Two companies, A and B, produce widgets. Each can produce 0, 1, 2, 3, or 4 widgets (they can’t
produce fractions of widgets). Let X be the number of units produced by A, and Y be the number of
units produced by B. Given X and Y, widgets will sell at a price equal to $(14-X-Y). Every widget costs
$5 to produce. The companies choose X and Y simultaneously, each trying to maximize profits.
a) (2 points) Derive an expression for A’s profits.
b) (2 points) Derive an expression for B’s profits.
c) (8 points) Draw a table representing this one-stage game, showing the players’ strategies and
payoffs.
In class we defined dominated strategies as those strategies that are never a best response. This
definition actually refers to strictly dominated strategies (i.e., for any action that the opponent might
take there is always another strategy that gives a higher payoff). There is a second type of dominated
strategies called weakly dominated strategies. These are strategies that give the same payoff as other
strategies for some actions that the opponent might take and give a lower payoff than other strategies
for all other actions the opponent might take (i.e. when a weakly dominated strategy is a best response
there is another strategy that is also a best response to the same opponent’s action)
d) (3 points) Does either company have a strictly dominated strategy?
e) (3 points) Draw the reduced game once strictly dominated strategies have been removed.
f) (3 points) In the reduced game, does either company have weakly dominated strategies? What are
they?
g) (3 points) If companies did not exclude the possibility of playing their weakly dominated strategies,
what are the possible Nash equilibria of the reduced game?
h) (3 points) Looking at the payoffs of the reduced game, does it make sense for either company to
play their weekly dominated strategies? Why or why not?
i) (3 points) Based on your answer to (h), what would be your prediction of the game?
a) To derive the expression for A's profits, we need to calculate the revenue and subtract the cost of production.
The revenue earned by A is equal to the price of each widget multiplied by the number of units sold, which is $(14 - X - Y) * X.
The cost of production is $5 per widget, so the total cost incurred by A is $5 * X.
Therefore, A's profits can be calculated by subtracting the cost from the revenue:
Profits of A = Revenue - Cost of production
= (14 - X - Y) * X - $5 * X
= (14X - X^2 - XY) - $5X
= 14X - X^2 - XY - $5X
= 14X - X^2 - $5X - XY
b) Similarly, to derive the expression for B's profits, we follow the same steps:
The revenue earned by B is $(14 - X - Y) * Y.
The cost of production is $5 per widget, so the total cost incurred by B is $5 * Y.
B's profits can be calculated by subtracting the cost from the revenue:
Profits of B = Revenue - Cost of production
= (14 - X - Y) * Y - $5 * Y
= (14Y - XY - Y^2) - $5Y
= 14Y - XY - Y^2 - $5Y
= 14Y - XY - $5Y - Y^2
c) The table representing this one-stage game is as follows:
| | X = 0 | X = 1 | X = 2 | X = 3 | X = 4 |
|--------|-------|-------|-------|-------|-------|
| Y = 0 | 0, 0 | 8, 6 | 20, 12| 30, 18| 38, 24|
| Y = 1 | 0, 0 | 6, 8 | 16, 15| 24, 21| 30, 26|
| Y = 2 | 0, 0 | 4, 10 | 8, 16 | 12, 21| 14, 25|
| Y = 3 | 0, 0 | 2, 12 | 4, 18 | 6, 21 | 6, 23|
| Y = 4 | 0, 0 | 0, 14 | 0, 20 | 0, 23 | 0, 24|
Each cell in the table represents the payoffs for A and B, respectively, depending on their choices of X and Y.
d) To determine if either company has a strictly dominated strategy, we need to compare the payoffs of each strategy with the payoffs of other strategies.
Upon examining the table, we can notice that for any X and Y, there always exists another strategy that gives a higher payoff for at least one player. Therefore, neither company has a strictly dominated strategy.
e) To draw the reduced game, we need to remove the strictly dominated strategies.
The reduced game table is as follows:
| | X = 1 | X = 2 | X = 3 |
|--------|-------|-------|-------|
| Y = 1 | 6, 8 | 16, 15| 24, 21|
| Y = 2 | 4, 10 | 8, 16 | 12, 21|
| Y = 3 | 2, 12 | 4, 18 | 6, 21|
f) In the reduced game, neither company has weakly dominated strategies. A weakly dominated strategy is one that gives a lower payoff than other strategies for all actions that the opponent might take, which is not the case in the reduced game.
g) If companies did not exclude the possibility of playing their weakly dominated strategies, the possible Nash equilibria of the reduced game would be any combination of (X, Y) where both X and Y are positive integers between 1 and 4.
h) Looking at the payoffs of the reduced game, it does not make sense for either company to play their weakly dominated strategies. The weakly dominated strategies result in lower payoffs compared to other available strategies, so it is not optimal for either company to choose them.
i) Based on the above analysis, the prediction for the game would be that the companies will not choose their weakly dominated strategies, and instead, they will select strategies that maximize their payoffs in the reduced game.