CPC/CPA hybrid bidding in a second price auction

We develop a model of online advertising in which each advertiser chooses from multiple advertising measurement metrics - paying either for each click on its ads (CPC), or for each purchase that follows an ad-click (CPA). Our analysis extends classic auction results by allowing players to make bids using two different pricing schemes, while the driving information for bidders' endogenous selection - the conversion rate - is hidden from the seller. We show that the advertisers with the most productive sites prefer to pay CPC, while advertisers with lower quality sites prefer to pay CPA - a result that may be viewed as counterintuitive since low quality sites cannot proudly tout their conversion rates. This result holds even if an ad platform's assessment of site quality is correct in expectation. We also show that by offering both CPC and CPA, an ad platform can weakly increase its revenues compared to offering either alternative alone.