Question

2 (yジ し3

write down the normal form of this game

Answer 1

Solution:

Clearly, this is a two player and 3 rounds game. Two players are Player 1 and Player 2; 3 rounds involve player 1 making a decision in the first round, then player 2 choosing one of the strategies in second round, finally if needed, player 1 makes a final choice in third round.

Denoting lamda by d and rho by p for ease of writing.

Then, player 1 has the following strategy set: {(X,l,d), (X,l,p), (X,r,d), (X,r,p), (E,l,d), (E,l,p), (E,r,d), (E,r,p)} and player 2 has the following strategy set: {L, M, R}. So, taking Player 1's profile horizontally, and player 2's profile vertically, the normal form game will have 8 rows (for 8 possible strategies of player 1) and 3 columns (for 3 possible strategies of player 2).

The payoffs can be written very easily noticing the branches of given game tree. Notice that if player 1 chooses X, no matter what player 2 chooses, payoff received is always (2, 1). Only if player 1 chooses strategy E in first round, further choice will fall on player 2 and accordingly further payoffs will be written. So, in normal form, we can represent the game in following manner:

			PLAYER 2
		L	M	R
	(X,l,d)	(2, 1)	(2, 1)	(2, 1)
	(X,l,p)	(2, 1)	(2, 1)	(2, 1)
	(X,r,d)	(2, 1)	(2, 1)	(2, 1)
PLAYER 1	(X,r,p)	(2, 1)	(2, 1)	(2, 1)
	(E,l,d)	(1, 2)	(3, 1)	(1, 3)
	(E,l,p)	(1, 2)	(3, 1)	(3, 1)
	(E,r,d)	(1, 2)	(1, 3)	(1, 3)
	(E,r,p)	(1, 2)	(1, 3)	(3, 1)