“Efficient Mechanism Design,” V. Krishna & M. Perry (2000)

Mechanism Design is the source of many of the most elegant results in economics. In this unpublished paper, Krishna and Perry show how powerful the famous Vickrey-Clarke-Groves mechanism is. The standard VCG just says that, to allocate goods, I should give them to the people with the highest value for those goods, and charge everyone a sum equal to the “externality” imposed on other bidders by my entrance into the mechanism. Such a mechanism makes it a dominant strategy for everyone to reveal their true private value, hence is efficient. This is an efficiency result, not a revenue maximization result.

Vickrey and Perry extend VCG by showing that VCG with an appropriate “basis” is the revenue-optimal mechanism among all efficient mechanisms, even when VCG is general enough to allow for complementaries among agents or consumption externalities. In particular, the basis is the “worst” surplus for agent i among his possible types given the mechanism. For instance, in a standard second-price auction, an agent with value 0 for the good gets expected surplus 0, so his basis is 0. The general VCG is just the VCG “externality” payment minus the participation constraint for the basis defined above. Since every agent is willing to participate in his worst case, then those agents are willing to participate in the mechanism always. From here, a short proof shows that generalized VCG is optimal among all efficient, incentive compatible, participation constraint satisfying mechanisms. Further, there exists an efficient, revenue optimal mechanism with balanced budgets (the sum of all transfers are zero) iff the sum of payments in VCG is greater than or equal to zero.

As an example, the authors show a simple proof of Mailath and Postlewaite’s 1990 result that no public project mechanism can pay for itself: government must force unwilling people to pay taxes to overcome information asymmetry. The intuition is that the designer does not know who really wants the project and who does not. He needs to charge the low value types a small enough value that they don’t veto the project, but he can’t charge high value types so much that they would pretend to be low value types. Note that the VCG externality is only positive if an individual is “pivotal” – that is, if his pretending to be a low type would switch the project from being constructed to not being constructed. The only agent that pays a VCG transfer pays exactly the cost of the project minus the sum of everyone else’s welfare had the project been constructed (call this Tj, and call individual i’s utility from completing the project Ti). Then c-Tj>=Ti must hold if the mechanism is efficient because the project is efficient to enact only if Tj+Ti>=c. It is known by Green and Laffont that the VCG mechanism gives transfers summing to less than the cost of the project (you can check this here: let there be two agents, c=1, Ti=.6 for both. Both agents are pivotal, and VCG payment is .4 for each, hence sum to .8<c). But we also know from above that VCG maximizes total payments among the class of efficient mechanisms satisfying participation constraints. So there exists no mechanism for eliciting private values for a public project that will pay for itself.

http://econ.la.psu.edu/~vkrishna/papers/vcg20.pdf

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