Fair Division II

Question 1

Adjusted Winner (AW)

Answer 1

efficient
envy-free
equitable with respect to bargainers’ announced preferences (not necessarily their true preferences).

> First choice for negotiations involving easily specified issues and well-defined goods
But not robust again untruthfulness

Question 2

Adjusted Winner Procedure

Answer 2

Let X be the sum of the points of all goods that Bob announces he values more than Carol. Let Y be the sum of the points of the goods that Carol announces she values more than Bob.

Assume X ≥ Y.
Assign the goods such that Bob initially gets all the goods where xi ≥ yi and Carol gets the others.
List the goods in an order G1, G2, etc. so that the following hold:
Bob values goods G1,…, Gr at least as much as Carol does (xi ≥ yi)
Carol values goods Gr+1, …, Gk more than Bob does (xi < yi)
x1/y1 ≤ x2/y2 ≤ … ≤ xr/yr, smallest ratio criterion
Transfer from Bob to Carol as much of G1 as is needed to achieve equitability (the point totals of the two players are equal). If equitability is not achieved, even after transferring all of G1 to Carol, continue with G2, G3, etc. This order, given by (c) smallest-ration criterion, ensures efficiency

Question 3

Proportional Allocation (PA)

Answer 3

like AW: envy-free and equitable outcomes.
Unlike AW: robust against false announcements in most situations.
optimal response by the players is to be nearly truthful.

Not efficient.

Question 4

Knaster’s Procedure

Answer 4

Players bid for items in an auction, and the highest bidder wins.
More successful players make side payments to
less-successful players, so that all players receive at least a proportional share in terms of items they won, money, or both.

> equitable (given same total valuation)
eny-free

Question 5

Combined Procedure

Answer 5

Combination of AW and PA

> PA as the default option,
PA provides a punishment option to encourage honest behaviour under AW.

Question 6

Vickrey Clarke Grooves Mechanism

Answer 6

The choice function maximizes overall welfare and the costs of each agent is equivalent to the opportunity costs, resulting in his participation

Independent of what the other player chooses , it is optimal, under this mechanism, for the player to announce his true type. CGM allows the efficient solution to be implemented. However, it does not have a balanced budget; If the good is provided (vA + vB > 1), both individuals together pay less than the cost of the public good, so that a third party has to put in some money.

> Every player pays for the externalities he puts on the others

> Pareto efficient

> not stable with coalitions

Question 7

Characteristics of Public Goods

Answer 7

NON-RIVALROUS: an additional user does not cause any extra cost either for the producer of the public good nor for the other uses (i.e. by decreasing the service quality). ——> Pure public good

NON-EXCLUDABILITY: consumers cannot be excluded from accessing and using the public good (i.e through payment). Public goods can therefore not be provided by private firms who sell their service.

Question 8

underprovided public good

Answer 8

NASH EQUILIBRIUM: in the Nash Equilibrium, the player with the higher marginal benefit pays everything and the other player just benefits and pays nothing.

Also, the equilibrium quantity of the public good is smaller than the efficient quantity; the public good is underprovided.