We solve the Bayesian sequential equilibrium of a general class of single-item first-price or all-pay auctions of incomplete information. Our main contribution is a general methodology for solving the optimal commitment problem, in closed form, for asymmetric continuous-type distributions.   Our approach consists of a number of innovations. We propose a modeling concept called equal-bid function to build a bridge between two players' strategies. Another concept called equal-utility curve transforms any commitment strategy into a weakly better continuous and everywhere directional-differentiable strategy.   The optimal commitment functions in these auctions reveal some important insights. When the player with commitment power (the leader) has low valuation, he bids passively. This is a credible way to alleviate competition and to enable collusion. We demonstrate, via concrete examples, that this is a credible way to threaten the follower so that the leader can secure a higher utility.
['Pingzhong Tang', 'Zihe Wang', 'Xiaoquan (Michael) Zhang']
Optimal Commitments in Asymmetric Auctions with Incomplete Information