4. A ruthless person kidnaps a child and demands a ransom of r>0 from the parents;...

 

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4. A ruthless person kidnaps a child and demands a ransom of r>0 from the parents; otherwise the kidnapper threatens to kill the child. Suppose the parents' value their child by an amount v>0 if alive and 0 if dead and suppose further that v>r. Finally, suppose the cost to the kidnapper of killing the child is k and the cost to the kidnapper of freeing the child is f. These costs to the kidnapper include the severity of the punishment if caught, the qualms of the kidnapper and whatever else might be relevant. Assume that, regardless of whether the ransom is paid or not, the kidnapper can free or kill the child. a. Write this game down in extensive form, with the parents moving first, deciding whether to pay the ransom or not, and then the kidnapper moving second, deciding whether to kill or free the child. Be sure to assign payoffs to the terminal nodes. b. Show that, regardless of whether k>f or f>k, in the subgame perfect equilibrium the parents never pay the ransom. (Incidentally, this is the logic for why governments such as the U.S. refuse to pay ransoms for kidnapped citizens). c. In reality, ransoms are often paid by parents. How might you modify the payoffs so that the subgame perfect equilibrium is for the parents to pay the ransom and for the kidnapper to free the child? Explain carefully. 4. A ruthless person kidnaps a child and demands a ransom of r>0 from the parents; otherwise the kidnapper threatens to kill the child. Suppose the parents' value their child by an amount v>0 if alive and 0 if dead and suppose further that v>r. Finally, suppose the cost to the kidnapper of killing the child is k and the cost to the kidnapper of freeing the child is f. These costs to the kidnapper include the severity of the punishment if caught, the qualms of the kidnapper and whatever else might be relevant. Assume that, regardless of whether the ransom is paid or not, the kidnapper can free or kill the child. a. Write this game down in extensive form, with the parents moving first, deciding whether to pay the ransom or not, and then the kidnapper moving second, deciding whether to kill or free the child. Be sure to assign payoffs to the terminal nodes. b. Show that, regardless of whether k>f or f>k, in the subgame perfect equilibrium the parents never pay the ransom. (Incidentally, this is the logic for why governments such as the U.S. refuse to pay ransoms for kidnapped citizens). c. In reality, ransoms are often paid by parents. How might you modify the payoffs so that the subgame perfect equilibrium is for the parents to pay the ransom and for the kidnapper to free the child? Explain carefully.