Risk aversion and the punishment paradox in a crime game

We study a simple game-theoretic model of criminal decision making as a coordination game between potential criminals, assuming that the expected punishment is perceived as lower than the gains from crime, for behavioral reasons. With three Nash equilibria (two in pure and one in mixed strategies), we invoke Pareto and risk dominance criteria to discuss equilibrium selection. The (completely-mixed) minmax strategies of the corresponding zero-sum game coincide with the unique mixed strategy Nash equilibrium of the original game. We show that, at the unique symmetric mixed strategy Nash equilibrium, the probability of committing a crime is positively related to the severity of punishment and negatively related to the reward from criminal activity and to the size of the population of potential criminals. We also analyze the effect of risk aversion on criminality and find that higher risk aversion increases the propensity for crime.