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 (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.** 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 (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.

- 3.1 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 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:**R**^{k}_{+}→**R**_{+} is called
distributive if it is non-decreasing and for any input
vectors x_{1},...,x_{m},
and non negative numbers
a_{1},...,a_{m} such that
x_{i}≤∑a_{i}x_{i} for all i=1,...,m,
it is satisfied that
∑a_{i}f(x_{i})≤f(∑a_{i}x_{i}).

- 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.