Rules are Made to Be Broken(ly Followed):

Probabilistically Modelling the Dynamics of

Escaping the Law

Rui-Jie Yew

April 22, 2024

Abstract

This research project presents a start to the task of modelling dynamics aris-

ing from the excludable nature of avoiding the law. Leveraging literature from

economics and law and an existing dynamical model for legal avoidance, I mo-

tivate the potential of this dynamic to be modelled using competitive Lokta-

Volterra equations. The system leverages probability for regulatory muta-

tions, which has relevance to concepts from CSCI 2540.

1 Introduction

In1696,KingWilliamIIIintroduceda“windowtax”onEnglishresidents. The

number of windows on your house above ten would leave you susceptible to

additional variable taxation. This tax intended to increase the property taxes

in line with the extravagance of the homes taxpayers resided in. But, instead,

residents covered up their windows, and homes were built with fewer windows.

Residents lowered their cost of compliance by having fewer windows than the

threshold that triggered additional taxes [5].

Today,theavoidanceoflegalrulescontinuesacrossarangeoflegaldomains–

from tax law to product safety to intellectual property [4]. In tax law, the

evasion oftaxesisnotlegal, butstructuringyourentityoryourtransactionsto

avoid taxes, on the other hand, may well be [3]. In intellectual property law,

existing patents have been designed around so as to achieve the same purpose

but with sufficiently different technical details to achieve a new patent [1].

Theadagegoesthat“rulesaremadetobebroken”,butperhapsthey’remade

to be brokenly followed. Given a set of rules, regulated entities may slither on

theboundariesofliability. Whenthisbehaviorisundesirablefromaregulator’s

perspective,itinducesaregulatoryresponse. Inthisresearchproject,Ievaluate

andextendthemodelpresentedin[6]ofthedynamicsofasystemofregulation

coming up against evolving strategies for legal avoidance. Namely, I consider

1

andmodelavoidanceofregulationsasanexcludable, rivalrous goodasdiscussed

in [7] by using competitive Lokta-Volterra equations. This model contains

importantprobabilisticaspects. Namely,IretainandandimplementtheMonte

Carlo protocols as detailed in [6].

2 Background

Ibuildonthemodelprovidedin[6],whichaimstocapturethedynamicsbetween

payday lenders and regulators. In particular, two main differential equations

capture the adversarial relationship between lenders and regulators:

(cid:20) (cid:18) (cid:19)(cid:21)

dL L

i = L g 1− T −(L f)−(L R d)+[L (Q )−L (Q )] (1)

dt i K i i i i−1 i−1,i i i,i+1

(cid:18) (cid:19)

dR R

i =P +R m 1− i (2)

dt i−1,i i i K

R

Thedynamicsthattheseequationsinducecanbesummarizedasfollows: as

the number of lenders grow in generation i, regulators mutate accordingly to

generation to get on their case and lenders mutate to generation i+1 to avoid

regulators.

L represents the lenders that live in generation i, and R represents the

i i

regulatorsthatliveingenerationi. Notably, wecanseethatK andK arethe

R

carrying capacities of lenders and regulators, respectively.

The terms Q and P represent a Monte Carlo protocol that determines

whether mutation occurs on the lender’s side and the regulator’s side, respec-

tively.

The Monte Carlo protocol for Q is defined as follows. Consider a uniform

random variable r between 0 and 1.

1

(cid:40)

r , if r ≤s∗R +(1−L )∗cc∗(L )

1 i i i

0, otherwise

Here, s is the rate of innovating ways around the law, while cc is the rate of

copycatting.

The Monte Carlo protocol for P is defined as follows. 1 Consider a uniform

random variable r between 0 and 1.

2

(cid:40)

r , if r ≤L /L

2 2 i T

0, otherwise

1Thedefinitionofthevariablemi aspartofthepaperwasnotclearandthesimulationin

theirMatlabfileanddoesnotappeartoincludethevariable. Thus, Ialsoomitthevariable

inmyimplementation.

2

2.1 Avoidance of the Law as an Excludable Good

Excludable goods, as described in the economics literature, are goods such that

peoplecanbeexcludedfromenjoyingthem. Rivalrousgoodsaregoodsthatcan-

notbeenjoyedbymultiplepeopleatonce. Methodsthatpromotetheavoidance

of the law can be considered excludable. Consider the use of peer-to-peer net-

works to download your favorite song as a way to avoid effects of copyright

law[7]. Thisuseisexcludablebecauseaccesstopeer-to-peersoftwareislimited

tothosewhoaretechnologicallysavvy. Moreover,developersofthismethodcan

controlandrevokeaccess,excludingcertainusersandusesoftheirtechnologies.

Methods to avoid the law may also be considered a rivalrous good. First,

when a firm employs an avoidance strategy the firm is typically the only en-

tity that benefits from the use of the avoidance strategy. When multiple firms

employ the same avoidance strategy, the avoidance strategy could lose value.

For instance, multiple firms employing the same strategy could lead to greater

regulatory detection because the externalities that may arise from the use of

these strategies may become compounded.

2.2 Lokta-VolterraandCompetitiveLokta-VolterraEqua-

tions

Lokta-Volterra TheLokta-Volterrasystemofdifferentialequationsisacanon-

icalsystemthatwasinventedtomodelthedynamicsbetweenpredatorandprey

in an ecological context [2]. It is written as follows:

dx

=x(a−by)

dt

dy

=y(−c+dx)

dt

Logistic Growth Another important equation to go over is the equation for

logistic growth:

dN N

=rN(1− )

dt K

The logistic equation models the situation where a population has a carrying

capacity of K and grows proportionally until it reaches that carrying capcity.

Lokta-VolterraCompetitiveModel TheLokta-Volterracompetitiveequa-

tions model inter-species competiive dynamics–building on both the logistic

equation and the canonical Lokta-Volterra model. They are written as follows:

(cid:18) (cid:18) (cid:19)(cid:19)

dx x +α x

1 =r x 1− 1 12 2

dt 1 1 K

1

(cid:18) (cid:18) (cid:19)(cid:19)

dx x +α x

2 =r x 1− 2 21 1

dt 2 2 K

2

3

Figure 1: This figure is from [7]. This paper focuses on modeling the dynamics

arising from the avoidance of the law, which is detailed on the right column.

where α represents the impact of the population of x on the population of

12 2

x and α represents the impact of the population of x on the population of

1 21 1

x . The term reflects a proportional relationship between your own species and

2

the other species.

3 Project Contribution

As part of this project, I re-implement and extend the model provided in [6],

whichIdetailinSection2. IimplementthemodelusingPythonandlibrariesin

Python, and I additionally incorporate legal theory about avoidance of the law

as detailed in [7]. As part of the model presented in [6], ways that mutations

can be learned (via innovating or copycatting) are treated the same. In prac-

tice,anentitytypicallybenefitsfrombeingthesole beneficiary ofwaystoget

around the law.2 Thus, in addition to investing in a particular strategy to get

around a law, there is likely also investment in excluding others from partak-

ing in the same strategy. I stratify copycatters and innovators as two different

groupsofactorswithinter-specific, competitive dynamics. Thisextendsthe

model provided in [6] to consider the dynamics between competitors and inno-

vators. I model this phenomenon by leveraging competitive Lotka-Volterra

equations.

2[7]discussescertainavoidancemechanismsasexcludablegoods.

4

Figure 2: I consider two strands of behaviors for each regulatory mutation.

4 Model

As part of the model presented in [6], ways that mutations can be learned (via

innovatingorcopycatting)aretreatedthesame. Aspartofthisresearchproject,

I implement the model presented in [6] from scratch using Python libraries. I

extendthemodelin[6]bystratifyingcopycattersandinnovatorsastwodifferent

groups of actors with dynamics involved. Unlike the model described in [6], in-

novatorsareharmedbycopycattersandalsoattempttoprotecttheiravoidance

strategies/ protect how they get around laws from copycatters. I model this

phenomenon by using competitive Lotka-Volterra equations. Competitive

Lotka-Volterra equations consider the population dynamics of species compet-

ing for common resources. In the case of regulatory strategy, copycatters have

a negative impact on innovators.

5 Code

Codeforthemodelcanbefoundatthefollowinglink: https://colab.research.

google.com/drive/1BmcBtfNF8_ov26kdANQ7FN7Ja0LCsTnj?usp=sharing.

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6 Conclusion and Future Work

This work presents a starting point for emphasizing and modelling the avoid-

ance of the law as an excludable, rivalrous good. However, an important next

stepisconsideringtheimplicationsofthisbehavioronregulatoryresponse. Sec-

ondary and indirect liability have been considered important points at which

liability is attributed and regulatory responses are targeted. There becomes

the question of whether regulators should focus efforts on pinpointing innova-

tors, which are typically high-resource firms. This is potentially reflected in

regulatory approaches that focus on deterrence [?].

7 Acknowledgements

I thank Daria Roithmayr and Fei Fang for generously sharing resources from

their paper [6]. This project additionally benefited from conversation with

Zheng Dai.

References

[1] Dan L Burk. Perverse innovation. Wm. & Mary L. Rev., 58:1, 2016.

[2] Josef Hofbauer and Karl Sigmund. The Lotka–Volterra equations for two

competing species, page 22–30. Cambridge University Press, 1998.

[3] Alfred Roman Ilersic. Tax avoision: The economic, legal, and moral inter-

relationships between avoidance and evasion. (No Title), 1979.

[4] Leo Katz. Why the law is so perverse. University of Chicago Press, 2011.

[5] Wallace E Oates and Robert M Schwab. The window tax: A case study in

excess burden. Journal of Economic Perspectives, 29(1):163–180, 2015.

[6] Daria Roithmayr, Justin Chin, Fei Fang, and Bruce Levin. The cat and

mouse of getting around the law. In Proceedings of the 2019 International

Conference of The Computational Social Science Society of the Americas,

pages 73–82. Springer, 2021.

[7] Tim Wu. When code isn’t law. Va. L. Rev., 89:679, 2003.

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