Nir Dagan / Teaching

Exercise 1 of Coalitional Games

GPEM, Universitat Pompeu Fabra, Academic year 1998/99

1. A TU game (N,v) is called convex if for all coalitions S and T, v(S)+v(T)-v(S∩T)≤v(S∪T).

1.1 Show that (xi)i∈N, x1=v(1), and for i>1 xi=v({1,...,i})-v({1,...,i-1) is a core utility profile.
1.2 Are all convex games market games? Are all market games convex? Prove your answer.

2. A TU game (N,v) is called simple if for every coalition S, v(S) is either 0 or 1, and v(N)=1. A coalition S with v(S)=1 is called a winning coalition. A player in a simple game is called a veto player if it belongs to all winning coalitions.

2.1 Show that a simple game has no veto players if and only if the core of the game is empty.
2.2 Let (N,v) be a simple game, and T be the set of veto players. What is the core of this game?

3. A game (N,v) is a weighted simple majority game if there exist non-negative numbers (qi)i∈N, qi≥0, ∑i∈Nqi=1 such that for all coalitions S, ∑i∈Nqi≠1/2, and v(S)=1 iff ∑i∈Nqi>1/2, and v(S)=0 otherwise.

3.1 Find conditions on the weights (qi)i∈N that are necessary and sufficient for the core of a weighted simple majority game to be non empty. Hint: Use the results of question 2 above.

4. Show that a TU market may not have a price equilibrium when all players have zero quantity of a given input. Hint: Consider Cobb-Douglas production functions.

5. A production function f:Rk+R+ is called distributive if it is non-decreasing and for any input vectors x1,...,xm, and non negative numbers a1,...,am such that xi≤∑aixi for all i=1,...,m, it is satisfied that ∑aif(xi)≤f(∑aixi).

5.1 Let f be a concave production function. Show that f is distributive if and only if it satisfies constant returns to scale.
5.2 Consider a TU game associated with a market in which all players have the same (possibly non-concave) distributive production function. Show that the core of this game is non-empty.