Use the following extensive-form game to answer the following questions. a. List the feasible strategies for player 1 and player 2. b. Identify the Nash equilibria to this game. c. Find the subgame perfect equilibrium.
a. Feasible strategy for player 1 are: {A,B}
Feasible strategy for player 2 are: {(ZX),(ZW),(YX),(YW)}
b. To find the Nash equilibrium of this game consider the following normal form of the game:
Player 1 (P1) has two strategies to choose from and Player 2 (P2) has 4 strategies available.
When P2 chooses ZX, P1 chooses B. When P2 chooses ZW, P1 chooses B only. And when P2 chooses YX or YW then P1 chooses A as this gives him better payoff.
Similarly, for P2, when P1 chooses A P2 is indifferent between ZW and YW. When P1 chooses B P2 is indifferent between ZX and ZW.
This gives us three pair of Nash equilibrium strategies:
(A, YW)
(B, ZX)
(B, ZW)
c. To find the sub-game perfect equilibrium we use the method of backward induction and we need to mark the sub-games (SG). The number of sub-games is equal to the number of decision nodes that is 3 sub-games will be there. The following chart shows the three sub-games:
In SG1 P2 is at the deciding node, he has to choose from W or X. W will give P2 a higher payoff.
In SG2 P2 chooses Z which gives him higher payoff from the available strategies Y and Z.
This gives the new game below:
Now P1 has to choose from A or B from which he chooses B.
Thus, the sub-game perfect equilibrium of this game is:
(B, ZW)