Consider a neighbourhood with 1000 residents. Each individual must choose (let’s assume simultaneously) whether to be a criminal or not. If an individual chooses to attempt a criminal act, their payoff is 200 if they succeed, but -300 if they are caught by the police. Whether or not they are caught, of course, depends on how many people choose to be criminals.
Let m represent the number of citizens who choose to to be criminals. With criminals out on the loose, the chance of any one criminal getting caught is 1/m. The chance of not getting caught is therefore m-1/m. The (expected) payoff to an individual who chooses a life of crime is therefore
(m-1/ m) 200 - (1/m) 300.
The payoff who to an ordinary individual who chooses not to engage in crime is 100.
-Is this game symmetric? Why or why not?
-What is the payoff to an individual choosing to be a criminal if m = 500? (Assume that m includes the individual whose payoff you’re calculating.) Also, what is the payoff to an ordinary individual who chooses not to engage in crime when m=500?