(a). Let us write normal form of this game.
Number of strategies of Player 1 = 4
Number of strategies of Player 2 = 3
Thus, payoff table will be 4x3
P1/P2 | L | C | R |
InT | (4,2) | (3,3) | (0,a) |
InB | (4,2) | (1,1) | (0,a) |
OutT | (2,2) | (2,2) | (2,2) |
OutB | (2,2) | (2,2) | (2,2) |
(b) a = 0 and we are asked to find the perfect sub-game
equilibrium.
One cannot apply backward induction in this game because it is not
a perfect information game. One can compute the subgame perfect
equilibrium, however. This game has two subgames: one starts after
Player 1 plays In; the second one is the game itself. The subgame
perfect equilibria are computed as follows.
First compute a Nash equilibrium of the subgame, then fixing the
equilibrium actions as they are (in this subgame), and taking the
equilibrium payoffs in this subgame as the payoffs for entering the
subgame, compute a Nash equilibrium in the remaining game.
Player 1 plays In, after which Player 2 plays L or C. When L is
selected, Player 1 gets utility 4 and Player 2 gets utility
2.
If Player 2 plays C, Player 1 will play T, getting utility of 3 and
Player 2 getting utility of 3 as well. These are the
subgames.
(c) a = 0 and we are asked to find the Nash equilibrium that is not
subgame perfect.
In this case, both try to maximise their payoffs. Player 2 gets
maximum utility if it selects C and Player 1 gets maximum utility
in case of InT. This is a Nash equilibria.
(d) If a = 3, equilibria become In/R/T or In/R/B.
1. Consider the following extensive form game with perfect information 1 Out 2 2 In 3...
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