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)?