Question

Three friends Alice(A), Bob(B), and Charlie(C) are planning to buy a TV.

TV’s are available in all sizes (size is the diagonal length of the display in

inches). Each friend i has a private valuation of the form

v_i(x) = a_ix − x^3/36

for a TV of size x, where i ∈ {A, B, C}

(a)

Use the VCG mechanism to decide which size of the TV should be

bought. Describe the size as a function of a_A, a_B, a_C.

(b)

Compute the payments for each friend under the VCG mechanism in

the situation where Alice has a_A = 10, Bob has a_B = 40, and Charlie

has a_C = 50.

Answer 1

a)

VCG mechanism:

outcomes:

$x^*(v)=arg\ max_x(\sum v_i(x))$

$x^*(v_{-i})=arg\ max_x(\sum_{j\neq i} v_j(x))$

transfers: 

Agent i receives

$t_i(v)=\sum_{j\neq i}v_j(x^*(v))-\sum_{j\neq i}v_j(x^*(v_{-i}))$

for a_A size of the TV:

x is root of equation $v_A(x) = a_Ax − x^3/36$

for a_B size of the TV:

x is root of equation $v_B(x) = a_Bx − x^3/36$

for a_C size of the TV:

x is root of equation $v_C(x) = a_Cx − x^3/36$

b)

for a_A = 10 :

$v_A(x) = 10x − x^3/36$

for a_B = 40 :

$v_B(x) = 40x − x^3/36$

for a_C = 50 :

$v_C(x) = 50x − x^3/36$

$\sum v_i(x)=100x-x^3/12$

$(\sum v_i(x))'=100-x^2/4=0$

$x=20$

outcome that maximizes the sum of values:

$x^*(v)=20$

for Alice(A):

$\sum_{j\neq i} v_j(x)=90x-x^3/18$

$\sum_{j\neq i} v_j(x)=90-x^2/6=0$

$x^*(v_{-A})=\sqrt{540}=23.24$

Alice receives:

$t_A=90\cdot20-20^3/18-(90\cdot23.24-23.24^3/18)=-38.72$

for Bob(B):

$\sum_{j\neq i} v_j(x)=60x-x^3/18$

$\sum_{j\neq i} v_j(x)=60-x^2/6=0$

$x^*(v_{-B})=\sqrt{360}=18.97$

Bob receives:

$t_B=60\cdot20-20^3/18-(60\cdot18.97-18.97^3/18)=-3.39$

for Charlie(C):

$\sum_{j\neq i} v_j(x)=50x-x^3/18$

$\sum_{j\neq i} v_j(x)=50-x^2/6=0$

$x^*(v_{-A})=\sqrt{300}=17.32$

Charlie receives:

$t_C=50\cdot20-20^3/18-(50\cdot17.32-17.32^3/18)=-21.79$

So, Alice has to pay 38.72, Bob has to pay 3.39, Charlie has to pay 21.79