The expected revenue maximizing auction and the efficient auction with complements

We analyze a problem of selling complementary items under asymmetric information between the seller and heterogeneous buyers. The seller tries to sell m complementary indivisible items to n potential buyers. Buyers are heterogeneous in two senses. First, each buyer values each item and each bundle of items differently and these are the private information of this buyer. In addition, different buyers are interested in different sets of items. We assume both buyers and the seller know ex-ante the probability distribution functions from which buyers’ valuations are drawn, and who are interested in which set of items. In other words, these two pieces of information are common knowledge among the seller and the buyers. When items are homogeneous, we call a buyer "a size k buyer" if his/her marginal utility is strictly positive up to kth units. We characterize the expected revenue maximizing auctions and efficient auctions. Unfortunately, no simple auction mechanism can be efficient and expected revenue maximizing at the same time. Therefore, efficiency and expected revenue maximization are two incompatible goals. Furthermore, the expected revenue maximizing auction systematically gives an advantage to a particular size of bidders. The underlying probability distribution functions decide which size of bidders is given an advantage over other size bidders. There are several important practical examples which our model can explain. The Federal Communications Commission (FCC) auctions of spectrum licenses for electromagnetic spectrum where local (regional) bidders and global (nationwide) bidders coexisted are explainable by our model. Further, the Japanese government considers auctioning arrival and departure slots at Tokyo International Airport where domestic and international arrival and departure slots are complements for All Nippon Airways and Japan Airways, but most low cost carriers such as Skymark, Solaseed Air, Air Do, and several others value only domestic arrival and departure slots. Our model is applicable to this problem.