Vickrey-Clarke-Groves mechanism

Vickrey-Clarke-Groves-Mechanisms (VCG-Mechanisms) are a generalization of the Vickrey auction.

VCG-Mechanism for the provision of public goods

The price of the public good is divided by the number of players, and evenly distributed among them. The utility to the players is the difference to the price. This difference will be the bids for the public good too. If the sum of all the bids is $ge$0, the public good will be supplied and all players have to pay their fraction. Otherwise (if the sum is <0) the good will not be supplied. Since the players could bid any amount, but still paying the same, the mechanism wouldn't work. Therefore there is an pay function to get the players to bid their utility:

0, & ext{if }hat sum ge 0 ext{ and}hat sum {}^{-i} ge 0\ 0, & ext{if }hat sum < 0 ext{ and}hat sum {}^{-i} < 0\ hat sum {}^{-i}, & ext{if }hat sum < 0 ext{ and}hat sum {}^{-i} ge 0\ - hat sum {}^{-i}, & ext{if }hat sum ge 0 ext{ and}hat sum {}^{-i} < 0end{cases}

i.e. if the sum of all bids except for player i are $ge$0 (<0), and the total sum with his bid<0 ($ge$0), he has to pay the absolute value of the sum of all bids except for his ($left|\; hat\; sum\; \{\}^\{-i\}\; ight|$), otherwise he doesn't have to pay anything. With this function the bids will be the utility.

The problem with this mechanism is that the paid amounts that exceed the price have to be destroyed, because otherwise there might be a point in bidding more than the utility. Furthermore there exists the risk of collusion, if all the bids are known.

References

Krebs, David M. (1990). "A Course in Microeconomic Theory". New York: Princeton University Press. ISBN 0691042640