Punishment and belief in free will

The surprising link between homicide rates and belief in free will.  Do we punish criminals because we believe they freely chose to commit the crimes and so deserve punishment? Or do we believe criminals acted freely because we desire to punish them? A reason to think we believe in free will because we desire to punish: “more crime-ridden places … also tend to believe more strongly in free will, presumably out of a desire to see criminals punished.”