Section A

A1 (30 points) Consider the game given by the extensive form below.

Pl 1

(6, 5, 4)

x3    x4

(6, 2, 4)

(6, 2, 0)

(4, 2, 10)    (-1, 6, 8)         (4, 6, 0)

Note that nodes connected by a dashed line constitute an informa- tion set.

(a) Find all pure strategy Nash equilibria of this game.                   [6]

(b) Assume player three plays l with probability γ . For which beliefs      over {x1 , x2 } is it sequentially optimal for player 2 to choose L     when he moves?                                                                     [4]

(c) For which beliefs over {x3 , x4 } is it sequentially optimal to play      l for player 3 when he moves?                                                 [4]

(d) For every pure strategy Nash equilibrium (see part a)), determine      whether it is also a Perfect Bayesian equilibrium.                       [5]

(e) Consider a behavior strategy, where player 1 plays M with prob-     ability ν > 0, T with probability τ > 0 and player 2 plays R     with probability ρ > 0. Which beliefs over {x1 , x2 } (for player     2) and {x3 , x4 } (for player 3) are consistent with this strategy?  [4]

(f) Which of the Perfect Bayesian equilibria from d) are sequen-      tial equilibria?  For each sequentlial equilibrium that you find,      demonstrate directly that it satisfies the definition.                    [7]

Notes:  The term “sequentially optimal” is to be understood as in the definition of a Perfect Bayesian equilibrium.

A2 (25 points) Consider the following 2-person, 2-goods exchange econ-

omy. Consumer 1’s preferences are represented by the function

u (x11 , x2 ) = x1 (x2 )2 ,

and her initial endowment is 10 units of both goods (i.e.   e1   = (10, 10)). Consumer 2’s preferences satisfy

u (x21 , x2 ) = 7(x1 )2x2

and she posseses 5 units of both goods (i.e. e2 = (5, 5)).

(a) What are the Walrasian equilibrium prices?  (You can assume

p >> 0.) What are the Walrasian equilibrium allocations?         [10] (b) Is (are) the allocation(s) you found in the core?                        [5]

Now consider a 3-person, 2-goods economy, where endowments and preferences of consumer 1 and 2 are as above.  Consumer 3 is en- dowed with 20 units of the first good and 10 units of the second good, (i.e. e3 = (20, 10)). Her preferences are given by the function u3 (x1 , x2 ) = 40x1 + 50x2 .

(c) Consider the allocation  that allocates 1  = (8, 12), to con- sumer 1, 2  = (7, 3) to consumer 2 and 3  = (20, 10) to con- sumer 3. Is this allocation in the core? If so, argue why. If not,

demonstrate how a specific coalition can block this allocation.      (Hint:  Consider consumers 2 and 3 to keep the mathematics      simple.)                                                                                 [5]

(d) Find one core allocation. (If you found the allocation from c) is      in the core, find another one.)                                                 [5]

Section B

B1 (15 points) Consider the 2-player game given below.  Player 1 can choose between T and B, player 2 chooses between R and L. The resulting payoffs are given by the table.

(a) Consider first the case where x = 0 (and this is common knowl-      edge).   Find the set of strategies (pure or mixed) which are      rationalizable, i.e. survive the iterated elimination of strategies      which are never a best response.                                              [4]

(b) Now consider the case where the value of x is unknown to player 1, but known to player 2. Assume that x can take only the values

0 (with probability 1/10) or 12 (with probability 9/10), and this      is commonly known. Write down a corresponding normal form      game that is suitable to find the (Bayes-)Nash equilibria of this      game.   Find all pure strategy (Bayes-)Nash equilibria of this      game.                                                                                    [8]

(c) What kind of game is this?  Which economic applications are      typically analysed with such games?                                         [3]