Vickrey–Clarke–Groves mechanism

In mechanism design, a Vickrey–Clarke–Groves (VCG) mechanism is a generic truthful mechanism for achieving a socially-optimal solution. It is a generalization of a Vickrey–Clarke–Groves auction. A VCG auction performs a specific task: dividing items among people. A VCG mechanism is more general: it can be used to select any outcome out of a set of possible outcomes.

There is a set ${\displaystyle X}$ of possible outcomes.

There are ${\displaystyle n}$ agents which have different valuations for each outcome. The valuation of agent ${\displaystyle i}$ is represented as a function:

which expresses the value it has for each alternative, in monetary terms.

It is assumed that the agents have Quasilinear utility functions; this means that, if the outcome is ${\displaystyle x}$ and in addition the agent receives a payment ${\displaystyle p_{i}}$ (positive or negative), then the total utility of agent ${\displaystyle i}$ is:

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