Exercise 1 of Coalitional Games

Graduate program in economics, Brown University, Academic year 1999/2000

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. Consider a market game (N,v) with one input and common non-decreasing continuous function f:R→R, that satisfies f(0)=0. Let v(S)= Max{∑i∈Sf(xi) | (xi)i∈S is an S-allocation}

2.1 Show that if f is convex or concave, the core is non-empty.
2.2 Find a function f (with one input) in which the core is empty.

3. 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.

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

4. 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.

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 3 above.

5. Show that a TU market may not have a price equilibrium when all players have zero quantity of a given input.