Near-Optimal Multi-Unit Auctions with Ordered Bidders
  We construct prior-free auctions with constant-factor approximation guarantees with ordered bidders, in both unlimited and limited supply settings. We compare the expected revenue of our auctions on a bid vector to the monotone price benchmark, the maximum revenue that can be obtained from a bid vector using supply-respecting prices that are nonincreasing in the bidder ordering and bounded above by the second-highest bid. As a consequence, our auctions are simultaneously near-optimal in a wide range of Bayesian multi-unit environments. 
Elias Koutsoupias, Stefano Leonardi, and Tim Roughgarden
cs.GT