Budget-Constrained Auctions with Unassured Priors: Strategic Equivalence and Structural Properties

In today's online advertising markets, it is common for an advertiser to set a long-period budget. Correspondingly, advertising platforms adopt budget control methods to ensure that an advertiser's payment is within her budget. Most budget control methods rely on the value distributions of advertisers. However, the platform hardly learns their true priors due to the complex environment advertisers stand in and privacy issues. Therefore, it is essential to understand how budget control auction mechanisms perform under unassured priors. This work answers this problem from multiple aspects. Specifically, we discuss five budget-constrained parameterized mechanisms: bid-discount/pacing first-price auctions, the Bayesian revenue-optimal auction and bid-discount/pacing second-price auctions. We consider the game among the seller and all buyers induced by these five mechanisms in the stochastic model. We restrict the discussion to efficient mechanisms in which the seller earns buyers' budgets sufficiently, and thus, leads to his high revenue. Our main result shows the strategic equivalence between the Bayesian revenue-optimal mechanism and the efficient bid-discount first-price mechanism. A broad equivalence among all these (efficient) mechanisms is also given in the symmetric case. We further dig into the structural properties of stochastic budget-constrained mechanisms. We characterize sufficient and necessary conditions on the efficient parameter tuple for bid-discount/pacing first-price auctions. Meanwhile, when buyers do not take strategic behaviors, we exploit the dominance relationships of these mechanisms by revealing their intrinsic structures.