Vickrey–Clarke–Groves auction

Vickrey -Clarke - Groves mechanisms ( VCG ) mechanisms are a generalization of the Vickreyauktion.

With this concept a class of mechanisms is called, whose members have the property that truthful bidding is a dominant strategy for the players.

VCG mechanisms can be applied if the utility function of the problem is quasi - linear, ie cash payments between agents are possible.

A mechanism is referred to as VCG mechanism if it satisfies the following two conditions:

• The selection function maximizes the overall benefit, and
• The payment of each agent corresponds to the opportunity costs that they incur in participating.

Example Vickreyauktion

Given the bidders with the types. is seen as good for the benefit of bidders. The selection function decides which bidder is awarded the contract, and satisfies the condition that one of the high bidder wins:

The opportunity cost for the winner correspond to the loss of profits that would have resulted in the adoption of the next highest bid, that is just the amount of the second highest bid, or 0 if no such exists:

Example Combinatorial auction

Tenderers are now offering for bundles of goods amount. Be the benefits, the agent draws from the bundle of goods. The type of an agent thus defines for each bundle of goods the particular benefits that:

The selection function of the VCG mechanism distributes the goods to the agents so that the total sum of the individual utility is maximized:

Denote

A possible selection function ( ie is the bundle of goods, the agent receives if the agents provide. ) and

The set of possible type vectors of the agents,

Thus solves the optimization problem

With the constraints

For.

For the payment function of the VCG mechanism applies to the designation

Example, provision of public goods

The price of the public good is shared equally by all players. Now the benefit of the player can be greater or less than this price, and the difference corresponds to the commandments ( and appreciation ). If the sum of all Bids 0 the public good is provided, the sum is <0, it is not provided. Thus, the true appreciation is offered, and not much more, so that the desired result is still a payment mechanism must be introduced which works as follows:

That if the sum of all bids without the player i 0 ( < 0), and the sum with its bid < 0 (0 ), it must pay the amount of the sum without him (), nothing else. This can be the bidding of true appreciation for the weakly dominant strategy. The paid as the price of the provision amount must, however, be destroyed, otherwise interdependencies arise that would affect the dominance of strategy. Thus, the mechanism is efficient, but not welfare Maximizing. Moreover, there is collusion if all bids are known.

Uniqueness of VCG mechanisms

Green and Laffont have shown in 1977 that, under the condition that the type space is path- connected, the VCG mechanisms are the only mechanisms that maximize the sum of individual benefit and where truthful game is a dominant strategy.

This theorem can also be derived from the envelope theorem.