Question

The extensive form of a 2pemon game as follones R. (a) What are the pure strategy sets for players I and II. (b) Derive the normal (strategie) form of the game? (e) Use backward induction to find the sub-game perfect Nash Equi- librium of the game. (d) Find the other Nash Equilibrium and explain why it is not sub game perfect.

Answer 1

Solution:

We are given an extensive form game with player 1 as the initiator or leader and player 2 as the follower.

a) Clearly, from the game tree, we see that player 1 always has 2 options to choose from. That is, player 1 will either carry out strategy L or will carry out strategy R. So, for player 1, pure strategy set becomes: {L, R}.

Similarly, for player 2, there are 4 options available to choose from, but they are conditioned with what strategy shall player 1 choose, or what action shall player 1 carry out. If player 1 chooses L, player 2 can choose either l or r, and if R is chosen initially, player 2 shall choose among L and R.

Thus, for player 2, pure strategy set will consist 4 elements: {(l, L), (l, R), (r, L), (r, R)}. Implication is that say strategy (l, L) means if player 1 had chosen L, player 2 would choose l, and had player 1 chosen R, player 2 would choose L (similarly, for remaining 3 pure strategies).

b) Writing the normal (or matrix) form game:

Now, that we have established the pure strategy sets for the two players, we can write the normal form of the given extensive game. We can write:

			Player 2
		(l, L)	(l, R)	(r, L)	(r, R)
Player 1	L
	R

Writing the payoffs using the implication already mentioned that player 1 simply chooses among L and R, for player 2 however, writing payoff for any strategy will consider decision already made by player 1. So, first (left-upper corner) entry means player 1 choosing L and thus, player 2 choosing l. In easy words, for player 2, strategy (x, y) means only x has to be considered if player 1 chooses L, and y can be ignored then while only y has to be considered if player 1 chooses R, and x can be ignored then. So, denoting any payoff by (a, b) meaning player 1 gets payoff of 'a' and player 2 gets payoff of 'b', we have

			Player 2
		(l, L)	(l, R)	(r, L)	(r, R)
Player 1	L	(4, 2)	(4, 2)	(2, 10)	(2, 10)
	R	(0, 0)	(5, 5)	(0, 0)	(5, 5)

c) Backward induction, as the name suggests, is a process to find the sub-game perfect Nash Equilibrium (SPNE) by going backwards that is, first considering the action taken by player 2, conditional on player 1 strategies, and finally choosing the best outcome(s) by player 1, among the choices made by player 2 in first place.

So, given the payoffs, had player 1 chosen strategy L, player 2 would choose r over l (as 10 > 2). And if player 1 had chosen strategy R, player 2 would choose R (as 5 > 0).

Thus, we know that player 1 will choose L or R based on the two payoff choices he/she has now: (2, 10) and (5, 5). Clearly, 2 < 5, meaning player 1 will choose strategy R as it gives a higher payoff.

Finally, SPNE for the given game becomes: (R, R) or more formally (R, (l, R))