. Consider the following game: a) Determine the unique rationalizable strategy f

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. Consider the following game:
a) Determine the unique rationalizable strategy for each player.
b) Determine the set of all Nash equilibria in pure strategies.
2. Consider the game expressed by the following matrix:
PLAYER 2
a b c d
PLAYER 1
A (0, 0) (4, 1) (1, 0) (0, 0)
B (4, 1) (0, 2) (2, 1) (1, 0)
C (2, 1) (0, 0) (5, 0) (0, 3)
D (2, 0) (1, 0) (0, 5) (0, 4)
PLAYER 2
a b
PLAYER 1
A (0, 100) (0, 1)
B (1, 0) (1, 1)
2
a) Determine the set of rationalizable strategies.
b) Suppose now that both players can tremble. In the current case, this means that
when a player adopts a strategy s, nature switches it to the other strategy s’ with a
probability ε = 0.1. The trembling probabilities are independent and identical for
each player. In other words, the probability that both of them simultaneously
tremble when adopting a strategy is ε×ε, the probability that none of them tremble
is (1-ε)×(1-ε) and so forth. Notice that there are four possible outcomes when
players commit to a strategy profile. Write the new game in strategic form and
identify the set of rationalizable strategies.
3. Consider the following normal form game:
a) Determine all of the Nash equilibria in pure strategies.
b) Does this game feature any Nash equilibria in mixed strategies? If so, identify them.
4. Three players (1, 2 and 3) are engaged in a simultaneous game. The strategy set for
players 1 and 2 is, respectively, S1 = {A, B} and S2 = {a, b}. Player 3 chooses the
strategic form underlying this game among the two options below:
Strategic Form I
PLAYER 2
a b c
PLAYER 1
A (3, 1) (0, 0) (1, 0)
B (0, 0) (1, 3) (1, 1)
C (2, 1) (0, 1) (1, 10)
PLAYER 2
a b
PLAYER 1
A (1, 3, 1) (1, 2, 2)
B (5, 2, -2) (2, 1, 2)
3
Strategic Form II
The payoffs are defined as (player 1 payoff, player 2 payoff, player 3 payoff).
a) Do any of the players feature a strictly dominant strategy? If so, which ones?
Explain.
b) Can you solve this game using iterated elimination of strictly dominated strategies?
If so, identify its solution.
5. Two individuals (1 and 2) simultaneously select a number between 0 and 1. Denote
these, respectively, by n1 and n2. If n1 < n2, player 1 obtains a payoff of (n1 + n2)/2 while player 2's payoff is 1 - (n1 + n2)/2. Conversely, if n1 > n2, player 1 obtains a payoff of 1
– (n1 + n2)/2 while player 2’s payoff is (n1 + n2)/2. Finally, when n1 = n2, both players get
a payoff equal to 1/2. Using the concept of best response, determine the Nash
equilibrium (or equilibria) for this game.
6. Two firms (1 and 2) simultaneously choose how much time to spend on research. This
is denoted, respectively, by t1 ≥ 0 and t2 ≥ 0. The payoff functions are expressed as
follows:
?!(?!, ?”) = ‘
? − ?! ?? ?! ≥ ?”
−?! ?? ?! < ?" ?"(?!, ?") = ' ? − ?" ?? ?" ≥ ?! −?" ?? ?" < ?! with V > 0.
a) Determine the best response correspondences for each firm, that is, BRi (t-i). Be
as comprehensive as possible.
b) Represent each best response correspondence on a graph and determine the
Nash equilibria for this game.
PLAYER 2
a b
PLAYER 1
A (2, 3, 2) (1, 2, 1)
B (3, 4, 1) (4, 3, 1)

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