# Final exam of Coalitional Games

December 15, 1998, GPEM, Universitat Pompeu Fabra, Academic year 1998/99

1. Consider a game (N,v) such that
there exist non-negative numbers a_i, i=1,...,n
and a non-decreasing function f:R->R, such that
v(S)= f(\sum(a_i)) where the sum is taken over all
players in S.

1.1 Show that if f is convex, the core is non-empty.
1.2 Give an example in which the core is empty.

2. Consider the solution concept that assigns to every
game (N,v) the intersection of the prekernel and the core.
Consider the set of all games whose set of players is a
subset of a given set U.

2.1. Define additivity (for multi-valued solution concepts)
For what cardinaties of U the above mentioned solution
is additive? Define the reduced game property and its
converse. For what cardinaties of U the above mentioned
solution satisfies these properties?

2.2. Give an axiomatic characterization of the above
mentioned solution that employs the reduced game property
and its converse. You may cite theorems studied in class
in your proofs as long as the theorems are stated clearly.

3. Consider bankruptcy rules that are defined for a domain
of all problems where the set of creditors is a subset of
a given finite set U.

For any problem (E,d) let (E,d)^i be (E',d') where
E' = Max{0,E-d_i} and d' = d|N\{i}.
In words, (E,d)^i is the problem that the creditors different
from i will face if i were payed in full and leave.

3.1 Show that there exists a unique solution concept f that
satisfies: f_i[E,d]-f_i[(E,d)^j] = f_j[E,d]-f_j[(E,d)^i]
for all i and j in (E,d), for all (E,d) with at least two
creditors.

3.2 What this solution concept assigns to the three examples
from the Talmud discussed in Aumann  and Maschler (1985)?