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.
This work is motivated by the results in @cite_12 . @cite_12 show how lotteries help in maximizing the revenue when designing envy-free prices. Here we address a similar type of question and aim at obtaining similar results for incentive-compatible lotteries.
@cite_12
1968517895
Randomized mechanisms, which map a set of bids to a probability distribution over outcomes rather than a single outcome, are an important but ill-understood area of computational mechanism design. We investigate the role of randomized outcomes (henceforth, "lotteries") in the context of a fundamental and archetypical multi-parameter mechanism design problem: selling heterogeneous items to unit-demand bidders. To what extent can a seller improve her revenue by pricing lotteries rather than items, and does this modification of the problem affect its computational tractability? Our results show that the answers to these questions hinge on whether consumers can purchase only one lottery (the buy-one model) or purchase any set of lotteries and receive an independent sample from each (the buy-many model). In the buy-one model, there is a polynomial-time algorithm to compute the revenue-maximizing envy-free prices (thus overcoming the inapproximability of the corresponding item pricing problem) and the revenue of the optimal lottery system can exceed the revenue of the optimal item pricing by an unbounded factor as long as the number of item types is at least 4. In the buy-many model with n item types, the profit achieved by lottery pricing can exceed item pricing by a factor of Θ(log n) but not more, and optimal lottery pricing cannot be approximated within a factor of O(ne) for some e > 0, unless NP ⊆ ∩Δ>0 BPTIME(2O(nΔ)). Our lower bounds rely on a mixture of geometric and algebraic techniques, whereas the upper bounds use a novel rounding scheme to transform a mechanism with randomized outcomes into one with deterministic outcomes while losing only a bounded amount of revenue.