Bilateral k+1-price auctions with asymmetric shares and values

- We study k+1-price auctions with asymmetric shares and asymmetrically distributed values.
- Bayesian Nash equilibria in continuous and strictly increasing pure strategies exist.
- Our proof combines results from the first-price and the double auction literature.
- If k∈(0,1), there is a continuum of equilibria.
- If k∈{0,1}, the equilibrium is unique.

We study a sealed-bid auction between two bidders with asymmetric independent private values. The two bidders own asymmetric shares in a partnership. The higher bidder buys the lower bidderʼs shares at a per-unit price that is a convex combination of the two bids. The weight of the lower bid is denoted by k∈[0,1]. We partially characterize equilibrium strategies and show that they are closely related to equilibrium strategies of two well-studied mechanisms: the double auction between a buyer and a seller and the first-price auction between two buyers (or two sellers). Combining results from those two branches of the literature enables us to prove equilibrium existence. Moreover, we find that there is a continuum of equilibria if k∈(0,1) whereas the equilibrium is unique if k∈{0,1}. Our approach also suggests a procedure for numerical simulations.