Three friends, Archie, Betty, and Veronica, are planning a party.

They disagree about how many people to invite. Each person i has a

quasilinear utility function of the form mi+ui(x) where mi is the number

of dollars that i has to spend and x is the number of guests invited to the

party. Suppose that for each i,

u_i = a_ix - 1/2 x²

Everyone knows the functional form of the others' utility functions and

knows his own value of ai but does not know anyone else's value. Let us

suppose that the actual values of ai are 20 for Archie, 40 for Betty, and

60 for Veronica.

(a) How many guests should be invited to maximize the sum of the three

persons' utilities?

(b) Suppose that the three friends decide to use the VCG mechanism to

determine the number of guests. If each plays his or her best strategy,

how many guests will be invited?

(c) In the VCG mechanism, if the amount of public good supplied is

x, Archie would receives a sidepayment equal to the sum of Betty's and

Veronica's utility for x. If Betty and Veronica play their best strategies

(without colluding) and if the amount of public good is x, this sidepay-

ment will be ___ . If everybody plays their best strategy, the

amount of this sidepayment in dollars is ____ .

(d) In addition to receiving sidepayments, the VCG mechanism requires

that each person must pay an amount equal to the maximum possible

sum of the other two persons' utilities. If Betty and Veronica play their

best strategies this amount is____ . On net, Archie has to pay the

di_erence between this amount and the sidepayment that he receives. If

everybody plays their best strategy, what is the net amount that Archie

must pay?

(e) If everybody plays their best strategy, what is the net amount that

Betty has to pay? What is the net amount that Veronica has

to pay?

(f ) Suppose that the party is organized not by just three people, but by a

dormitory with 21 residents. All of these residents have utility functions

of the same form as Archie, Betty, and Veronica. Seven of them have

ai = 20, seven have ai = 40, and seven have ai = 60. In order to

maximize the sum of the residents' utilities, how many guests should be

invited? If there were only six persons with ai = 20,

seven with ai = 40 and seven with ai = 60, how many guests would have

to be invited in order to maximize the sum of utilities?

(g) If everybody plays their best strategy in the VCG game, then after all

sidepayments and taxes are collected, how much net tax will each of the

people with ai = 20 have to pay? How much net tax will each of

the people with ai = 40 have to pay? How much net tax will

each of the people with ai = 60 have to pay?