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.
Hart and Nisan @cite_5 use a model similar to ours (risk-neutral bidders and lottery offers) in their study of the optimal revenue when selling multiple items.
@cite_5
2088970293 1495768673 2275043201 2169604085
Abstract Consider the revenue-maximizing problem in which a single seller wants to sell k different items to a single buyer, who has independently distributed values for the items with additive valuation. The case was completely resolved by Myerson’s classical work in 1981, whereas for larger k the problem has been the subject of much research efforts ever since. Recently, Hart and Nisan analyzed two simple mechanisms: selling the items separately, or selling them as a single bundle. They showed that selling separately guarantees at least a fraction of the optimal revenue; and for identically distributed items, bundling yields at least a fraction of the optimal revenue. In this paper, we prove that selling separately guarantees at least fraction of the optimal revenue, whereas for identically distributed items, bundling yields at least a constant fraction of the optimal revenue. These bounds are tight (up to a constant factor), settling the open questions raised by Hart and Nisan. The results are valid for arbitrary probability distributions without restrictions. Our results also have implications on other interesting issues, such as monotonicity and randomization of selling mechanisms. In this paper, we introduce a novel approach for reducing the k-item n-bidder auction with additive valuation to k-item 1-bidder auctions. This approach, called the Best-Guess reduction, can be applied to address several central questions in optimal revenue auction theory such as the relative strength of simple versus complex mechanisms, the power of randomization, and Bayesian versus dominant-strategy implementations. First, when the items have independent valuation distributions, we present a deterministic mechanism called Deterministic Best-Guess that yields at least a constant fraction of the optimal revenue by any randomized mechanism. This also gives the first simple mechanism that achieves constant fraction optimal revenue for such multi-buyer multi-item auctions. Second, if all the nk valuation random variables are independent, the optimal revenue achievable in dominant strategy incentive compatibility (DSIC) is shown to be at least a constant fraction of that achievable in Bayesian incentive compatibility (BIC). Third, when all the nk values are identically distributed according to a common one-dimensional distribution F, the optimal revenue is shown to be expressible in the closed form Θ(k(r + ∫0mr(1 − F(x)n)dx)) where r = supx≥0 x(1 − F(x)n) and m = lk nr; this revenue is achievable by a simple mechanism called 2nd-Price Bundling. All our results apply to arbitrary distributions, regular or irregular. Optimal mechanisms have been provided in quite general multi-item settings [ 2012b, as long as each bidder's type distribution is given explicitly by listing every type in the support along with its associated probability. In the implicit setting, e.g. when the bidders have additive valuations with independent and or continuous values for the items, these results do not apply, and it was recently shown that exact revenue optimization is intractable, even when there is only one bidder [ 2013]. Even for item distributions with special structure, optimal mechanisms have been surprisingly rare [Manelli and Vincent 2006] and the problem is challenging even in the two-item case [Hart and Nisan 2012]. In this paper, we provide a framework for designing optimal mechanisms using optimal transport theory and duality theory. We instantiate our framework to obtain conditions under which only pricing the grand bundle is optimal in multi-item settings (complementing the work of [Manelli and Vincent 2006]), as well as to characterize optimal two-item mechanisms. We use our results to derive closed-form descriptions of the optimal mechanism in several two-item settings, exhibiting also a setting where a continuum of lotteries is necessary for revenue optimization but a closed-form representation of the mechanism can still be found efficiently using our framework. 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.