Rocco and Sasha are roommates. They are considering buying a new TV for their apartment. Rocco's valuation of the TV is $165 and Sasha's valuation of the TV is $180. Neither roommate knows the other's valuation, but each will be asked to report a valuation. If the reported values exceed the total cost of the TV, then they will purchase it and each pay half. The price of the TV is $300.
The roommates decide to use a Vickrey-Clarke-Groves (VCG) mechanism. This mechanism specifies that Rocco's payoff is given by (vr−cr)x+(rs−cs)x−maxy(rs−cs)y(vr−cr)x+(rs−cs)x−maxy(rs−cs)y, where vrvr is Rocco's valuation, crcr is Rocco's cost share, rsrs is Sasha's reported valuation, cscs is Sasha's cost share, xx is an indicator variable that is 1 if the TV is purchased and 0 if it is not, and a final term measures Sasha's maximum utility if Rocco is not included in the decision (note: in this case, this term equals 0).
Suppose that Sasha makes an untruthful report of a valuation of $80. What value should Rocco report? $
Suppose, instead, that Sasha reports a truthful valuation of $180. What value should Rocco report? $