Homework Assignment 3

Mechanism Design and Social Choice, summer ’09 Prof. Dr. Benny Moldovanu

Wirtschaftstheorie II Lennestr. 37, 53113 Bonn

Konrad Mierendorff Office hours: Montag 11:45 - 12:45 Lennestr. 37, 2. Stock mierendorff@uni-bonn.de

Homework Assignment 3

This exercise sheet will be discussed Mai 7.

Please prepare solutions for the exercise session.

1. Show that the Shapley Value satisfies the Pareto property.

2. The Shapley Value is the only cooperative solution that satisfies the Pareto axiom (P), Symmetry (S), the dummy axiom (D) and additivity (A)¹

(OR², Ex. 294.2) Show the following results, which establish that if any of the three axioms (S), (D) and (A) is dropped, then there is a solution satisfying all other axioms, that is different from the Shapley Value.

(a) For any cooperative game v and any i ∈ N, let ψ_i(v) be the average marginal contribution of player iover all the (|N| −1)! orderings ofN in which player 1 is first. Thenψ satisfies (D) and (A) but not (S).

(b) For any cooperative gamev and anyi∈N, let ψ_i(v) = ^v(N_|N|⁾. Then ψ satisfies (S) and (A) but not (D).

(c) For any cooperative game v, and any i∈N, let D(v) be the set of dummies in v and let

ψ_i(v) = ( 1

N\D(v)

³

v(N)−P

j∈D(v)v({j})

´

ifi∈N\D(v)

v({i}) ifi∈D(v).

Then ψsatisfies (S) and (D) but not (A).

3. In a weighted majority game, each playeri∈N has a number of votesw_i. A coalition S⊂N wins, if the sum of votesw(S) =P

i∈Sw_i exceeds some quota q:

v(S) = (

1, ifw(S)≥q, 0, otherwise.

Consider a parliament with n parties. Two of them have one third of the seats and the other n−2 share the remaining third equally. Model this situation as a weighted majority game.

(a) Show that the Shapley value payoff of each of the large parties converges to ¹₄ as n→ ∞. (Hint: Assume thatnis even and q= ¹₂.)

(b) Is it desirable according to the Shapley value for the small parties to form a single united party?

1In MWG, additivity is called linearity.

2Osbourne, RubinsteinA course in game theory.

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