Question

Consider the following game that has two players. Player A has three actions, and player B has three actions. Player A can either play Top, Middle or Bottom, whereas player B can play Left, Middle or Right. The payoffs are shown in the following matrix. Notice that a payoff to player A has been omitted (denoted by x).

Player B

  

	Left	Middle	Right
Top	(-1,1)	(0,3)	(1,10)
Middle	(2,0)	(-2,-2)	(-1,-1)
Bottom	(x,-1)	(1,2)	(3,2)

(player A)

Both players choose their actions separately and simultaneously. Each must decide what to do knowing the available actions and payoffs but not what the other will actually choose. Each player believes the other to be rational and self-interested.

a) Determine the range of values of x such that Player A HAS a strictly dominant strategy.

b) List Player B’s best responses to Player A’s actions.

c) Suppose that Player A has a strictly dominant strategy. Find all the Nash equilibrium/equilibria to the game.

d) Suppose that Player A does NOT have a dominant strategy. Find all the Nash Equilibria

Answer 1

(a) x ≥ 3 (x should be greater or equal to 3).

reason: In response to Player B's Middle and Right, Player A will choose strategy Bottom. So strategy Bottom is likely to be a dominant strategy for  Player A. If he should choose strategy Bottom against Player B's move Left too, it shoud be the highest value in the column Left for him. It should be greater than 2. So, x ≥ 3.

(b) When Player A moves Top, Player B will move Right as it has the highest payoff in that row for him (10>3>1). When Palyer A moves Middle, Player B will move Left as that is th highest payoff foor him in that row (0>-1>-2). If Player A moves Bottom (whatever the value of X, Player B will be indifferent between Middle and Right as both have the same payoff for him (2=2>-1).

(c) If Player A has a strictly dominant strategy, that strategy will be Bottom. Against Bottom, Player 2 will choose Middle or Right (as he is indifferent between the two). Thus Nash equilibria are (Bottom, Middle) and (Bottom, Right) with payoffs (1, 2) and (3, 2) respectively.

(d) If Player A has no dominant stratgy, then Nash equilibria are:
(Middle, Left), (Bottom, Middle) and (Bottom, Rght)
with payoffs (2, 0), (1, 2), and (3, 2) respectively.

reason:

When X is less than 2 (that is when Player 1 has no dominant strategy), Player 1 will choose Middle against Player 2's Left, Bottom against Player 2's Middle, and Bottom against Player 2's Right. Player 2 will choose Right against Player 1's Top, Left against Player 1's Middle and either Middle or Right against Player 1's Bottom. Thus, we see that player 1 and 2 coincide in 3 places: (Middle, Left), (Bottom, Middle) and (Bottom, Right).