# Locke & Judicial Disincentives

This is the basis for the great law of nature, Whoever sheds man’s blood, by man shall his blood be shed. Cain was so fully convinced that everyone had a right to destroy such a criminal that after murdering his brother he cried out ‘Anyone who finds me will slay me’—so plainly was this law written in the hearts of all mankind.

For the same reason a man in the state of nature may punish lesser breaches of the law of nature. ‘By death?’ you may ask. I answer that each offence may be punished severely enough to make it a bad bargain for the offender, to give him reason to repent, and to terrify others from offending in the same way.

— Locke, Second Treatise of Government

## Justice - Addition via Subtraction¶

A canonical example often used to explore the concept of morality in a Game Theory framework is the "Hawk-Dove Game". Imagine two birds simultaneously discover $S$ seeds. Each bird has the opportunity to be either cooperative (a "Dove") or aggressive (a "Hawk"), with the following outcomes:

• if both cooperate, they share and each get $\frac{S}{2}$ seeds
• if one cooperates and the other is aggressive, the aggressive bird gets all $S$ seeds and the cooperative bird gets none
• if both are aggressive, then they fight, which hurts them both, before they are finally able to separate the seed pile. Some of the seeds they get are needed to heal their injuries, so they each effectively only get $\frac{S-C}{2}$ seeds.

This is often visualized as follows:

Dove Hawk
Dove $(\frac{S}{2}, \frac{S}{2})$ $(S, 0)$
Hawk $(0, S)$ $(\frac{S-C}{2}, \frac{S-C}{2})$

One way of thinking about the outcomes is to imagine either party as purely self-interested, pursuing the best outcome in this one game. With that criterion, each bird should be aggressive -- if the other bird is cooperative, then being aggressive gets the whole pile, instead of half, whereas if the other bird is aggressive, then being aggressive at least get some amount. This is a common setup in analysing cooperation -- were either to not cooperate, they could almost always gain some benefit.

Imagine a third party comes along and proposes to the two birds that he will fight against whichever bird is aggressive, or both if they are both aggressive, so that the payoff now looks like this:

Dove Hawk
Dove $(\frac{S}{2}, \frac{S}{2})$ $(S-J, 0)$
Hawk $(0, S-J)$ $(\frac{S-C-J}{2}, \frac{S-C-J}{2})$

If this party is strong enough (imagine $J = S$), then we end up with the following:

Dove Hawk
Dove $(\frac{S}{2}, \frac{S}{2})$ $(0, 0)$
Hawk $(0, 0)$ $(\frac{-C}{2}, \frac{-C}{2})$

In this setup, no matter what the other bird does, each bird should always be a Dove. If the other bird cooperates, they both win. If the other bird is aggressive, then at least they aren't getting hurt.

Overall, this means it's much more likely for both birds to cooperate and share the seeds. The overall situation is actually now much better for both birds, even though nothing has been added. This is how a monopoly of force can enhance cooperation between members in a society -- by making defecting a much less attractive possibility. Locke points out the condition that must be satisfied: "...each offence may be punished severely enough to make it a bad bargain for the offender, to give him reason to repent, and to terrify others from offending in the same way."

"If your right eye makes you stumble, tear it out and throw it from you; for it is better for you to lose one of the parts of your body, than for your whole body to be thrown into hell. If your right hand makes you stumble, cut it off and throw it from you; for it is better for you to lose one of the parts of your body, than for your whole body to go into hell."
-- Jesus, Matthew 5:29-30

Plato government virtue...

## Alternative Frameworks of Punishment¶

### Punitive¶

It is, of course, not always the case that crimes are committed due to rational self-interest -- crimes of passion are common and, for many people in many situations, their passions may overwhelm any disincentive.

! fear-inducing; intentionally cruel

### Virtue-Inducement (this fmk)¶

! proper object of operation of the government is the meme towards virtue / lack of virtue, not the individual. Xpian "my kingdom is not of this world" / forgiveness and prospective justice

## Least Sufficient Disincentive¶

Imagine a crime that has a payoff to the criminal of $X$, and that there is some chance of the criminal getting punished $p$. For the crime to not be in the best interest of the criminal, the expected value of the punishment $J$ has to be at least the value of the crime, that is: $$\mathbb{E}(J) = pJ \geq X$$

Note that $p$ is the probability that (1) the criminal is caught and (2) the criminal being found guilty. This means that, if there are improvements for either the capture rate for crimes or the true positive rate of the justice system, then theoretically the required sentences should be decreased. I've never heard anyone make this claim for either police or judicial efficiency, but it's very important--a more effective police or judicial system minimizes the amount of punishment the government needs to inflict.

Imagine a world where, if you committed a robbery, you could be absolutely sure it would be found out and you would be punished. In that system, if the punishment was only to return all the money and one additional dollar, then you would still never have a rational reason to rob. If the lower bound on the amount of punishment that needs to be inflicted for a crime with payoff $X$ is $X$, then we can define an efficiency cost $C_{eff}$ to the judicial system:

$$C_{eff} \triangleq \frac{X}{p} - X = X (\frac{1}{p} - 1) = X \frac{1 - p}{p}$$

That is, the efficiency cost of a judicial system is the payoff of the crime ($X$) times the odds ratio of getting away with it ($\frac{1-p}{p}$).

More exactly, if for this specific crime, given a best possible $p$ as $p^*$:

$$C_{eff} \triangleq \frac{X}{p} - \frac{X}{p^*} = X (\frac{1}{p} - \frac{1}{p^*}) = X \frac{p^* - p}{pp^*}$$

Consider now the case of punishments for attempted crimes. While the above dealt with the punishment in case of a success, punishment for attempted crimes should consider the case of failure. Imagine a success probabilty $s$, and a probabilty $q$ of being punished if you fail. With success probability $s$, the expected number of attempts until success is $\frac{1}{s}$, so, imagining $\frac{1}{s}$ attempts before a success, each of which could result in punishment:

$$\mathbb{E}(J_{\text{attempt}}) \cdot \frac{1}{s} \geq X$$$$q J_{\text{attempt}} \geq s X$$$$J_{\text{attempt}} \geq \frac{s}{q} X$$

Note that if $s$ is close to $1$ (that is, the crime is almost always successful), then the punishment for the attempt is close to the punishment for the completion of the act. However, if $s$ is close to $0$ -- that is, a crime is attempted that has almost no chance of success -- then it only takes a slap on the wrist to disincentivize further action.

Given some ideal success rate $s^*$ and some ideal punishment rate $q^*$, the efficiency cost is:

$$C_{eff} \triangleq \frac{s}{q}X - \frac{s^*}{q^*}X = X\frac{sq^* - s^*q}{qq^*}$$

Note that if it were possible to make the crime entirely preventable ($s^* = 0$), then the entirety of the punishment would be unnecessary.

The reduction of the success rates for crimes, and the ability to punish criminals after failed attempts are now two additional variables which, theoretically, should reduce the amount the government must punish its citizens. There is, of course, another variable in this equation that we have been ignoring all this time -- that of the payoff $X$ of the crime. Reduction of $X$ could be achieved in the economic sense by making it harder to resell stolen goods, but there's also a cultural element -- the lower the status of being a criminal, the less desirable successfully committing a crime is.

Under consideration:

• why independent punishment systems - does this result in double counting
• if s = 1 and q < p; J_attempt > J_success, which seems counterintuitive

## Variation Among Citizens¶

Parameter Variation ! note also that $p$, $s$, and $X$ might vary among members of the population -- should the value chosen be max/min? Or some number of standard deviations away? Is that variance the point of judicial sentencing?

#### Non-linear utility functions¶

Condition on wealth with log payoff -- behavior in extremes?

## The Cost of Punishing Innocence¶

?? SNR and false positives ??

to do:

• finish variation among citizens
• finish cost of punishing innocence
• finish alternative frameworks
• collect data on differential punishments (by geo, by time)
• collect data on crimes and success rates
• calculate implicit payoffs
• analyze comparative deltas in punishments; compare to model
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