  Containing the spreading of crime is a major challenge for society. Yet,
since thousands of years, no effective strategy has been found to overcome
crime. To the contrary, empirical evidence shows that crime is recurrent, a
fact that is not captured well by rational choice theories of crime. According
to these, strong enough punishment should prevent crime from happening. To gain
a better understanding of the relationship between crime and punishment, we
consider that the latter requires prior discovery of illicit behavior and study
a spatial version of the inspection game. Simulations reveal the spontaneous
emergence of cyclic dominance between ''criminals'', ''inspectors'', and
''ordinary people'' as a consequence of spatial interactions. Such cycles
dominate the evolutionary process, in particular when the temptation to commit
crime or the cost of inspection are low or moderate. Yet, there are also
critical parameter values beyond which cycles cease to exist and the population
is dominated either by a stable mixture of criminals and inspectors or one of
these two strategies alone. Both continuous and discontinuous phase transitions
to different final states are possible, indicating that successful strategies
to contain crime can be very much counter-intuitive and complex. Our results
demonstrate that spatial interactions are crucial for the evolutionary outcome
of the inspection game, and they also reveal why criminal behavior is likely to
be recurrent rather than evolving towards an equilibrium with monotonous
parameter dependencies.
