Question
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
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,
ui = aix - 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?
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