Question

Consider the following strategic-form simultaneous game. Player 1’s payoffs are listed first, in bold

		Player 2
		W	X	Y	Z
Player 1	A	2 , 500	30 , 400	200 , 200	4 , 4
	B	5 , 50	20 , 40	10 , 20	4 , 4
	C	1 ,   0	1 ,   4	3 ,   2	4 , 4

1. Does either player have any weakly dominant strategies? If yes, list them. Briefly explain.
2. Does either player have any weakly dominated strategies? If yes, list them. Briefly explain.
3. List all pure-strategy Nash equilibria of the above game.
4. What does it mean for the payoffs in the above game to be “common knowledge”?

Answer 1

a. Yes, the players have weakly dominant strategies.

The weakly dominant strategy for Player 1 is strategy A. The weakly dominant strategy for Player 2 is strategies W, X and Y

b. Yes, the players have weakly dominated strategies

The weakly dominated strategy for Player 1 is strategy B. The weakly dominated strategy for Player 2 is strategies X, Y and Z

In this given game, The weakly dominant strategy pair of Player 1 is A and B where A weakly dominates B as the outcome of player 1 upon playing strategy A are (2,30,200,4) and that upon playing strategy B are (5,20,10,4). Hence, playing A is beneficial on two occasion over playing B. In case of other strategies of player 1, A strictly dominates C and B strictly dominates C.

The payoff of Player 2 for all of its strategy are:

W={500,50,0}

X={400,40,4}

Y={200,20,2}

Z={4,4,4}

It is observed that the Strategy W weakly dominates Strategy X,Y and Z as its payoff is higher in all cases except for the third case where player 2’s payoff is zero.

Strategy X weakly dominates strategy Z for player 2 as the third output is same for both strategies. Strategy Y also weakly dominates Strategy Z for player 2 as third payoff is higher in Z strategy

c. In the given game,

When player 1 plays:

Strategy A, player 2’s best response is Strategy W (as 500>400>200>4)

Strategy B, player 2’s best response is Strategy W (as 50>40>20>4)

Strategy C, player 2’s best response is indifferent between Strategies X and Z as both have same payoff i.e. 4

When player 2 plays:

Strategy W, player 1’s best response is Strategy B (as 2>5>1)

Strategy X, player 1’s best response is Strategy A (as 30>20>1)

Strategy Y, player 1’s best response is Strategy A (as 200>10>3)

Strategy Z, player 1’s best response is indifferent between all three Strategies A,B and C as all have same payoff i.e. 4

Therefore, the pure strategy Nash Equilibria for the above game is the one When player 1 plays Strategy B and player 2’s best response is Strategy W and When player 2 plays Strategy W, player 1’s best response is Strategy B i.e. (Strategy B, Strategy W) = (5,50)

d. The payoffs in the above game to be “common knowledge” means that the payoff function and strategy spaces of both the players are known by each other that is both the players in this game have complete information about each other’s payoff.