  We are interested in the problem of optimal commitments in rank-and-bid based
auctions, a general class of auctions that include first price and all-pay
auctions as special cases. Our main contribution is a novel approach to solve
for optimal commitment in this class of auctions, for any continuous type
distributions. Applying our approach, we are able to solve optimal commitments
for first-price and all-pay auctions in closed-form for fairly general
distribution settings. The optimal commitments functions in these auctions
reveal two surprisingly opposite insights: in the optimal commitment, the
leader bids passively when he has a low type. We interpret this as a credible
way to alleviate competition and to collude. In sharp contrast, when his type
is high enough, the leader sometimes would go so far as to bid above his own
value. We interpret this as a credible way to threat. Combing both insights, we
show via concrete examples that the leader is indeed willing to do so to secure
more utility when his type is in the middle. Our main approach consists of a
series of nontrivial innovations. In particular we put forward a concept called
equal-bid function that connects both players' strategies, as well as a concept
called equal-utility curve that smooths any leader strategy into a continuous
and differentiable strategy. We believe these techniques and insights are
general and can be applied to similar problems.
