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Dear Michigan students, friends and colleagues: I have enclosed a draft of an evolutionary dynamics project, pitched for a crowd that is relatively comfortable with differential equations. If it s been
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Dear Michigan students, friends and colleagues: I have enclosed a draft of an evolutionary dynamics project, pitched for a crowd that is relatively comfortable with differential equations. If it s been awhile since you thought much about calculus, you might find the supplement to be a relatively gentler (though longer) introduction to this work. If you still find the supplement rough going, worry not. My presentation will walk you through the project even more slowly. Two notes with regard to reading. First, you may want to look at the figures and graphs online, as they are in color. Second, the last part of the supplement includes robustness checks that are for an older version of the model: we are in the process of re-running those. Looking forward to our discussion soon! Best, Daria 1 CAT AND MOUSE: THE EVOLUTIONARY DYNAMICS OF GETTING AROUND THE LAW Daria Roithmayr, Justin Chin, Fei Fang and Bruce Levin Abstract Regulators in many fields of law tax shelters, payday lending, intellectual property--play a costly game of cat and mouse with those who innovate to escape regulation. Payday lenders, for example, have developed a series of innovative loopholes to avoid regulation, each more ingenious than the last; most recently, lenders have partnered with Indian tribes to claim immunity from state regulation altogether. Insight into this ever-evolving game of innovation comes from an unexpected place: recent evolutionary dynamics work in biology on adaptive drug resistance. Math models have used coupled differential equations to help public health officials optimize the timing and strength of drug treatment. Borrowing from these models, we develop a simple model to describe the dynamic cat and mouse relationship, with an eye towards advising regulators about how to optimize their strategy under constrained resources. We counsel regulators that in many instances, the optimal strategy surprisingly is to respond quite slowly to the latest innovative loophole or to weaken the strength of regulation. These counterintuitive results show that the evolutionary dynamics of biological systems can potentially shed light on the dynamics of social systems, as actors innovatively adapt to each other in new and often unpredictable ways. 2 In many fields of law, actors who are regulated are constantly hunting for new ways to escape regulation. Taxpayers are perpetually creating new tax shelters. Intellectual property pirates are always inventing new versions of file-sharing technology that aren t covered by the regulation. High frequency traders write new and innovative algorithms that actually use regulation to bait the trap for unsuspecting institutional traders. In response to these innovators, regulators must continually return to the drawing board to address the latest innovation, a process that is costly and wasteful. The cat-and-mouse problem emerges most often in fields in which the financial stakes are high and in which hackers can innovate with the stroke of a pen. 1 This regulatory arms race plays a central role in payday lending. Over the last two decades, predatory payday lenders have generated new and increasingly creative strategies to escape the reach of state (and now federal) regulation. Payday lenders latest ingenious innovation has been to partner with Indian tribes to be able to claim immunity from all state regulation. 2 In turn, regulators have had to enact new regulation to re-target the lenders, which then triggers further lender innovation, and so on. In this paper, we investigate the dynamics of this game between regulators and those they regulate, with an eye towards giving advice to the regulator. Insight comes from an unexpected place: recent work in biology on adaptive resistance. Biologists have modeled the cat and mouse game between adaptive pathogens on one side and the human immune system or drug treatments on the other. These models use coupled differential equations to describe the co-evolving relationship between pathogens that mutate, and immune systems that mutate in response, triggering further mutation in the pathogens and so on. Such models have helped public health officials to optimize their regulatory strategy against continually evolving pathogens. 3 1 For example, tax shelter promoters can create a new shelter merely by drafting a new legal partnership. Payday lenders can draft a simple legal agreement with an Indian tribe. (Buell, Katz). In HFT, traders have baited front-running traps using the SEC rule requiring rerouting of orders to exchanges with the lowest price. Peter Henning, High Frequency Trading Falls Through the Cracks of Criminal Law, The New York Times, April 7, For digital piracy, see Minku Kang, Justin Chin and Brandon Hartstein, Digital Copyright Piracy and Enforcement: An Arms Race Model (unpublished, on file with the authors). Innovative workarounds are very common in high frequency trading, tax compliance, cybersecurity and spam regulation, and intellectual property regulation. 2 Nathalie Martin and Joshua Schwartz, The Alliance Between Payday Lenders and Indian Tribes: Are Both Tribal Sovereignty and Consumer Protection at Risk? 69 Wash. & Lee L. Rev. 751 (2012). Adam Mayle, Usury on the Reservation: Regulation of Tribal-Affiliated Payday Lenders, 31 Rev. of Banking & Fin. Law 1053 (2011). 3 See, e.g., Bruce R. Levin, Fernando Baquero and Paul Johson, A model-guided analysis and perspective on the evolution and epidemiology of antibiotic resistance and its future, 19 Current Opinion in Microbiology 83 (2014); Abel zur Wiesch P, Kouyos R, Abel S, Viechtbauer W, Bonhoeffer S (2014) Cycling Empirical Antibiotic Therapy in Hospitals: Meta-Analysis and Models. PLoS Pathog 10(6): e doi: /journal.ppat ; Robert E. Beardmore and Rafael Pena-Miller, Rotating antibiotics selects optimally against resistance in theory, 7(3) Math. Biosciences and Engin. 527 (2010). Similar models have guided policymakers in managing insecticide-resistant mosquitos that cause malaria and the evolution of pesticide-resistant genetically modified crops Simon A. Levin, Ecological issues related to the release of genetically modified organisms in the environment, in Introduction of Genetically Modified Organisms into the Environment (H. A. Mooney and G. Bernardi eds. 1990); Andrew F. Read, Penelope A. Lynch and Matthew B. Thomas, How to Make Evolution-Proof Insecticides, 7 PLoS Biology e (2009). (Levin, Baquero and Johnson 2014; zur Weisch et al. 2014; Beardomre and Pena-Miller 2010; Levin 1990; Read, Lynch and Thomas 2009.) 3 Drawing on this literature, we develop a simple mathematical model to investigate this regulatory arms race. As in biological models, we use coupled differential equations to describe the reciprocal influence that law exerts on economic behavior, and in turn, that behavior exerts on law. We then conduct numerical simulations to optimize the timing and strength of a regulator s strategy in the face of ever-evolving legal resistance. More specifically, we counsel the regulator about how to allocate resources towards minimizing both predatory lending and the need to reregulate. Based on our toy model, we make two counterintuitive suggestions to the regulator. First, a regulator who cares as much about minimizing prohibited behavior as about the need to constantly retarget its regulation should slow the rate at which it adapts to escaping innovators. Second, a regulator who faces budget constraints that make minimizing costly re-regulation a top priority can do so by weakening its regulations. As these surprising prescriptions suggest, dynamic models of biological systems can potentially shed light on the hard-to-forecast dynamics of social systems, as actors innovatively adapt to each other in new and often unpredictable ways. Co-evolutionary Model of Relationships Between Regulation and Regulated Actors We develop a probabilistic model of coupled differential equations to describe one regulator interacting with multiple regulated actors. Regulated actors continually innovate to escape each new regulation, and the regulator must continually innovate to re-regulate the escaping innovators. The history of payday lending legislation in Maryland illustrates the arms race dynamics that we model and the deadweight costs such an arms race imposes. 4 In the late 1990s, lenders unsuccessfully tried to obtain an exemption from Maryland's usury limit on interest rates, but consumer lenders quashed those efforts. 5 In response, lenders began partnering with out of state and then national banks to be able to claim the right to charge the interest rate of the bank s home state. Maryland legislators countered with regulation making each of these moves illegal. Lenders then tried a series of innovations, each countered by retargeting regulation: adding exorbitant service fees on the loans, offering tax refund anticipation loans, and most recently, partnering with a federally recognized Indian tribe to claim immunity from all state regulation. 6 Federal regulators have now stepped in to regulate at the national level. To stay one step ahead of lenders, federal officials now closely monitor consumer complaints with the goal of mounting a 4 The history of the Maryland payday lending arms race is chronicled at length in Gomez v. Jackson Hewitt, 427 Md A.3d 443 (2011). 5 Gomez v. Jackson Hewitt, 427 Md A.3d 443 (2011). 6 Courts have yet to rule on the validity of the rent-a-tribe strategy: a Maryland federal court recently ruled that it would be more appropriate for the Maryland Commission on Financial Regulation to determine whether a tribal partnership entitled lenders to avoid state regulation. The parties then settled shortly after the federal court's ruling but before the commission could take up the question. See Adam Mayle, Usury on the Reservation: Regulation of Tribal-Affiliated Payday Lenders, 31 Rev. of Banking & Fin. Law 1053 (2011). See also Western Sky Financial v. Maryland Comm r of Fin. Reg., 2012 WL (unpublished). See Adam Mayle, Usury on the Reservation: Regulation of Tribal-Affiliated Payday Lenders, 31 Rev. of Banking & Fin. Law 1053 (2011). 4 response as quickly as possible. 7 They also respond with the strongest regulation possible. But, as we suggest below, a quick and strong response might actually be counterproductive, for the same reason that immediate and high-strength antibiotic treatment might actually encourage drug resistance. The dynamic model that we propose borrows from a recent literature in biology that uses coupled differential equations to map the evolutionary dynamics of drug resistance in bacteria. 8 These equations describe the dynamic evolution over time of bacteria and their regulators: viruses that are genetically engineered using the bacteria s own DNA to attack the bacteria. The mathematical model we propose here is similar to predator-prey models and is a variation of the population dynamics of lytic phage and bacteria found in (1). More details are available in the Appendix SI. We model a coupled equation system of multiple lenders and a single state regulator. We denote a population of strategy-specific predatory lenders of generation i as L , where L is the number of lenders in a strategy-specific population. The subscript i represents the particular strategy that lenders pursue (and a matching subscript for the regulator represents the matching state regulation that can effectively regulate this generational strategy.) The value of L changes over time, as ordinary business forces affect the market. The population of lenders grows logistically at rate g until it reaches market saturation level K, and lenders exit the market at rate f owing to business failure. In addition, the market loses lenders as the state regulator closes predatory lenders down (at lender closure rate d). The population dynamics equation that represents the evolving density of lenders is thus: $% & = L $' g 1 % -. L f L R d [1] In Equation 1, the first term on the right in brackets represents logistic lender growth. 9 The second term in parentheses represents lender failure for business reasons, and the third term models the closure or death of lenders from regulatory attack. In Equation 2 below, regulation matures over time in tandem with the lender population. $4 5 $' = R 6 m 6 ( ) [2] Here, the term R 6 represents a single strategy-specific regulator (state or federal). Because we have only a single regulator, the number of regulators does not change over time as it does for lenders. Rather, R 6 depicts the evolution of the regulation s maturity from 0 to 1, as it progresses 7 CFPB conversation. 8 I.G. de Pillis and A. Radanskaya, A Mathematical Tumor Model with Immune Resistance and Drug Therapy: An Optimal Control Approach, 3 J. Theoretical Medicine 79 (2001). Bruce Levin, Sylvain Moineau, Mary Bushman and Rodolphe Barrangou, The Population and Evolutionary Dynamics of Phage and Bacteria with CRISPR-Mediated Immunity, 9(3) PLoS e (2013); Bruce Levin, Nasty Viruses, Costly Plasmids, Population Dynamics and the Conditions for Establishing and Maintaining CRISPR- Mediated Adaptive Immunity in Bacteria, 6(10) PLoS Genet e (2010). 9 L ; represents the total number of lenders in the population, consisting of all strategies/generations. 5 from early glimmer of an idea through to committee report or proposed regulation, and finally to well-settled regulation fully clarified by jurisprudence. The maturation value of R, ranging from 0 to 1, reflects this life cycle of regulation, and a more mature regulation can close lenders more efficiently, as reflected in Equation 1. Regulation matures logistically at a rate m. More specifically, a single generation of strategy specific regulation of type i matures logistically at rate m towards full maturity at an R 6 of 1. Thus, Equation 2 represents the dynamics of a strategy-specific regulator developmentally maturing from 0 to 1. The system of lenders and a state regulator described by the above two equations converges to a steady state of lenders under regulatory pressure by a mature regulation, represented by Li = (1 (d + f)/g). The derivation of this equilibrium point is provided in the Appendix SI. We now extend the model to include the evolution, via innovative behavior, by both lenders and the state regulator. Lenders mutate or innovate with some probability to discover strategies that enable them to escape the regulator. The regulator s ability to close down lenders creates selection pressure that favors innovative predatory lenders, who outcompete lenders using the older strategy. Put more formally, in proportionate response to lender closure rates, lenders discover and adopt with some probability an innovative strategy that enables them to escape regulation. Introducing this new mutant lender strategy into the model is fairly straightforward. Proportionate to the closure of lenders, some fraction of innovative lenders from the previous generation i-1 will develop a new mutant innovative strategy i to escape regulation. Likewise, some fraction of lenders will copy those innovators. Pursuant to innovation and copying, populations of generation i will switch to a next-generation innovative strategy, i + 1, at rate Q , =9. Thus, each generation i loses lenders as those lenders escape to the next-generation strategy i+1. In addition, each generation also gains lenders from the previous generation, as they switch to the instant strategy i from an earlier, outdated strategy i-1. The equation describing the evolution of lenders via innovation over time is modified to include lender innovation, which is described as a sort of mutation: $% & = L $' g 1 % -. L f L R d + [L A9 Q A9, L Q , =9 ] [3] In Equation 3, innovative lenders are added from the last generation and also lost to the next generation, as evolution continues. More specifically, the fourth term (in brackets) represents this addition of mutant lenders from the previous generation and the subtraction of mutant lenders from the instant generation who have innovated to join the subsequent generation. In response to lender innovation, the regulator adaptively innovates as well. In particular, the regulator discovers with some probability p an innovative strategy to re-regulate the escaping lenders. Here, p represents the resources and effort that the regulator devotes to innovating to discover a new regulatory strategy. The regulator s probability of discovering how to re-regulate is a stochastic term, and is proportional to the relative fraction of escape lenders, % 5, as well as the %; maturity of the existing regulation R . We thus represent the probability of the actual event of adaptation, adjusted for changing conditions and for uncertainty, as P A9, . This term is a random variable drawn from a distribution described in more detail in the SI/Appendix. 6 The modified dynamics for the regulator, which now includes the birth of a new regulation, is written thus: $4 5 $' = P 6A9,6 + m R 6 (1 4 5 E F ) [4] In Equation 4, the first term is the new birth term that represents the probability that an earliergeneration regulation will adapt to re-regulate the latest generation of mutant lenders. As before, the second term is the growth or maturation term. In our model, regulation does not regress in maturity. Regulation is additive, meaning each new generation is able to address lenders with a matching subscript and all lenders from earlier generations. The Regulator s Strategy: Strength and Timing In managing drug resistance, public health officials must balance the need to eliminate harmful pathogens with the need to minimize the number of new drugs that researchers must discover to control newly-resistant pathogens. Optimizing regulation in the face of evolving lenders poses the same central challenge for the regulator. The regulator must minimize levels of predatory lending at the same time it minimizes the cost of retargeting regulation to effectively recapture escaped lenders who have innovated their way out of regulatory reach. The regulator controls two key parameters: regulatory speed and regulatory strength. First, the regulator decides with what speed to respond to a new lender innovation, represented by the control parameter p. 10 This parameter represents the maximum probability in the next time step that the regulator will adapt to re-regulate the escaped lenders, assuming that all lenders have escaped to the next generation. The inverse of this term 9 specifies the amount of time that the G regulator allows to elapse before beginning the process of developing a regulatory response. Regulators typically control the rate at which they can adapt to lender innovations by hiring more lawyers and legislative assistants to improve the regulator s ability to retarget regulation to recapture escaping lenders. As we will discuss, budget constraints often impede a regulator s ability to re-regulate escaped lenders. Second, the regulator decides how strong to make the regulation, as measured by the rate at which the regulation closes predatory lenders. This parameter is d, the lender death or closure rate, measured in lenders closed per given period of time. Here, regulators can choose whether to adopt very strong regulation, for example, interest rate caps that prohibit lending over a threshold interest rate, or relatively weaker regulation, for example, roll-over prohibitions that limit the number of times a borrower can roll their loan over to a subsequent borrowing period. 11 These control parameters affect the rise and fall of payday lender generations, and the emergence and maturation of re-regulation. 10 Importantly, p is the regulatory control set by the regulator, who chooses for example how many lawyers to assign to the task of re-regulation, as noted above. But adapting is uncertain, and the actual probability of adapting may not correspond
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