Find all Nash Equilibria. There is a rough neighborhood with n>= 2 residents. Ea

Question

Find all Nash Equilibria.

There is a rough neighborhood with n>= 2 residents. Each resident has to decide whether to engage in the crime of theft. If an individual chooses to he a thief and is not caught by the police, he receives a payoff of W'. If he is caught by the police, his payoff is 7.. If he chooses not to commit theft, he receives a zero payoff. Assume that W > 0 > Z. All m residents simultaneously decide whether or not to commit theft. The probability of a thief being caught equals where m is the number of residents who choose to engage in theft. Thus, the probability of being caught is lower when more crimes arc committed and the police have more crimes to investigate. The payoff from being a thief, given that m - I other people have also chosen to be thieves, is then (m-1/m) W + (1/m)Z.

Explanation / Answer

m chooses to be a thief n chooses not to be a thief Payoff (P1) when theif and not being caught by the police = (m-1/m)*W Payoff (P2) when theif and being caught by the police = Z/m Payoff (P3) when not a thief and not being caught by the police = 0 Payoff (P4) when not a thief and being caught by the police = 0 since W>0>Z therefore P1>P3=P4>P2 hence Nash Equilibria is P1 = (m-1/m)*W

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