Question

(6 points) Consider the simultaneous game below: Left Center Right Top 22 14 13 Mddle 4.0 330,2 Bottom 3 20 22 (a) (1 point) Show that there is not equilibrium in dominant strategies. (b) (5 points) Write the best response functions and find all Nash equilibrium of the game.

Answer 1

(a). We can see that Player 1 has two weakly dominant strategies - Middle and Bottom (sum of payoffs is greater).
Player 2 has two weakly dominant strategies as well - Center and Right. Again, in the 2x2 matrix, we see that both can either play safe (2,2/3,3) or punish the other (0,2/2,0). No equilibrium is attained.

(b). Another way of thinking about Nash equilibrium is by thinking about player i’s best-response function B_i(σ_−i) = { Player i’s best-response to σ_−i }. A Nash equilibrium occurs where, for each player i, B_i(σ^∗_−i ) = σ^∗.

	Left	Center	Right
Top	2,2	1,4	1,3
Middle	4,0	3,3	0,2
Bottom	3,1	2,0	2,2

The best responses are underlined in each case:
When Player 1 plays TOP, Player 2 should play CENTER.
When Player 1 plays MIDDLE, Player 2 should play LEFT.
When Player 1 plays BOTTOM, Player 2 should play RIGHT.

Similarly,
When Player 2 plays LEFT, Player 1 should play MIDDLE.
When Player 2 plays CENTER, Player 1 should play TOP.
When Player 2 plays RIGHT, Player 1 should play BOTTOM.

Is there any intersection in these strategies such that payoff is also increased? Yes, the Nash equilibria is attained when both play Middle and Center, resulting in a pay-off of (3,3).

Hope this helped!