game theory

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?

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asked by Phil

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