FIGURE 2.1 Extensive Form of the Kidnapping Game Guy TABLE 2. KIDNAPPING GAME AN

Question

FIGURE 2.1 Extensive Form of the Kidnapping Game Guy TABLE 2. KIDNAPPING GAME AND PAYOFFS Guy (Violent) Guy Vivica Do not kidnap Kidnapping ransom is paid, Orlando is killed Kidnapping, ransom is paid, Orlando is released Kidnapping, ransom is not paid, Orlando is killed Kidnapping, ransom is not paid, Orlandio is released 5 Vivica Guy 3 Vivica 5 Pay Do not pay ransom Guy Guy Release Kill Consider the Kidnapping game in figure 2.1, Solve for all Nash Equilibria. Identifv which equilibria are subeame perfect Nash equilibria. Please explain.

Explanation / Answer

Let's try to solve this question using backward induction-

The payoffs shall be written as (Guy, Vivica)

Part 1- At nodes where the guy needs to decide between Kill and Release

Left Node > Release (5,3) has higher payoffs for the guy than Kill (4,1)

Part 2- At nodes where the guy needs to decide between Kill and Release

Right Node > Kill (2,2) has higher payoffs for the guy than Release (1,4)

Part 3- Comparing Left and Right nodes for Vivica's payoff -

(5,3) vs (2,2)

Viviva is better off with 'Pay ransom with a payoff of 3 vs 2(Don't pay)

Part 4- At this stage,we are at the top of the decision tree-

Guy-

do not kidnap (3,5) vs Kidnap (5,3)

He is better off with kidnap.

Subgame perfect nash equilibria -

This means we need to look at smaller games within the decision tree. Any outcome where the equilibria was such that the payoffs made both the players better off (i.e., they had no incentive to deviate), that will be called subgame perfect nash equilbria.

Notice Part 1 and Part 3 of the decision tree (mentioned above in the answer).

These are sub game perfect nash equilibrias. The players have no incentive to choose the alternate option available.

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