Question

1.     Consider the following game, which illustrates Nobel Laureate Thomas Schelling's solution to the problem of how a kidnapped person can induce a kidnapper to release him after the person has learned the identity of the kidnapper.

     The game starts off with the kidnapper, say Alex, deciding whether or not to kidnap Bart. If he decides not to kidnap Bart, the game ends. Alex gets a payoff of 3, while Bart receives a payoff of 5. However, if Alex decides to go through with the kidnapping, he is successful, but there is a scuffle in which he is unmasked, so that his identity is revealed to Bart. This creates another problem for Bart, since Alex would now be inclined to kill Bart since Bart, if released could identify Alex to the police. Bart then decides whether to reveal something compromising/incriminating about himself to Alex that is not known to the rest of the world. (For example he could have embezzled funds from the company he works in or has had an extramarital affair his wife does not know about. Effectively, he is offering this secret as a "collateral" which Alex can use to ensure his silence). Whether or not Bart decides to tell his secret, Alex has to decide to either kill him or release him. If he kills Bart the game ends and payoffs to Alex and Bart are respectively 4 and 0. If he releases Bart, then Bart has to decide whether to tell the police or not. If Bart does not tell the police the game ends and payoffs to Alex and Bart are respectively 5 and 2. If Bart chooses to tell the police, however, there are two scenarios to consider. If he hasn't revealed his incriminating secret to Bart, the game ends and the payoffs to Alex and Bart are 1 and 4 respectively. But if he has revealed his secret to Alex, then Alex has to decide whether to reveal that secret to the world. If he does then the game ends and payoffs are 2 to Alex and 1 to Bart. If he doesn't the game ends with payoff of 1 to Alex and 3 to Bart.

(a) Draw the extensive (tree) form of the game. How many strategies do Alex and Bart have? Write them down. {Hint: In doing so do not forget that a strategy is a complete plan of action.}

(b) Now present the game in strategic form. {Hint: The payoff table is pretty big! J } Identify all of the Nash equilibria in this game.

(c) Which of these equilibria is the subgame perfect Nash Equilibrium (SPNE)? Explain and illustrate your answer using backward induction on the game tree.

(d) Pick any two of the other Nash equilibria that are not SPNE. Discuss the credibility problems associated with these equilibria.

Answer 1

a) The extensive tree is given as below.

Alex has 5 different strategies while Bert has 5 different strategies. However, Bert's 5 strategies are shown as 7 in the payoff table since there is a decision where Bert reveals secret or Doesn't reveal secret and then takes a further strategy. But If Bert is killed before the next level of strategy, then it needs to be terminated. So they are shown seperately as well as in combination with future strategies.

b) The payoff table is given below

		BERT
		No Action	Don't Reveal Secret to Alex and Tell Police	Don’t Reveal Secret to Alex and Don't tell Police	Reveal Secret to Alex and Tell Police	Reveal Secret to Alex and Don't tell Police
ALEX	Don't Kidnap	3,5	0,0	0,0	0,0	0,0
Kidnap and Kill	0,0	4,0	4,0	4,0	4,0
Kidnap and Release	0,0	1,4	5,2	0,0	5,2
Kidnap, Release, Don't Reveal Secret to World	0,0	0,0	0,0	1,3	0,0
Kidnap, Release, Reveal Secret to World	0,0	0,0	0,0	2,1	0,0

The Nash equilibrium are Alex Doesn't Kidnap Bert and Bert does not do anything else. The payoff is 3,5

Alex Kidnaps and Kills Bert irrespective of whether Bert tells the secret or not. So the two combined is shown as a Nash equilibrium with the payoff being 4,0

Alex Kidnaps and releases Bert. Bert Tells the police without revealing any secret to Alex beforehand. The payoff is 1,4

Alex Kidnaps and Releases Bert. Bert Tells the police after telling the secret to Alex. Alex reveals the secret to the world. The payoff will be 2,1.

The Nash Equilibrium are marked in the table.

c) Let us start from the end.

If Alex knows the secret and reveals it to the world, Alex will get a payoff of 2 instead of 1 if he doesn't reveal the secret. In this case, Bert will have a payoff of 1. So if Bert doesn't tell the police, he will have a payoff of 2/ Now Alex knows that Bert will not tell the police. So he will have a payoff of 5 if he releases Bert and a payoff of 4 if he kills Bert. So he will release Bert if he is told the secret and the payoff will be 5,2

Now if Alex doesn't know the secret, and if Bert tells the police to get a better payoff of 4, Alex will get a payoff of 1. This is if he releases Bert. If he kills Bert, then Alex will get a payoff of 4. So Alex will kill Bert if he doesn't know the secret and the payoff will be 4,0.

Now Bert realises that if he tells the secret, he will get a payoff of 2 and if he doesn't tell, his payoff will be 0. So he decides to tell the secret and the payoff will be 5,2.

Now Alex Sees that if doesn't kidnap Bert, his payoff will be 3 while if he kidnaps Bert, he will get a payoff of 5. So it is better that he kidnaps Bert.

So the Nash equilibrium will be when Alex Kidnaps Bert, Bert Tells the secret to Alex. Alex Releases Bert but Bert does not go to the police. The payout will be 5,2

d) The other 2 nash equilibria are not SPNE. By backward integration, we see that all these options are eliminated and since it is a sequential game, so even though the payoff will be showing a Nash Equilibria, it will not be a Sub Game Perfect Nash Equilibrium.

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