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Exercises in Game Theory
by
Rod Garratt and John Wooders
This document contains questions that are appropriate for advanced undergraduate
classes or graduate classes in game theory. They have each been used at least once in
graduate game theory courses taught by us at the University of Arizona and the
University of California, Santa Barbara. We created each of the questions, although some
are based on published works or the research of our colleagues.
1
The questions are
grouped according to the equilibrium concepts they use. Answers and recommendations
for extensions are available upon request.
Nash Equilibrium, Subgame Perfect Equilibrium
1
.
There are two baseball teams that are preparing for a three game series. Each team has
three pitchers. Team 1 has an Ace, a mediocre pitcher and a scrub. Team 2 has two
mediocre pitchers and a scrub. A pitcher can only be used once in the series. The
probabilities of winning the game depending on the pitcher match-ups are as follows.
Match up
Outcome
Ace versus mediocre pitcher
Ace wins with probability .7
Ace versus scrub
Ace wins with probability .9
Mediocre pitcher versus scrub
Mediocre pitcher wins with probability .6
Same versus same
Each wins with probability .5
Assume a win is worth 1 and a loss is worth 0.
(a)
Suppose the three games are played sequentially. The pictures for each game are
chosen simultaneously, but after a game is played it is common knowledge what
pitchers were used.
Remember each team can only use each pitcher once. Write
out the extensive form of the game. (The extensive form has 18 terminal nodes.)
(b)
Calculate the set of subgame perfect equilibria.
(c)
Draw the normal form representation of the extensive form. Find all the Nash
equilibria.
(d)
Would either team prefer a system whereby each team’s pitcher choices for the
entire series had to be posted simultaneously before the series began?
(e)
Compare the set of Nash equilibria of the game to the set of subgame perfect
equilibria.

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