard could be collapsed into three categories: common (initial codes 1 and 2), mixed (initial codes 3, 4, and 5), and independent (initial codes 6, 7, and 8). Independent values for interest rates most often took the form of pre-petition contract rates (a rate bargained for at some point in time historically distant from the time of the adjudicated valuation) or statutory rates (what a legislature decreed was acceptable compensation for the time value of money at some time historically distant from the time of the adjudicated valuation). As placeholders for value, both pre-petition contract rates and statutory rates assume that the found value should be something other than (or in addition to) what a willing buyer, in open competition with many other willing buyers, would pay a willing seller, in open competition with other sellers, for this item in these circumstances at this time. 60. Represented mathematically, a "common" valuation is a draw from some probabilityS2 1 distribution H(vIV) where v, = v, V , . . . v. The only difference between probability distribu- tion H (common valuation) and probability distribution F (independent valuation) is that the H quantity contains a term "V" representing the reference point by which all components of v, adjust their valuations. On this theory of value, each component of v, is an estimate of "V," the item's true value. These estimates stand to improve as each valuer learns the valuations of others. This true worth theory of value is the one employed by Bob Lawless and Steve Ferris. See Lawless & Ferris, supra note 9, at 11-12. 61. MILGROM, supra note 56, at 162. In our database, common value for valuing principal was most often manifested as valuation models constructed by expert witnesses and which utilized elements such as liquidation models, comparables, and discounted cash flow. Common values for interest rates most often involved models utilizing elements of formal risk analysis. 62. Specifically, a multinomial logistic analysis, a likelihood ratio test, and a Wald test all showed that the conditional distribution of the valuation standard could be collapsed from eight categories to three. See infra Appendix (directing the reader to website access of documents). The fact that the research project went fishing for eight categories and ended up with three is, itself, an interesting finding. This finding may verify, with statistical analysis of real-world bankruptcy data, the robustness of auction theory's three-category conceptualization of valuation. Patterns in a Complex System 377 Figure 6 shows the courts' valuation standard (CtVS) in the 180 obser- vations of the database, as coded on an eight-part continuum ranging from "1" (extreme common value) to "8" (extreme independent value). We see that the valuations performed by bankruptcy courts in the database were skewed toward the common pole of the continuum, with most of the valuations coded at "3" (common, but not extremely common).63 FIGURE 6 COURT'S VALUATION STANDARD: FREQUENCY 70 # of 60 --- - -- ctVs Observations Percent z 1 16 8.9 50 2 32 17.8 3 63 35 40- - - - 4 26 14.4 0 30 -- 5 19 10.6 6 10 5.6 Se20 - - -- 7 6 3.3 8 8 4.4 10 - TOTAL 180 100 1 2 3 4 5 6 7 8 Court's Valuation Standard Table A shows the distribution of the CtVS, broken out by whether the valuation contained a finance element-a calculation or discussion of the time value of money. TABLE A COURT VALUATION STANDARD BY FINANCE ELEMENT Frequency Mean Standard Deviation Finance Element 74 3.20 1.48 No Finance Element 106 3.75 1.86 The presence of a finance element in the valuation tends to drive the CtVS closer to the common value.' As discussed in Part I.C.3, the observed cor- 63. These data tend to support a general impression that the cases in the database showed a rebuttable presumption in favor of more common-based valuation, a presumption which had to be overcome by the party seeking a more independent-based valuation. 64. The t-test shows that t = 2.0914 with a probability of 0.0379 and that the observed difference is due to chance alone. So, the difference is statistically significant at the 5 percent level. See also the regression infra tbl.C. 378 53 UCLA LAW REVIEW 357 (2005) relation between common valuation and the presence of a finance element is not only a feature of the sample (our database), but is also strong enough to allow an inference that the same correlation is true across the entire population. Table B gives the mean and standard deviation of the CtVS for each of the federal circuits. TABLE B COURT VALUATION STANDARD BY CIRCUIT Circuit Mean Standard Deviation Frequency 1st 3.38 1.30 8 2d 3.96 2.01 27 3d 2.88 1.36 17 4th 3.40 1.96 15 5th 4.80 1.37 15 6th 3.17 1.63 24 7th 2.91 1.04 11 8th 3.32 1.95 19 9th 3.23 1.24 13 10th 3.50 2.43 6 1lth 3.79 1.77 24 D.C. 3 n/a 1 *All Circuits 3.52 1.73 180 The Second, Fifth, and Eleventh Circuits have means that are statistically greater than "3 . 6' There is some geographic distribution in the CtVS. Elimi- nating the circuits with fewer than ten observations (the First Circuit-Maine, Massachusetts, New Hampshire, Rhode Island, and Puerto Rico; the Tenth Circuit-Colorado, Kansas, New Mexico, Oklahoma, Utah, and Wyoming; and the D.C. Circuit), we can divide the data into a nonsouthern circuit group (the Second-Connecticut, New York, and Vermont; the Third-Delaware, New Jersey, Pennsylvania, and the Virgin Islands; the Sixth-Kentucky, Michigan, Ohio, and Tennessee; the Seventh-Illinois, Indiana, and Wisconsin; the Eighth-Arkansas, Iowa, Minnesota, Missouri, Nebraska, 65. A t-test confirms these results at the 5 percent level. This means that there is only a 5 percent chance that the observed correlation could have occurred by chance. The probability that the mean of the Second Circuit is not "3" is 0.0195; the Fifth Circuit's probability is 0.0002; and the Eleventh Circuit's is 0.0387. Descriptive statistics (frequency charts and tables reporting mean and standard deviation) for the CVS and DVS variables are maintained on the author's website. See infra Appendix. Patterns in a Complex System 381 The model shows that the explanatory variables DVS, CVS, and FINANCE predict CtVS variance at a statistically significant level. The pseudo-r-squared" is remarkably good for social science research, showing that the independent variables explain 47.5 percent of the dependent variable's observed variance. Overall, it appears that CVS, DVS, and FINANCE very usefully explain and predict the bankruptcy courts' valuation standard. The odds ratios are quantities that help explain the model's results. An odds ratio74 gives the odds that the dependent variable will be in one category versus another for every one-unit change in the independent variable. 7 ' For example, if CVS increases by one unit, the odds of CtVS being mixed becomes 2.347 times higher than the odds of CtVS being common. Likewise, when CVS increases by one unit, the odds of CtVS being independent becomes 2.347 times higher than the odds of CtVS being either common or mixed. The results of the ordered logit models, as represented in Table C, sup- port two interpretations. First, courts are much more likely to use a common valuation standard when the valuation contains a finance element (a calcu- lation for the time value of money). 6 Why would the presence of a finance a bell-shaped distribution) except for the two extreme tails of 0.5 percent each. Adding "circuit" as an independent variable to the regression did not change the coefficients substantially. 72. Ordered logit models assume proportional odds-the effect of the independent variable on the dependent variable is assumed to be proportional for all comparisons of categories. In other words, the slopes coefficients are identical across the estimated equations. In this case, two equations are estimated for each of the binary logistic regressions, yielding the coefficients denoted in Table C as "cut 1" and "cut 2." Though the slopes are the same, the probability curve is shifted to the left or right depending on the categorical comparison. 73. The r-squared is a statistical measure of how well the model fits the data. In the case of our ordered logit model, a pseudo-r-squared is used. Roughly speaking, the r-squared tells us some- thing about how much of the variation in the dependent variable is explained by the independent variables in the model. 74. The odds ratios are the bracketed quantities in Table C. We get the odds ratio for a given independent variable by exponentiating its coefficient. For example, we exponentiate the FINANCE coefficient by taking base e (roughly 2.718) and raising it to the power of -1.613 (FINANCE's coefficient), yielding an odds ratio of 0.199. 75. The odds of X happening is the probability of X happening divided by the probability of X not happening. An odds of "1" indicates that X stands a 50/50 chance of happening. An odds of greater than "1" indicates that the probability of X happening is greater than the probability of X not happening. And an odds of between "0" and "1" indicates that the probability of X not happening is greater than the probability of X happening. 76. This finding can be discerned from Table C, which reports an odds ratio of 0.199 for the finance variable. The odds ratio of 0.199 means that, holding all other variables constant and looking just at the relationship between CtVS and FINANCE, if the observed valuation possessed a finance element, then (1) the odds of CtVS being mixed is 0.199 times the odds of CtVS being common; and (2) the odds of CtVS being independent is 0.199 times the odds of CtVS being either mixed or common. The data show that the presence of a finance element in the valuation pushes the courts' valuation standard sharply toward the common pole. This finding might be explained away as a meaningless coding error if cases containing a finance calculation had been automatically 382 53 UCLA LAW REVIEW 357 (2005) element tend to drive valuations toward common standards? One possible expla- nation is that judges deploy common valuation standards in high-complexity cases," but use more independent valuations in lower-complexity cases. Second, the courts' valuation standard correlates strongly with the debt- ors' 78 and creditors'7 1 valuation standard. From this statistical result, we can verify the unsurprising8° proposition that lawyers frame the parameters of the valuation dispute, and that judges typically work within those parameters." Since the data on parties' valuation standards is filtered through the courts' interpretation, we cannot exclude an important alternative interpretation for the correlation between courts' and parties' valuation standards: The court may receive multiple valuation standards from the parties, but report only those standards that it ultimately agreed with. 2 coded closer to common than to independent. The codes, however, demonstrate no such tendency. Observations with finance elements show codes that span the range of valuation standards. See infra Appendix (web access to spreadsheet of coding results, column BZ). Generally, a statistical model cannot return sensible results unless there is meaningful variation in the data. Such variation would be foreclosed by a coding error classifying finance element cases as common. Note also that many cases that were coded closer to the independent pole contained a finance element. See infra note 78. 77. See generally Elizabeth Warren, Vanishing Trials: The Bankruptcy Experience, 1 J. EMPIRICAL LEGAL STUD. 913, 930-37 (2004) (contrasting high-complexity/low-volume bankruptcy cases with low-complexity/high-volume cases). In our database, cases using common valuation standards tended to involve competing and complicated models of the asset's valuation, constructed from the ground up by dueling experts. Nothing in the database supports the proposition that common valua- tion standards are used exclusively in business bankruptcy. Note, for example, the distribution illustrated in Figure 6. Valuations in business cases, including valuations with a finance component, can and do utilize standards closer to independent private value. The operative category for predict- ing the occurrence of common versus private valuation standards seems to be high complexity versus low complexity rather than business versus nonbusiness cases. 78. Table C shows an odds ratio of 2.541 for DVS. This means that, holding all other variables constant and looking just at the relationship between CtVS and DVS: (1) As DVS increases by one unit, the odds of CtVS being mixed is 2.541 times higher than the odds of CtVS being common; and (2) a one unit increase in DVS makes the odds of CtVS being independent 2.541 times higher than the odds of CtVS being either common or mixed. 79. Table C shows an odds ratio of 2.347 for CVS. This means that, holding all other variables constant, and looking just at the relationship between CtVS and CVS: (1) As CVS increases by one unit, the odds of CtVS being mixed is 2.347 times higher than the odds of CtVS being common; and (2) a one unit increase in CVS makes the odds of CtVS being independent 2.347 times higher than the odds of CtVS being either common or mixed. 80. It should be noted that unsurprising is not unimportant. The empirical verification of an anecdotally or experientially familiar proposition is part of what we should expect from applying the scientific method to legal studies. See supra note 17 and accompanying text (explaining the evidence-based law approach to doctrine). 81. Of course, the observed correlation runs both ways: The values of DVS and CVS may explain CtVS. But it is equally likely that debtors and creditors select their valuation approaches based on their perception of courts' tendencies. Thanks to Bill Whitford for bringing this inter- pretation to my attention. 82. I thank Bill Whitford for this point. D. Valuation Models A valuation model is a particular argument or showing about the value of an item. The most common valuation models in the database were: * Comparables, in which the modeler estimates an asset's value by looking to the known values of other, ostensibly similar assets." Comparables are a common model for valuing residential real estate, in which a price of a house up for sale is compared to the recent sale prices of other houses of a similar location, size, and style. " Discounted cash flow (DCF) models, which estimate the pre- sent value of future expected cash receipts and expenditures.84 A DCF model is typically generated by a financial professional who is introduced as an expert witness. The financial expert will estimate the firm's future cash flows and then discount those expected future returns to present value through the use of a dis- count rate. * Liquidation (also called cost, in some circumstances), in which the modeler estimates an asset's value by looking to what price the asset would fetch if sold.85 * Testimony by a current or prospective user of the asset regarding the valuation aspects of the intended use. " A pre-petition contract rate agreed upon by the parties (used in the finance, or time value, aspect of the valuation). " An accounting made for the risk profile of the party acquiring the asset (again used for the finance aspect of the valuation- higher risk yields an increased interest rate). * General observations about market and economic conditions affecting the asset's value. In addition to these categories, we also coded as unknown those observa- tions in which the language in the opinion could not support the assigning of a code. 83. See, e.g., MARK GRINBLATT & SHERIDAN TITMAN, FINANCIAL MARKETS AND CORPORATE STRATEGY 368 (1998). 84. See, e.g., RICHARD A. BREALEY & STEWART C. MYERS, PRINCIPLES OF CORPORATE FINANCE 73-77 (5th ed. 1996). 85. See, e.g., GRINBLATT & TITMAN, supra note 83, at 560. 383Patterns in a Complex System 386 53 UCLA LAW REVIEW 357 (2005) E. Self-Organization These data are useful for testing the proposition that the U.S. bank- ruptcy legal system might be a self-organizing system-a system whose con- tent contains patterns not fully explained by exogenous dynamics." This subpart presents the evidence for self-organization as observed in three categories of variables: (1) the use of valuation standards by courts, debtors, and creditors; (2) the use of valuation models by debtors and creditors; and (3) courts' acceptance of experts. 1. Valuation Standards If a system's content self-organizes, we should be able to measure a decline over time in the variability of some of the system's variables. That is, we treat variation itself as a structural variable to be measured and observed." The measure of a variable's variation is its standard deviation. Just as the mean, median, and mode say something about the central tendency of a distribution, the standard deviation shows how that central tendency was achieved-with relatively larger standard deviations signifying a wide range of variation, and relatively smaller standard deviations signifying a smaller range of variation. If the standard deviation of a variable declines over time, this may indicate a sort of learning or patterning of system content over time. Such patterns, unless explained by extra-system forces-such as control by appellate courts or Congress-may constitute evidence of self-organization. 90. See Trujillo, supra note 1, at 519-43 (explaining the process of self-organization). 91. The field of sociolinguistics has contributed much to my understanding of the construc- tion of variation as a variable in order to study the change of a social artifice over time. See generally ALESSANDRO DURANTI, LINGUISTIC ANTHROPOLOGY 79 (1997) ("Whereas the realization of such variability convinced formal grammarians to ignore it by establishing an idealized homogene- ity... sociolinguists decided to face variability and make it the subject matter of their investigation.") (emphasis added); J.K. CHAMBERS, SOCIOLINGUISTIC THEORY: LINGUISTIC VARIATION AND ITS SOCIAL SIGNIFICANCE 11 (1995) ("[Ihe admission of the variable as a structural unit in linguistic analysis.., represents a breakthrough of considerable magnitude in linguistic theory .... ). 53 UCLA LAW REVIEW 357 (2005)386 Patterns in a Complex System Figure 7 shows the standard deviation of the CVS over time." FIGURE 7 CVS: DESCRIPTIVE 3 .50 - - - - - - - - - - - - - - - - - - - 3 .05 - -- - - - - - - - -- - - -- - - 2 5. - - - - - - - - - - - - - - - - --0 0 .0 1 . . I I ,I Year Year SD 1982 1.8708 1984 3.559 1985 3.12517 1986 3 1987 1.8172 1988 2.5 1989 1.37032 1990 1.58114 1991 1.98206 1992 1.37069 1993 1.14354 1994 1.76777 1995 0.97183 1996 0.57735 1997 0.8165 1998 0.57735 As a matter of descriptive statistics, it appears that the variability in the CVS declined markedly over time. We can see whether it is proper to infer that the decline in the creditors' standard deviation, observed in the sample, is also a characteristic of the population. We do this by performing a standard linear regression.93 The results of the regression are displayed in Figure 8. 92. In Figures 7-12, the y-axis shows the standard deviation of the relevant valuation standard, as measured on the "1" (common) through "8" (independent) scale. The x-axis shows the year. 93. We can use a linear regression here, but we could not for the model of valuation standards presented in Table C. Here our dependent variable is the sample variance-a variable measured in cardinal numbers. It was the ordinal measure of the dependent variable in Table C that required the use of an ordered model. See infra Appendix. The formula for this standard regression is y = A + Bx, where y is the dependent variable (the variance of CVS), A is the y-intercept of the line, B is the slope of the line, and x is the independent variable (the year). This model estimates a line describing the relationship between the dependent and independent variables in the population. A negative slope provides evidence of a negative relationship between variation and time. . _ _0 _ 53 UCLA LAW REVIEW 357 (2005) FIGURE 8 CVS: INFERENTIAL _ • Y 0.04 14 e11 11 4,l le 1- le 41e le le I- o 4 01 le Year The line estimated by the linear regression model explains the behavior of the sample data quite well,94 allowing us to conclude that the CVS variable does organize over time. Figure 9 shows the standard deviation of the DVS over time. FIGURE 9 DVS: DESCRIPTIVE 4.0 3.0 05 Year Year SD 1982 2.886751 1984 3.544949 1985 3.207135 1986 2.828427 1987 2.219933 1988 1.889822 1989 0.996205 1990 1.242118 1991 2.00454 1992 1.785165 1993 1.57181 1994 1.874755 1995 1.705947 1996 0.5 1997 2.097618 1998 2 94. r2 = 0.6719, which is quite strong given the limitations of social science data. 388 -- - - - - - - 41 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - & - - - - - - - - - - - - - - - - - - - - - - - - - - -------------- -------------------0-----------0 ------------------------------------------- -------------------------------------------------1.0+ Patterns in a Complex System While the valuation standards for the debtors and the creditors seem to show some patterning over time, the courts' valuation standard shows no such organization. Figure 11 shows the standard deviation of the CtVS over time. FIGURE 11 CTVS: DESCRIPTIVE Year 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 SD 1.2247 0.7071 2.9761 2.7646 2.8284 1.5579 1.5811 0.9535 1.2071 1.4367 1.0271 1.088 1.9183 1.1983 2.63 2.0976 2.0736 The eye detects no systematic decline in the sample variance, and this con- clusion is borne out by the regression shown in Figure 12. FIGURE 12 CTVS: INFERENTIAL 4.0 3- - - Predicted Y 3.0 - - 2.0 - - ---- 15 -- 0 - 1.0 - 0 - - - _. 0.5 0.0 392 53 UCLA LAW REVIEW 357 (2005) We can safely conclude from Figure 12 that the CtVS shows no organization or patterning over time.9 Thus, while we observe some ordering phenome- non in the valuation standard variable among debtors and creditors, we observe no such phenomenon among the courts." 2. Parties' Valuation Models In a self-organizing system, we would also expect to see forms introduced from outside the system, as well as a process of competition by which some of these forms achieve normative status.1 ' We perform time-series analyses of the parties' use of particular valuation models to test for the presence of such form innovation and norm emergence. 99. The slope of the regression line is not negative, and the r2 is 0.0003. One simple conclusion we can draw from this absence of a decline in courts' variance is that any ordering that we observe in this area of valuation is likely not exogenous (that is, in response to congressional or appellate court directives). If there were exogenous ordering, we would likely see a decline in court variance. 100. Figures 7-12 organize party-based variables and court-based variables along the matrix of time. When time is the explanatory variable, we see that party-based variables and court-based variables perform differently. This is an intriguing finding, perhaps disclosing an important mathematical/structural feature of legal system dynamics. Much further analysis (probably using a model generating a large number of observations over a long period of time, thus overcoming the profound limitations of the instant database) will be needed before reaching any reliable conclusions. As a tentative first pass, consider the problem from two related perspectives: First, why would parties and courts behave in different ways? Using the language of dynamical systems we can say that we have observed an attractor, or pattern, among the CVS and DVS variables. This attractor is explained by the parameters of the system, which parameters must themselves be off the attractor. That is, the parameters determine the shape of the attractor, but are not contained within the plot of the attractor. Here the attractor, the pattern we observe, is reported litigant behavior. Courts, and more specifically, court allocation of the scarce resource of official recognition of party-introduced forms, are part of the system parameters that determine the shape of the attractor. Second, why would we see parties' variance shrink over time, but not courts' variance? Suppose a dynamics in which two opposing networks of parties (Network A and Network B) work independently to develop forms that will compete for the scarce resource of official recognition allocated by courts. Within each of the two networks, we would anticipate observing the emergence of norms, with the norms of Network A differing from the norms of Network B. The two networks then pose their norms before a court, which picks between the two. This process, over many iterations, would likely produce patterns similar to the patterns produced by self-organizing critical systems-we would see clustering of system content (a decline in party variance), with no long-term predictability of the size, shape, or location of those clusters (no decline in court variance). Conversations with Bill Clune and Clint Sprott have contributed much to the early development of these points. 101. See Sprott, supra note 1, at 329; J.C. Sprott et al., Coexistence and Chaos in Complex Ecologies, 335 PHYSICS LETTERS A 207-212 (2005), available at http://sprort.physics.wisc.edu/pubs/paper293.pdf. Patterns in a Complex System 393 Figure 13 shows the debtors' use of DCF models during the years of the sample.' 2 FIGURE 13 DEBTORS' USE OF DCF VALUATION MODELS 0.6 0 .5 - - - - - --- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0 .2 - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0 .1 - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0.0 044 Year Mean 1982 0 1984 0.2 1985 0 1986 0.2 1987 0.31 1988 0.2 1989 0.42 1990 0.14 1991 0.25 1992 0.43 1993 0.2 1994 0.43 1995 0.39 1997 0.4 1998 0.5 This figure shows simple descriptive evidence that debtors increased their use of DCF models during the years of this sample. A mean of "0" indicates that debtors did not use DCFs at all in the given year, while a mean of "1" would indicate that debtors used DCFs in every instance of a given year. The data tell a story of the DCF steadily gaining usage among debtors' networks, until the point where it becomes a regular "arrow in the quiver" of debtors' valuation arguments. 102. In Figures 13-16, the variable is a binary and so only the mean is presented. Where the variable is a binary, the standard deviation is simply a transformation of the mean, and so the stan- dard deviation adds no information to the mean's description of the distribution. 396 53 UCLA LAW REVIEW 357 (2005) Creditors' data show a trendline that is essentially flat, indicating that comparables were part of the creditors' arsenal throughout the sample period. Noting the same period from 1990-1993 that we saw in the debtors' data, we can conclude that parties' use of comparables rose to a normative level before tapering off somewhat during the mid-1990s. Generally speaking, we can say that the introduction of such forms as the DCF and comparables support a subjective impression of the database that, following some initial unfamiliarity, bankruptcy actors became more sophisticated in their handling of valuation disputes over time.) 4 Such a process of adaptation or learning is a form of self-organization. Adaptation, including the emergence of new systemic features, occurs as sys- tem actors acquire expertise by importing valuation forms from outside the legal system, and gradually coming to master those forms.' From a courts- based perspective, one could say that the judges' exercise of their discretion gradually became more transparent. From a clients-based perspective, one could 106say that the risk--defined as the degree of dispersion of possible outcomes 6- of submitting a dispute to adjudication declined over time. Note that an increase in the valuation sophistication of bankruptcy actors does not mean that those actors move closer to a common valuation standard. The phenomenon of rising sophistication in conducting valuation determinations-for example, the importation of technical financial models into the courtsl7-seems largely independent of whether the valuer uses a common or independent theory of value."l4 104. See, e.g., Butler, supra note 9, at 342 (noting an increase in bankruptcy valuation proceedings as "bankruptcy practitioners become more practiced in their art"); Fortgang & Mayer, supra note 9, at 1061 (sketching basic finance principles for bankruptcy valuation proceedings). The articles from Butler (a practitioner in a South Carolina law firm) and Fortgang and Mayer (practitioners at Wachtell, Lipton, Rosen & Katz in New York City) are themselves data indicating the bar's gradually increasing financial sophistication in trying valuation disputes. 105. This learning occurs first as judges become educated by experts about financial valuation methods, and then, as similar issues recur multiple times, deepening the courts' experience and increasing their confidence to criticize and depart from experts' models. See ELIOT FREIDSON, PROFESSIONAL POWERS: A STUDY OF THE INSTITUTIONALIZATION OF FORMAL KNOWLEDGE, at xi (1986) ("[W]hile the institutionalization of knowledge is a prerequisite for the possibility of its connection to power, institutionalization itself requires the transformation of knowledge by those who employ it."); Trujillo, supra note 1, at 536-39, 547-53 (discussing intersystem "transportation- cum-distortion" of forms). 106. See, e.g., WILBUR G. LEWELLEN, THE COST OF CAPITAL 17 (1969). 107. See, e.g., Oliver E. Williamson, Chester Barnard and the Incipient Science of Organization, in ORGANIZATION THEORY: FROM CHESTER BARNARD TO THE PRESENT AND BEYOND 172 (Oliver E. Williamson ed., expanded ed. 1995) (discussing "managerial norms"). 108. For example, the DCF is a sophisticated valuation technique, yet is based entirely on projections of an individual firm's peculiar use of property. 3. Summary of Findings The data offer preliminary support for the conclusion that some aspects of the U.S. bankruptcy legal system show a tendency to self-organize. Con- clusive evidence of self-organizing dynamics in a legal system could have sub- stantial jurisprudential significance. We know that simple deterministic dynamics do not explain the data we observe in legal systems. Since the decline of legal formalism, the dominant mode of explanation has been to attribute a randomness, or nondeterminism, to legal system dynamics and to suggest that any observable patterns are due to exogenous ordering-such as decisionmaker bias-that affect legal ordering intersystemically.' ° Evidence of self-organizing dynamics suggests the possi- bility that at least some of the patterns we observe are generated by deter- ministic dynamics operating intrasystemically. Legal realism and critical legal studies moved legal explanation from simple determinism to nondeterminism plus exogenous ordering. Now, evidence of self-organization may move legal explanation back in the direc- tion of a complex determinism. Self-organization suggests that at least some patterns in legal data are generated by a complex and nonlinear deterministic dynamics-in other words, chaos."' Put another way, the data in this study may help to falsify quantitatively the jurisprudential hypothesis that legal dynamics are random rather than chaotic. Any such argument based on these data is, however, far from conclusive because of the low number of observations-180, with some variables pos- sessing fewer than 180 observations. These data simply are not sufficient to take advantage of all that time-series analysis has to offer in substantiating robust conclusions about the dynamics of the legal system."I 109. The Efficient Capital Markets Hypothesis (ECMH), as a form of financial explanation, follows a similar course. The ECMH attributes a randomness to the trade-to-trade movements of stock prices and argues that only exogenous forces-the surprise of new information--explain price movements. See, e.g., Eugene F. Fama, Efficient Capital Markets: A Review of Theory and Empirical Work, 25 J. FIN. 383 (1970). 110. See SPROTT, supra note 1, at 20. Sprott notes that: Chaotic systems ... have a number of characteristics: 1. They are aperiodic (they never repeat). 2. They exhibit sensitive dependence on initial conditions (and hence they are unpredictable in the long term). 3. They are governed by one or more control parameters, a small change in which can cause the chaos to appear or disappear [and] 4. Their governing equations are nonlinear. Id. 111. See id. at 236-38 (discussing time-delayed space embedding). Patterns in a Complex System 397 This research on legal systems' self-organization can take at least three possible future directions. First, we can go about collecting much more data, in order to get the number of observations up to the point that will support robust conclusions. Second, we can pursue a number of methodologies that have been developed to assist in the analysis of low-observation datasets." '2 Finally, we can construct a mathematical model of the system dynamics we want to explain, and that model will generate sufficient observations to support a full time-series analysis. CONCLUSION This Article has presented some initial results from the bankruptcy cramdown valuation database. It also has set out to explain both the doctrine of bankruptcy valuation, and what that doctrine might indicate about the dynamics of the bankruptcy legal system. Major findings of this study include the following: 1. Courts tend to split the difference in valuations much less fre- quently than we had expected (Figures 1-2). 2. Parties' valuations, initially close together, have moved further apart over time (Figures 3-5). 3. Bankruptcy courts are much more likely to use a common valua- tion standard when the valuation includes a calculation for the time value of money (Table C). 4. There seems to be some geographic distribution of courts' accep- tance of valuation models, with courts in southern circuits more likely to accept soft valuation models, such as owner testimony or evidence of general market conditions, and nonsouthem circuit courts more likely to accept hard valuation models, such as dis- counted cash flow models, comparables, or liquidation (Table E). 5. There is evidence that bankruptcy system content tends to self- organize. Specifically, variation decreases over time (Figures 7-12), and we see evidence of form transportation and norm emergence (Figures 13-16). Areas of future research based on these data may include an exploration of the relationship of valuation models to success, to find out whether certain arguments about valuation achieve client victory more often than others. 112. See, e.g., John C. Gallant et al., Estimating Fractal Dimension of Profiles: A Comparison of Methods, 26 MATHEMATICAL GEOLOGY 455 (1994); Alberto Malinvemo, A Simple Method to Estimate the Fractal Dimension of a Self-Affine Series, 17 GEOPHYSICAL RES. LETTERS 1953 (1990). 398 53 UCLA LAW REVIEW 357 (2005) Patterns in a Complex System Difference of +/-4 Total Number of Cases Recoded 150 Total Number of Cases with a Difference of +/-6 0 Percentage Difference 0.00% Percentage Same 100.00% Difference of +-5 Total Number of Cases Recoded 150 Total Number of Cases with a Difference of +/-5 0 Percentage Difference 0.00% Percentage Same 100.00% Difference of +/-6 Total Number of Cases Recoded 150 Total Number of Cases with a Difference of +/-6 0 Percentage Difference 0.00% Percentage Same 100.00% Difference of +-7 Total Number of Cases Recoded 150 Total Number of Cases with a Difference of +/-7 0 Percentage Difference 0.00% Percentage Same 100.00% Difference of +/-8 Total Number of Cases Recoded 150 Total Number of Cases with a Difference of +/-8 0 Percentage Difference 0.00% Percentage Same 100.00% Difference When Code Assigned No Value and Recode Assigned Value Total Number of Cases Recoded 150 Total Number of Cases with a Difference 7 Percentage Difference 4.67% Percentage Same 95.33% Total Difference Total Number of Cases Recoded 150 Total Number of Cases with a Difference 15 Percentage Difference 10.00% Percentage Same 90.00% The original coding process yielded 180 observations. The recode took fifty observations (27.8 percent of the observations in the database) and independently assigned codes for the three valuation standard variables- CtVS, DVS, and CVS. This process thus yielded 150 recodes. Table 1-A shows that, of the 150 recodes, 135 received the same code as in the initial coding process, resulting in a recode reliability rate of 0.9, which is well within the boundaries of reliability for social science research.' B. Ordered Logit Model Table C reports the results of an ordered logit model. This section explains the model and why such a model was needed. How one measures the dependent variable determines the selection of the appropriate statistical model. A very common statistical model is the least-squares regression. Least-squares regressions, however, require that the dependent variable be presented in interval measures-cardinal numbers that specify the distance between each value. For example, the dependent variable height is expressed in interval measures because differences between values (say between 5'8" and 5'10") are fixed and meaningful. The present study tries to explain how bankruptcy courts approach valuation. This dependent variable cannot be expressed with interval measures. We measure standards of valuation based on a set of conceptual categories. The values of our dependent variable are thus expressed not car- dinally (with numbers such as 1 and 2, or 5'8" and 5'10"), but rather ordinally (with numbers such as 1st and 2nd). While the values of cardinal numbers are known absolutely, the values of ordinal numbers are known only by the relationships among the values, for example, the second largest amount, so that the distances between ordinal categories are not quantified. Ordinal categories present a special problem for statistical model building. If data are presented in interval measures, the difference between one category and another is quantified by definition. But the statistician must take special steps to quantify the difference between ordinal categories. Because the data in this study are ordinally measured, we need a statistical model equipped to estimate the intercepts-where one category ends and the next begins-for each category. This study uses an ordered logit model, which estimates the intercepts for each category of the dependent variable. 113. See PAUL E. SPECTOR, SUMMATED RATING SCALE CONSTRUCTION: AN INTRODUCTION 65-67 (Sage University Paper Series: Quantitative Applications in the Social Sciences No. 07-082, 1992). 53 UCLA LAW REVIEW 357 (2005)402 Patterns in a Complex System. 403 For an ordinal variable withJ categories, let the odds that the dependent variable (the y term in the equation below) will have a value less than or equal to a given intercept m, versus the odds that the dependent variable will have a value greater than a given intercept m, be: (2 (X) = Pr(y mkIx)/Pr(y > mjx) where x is a vector of explanatory variables, and we estimate for J-1 thresholds. This equation merely defines the odds of an event-that the dependent variable will have a certain value relative to the independent variable-as a ratio of the probabilities for that event. The log of the odds as defined in the first equation is thus equal to: ln _ (X) = r + x'mkl mt Mk where r states the estimated thresholds, y estimates the intercepts along the y-axes, and 8 (as a vector of the coefficients) states the slope of the dependent variable as it moves along the x-axis. This equation predicts the value of the dependent variable given certain independent variables. Figure 1-1 graphically represents the parameters and predictions of the model. FIGURE 1-1 ORDERED LOGIT MODEL OF CTVS: GRAPH AND EQUATION I 01 M , I .~ m m Common Mixed Independent x - values of independent variables