There has been much recent work on the revenue-raising properties of truthful mechanisms for selling goods to selfish bidders. Typically the revenue of a mechanism is compared against a benchmark (such as, the maximum revenue obtainable by an omniscient seller selling at a fixed price to at least two customers), with a view to understanding how much lower the mechanism's revenue is than the benchmark, in the worst case. We study this issue in the context of lotteries , where the seller may sell a probability of winning an item. We are interested in two general issues. Firstly, we aim at using the true optimum revenue as benchmark for our auctions. Secondly, we study the extent to which the expressive power resulting from lotteries, helps to improve the worst-case ratio. We study this in the well-known context of digital goods , where the production cost is zero. We show that in this scenario, collusion-resistant lotteries (these are lotteries for which no coalition of bidders exchanging side payments has an advantage in lying) are as powerful as truthful ones.
Myerson's classic result provides a full description of how a seller can maximize revenue when selling a single item. We address the question of revenue maximization in the simplest possible multi-item setting: two items and a single buyer who has independently distributed values for the items, and an additive valuation. In general, the revenue achievable from selling two independent items may be strictly higher than the sum of the revenues obtainable by selling each of them separately. In fact, the structure of optimal (i.e., revenue-maximizing) mechanisms for two items even in this simple setting is not understood. In this paper we obtain approximate revenue optimization results using two simple auctions: that of selling the items separately, and that of selling them as a single bundle. Our main results (which are of a "direct sum" variety, and apply to any distributions) are as follows. Selling the items separately guarantees at least half the revenue of the optimal auction; for identically distributed items, this becomes at least 73 of the optimal revenue. For the case of k > 2 items, we show that selling separately guarantees at least a c log2(k) fraction of the optimal revenue; for identically distributed items, the bundling auction yields at least a c log(k) fraction of the optimal revenue.