# An Approach to a Legal System with Utilitarian Members

Definition 1.  A static legal system is a collection of individuals together with a collection of laws.  (Mathematically, it’s a set $P$, whose elements are called persons, together with a unary operation $\varnothing\in P$, called the null person, and a collection of infinitary $\{0,1\}$-valued maps $\{l_n:P^\infty\to\{0,1\}\}$, called laws, such that all but finitely many of the terms in the domain are $\varnothing$).

For example, a law $l_n$ defined on persons $p_1,...,p_k$ may evaluate $l_n(p_1,...,p_k,\varnothing,...)=1$ , meaning that law $l_n$ pertaining to individuals $p_1,...,p_k$ is legal (or illegal if it returned value $0$).   To accommodate the influence that individual complexity has in a  legal system, we could make the additional assumption that each person is a static legal system with persons and laws respectively replaced by perceptions and thoughts.  Let $p$ denote the set of perceptions,  the thoughts have the form $\{t_n:p^\infty\to\{0,1\}\}$ such that all by finitely many of the perceptions are null perceptions.  The value the thought takes determines whether or not a behavior is executed.  But what is a behavior?  A behavior executed from a thought is a model that satisfies the perceptions on which that thought was defined.  Hence we could write

$\displaystyle t_n(p_1,...,p_k,\varnothing,...)=1\Rightarrow\left( B(t_n)\vDash\{p_1,...,p_k\}\right)$

where $p_i$ are perceptions.  In simple terms, this just means that you only act in agreement with your perceptions (simple thoughts that are assumed true).  To reconnect back to the original static legal system, we may now say that laws dictate the legality of individual or group behavior.  A person is simply a collection of perceptions together with thought functions.  Hence we may say that a law defined on a thought function $t$ is in turn defined on all behaviors $B(t)$.  We simply set $l(B(t))=l(t)$.

Let us now assume every person behaves in a utilitarian manner.  By this I mean that every person has a function $u_p:\mathcal{B}\to\mathbb{R}$ where $\mathcal{B}$ is the set of legal behaviors of an individual that satisfies the following conditions for all thoughts $t$ and perceptions $\pi_1,...,\pi_k$ such that $B\vDash\{\pi_1,...,\pi_k\}$ (we also omit null perceptions for convenience):

1. $u_p(B)\leq 0\Rightarrow t(\pi_1,...,\pi_k)=0$ and
2. $u_p(B)>0\Rightarrow t(\pi_1,...,\pi_k)=1$.

Hence, assuming legality of the behavior, the above implications are equivalences  (i.e. a legal behavior is committed if and only if it has positive utility).  We could extend the utility functions to illegal behaviors, and the legal utility would be defined as

$\lambda(B)=u(B)+u(B')$

where $B'$ is the cobehavior (or punishment) of $B$.  For legal behaviors $B$, we have $B'=\varnothing$ and hence $\lambda(B)=u(B)$.  But we will assume all committed behaviors are legal for simplicity.

Sometimes groups of individuals will work together as a firm (or corporation) and execute aggregate legal behaviors of positive aggregate utility.  The aggregate behavior and utility are defined as some function of the behavior and utility functions of members of the firm.

Example 2.  Suppose $B=\sum_iB_i$ is a corporate behavior which is a formal sum of behaviors of its members.  Define the aggregate utility $U_F$ of a firm $F$ as

$\displaystyle U_F(B)=\sum_i u_i(B_i)$

where $u_i$ is the utility function of the person committing behavior $B_i$.  Perhaps if a member contributes $n$ behaviors per day, then they may say the utility of a behavior is $S_p/(365n)$ where $S_p$ is the salary of person $p$.  In this case the corporate utility summed over behaviors throughout the year would be their (in economic terms) normal profit (assuming no other expenditures), but it isn’t a very useful value since it does not indicate economic profits, which is really their utility in capitalistic economies.

Definition 3.  A dynamic legal system is collection of static legal systems.  A dynamic legal system $P=\{P_t\}_{t\in\mathbb{R}}$ is continuous if for every static legal system $P_t$ there is some $\varepsilon>0$ such that the cardinalities of sets of laws for all static legal systems $\{P_{t-\varepsilon},...,P_t,...,P_{t+\varepsilon}\}$ differ by at most $1$ (i.e. laws don’t change too fast).

A token economy is a special kind of dynamic legal system, indexed by time, where we define the utility of a behavior to be $1$ if it gives the person $1$ token.  What makes a token economy special is conservation of utility.  This means that

$\displaystyle\sum_{ij} u_i(B_{ij})=0$

where $B_{ij}$ is the $j$th behavior committed by person $p_i$ for all static legal systems $P_t$ (i.e. the sum of token changes at any time is equal to $0$).

In particular it implies that any behavior $B_p$ of person (or firm) $p$ with utility $u_p(B_p)$ has a coperson (or cofirm) $p'$ such that $B_p$ is a cobehavior to person $p'$ and $u_p(B_p)+u_{p'}(B_p)=0$.

Note:  It’s definitely choppy in some areas and could use revision, but I thought I’d throw it out first.

Let me briefly summarize the post.  A legal system is a collection of people, where each person is, for all intensive purposes, a collection of behaviors, together with a collection of laws that govern those behaviors.  We can further assume that behaviors are committed provided that they are of positive utility.  The behaviors that take negative values are cobehaviors (behaviors executed by other persons for whom the utility is positive and equal in magnitude).  In actuality when one purchases something, although they lose money, they typically receive something else useful in return; however, our study is limited strictly to currency, in which case they lose from a purchase.  In this case we can think of the economy of currency/tokens as an approximation to a fluid that changes its density through transactions; whereas the individual who loses/destroys tokens in exchange for a product can simply think of the product as a manifestation of the tokens exchanged for it.