Zero tolerance and behavior control

Suppose that a given aspect of societal behavior can be quantified according to a numerical value x, so that values of x greater than some threshold x_legal are illegal, and more harmful to society as x grows: examples of this are speeding and alcohol driving limits, tax evasion, etc.

What determines the choice of behavior, i.e. the preferred x, for a given individual? If we denote by u(x) the utility of x for the individual, this will choose the x at which u(x) is maximum. In the absence of any law regulating this behavior, u(x) is composed of two kind of components:

1. Positive factors that make the individual utility grow as x increases: Pleasure from speeding, money evaded, etc.
2. Negative factors detracting from the utility as x increases: Fear of an accident, moral concerns, etc. By definition, law penalties are not included here.
   

To simplify things, let us assume that positive and negative factors contribute in such a way that u(x) increases up to a single maximum at x_max and then decreases from that point on --in other words, u(x) is a unimodal function. When there is no law enforcement, if x_max ≤ x_legal the individual is a law-abiding citizen, otherwise she is a law offender.

Fig. 1: unimodal utility function.

Consider now the introduction of a law penalty p(x) which is null for x ≤ x_legal and non-decreasing for x ≥ x_legal. Under this new scenario, the resulting individual utility is

w(x) = u(x) − p(x).

For law-abiding individuals, the penalty has no effect and thus the resulting behavior is unaffected. For law offenders, the utility of their original position decreases; if p(x) is high enough for all x > x_legal, the maximum of w(x) will be reached then at x_legal and the former offender will opt for law abidance.

There are several approaches to designing an effective penalty function. A zero-tolerance policy is one for which even small offenses on the vicinity of legal behavior are punished severely and without exception. Within our model, a zero-tolerance penalty has the form:

p(x) = p₀ + r(x), x > x_legal,

where p₀ > 0 and r(x) is a positive, non-decreasing function (for x > x_legal) which we call the residual penalty. p₀ is the minimum penalty an offender will get regardless of the degree of the offence. The greater p₀, the tougher the policy is. On the other hand, we say that p(x) is a progressive penalty if it is nondecreasing and continuous (zero-tolerance penalties are obviously not continuous at x_legal). Please note that the characterizations given here for zero-tolerance and progressive penalties are my own and should not be taken as universally known or accepted. The following two figures show the effect of a zero-tolerance and a progressive penalty on the utility function of a law offender.

Fig. 2: impact on utility of a zero-tolerance penalty.

Fig. 3: impact on utility of a progressive penalty.

In both cases the penalty makes the individual opt for a legal position, but the zero-tolerance policy does it at the cost of stricter than necessary punishment for small offences, as stated by the following

Fact. For any zero-tolerance penalty p(x) succesfully enforcing law on a utility function u(x) there exists an equally effective progressive penalty p'(x) which is less severe than p(x), i.e. p'(x) ≤ p(x) for all x.

(We will formally state and prove this rather simple result in a later entry. Informally speaking, we can transformate p(x) into a progressive p'(x) in such a way that the gap u(x_legal) − w(x) tends to 0 as x approaches x_legal --for a zero-tolerance penalty this gaps tends to some p₀ > 0.)

The fact above is the main reason why I find it hard to justify the alleged benefits of zero-tolerance approaches to illegal behavior control as compared with traditional progressive penalties.