Question

1. (25 points) The following is a simplified duopoly model of competition between two firms. Each firm is restricted to producing 25, 35, 50 or 100 units of output. The details of how the payoffs are derived are unimportant because payoffs are all given in the table below.

                                                                                  FIRM 2

		25	35	50	100
	25	125, 125	100, 140	63, 125	-63, -250
FIRM 1	35	140, 100	105, 105	53, 75	-123, -350
	50	125, 63	75, 53	0, 0	-250, -500
	100	-250, -63	-350, -130	-500, -250	-900, -900

1. (5 points) Firms simultaneously choose the quantity of outputs to produce, and then profits are realized. Find if there are any dominant strategies for the firms or not. In your response, make sure that you define what a dominant strategy is. Also describe the entire process that helped you in making that determination.

2. (5 points) Find the Nash equilibrium(s) in the game. Justify whether it is a Nash equilibrium or not. Describe in words.

3. (7.5 points) Now assume that FIRM 1 is the Stackelberg leader in this market. And FIRM 2 is the follower. Being the leader, FIRM 1 makes the first move in choosing the quantity of output, followed by FIRM 2. Draw the extensive form or the game tree for this sequential form game.

(7.5 points) Using the game tree, now determine the sub-game perfect Nash equilibrium(s). Describe the process that helps you in determining it

Answer 1

An underline below a respective payoff indicates a chosen payoff.

a) A strategy is a dominant strategy if choosing it leads a player to better outcomes than the other strategies that they can choose.  

If firm 1 chooses 25, firm 2 will choose 35

If firm 1 chooses 35, firm 2 will choose 35

If firm 1 chooses 50, firm 2 will choose 25

If firm 1 chooses 100, firm 2 will choose 25

If firm 2 chooses 25, firm 1 will choose 25

If firm 2 chooses 35, firm 1 will choose 35

If firm 2 chooses 50, firm 1 will choose 25

If firm 2 chooses 100, firm 1 will choose 25

(Represented by underlines below payoffs)

Clearly there is no strategy which is dominant, strategy of a player will depend on other players' strategy.

b) Nash equilibrium for the game is when Player 1 chooses 35, and Player 2 chooses 35. At this strategy, neither player would have any incentive to deviate from their strategy. As can be seen above,

If firm 2 chooses 35, firm 1 will choose 35

If firm 1 chooses 35, firm 2 will choose 35

c) The image shows the extensive tree representation of the game.

If firm 1 chooses 25, firm 2 will choose 35

If firm 1 chooses 35, firm 2 will choose 35

If firm 1 chooses 50, firm 2 will choose 25

If firm 1 chooses 100, firm 2 will choose 25

With this knowledge, firm 1 must pick between its respective payoffs considering what firm 2 would do. Therefore, it would pick the highest payoff of 105, and play the strategy 35.