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							                                   Simon Fraser University
                                   Department of Economics


    Economics ECON - 383-3                                                            Daniel Monte
    Selected Topics in Game Theory                                                    Summer 2009


                        Problem Set 2: Due Wed, July 29th, in class

   1) Consider the following stage game:

                                       L              R
                                   T   M; M             L; M +
                                   B   M+ ; L         0; 0
    Assume that M > 0 and L > 0 and that the game above is played in…nitely often. Players
discount the future with a discount factor < 1.
                                                  1
    a) For part a, assume that       > 0 and = 2 : Describe a strategy pro…le and …nd for which
values of     (as a function of the parameters) would this strategy induce a subgame perfect Nash
equilibrium in which each player’ repeated game payo¤ is 1M .
                                   s
    Answer:
    Strategy
      1 : play T in the initial period and play T if all previous periods have been (T; L). play B
otherwise.
      2 : play L in the initial period and play L if all previous periods have been (T; L). play R
otherwise.
     M
    1       M+

                                                 M
                                                    1
    b) Now assume that       < 0. For what values of would the repeated play of T and L be the
outcome of a subgame perfect Nash equilibrium in the in…nitely repeated game?
    Answer:
    Consider the following strategy pro…le: 1 : play T regardless of the history, 2 :play L regardless
of the history. This strategy pro…le induces a SPNE for any .
    2) Consider the following simultaneous game between 2 players. There are two states of the
world and only player 1 knows which state it is. Player 2 knows only that with probability the
state of the world is s1 and with probability 1      it is state s2 .


                            player 2                                           player 2
                             L     R                                          L     R
                 player 1 T 2; 2     1; 1                      player 1 T     4; 4 1; 3
                          B 3; 0 0; 1                                   B     2; 2 0; 3
                                 s1                                             s2

   a) If player 2 knew exactly which state of the world was the true one, what would be the Nash
Equilibria of the two games?
   Answer:

                                                  1
    (B; R) in game of s1
    (T; L) in game of s2
    b) If player 2 does not know which state is the true one, but only knows that with probability
  the state of the world is s1 and with probability 1       it is state s2 . What is the Bayesian Nash
Equilibrium of the game? (Hint: are there dominated strategies?)
    Answer:
    Note that T is strictly dominated by B in s1 and B is strictly dominated by T in s2 . Thus,
player 1 will play B if s1 and T if s2 .
    Player 2:
    U (L) = 0 + (1       )4
    U (R) = 1 + (1       )3 = 3 2
    U (L) > U (R) if
    4 4 >3 2
    2 <1
      <1 2
    If < 1 , then BNE is:
            2
    player 1 will play B if s1 and T if s2 player 2 plays L
    If > 1 , then BNE is:
            2
    player 1 will play B if s1 and T if s2 player 2 plays R
    3) The game below is played for two periods, where both players discount the second period
using a discount factor < 1.

                                          L       M       R
                                      A   5; 5    0; 6      1; 4
                                      B   6; 0    2; 2      1; 0
                                      C   4; 1    0; 1    0; 0
    Construct a strategy for the repeated game above that induces a subgame perfect Nash equi-
librium in which players play A and L in the …rst period? For what values of will this strategy
indeed induce the SPNE?
    Answer:
      1 :play A in the initial period, play B in the second period if in the …rst period the outcome
was (A; L), play C if in the …rst period the outcome was di¤erent then (A; L)
      2 :play L in the initial period, play M in the second period if in the …rst period the outcome
was (A; L), play R if in the …rst period the outcome was di¤erent then (A; L).
    5+ 2>6+ 0
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