Mafia Game
In Chapter 4, we described the Prisoner's Dilemma Game in which two prisoners had to choose between ratting out their partner or keeping quiet. The Nash equilibrium in this game involved both prisoners' deciding to talk even though they would both have been better off keeping quiet. If any of you watch episodes of Law and Order, TV's longest running crime series, this scenario will probably be very familiar to you. However, many of you will probably be aware that there are certain types of criminals in the real world who rarely talk or rat out their accomplices. In particular, it is well known that members of the Mafia or Cosa Nostra rarely provide incriminating evidence against their accomplices. Why is this? The answer is that the Mafia organization imposes a cost, often physical and deadly, on anyone who talks to the police. This additional cost changes the structure of the strategic situation in which the two prisoners find themselves. Rather than playing the traditional Prisoner's Dilemma, we can think that the prisoners are playing a Mafia Game with a payoff matrix like the one shown in Figure 1, below; "c" is the cost imposed by the Mafia on a suspect who talks.
Figure 1: Mafia Game

-In the Mafia Game, what is the minimum cost that the Mafia needs to impose on members who talk in order for the Nash equilibrium to be one in which both suspects keep quiet?

Question

Mafia Game
In Chapter 4, we described the Prisoner's Dilemma Game in which two prisoners had to choose between ratting out their partner or keeping quiet. The Nash equilibrium in this game involved both prisoners' deciding to talk even though they would both have been better off keeping quiet. If any of you watch episodes of Law and Order, TV's longest running crime series, this scenario will probably be very familiar to you. However, many of you will probably be aware that there are certain types of criminals in the real world who rarely talk or rat out their accomplices. In particular, it is well known that members of the Mafia or Cosa Nostra rarely provide incriminating evidence against their accomplices. Why is this? The answer is that the Mafia organization imposes a cost, often physical and deadly, on anyone who talks to the police. This additional cost changes the structure of the strategic situation in which the two prisoners find themselves. Rather than playing the traditional Prisoner's Dilemma, we can think that the prisoners are playing a Mafia Game with a payoff matrix like the one shown in Figure 1, below; "c" is the cost imposed by the Mafia on a suspect who talks.
Figure 1: Mafia Game
 
-In the Mafia Game, what is the minimum cost that the Mafia needs to impose on members who talk in order for the Nash equilibrium to be one in which both suspects keep quiet?

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The Answer of Mafia Game
In Chapter 4, we described the Prisoner's Dilemma Game in which two prisoners had to choose between ratting out their partner or keeping quiet. The Nash equilibrium in this game involved both prisoners' deciding to talk even though they would both have been better off keeping quiet. If any of you watch episodes of Law and Order, TV's longest running crime series, this scenario will probably be very familiar to you. However, many of you will probably be aware that there are certain types of criminals in the real world who rarely talk or rat out their accomplices. In particular, it is well known that members of the Mafia or Cosa Nostra rarely provide incriminating evidence against their accomplices. Why is this? The answer is that the Mafia organization imposes a cost, often physical and deadly, on anyone who talks to the police. This additional cost changes the structure of the strategic situation in which the two prisoners find themselves. Rather than playing the traditional Prisoner's Dilemma, we can think that the prisoners are playing a Mafia Game with a payoff matrix like the one shown in Figure 1, below; "c" is the cost imposed by the Mafia on a suspect who talks.
Figure 1: Mafia Game
 
-In the Mafia Game, what is the minimum cost that the Mafia needs to impose on members who talk in order for the Nash equilibrium to be one in which both suspects keep quiet?

Mafia Game in Chapter 4, We Described the Prisoner's Dilemma Game in Game