  In today's online advertising markets, it is common for advertisers to set
long-term budgets. Correspondingly, advertising platforms adopt budget control
methods to ensure that advertisers' payments lie within their budgets. Most
budget control methods rely on the value distributions of advertisers. However,
due to the complex advertising landscape and potential privacy concerns, the
platform hardly learns advertisers' true priors. Thus, it is crucial to
understand how budget control auction mechanisms perform under unassured
priors.
  This work answers this problem from multiple aspects. We consider the
unassured prior game among the seller and all buyers induced by different
mechanisms in the stochastic model. We restrict the parameterized mechanisms to
satisfy the budget-extracting condition, which maximizes the seller's revenue
by extracting buyers' budgets as effectively as possible. Our main result shows
that the Bayesian revenue-optimal mechanism and the budget-extracting
bid-discount first-price mechanism yield the same set of Nash equilibrium
outcomes in the unassured prior game. This implies that simple mechanisms can
be as robust as the optimal mechanism under unassured priors in the
budget-constrained setting. In the symmetric case, we further show that all
these five (budget-extracting) mechanisms share the same set of possible
outcomes. We further dig into the structural properties of these mechanisms. We
characterize sufficient and necessary conditions on the budget-extracting
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.
