Most of us would like to live in a world where crimes are reported and dealt with, but we’d sure prefer to have others bear the burden of reporting a crime. Suppose a crime is witnessed by N people, and suppose the cost of picking up the phone and reporting a crime is c > 0.
A: Begin by assuming that everyone places a value x > c on the crime being reported, and if the crime goes unreported, everyone’s payoff is 0. (Thus, they payoff to me if you report a crime is x, and the payoff to me if I report a crime is (x −c).)
(a) Each person then has to simultaneously decide whether or not to pick up the phone to report the crime. Is there a pure strategy Nash equilibrium in which no one reports the crime?
(b) Is there a pure strategy Nash equilibrium in which more than one person reports the crime?
(c) There are many pure strategy Nash equilibria in this game. What do all of them have in common?
(d) Next, suppose each person calls with probability δ < 1. In order for this to be a mixed strategy equilibrium, what has to be the relationship between the expected payoff from not calling and the expected payoff from calling for each of the players?
(e) What is the payoff from calling when everyone calls with probability δ < 1?
(f) What is the expected payoff from not calling when everyone calls with probability δ?
(g) Using your answers to (d) through (f ), derive δ as a function of c, x and N such that it is a mixed strategy equilibrium for everyone to call with probability δ. What happens to this probability as N increases?
(h) What is the probability that a crime will be reported in this mixed strategy equilibrium?
(i) True or False: If the reporting of crimes is governed by such mixed strategy behavior, it is advantageous for few people to observe a crime—whereas if the reporting of crime is governed by pure strategy Nash equilibrium behavior, it does not matter how many people witnessed a crime.
(j) If the cost of reporting the crime differed across individuals (but is always less than x), would the set of pure Nash equilibria be any different? Without working it out, can you guess how the mixed strategy equilibrium would be affected?
B: Suppose from here on out that everyone values the reporting of crime differently, with person no’s value of having a crime reported denoted x n. Assume that everyone still faces the same cost c of reporting the crime. Everyone knows that c is the same for everyone, and person n discovers x n prior to having to decide whether to call. But the only thing each individual knows about the x values for others is that they fall in some interval [0, b], with c falling inside that interval and with the probability that x n is less than x given by P(x) for all individuals.
(a)What is P (0)? What is P (b)?
(b) From here on out, suppose that P(x) = x/b. Does what you concluded in (a) hold?
(c) Consider now whether there exists a Bayesian Nash equilibrium in which each player n plays the strategy of reporting the crime if and only if x n is greater than or equal to some critical value y. Suppose that everyone other than n plays this strategy. What is the probability that at least one person other than individual n reports a crime?
(d) What is the expected payoff of not reporting the crime for individual n whose value is x n? What is the expected payoff of reporting the crime for this individual?
(e) What is the condition for individual n to optimally not report the crime if x n < y? What is the condition for individual n to optimally report the crime when x n ≥ y?
(f) For what value of y have we identified a Bayesian Nash equilibrium?
(g) What happens to the equilibrium probability of a crime being reported as N increases?
(h) How is the probability of a crime being reported (in this equilibrium) affected by c and b? Does this make sense?