Game Theory: Truthful Mechanism Design

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Self-revelation in Auctioning using Game Theory
Application of Game Theory in Auction design has been known for quite some time now. Vickrey Auction or Second price sealed bid auction is one particularly elegant and powerful instance of the same. It is very much like a normal auction with a slight twist which does the trick. The bidder with the highest bid amount wins the auction but he pays the second-highest bid. This rule gives the bidders an incentive to bid their true value. This sounds too naive to be effective but lets see how it works. For the sake of argument , suppose there are only 2 players(A and B) whose real valuation for the item being auctioned is 100 and 110 respectively.
Assume that B bids an amount(say VB ) greater than 110 i.e VB >100
Possibility 1: VA < 110 < VB ,As per the terms of second price auction B wins the auction with a payoff 110- VA, which is the same payoff that he would have received had he bid 110.
Possibility 2: VA > VB >110, B loses the bid and his payoff equals 0.This is again the same payoff he would have received on bidding 110.
Possibility 3: VB > VA >110, B wins the bid but his payoff (110- VA ) is negative.Thus he would have been better off if he had bid 110.
Now Assume that B bids an amount less than 110 i.e VB < 100
Possibility 1: VA >110> VB ,B loses the item under the terms of auction with a payoff of zero.
Possibility 2: 110> VB > VA , B wins the item with a payoff of 110- VA.He would have received the same payoff if he had bid 110.
Possibility 3: 110> VA > VB ,B would lose the bid in this case.If he had bid 110 he would have won the bid with a positive payoff of 110-VA.
We find that truthful bidding dominates all the possible cases under over- bidding and underbidding.The same argument can be generalized to an Auction with N(N>2) number of bidders. Companies like Google,Yahoo,ebay etc use a slight variant of this strategy called generalized

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