Cooperative or vindictive: Bidding strategies in sponsored search auction

We analyze the economic stability and dynamic manipulation of vindictive strategies in conjunction with forward-looking cooperative bidders in Sponsored Search Auction. We investigate different vindictive strategies of different rationalities: malicious, conservative and selective. In a malicious vindictive strategy, the bidder forces his competitors to pay more by bidding just one cent lower of his competitor's bid. We show that Nash Equilibrium is vulnerable even there is one malicious vindictive bidder. However, on bidder's perspective, he has not much incentive to use a malicious vindictive strategy. A conservative vindictive bidding strategy makes a bidder never sacrifices his own benefit to take revenge on his competitor. Under this strategy, we prove that there always exists an output truthful Nash Equilibrium. However, it may not always be the unique equilibrium. Lastly, we investigate a selective vindictive strategy that a bidder rationally chooses to bid cooperatively or vindictively. The bidder takes a vindictive strategy only if the bidder who gets one position higher has a larger private value. We prove that selective vindictive strategy always results in a unique truthful Nash Equilibrium in conjunction with forward looking cooperative bidders. Interestingly, forward looking strategy gives the same payment as VCG mechanism if all the bidders takes it. However, the bidder prefers selective vindictive strategy while the auctioneer's revenue reach maximum when all the bidders takes the selective vindictive strategy.