Nir Dagan / Teaching

Final exam of Coalitional Games

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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)?

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Nir Dagan / Contact information / Last modified: December 21, 1998.