If a group of individuals subject to some rule face an enforcement mechanism that is limited in its capacity, their rates of rule-violation will be interdependent.

Imagine a classroom of well-behaved children. When Johnnie throws a spitball at Suzie, Ms. Jones can give him her full attention, and Johnnie learns (and the others learn vicariously) that he can’t get away with throwing spitballs in Ms. Jones’s class.

Now imagine a classroom full of unruly children. When Johnnie throws a spitball at Suzie, Ms. Jones is too distracted by the need to break up the fight between Dick and Fred to have time to rebuke Johnnie, let alone the six others who are acting out at the same time. Johnnie and the others learn that they can get away with almost anything in Ms. Jones’s class.

Thus both the well-behaved and the ill-behaved classroom are self-sustaining situations. Indeed, they can be two equilibria of the same system: the very same children with the very same teacher may wind up either well-behaved or ill-behaved as the result of random accidents at the beginning of the period. This is an instance of the classic “tipping” model introduced by Thomas Schelling and popularized by Malcolm Gladwell.

In such a situation, the following statements can be rigorously demonstrated using fairly minimal behavioral assumptions, as Beau Kilmer and I show in a paper called “The Dynamics of Deterrence” (published in the *Proceedings of the National Academy of Sciences*, and adapted as Chapter 4 of *When Brute Force Fails*):

- Increasing enforcement capacity can lower not only violation rates but the volume of punishment actually administered.
- A temporary increment to enforcement capacity can have a lasting impact on violation rates if it succeeds in “tipping” the system from its high-violation to its low-violation equilibrium.
- Even if increased enforcement capacity cannot be obtained, the same effect can be created by “dynamic concentration”: focusing enforcement effort on a subset of offenders until their behavior comes under control, and then using the enforcement capacity freed up by reduced violation rates among that initial focus group to expand the size of the group.

The logic of these claims can be illustrated in a simple two-person game. Let Al and Bob by rational, risk-neutral actors, both be subject to some rule, and let the cost of complying with the rule be $10 while the penalty for non-compliance is $15. Assume that Al and Bob are not conscientious about the rule: each treats a penalty dollar and a compliance-cost dollar as of equal value.

Let Al have the first move: he either complies or violates, and then Bob chooses whether to comply or violate.

Assume that the enforcement system is constrained to be able to punish only one violation each round. Thus if Al alone violates, he is punished with certainty; if Bob alone violates, Bob is punished with certainty. But if both violate, each is punished with probability ½.

What should Al do? If he complies, he pays $10. If he violates, he pays $15 with certaintyif Bob complies, but pays $15 with probability ½ if Bob also violates. Since Al is risk-neutral, he values that ½ chance of a $15 penalty at $7.50. So Al wants to do whatever Bob does: he would prefer to violate, if Bob also violates, but would prefer to comply, if he thinks Bob will comply.

Under the standard “common knowledge” assumptions of game theory, Al can predict Bob’s behavior by assuming that Bob will act rationally. If Al violates, then Bob’s choice is between violating also, paying an expected cost of $7.50, or complying, paying a cost of $10. Thus Bob ought rationally to violate in that situation, and Al can be confident that he will do so.

Since Al prefers to violate if Bob violates also, and since Al knows that Bob will violate if Al violates, Al will indeed violate, as will Bob. Thus the score for each round is: violations 2, sanctions 1.

If the sanctions capacity constraint is relaxed so that two punishments per round can be given, then both Al and Bob know that violation will lead to certain punishment. Therefore, they both comply, and neither is punished: violations 0, sanctions 0. This illustrates the claim that greater sanctions capacity can lead not just to lower violation rates but to less actual punishment: a convincing threat never has to be carried out.

But – and this is the key point – it is not necessary to relax the constraint. A strategic enforcement authority can bring both Al and Bob in to compliance by establishing a priority order for punishment.

Say the enforcer announces that, if both Al and Bob violate, it is Al who will be punished. In that case, Al will certainly comply. But once Al has complied, Bob will also face certain punishment if he violates, so Bob will comply as well. Violations 0, sanctions zero. (The same is true if Bob is given priority; Al knows that Bob will comply, and therefore that if Al violates he will be punished.)

And the system can be extended to Charlie and Dan and Edgar: in theory to countably many potential violators. If no one wants to be the first violator, then no one will violate at all.

That, in a nutshell, is the key to having less crime and less punishment. The trick is making it work in practice. The HOPE probation-enforcement experience and the High Point low-arrest drug crackdown illustrate the feasibility of building actual enforcement programs on this basic principle.

*[graf staring “What should Al do?” edited to fix error]*