Optimal linear-payment auction design with aftermarket collaboration

Abstract: This paper studies optimal auction design when valuations depend endogenously on post-auction collaboration between the seller and the winning bidder. Both parties exert non-contractible efforts after the auction, generating a double moral hazard problem alongside adverse selection. We analyze two role structures -- winner-pivotal and seller-pivotal collaboration -- and characterize optimal direct mechanisms using linear payment schemes that combine cash transfers with proportional value sharing. The optimal mechanism allocates the asset to the bidder with the highest virtual surplus, employs a deterministic value-sharing rule, and achieves full type revelation through the signal realization rule. Comparing the two scenarios yields three main findings. First, regarding value sharing, the seller secures a strictly higher share under seller-pivotal collaboration: for sufficiently low-type winners, the seller extracts the entire value, whereas under winner-pivotal collaboration every winner must retain a positive share to sustain his critical effort. Second, regarding effort exertion, the pivotal party always exerts higher post-auction effort than the supporting party, and each party exerts greater effort when pivotal than when providing support. Third, seller-pivotal collaboration yields strictly higher seller revenue than winner-pivotal collaboration for any type distribution. Finally, these optimal mechanisms can be implemented through ascending auctions with endogenously determined linear contracts.