Footnotes

• ¹To whom correspondence should be addressed. E-mail: kleiman{at}ucla.edu

• Author contributions: M.K. designed research; M.K. and B.K. performed research; B.K. analyzed data; and M.K. and B.K. wrote the paper.

• The authors declare no conflict of interest.

• This article contains supporting information online at www.pnas.org/cgi/content/full/0905513106/DCSupplemental.

• ↵† A randomized controlled trial of this program is underway, and the preliminary results are promising (9).

• ↵‡ Variously attributed to Nimzowitsch, Tartakower, and Eisenbach.

• ↵§ There are alternative explanations and differential crime rates in seemingly similar communities. See SI Text for a review of these explanations.

• ↵¶ Previously published theoretical models suggest that focused enforcement with announcement can outperform a strategy of random enforcement (21). Using computational modeling, this paper builds on these efforts by (i) focusing on repeated game settings, (ii) relaxing the assumption that potential offenders have perfect information about the probability of being punished, (iii) allowing those subject to a rule to update their subjective probability of being punished in Bayesian fashion, and (iv) showing that dynamic concentration as a punishment-allocation strategy can reduce both offense levels and punishment levels, compared to equal-probability punishment. The same game has also been approached from the other side, focusing on the problem of political dissenters who need to coordinate the time of their dissent to minimize the punishment risk faced by each 1 of them (22). van Baal's book on computer simulations of criminal deterrence (23) does not address our main concern about how to choose who to sanction when the number of offenders exceeds the number of available sanctions.

• ↵‖ If P < C, then violation is a dominant strategy for both players; if P > 2C, then compliance is stochastically dominant for both players, because in that case P/2, the expected value of punishment, is greater than the cost of compliance even if both players were to violate. So only the intermediate case where C is between P and 2P (more generally, between P and nP) is strategically complex.

• ↵†† However, if there is any player who attributes to each other player a nonzero probability of defection, then that player's subjective estimate of the probability that someone else will defect, and therefore that player's incentive to defect, tends toward unity as n grows.

• ↵†† In Fig. 2, the tipping point for random sanctioning is ≈20% higher than it is for dynamic concentration; the comparable number for Fig. S3 is ≈33%.

• ↵§§ That adding sanctions capacity can, mathematically, reduce actual sanctions use does not imply that any actual increase in sanctions capacity, e.g., building more prisons, will lead to less punishment. That depends on circumstances and on how the additional capacity is used; using more prison cells to impose longer terms does not increase certainty and therefore will increase, rather than decrease, total punishment actually inflicted.