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 equilibriums 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) A's profits can be derived by subtracting the cost of production from the revenue earned:

Profits (A) = Revenue (A) - Cost of production (A)
= Price per widget * X - Cost per widget * X
= (14 - X - Y) * X - 5 * X
= 14X - X^2 - XY - 5X

b) B's profits can also be derived in a similar manner:
Profits (B) = Revenue (B) - Cost of production (B)
= Price per widget * Y - Cost per widget * Y
= (14 - X - Y) * Y - 5 * Y
= 14Y - XY - Y^2 - 5Y

c) The table representing this one-stage game can be shown as follows:

| Y = 0 | Y = 1 | Y = 2 | Y = 3 | Y = 4 |
--------------------------------------------------
X = 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 |
--------------------------------------------------
X = 1 | 9, 5 | 8, 4 | 7, 3 | 6, 2 | 5, 1 |
--------------------------------------------------
X = 2 | 12,10 | 10, 6 | 8, 4 | 6, 2 | 4, 0 |
--------------------------------------------------
X = 3 | 12,15 | 11,11 | 9, 7 | 7, 3 | 5, 1 |
--------------------------------------------------
X = 4 | 12,20 | 12,16 | 10,12 | 8, 8 | 6, 4 |
--------------------------------------------------

Each cell represents the payoff for A (Profits A, Profits B) when A chooses a particular value for X and B chooses a particular value for Y.

d) To determine if either company has a strictly dominated strategy, we need to compare the payoffs for each strategy. A strategy is strictly dominated if there is another strategy that always gives a higher payoff regardless of what the opponent chooses.

For company A, strategy X=4 is strictly dominated by X=3. This is because for any value of Y chosen by company B, the payoff for A is always higher when A chooses X=3 compared to X=4.

e) After removing the strictly dominated strategy X=4 for company A, the reduced game table would look as follows:

| Y = 0 | Y = 1 | Y = 2 | Y = 3 | Y = 4 |
--------------------------------------------------
X = 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 |
--------------------------------------------------
X = 1 | 9, 5 | 8, 4 | 7, 3 | 6, 2 | 5, 1 |
--------------------------------------------------
X = 2 | 12,10 | 10, 6 | 8, 4 | 6, 2 | 4, 0 |
--------------------------------------------------
X = 3 | 12,15 | 11,11 | 9, 7 | 7, 3 | 5, 1 |
--------------------------------------------------

f) In the reduced game, neither company has weakly dominated strategies. A weakly dominated strategy would give the same payoff as another strategy for some opponent actions and a lower payoff for all other opponent actions.

g) Since there are no weakly dominated strategies in the reduced game, we can analyze the possible Nash equilibria. A Nash equilibrium occurs when neither player can improve their payoff by unilaterally deviating from their strategy.

From the reduced game table, we can observe that there are multiple Nash equilibria:
- (X=0, Y=0)
- (X=1, Y=1)
- (X=2, Y=2)
- (X=3, Y=3)

h) Based on the payoffs of the reduced game, it does not make sense for either company to play their weakly dominated strategies. The weakly dominated strategies have lower payoffs compared to the other available strategies.

i) Based on the analysis, the prediction for the game would be that the players will likely choose strategies where the payoffs are maximized for both companies. This would result in a Nash equilibrium where both players choose the strategies X=3 and Y=3.

a) To derive an expression for Company A's profits, we need to calculate the revenue and the cost. The revenue is given by the selling price of each widget, which is $(14-X-Y), multiplied by the number of units produced by Company A, X. So, the revenue for Company A is (14-X-Y) * X.

The cost is the cost per widget, which is $5, multiplied by the number of units produced by Company A, X. So, the cost for Company A is 5 * X.

Therefore, Company A's profits can be calculated by subtracting the cost from the revenue:

Profit A = (14-X-Y) * X - 5 * X

b) Similarly, to derive an expression for Company B's profits, we use the same approach. The revenue for Company B is (14-X-Y) * Y, and the cost for Company B is 5 * Y. Therefore:

Profit B = (14-X-Y) * Y - 5 * Y

c) To draw the table representing the game, we need to consider the possible strategies and payoffs for each company. Since both companies can produce 0, 1, 2, 3, or 4 widgets, there are 5 strategies for each company.

The table will look as follows:

| | 0 widgets | 1 widget | 2 widgets | 3 widgets | 4 widgets |
|:------- |------- |------- |------- |------- |------- |
| A | Payoff A | Payoff A | Payoff A | Payoff A | Payoff A |
| B | Payoff B | Payoff B | Payoff B | Payoff B | Payoff B |

For each combination of strategies (X, Y), we can calculate the payoffs using the expressions derived in parts a) and b).

d) To determine if either company has a strictly dominated strategy, we need to compare the payoffs for each strategy to see if there is always another strategy that gives a higher payoff.

For Company A, we would compare the payoffs of each strategy and see if there is always another strategy that gives a higher payoff. If there is, then that strategy is strictly dominated.

e) To draw the reduced game, we need to remove the strictly dominated strategies from the table and only include the remaining strategies.

f) In the reduced game, we can check if either company has weakly dominated strategies. Weakly dominated strategies are those that give the same payoff as other strategies for some actions that the opponent might take, but always give a lower payoff for all other actions the opponent might take.

g) If we consider the reduced game without excluding the weakly dominated strategies, we can identify the Nash equilibria. A Nash equilibrium occurs when no player can unilaterally change their strategy to improve their payoff.

h) Examining the payoffs of the reduced game, we can determine if it makes sense for either company to play their weakly dominated strategies. This analysis would involve comparing the payoffs and considering the strategic interactions between the companies.

i) Based on the analysis of weakly dominated strategies and the payoffs of the reduced game, we can make a prediction about the likely outcome of the game. This prediction would involve considering the rational choices of the companies and the potential Nash equilibria.