# Fairness of Governance III

Some time ago we investigated the equality of benefits. Roughly speaking let us consider degenerate real world actions into discretely selectable choices of action $a\in A$ given individual $x$, who has observable features $f(x)$ and protected feature $p(x)$. Suppose the company has to choose among a set of actions to take $a \in A$. What is a workable definite of fairness or equality in such a decision making effort with respect to protected properties $p$?

Let god bestow upon us, a neutral third party, with a utility functor u whose evaluation on the individual $u(x)$ results in a function $u(x)(a)$ is the utility of company taking action a to individual $x$, $u(x)(b)$ is the utility to individual $x$ of company taking action $b$.

Let $f$ be the decision process of company $g$, $g(x)$ is the decision company makes, some $a$ for the individual $x$. Then the right thing to do

$g(f(x)) = argmax_{a\in A}(u(x)(a)) = g(f(x), p(x))$

Specifies what it means to perform action $A$ indiscriminately with respect to $p$.

Suppose the protected properties $p()$ has domain in a space $M$. These are the values of protected attributes that we choose to strive for equality. For example M could be cartesian product of age, sex, race, birthplace, religion and political party.

$E(u(x)(g(f(a)))| p(x)) = c\ \forall p(x) \in M$

That the expected population utility for each variation of protected property is identically some constant $c$.

But such matter are purely to determine what a company does in consideration of its customers. What should a government do? For example, in a sentencing scenario as described in ProPublica’s Machine Bias? There are other costs more primal to the considerations: prison cost money to build, can justice and correctional actions be served with less prisoners ?

This matter is completely different from what we have considered above where corporations have, purely, the intent to service their customers utilities, in an equitable way with respect to that utility and protected and sensitive attributes.(OMG I have drank too much customer-centric-corporation-philosophy koolaid from my present employer) In this case, the government is trying to optimize for cost of operation–it is profit maximizing if we state it positively.

The part of our government in question is the justice system, aka the courts. It optimizes some “global” idealized justice $J$ Such that we can evaluate such a utility which can best described as “society’s utility in justice” or the “cost of injustice.” What this cost is in material-real-world units is hard to say, however let’s suppose it can be quantified deterministically in the same units. $J$ is functor mapping individuals to the justice of an action the government takes: $J(x)(a)$, or example, would evaluate very negatively if $x$ is innocent and $a$ is imprisonment. We skip innumerable details here regarding the process of due process, as well as the all-eventual-worlds analysis regarding later actions of $x$–god-oracle has given us an instantaneous justice function which we shall use.

The government in order to take action $a$ incurs a material-real-world costs, such as building prisons, let’s call these $C(x)(a)$ for the situation of acting on $x$.

Taking the action yields a utility of $R(x)(a)$. $R$ is the cost to the society after action $a$ is taken. For example: if a criminal is sentenced to no prison time and commits a crime, the damage of that crime, to society, is the cost $R$ (Result or Recidivism)

So, therefore, our rational government seeks to maximize its constituent utility subject to some constraints:

Maximize:

$argmax_{a \in A}(\sum_{x\in X}{J(x)(a) - C(x)(a) - R(x)(a)})$

With the constraint:

$J(x)(a) = c\ \forall x \in X$

(Some population $X$)

If the decision process can only be quantified probabilistically with some distribution of actions

Maximize:

$E_{a,x}(J(x)(a) - C(x)(a) - R(x)(a))$

With the constraint:

$E_{a,x}(J(x)(a)|p(x)) = c\ \forall p(x) \in M$

$M$ is the space of protected properties. Hard to see the link? Consider if $C$, $R$ or even $J$ are actually individually functor of $x$ through the two observation functions $f(x), g(x)$, such as in situations of automated intelligent machines, perhaps trained using machine learning technology.

Do these writings then have some more meaning?

What is the cost of injustice to society? Do we fear that we may lock up Einstein, Martin Luther King Jr., or Barack Obama? (That their $R$ for some $a$ are very large to the society?) what is the true cost of injustice? Perhaps it can be reduced to the legal costs and reparation costs due to a lawsuits from the aclu, naacp (what are some other litigious minority individual protection organizations?) what is the cost of injustice when government wrongly accuse, convict and imprison someone? Is wrongful deprivation of many important human rights: rights to privacy, for one, right of property for another, and right to pursuit of happiness for yet another; is the deprivation of an individual’s human right an insufferable injustice? What is the cost of injustice ?

What is the cost of $R$? What happens when a drunken driver, having been insufficiently rehabilitated, drives drunk and causes a major injury or death? What is the $R$ of a flying bullet? Or leaked cypher keys? Or even some “minor” trade secret?

Personally, the best I can imagine is $min(R)=max(J)$ the worst social injustice against a person is the greatest crime a person can commit.