Editing Mechanism design (section)

===Vickrey–Clarke–Groves mechanisms===
{{main|Vickrey–Clarke–Groves mechanism}}

The Vickrey (1961) auction model was later expanded by {{harvs|txt|last=Clarke|year=1971|author-link=Edward H. Clarke}} and Groves to treat a public choice problem in which a public project's cost is borne by all agents, e.g. whether to build a municipal bridge. The resulting "Vickrey–Clarke–Groves" mechanism can motivate agents to choose the socially efficient allocation of the public good even if agents have privately known valuations. In other words, it can solve the "[[tragedy of the commons]]"—under certain conditions, in particular quasilinear utility or if budget balance is not required.

Consider a setting in which <math>I</math> number of agents have quasilinear utility with private valuations <math>v(x,t,\theta)</math> where the currency <math>t</math> is valued linearly. The VCG designer designs an incentive compatible (hence truthfully implementable) mechanism to obtain the true type profile, from which the designer implements the socially optimal allocation
:<math> x^*_I(\theta) \in \underset{x\in X}{\operatorname{argmax}} \sum_{i \in I} v(x,\theta_i) </math>

The cleverness of the VCG mechanism is the way it motivates truthful revelation. It eliminates incentives to misreport by penalizing any agent by the cost of the distortion he causes. Among the reports the agent may make, the VCG mechanism permits a "null" report saying he is indifferent to the public good and cares only about the money transfer. This effectively removes the agent from the game. If an agent does choose to report a type, the VCG mechanism charges the agent a fee if his report is '''pivotal''', that is if his report changes the optimal allocation ''x'' so as to harm other agents. The payment is calculated
:<math> t_i(\hat\theta) = \sum_{j \in I-i} v_j(x^*_{I-i}(\theta_{I-i}),\theta_j) - \sum_{j \in I-i} v_j(x^*_I (\hat\theta_i,\theta_I),\theta_j) </math>
which sums the distortion in the utilities of the other agents (and not his own) caused by one agent reporting.