Answers to Sample Exam (Dec 2007)

							      ECON3014: Game Theory. A¢ liate Exam, December 2007.
                          Answers

     1.
                                          L           C        R
                                  U       1,-2        -2,1     0,0
                                  M       -2,1        1,-2     0,0
                                  D       0,0         0,0      1,1


   a) (D,R) is a pure strategy Nash equilibrium (you need to show why).
(U,L) is not a NE since player 2 can do better by deviating to C. Similarly, for
every other pure strategy pro…le, some player can do better by deviating.

     b) Let p be the probability that U is played, let M be played with prob
(1    p): For 2 to be indi¤erent between L and R,

                                2p + (1         p) = p        2(1     p)        (1)

                                          1
                                            p=                              (2)
                                          2
   So the payo¤ of 2 in this candidate equilibrium is 0:5 < 0: But then playing
R is better since it gives a payo¤ of 0. So there is no such mixed equilibrium
where both players randomize across their …rst two strategies.

                     s
    c) Solving for 2’ optimal strategy. must play C after U, L after M and R
after D. So 1 chooses R:

   4. Suppose that each bidder bids a constant fraction of his valuation. Let
                     s
w be my opponent’ valuation, and v be my own. My expected payo¤ from
bidding b given v is

                                      Z    b=
                     U (b; v)    =               [v     0:5b        0:5 w] dw   (3)
                                      0
                                                                         b=
                                                                    w2
                                 =    vw          0:5bw                         (4)
                                                                    4    0

                                           vb         0:5b2      0:5b2
                           U (b; v) =                                           (5)
                                                                  2
     di¤erentiate to get …rst order condition for b :
                                        2
                                          v b=                                  (6)
                                        3
                                        2
     So a symmetric equilibrium has b = 3 v for both bidders.

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    2. Two people select a policy that a¤ects them both by alternately vetoing
policies until only one remains. First person 1 vetoes a policy. If more than
one policy remains, person 2 then vetoes a policy. If more than one policy still
remains, person 1 then vetoes another policy. The process continues until only
one policy has not been vetoed.
    a) Suppose there are three possible policies, X, Y, and Z. Person 1 prefers
X to Y to Z, and person 2 prefers Z to Y to X.
    In stage 2, 2 will choose his preferred option. So if 1 vetoes X, the outcome
is Z, and if he vetoes Y the outcome is Z, and if he vetoes Z the outcome is Y.
So the subgame perfect equilibrium has one vetoing Z.
    b) Suppose that there are three possible policies, X, Y, and Z. Person 1
prefers X to Y to Z, and person 2 prefers Y to X to Z. Find the subgame perfect
equilibrium.
    Now if 1 vetoes X, the outcome is Y, and if he vetoes Y the outcome is X,
and if he vetoes Z the outcome is Y. So the subgame perfect equilibrium has
one vetoing Y.

    3. Regardless of ; if player 2 chooses L; 1 must choose the strategy (B,T).
If 2 chooses R, 1 must choose (T; B) : So these are the candidate equilibria.
    If 1 chooses (B,T),
    supppose that 1 chooses BT.,
       s
     2’ payo¤ from L is

                          u2 (L; BT ) = 1 + 4(1      )                        (7)

   while his payo¤ from R isf

                       u2 (R; BT ) = 0 + 0(1      ) = 0:;                     (8)
   So the payo¤ from L is greater than payo¤ from R for every between 0
and 1.
   So regardless of ; there is a pure strategy equilibrium: (B,T) for player 1,
L for player 2.
   Now let us consider an equilibrium where 1 plays (T,B).


                          u2 (L; T B) = 2 + 0(1      );                       (9)

                          u2 (R; T B) = 3 + 2(1      )                      (10)
   Since u2 (R; T B)  u2 (L; T B) for every 2 [0; 1]; there is a pure strategy
equilibrium where 1 plays (T,B) and 2 plays R, for every value of :




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