Question

Solve the following games: No explanations are necessary. Just the solutions.

a) Find all Nash equilibria in pure strategies for the following games. Describe the steps that you used in finding the equilibria. 


		Colin
		Left	Mid	Right
Rowena	Up	5,3	7,2	2,1
	Straight	1,2	6,3	1,4
	Down	4,2	6,4	3,5

b) Find all pure-strategy Nash equilibria for the following game. Describe the process that you used to find the equilibria. Use this game to explain why it is important to describe an equilibrium by using the strategies employed by the players, not merely the payofs received in equilibrium.

		Colin
		Left	Center	Right
Rowena	Up	1,2	2,1	1,0
	Level	0,5	1,2	7,4
	Down	-1,1	3,0	5,2

c) Consider the following game table:

		Colin
		N	S	E	W
Rowena	Earth	1,3	3,1	0,2	1,1
	Water	1,2	1,2	2,3	1,1
	Wind	3,2	2,1	1,3	0,3
	Fire	2,0	3,0	1,1	2,2

1. Does either Rowena or Colin have a dominant strategy? Explain why or why not.
2. Use iterated elimination of dominated strategies to reduce the game as much as possible. Give the order in which the eliminations occur and give the reduced form of the game. 

3. Is this game dominance solvable? Explain why or why not. 

4. State the Nash equilibrium (or equilibria) of this game. 


d) Find all Nash equilibria in pure strategies for the following games. First, check for dominant strategies. If there are none, solve the games using iterated elimination of dominated strategies. Explain your reasoning.

		Colin
		Left	Mid	Right
Rowena	Up	5,2	1,6	3,4
	Straight	6,1	1,6	2,5
	Down	1,6	0,7	0,7

Answer 1

a) The game matrix:

Nash equilibria are (Up, Left) and (Down, Right) with payoff (5, 3) and (3, 5) respectively.
Reason: If best response of each player is marked in each cell, the highlighted values in each cell represent that best response. In the cells (Up, Left) and (Down, Right), the best responses of each player coincide. Hence (Up, Left) and (Down, Right) are  the Nash equilibria.

b) The game matrix:

  

In this game there is only one Nash equilibrium - (Up, Left) with payoff (1, 2).
This is because of two reasons: (i) best response of each player and the resulting payoff, (ii) because of eliminating the dominated strategies of each player.
(i) If we mark the best response of each player in the cells, we see that the highlighted payoffs are the nest responses. And the resulting strategies are (Up, Left).
(ii) Eliminating dominated strategies: We see that for Colin, Center is dominated by Left. We eliminate Center.
For Rowena, now, Down is dominated by Level. We eliminate Down
For Colin, Right is dominated by Left. We eliminate Right.
For Rowena, Level is dominated by Up. We eliminate Level.
So, the cell left is (Up, Left) with payoff (1, 2).
Thus, both ways lead up to the same result.

c) The game matrix:

     

1. No, neither Colin nor Rowena has a dominant strategy. Because each player plays a different strategy in response to the other player's strategy.
2. No, this game is not solvable by elimination of dominated strategies. There are two equilibria. The reduced form is :

  

3. Order of elimination:
For Colin, S is dominated by E. We eliminate S.
For Rowena, Earth is dominated by Fire. We eliminate Earth.
For Colin, N is dominated by E. We eliminate N.
   For Rowena, Wind is dominated by Water. We eliminate Wind.
There are no more eliminations possible as there are two Nash equilibria.

4. Nash equilibria: (Water, E) and (Fire, W) with payoffs (2, 3) and (2, 2) respectively.

d. The game matrix:

1. Pure strategy Nash equilibria: (Up, Mid) and (Straight, Mid) with payoffs (1,6) and (1,6) respectively.
2. There are no dominant strategies for either player.
3. Elimination of dominated strategies:
For Rowena, Down is dominated by Straight. We eliminate Down.
For Colin, Left is dominated by Mid. We eliminate Left.
Now there are only weakly dominated strategies left.
So, applying best response, we get Nash equilibria - (Up, Mid) and (Straight, Mid) with payoffs (1,6) and (1,6) respectively.