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 (x
_{i})_{i∈N}, x_{1}=v(1), and for i>1 x_{i}=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∈S}f(x_{i}) | (x_{i})_{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 (q_{i})_{i∈N},
q_{i}≥0, ∑_{i∈N}q_{i}=1
such that for all coalitions S,
∑_{i∈N}q_{i}≠1/2, and
v(S)=1 iff ∑_{i∈N}q_{i}>1/2,
and v(S)=0 otherwise.

- Find conditions on the weights
(q
_{i})_{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.