Optimal complementary auctions

This paper considers the situation where two products are sold by the same seller, but to disjoint sets of potential buyers. Externalities may arise from each market outcome to the other. The paper examines the nature of the seller's optimal mechanism, and, for example in the case of positive externalities, it is shown that the allocation decision in either market depends on the highest types in both markets. The optimal mechanism can be implemented by an indirect mechanism that essentially charges winning bidders for the value of their externalities. The analysis is applied to the sale of public sector franchises including exploration and development rights for oil and gas tracts

Draft Auctions

We introduce draft auctions, which is a sequential auction format where at each iteration players bid for the right to buy items at a fixed price. We show that draft auctions offer an exponential improvement in social welfare at equilibrium over sequential item auctions where predetermined items are auctioned at each time step. Specifically, we show that for any subadditive valuation the social welfare at equilibrium is an $O(\log^2(m))$-approximation to the optimal social welfare, where $m$ is the number of items. We also provide tighter approximation results for several subclasses. Our welfare guarantees hold for Bayes-Nash equilibria and for no-regret learning outcomes, via the smooth-mechanism framework. Of independent interest, our techniques show that in a combinatorial auction setting, efficiency guarantees of a mechanism via smoothness for a very restricted class of cardinality valuations, extend with a small degradation, to subadditive valuations, the largest complement-free class of valuations. Variants of draft auctions have been used in practice and have been experimentally shown to outperform other auctions. Our results provide a theoretical justification

Aversion to price risk and the afternoon effect

Many empirical studies of auctions show that prices of identical goods sold sequentially follow a declining path. Declining prices have been viewed as an anomaly, because the theoretical models of auctions predict that the price sequence should either be a martingale (with independent signals and no informational externalities), or a submartingale (with affiliated signals). This paper shows that declining prices, the afternoon effect, arise naturally when bidders are averse to price risk. A bidder is averse to price risk if he prefers to win an object at a certain price, rather than at a random price with the same expected value. When bidders have independent signals and there are no informational externalities, only the effect of aversion to price risk is present and the price sequence is a supermartingale. When there are informational externalities, even with independent signals, there is a countervailing, informational effect, which pushes prices to raise along the path of a sequential auction. This may help explaining the more complex price paths we observe in some auctions

Sequential item pricing for unlimited supply

We investigate the extent to which price updates can increase the revenue of a seller with little prior information on demand. We study prior-free revenue maximization for a seller with unlimited supply of n item types facing m myopic buyers present for k < log n days. For the static (k = 1) case, Balcan et al. [2] show that one random item price (the same on each item) yields revenue within a \Theta(log m + log n) factor of optimum and this factor is tight. We define the hereditary maximizers property of buyer valuations (satisfied by any multi-unit or gross substitutes valuation) that is sufficient for a significant improvement of the approximation factor in the dynamic (k > 1) setting. Our main result is a non-increasing, randomized, schedule of k equal item prices with expected revenue within a O((log m + log n) / k) factor of optimum for private valuations with hereditary maximizers. This factor is almost tight: we show that any pricing scheme over k days has a revenue approximation factor of at least (log m + log n) / (3k). We obtain analogous matching lower and upper bounds of \Theta((log n) / k) if all valuations have the same maximum. We expect our upper bound technique to be of broader interest; for example, it can significantly improve the result of Akhlaghpour et al. [1]. We also initiate the study of revenue maximization given allocative externalities (i.e. influences) between buyers with combinatorial valuations. We provide a rather general model of positive influence of others' ownership of items on a buyer's valuation. For affine, submodular externalities and valuations with hereditary maximizers we present an influence-and-exploit (Hartline et al. [13]) marketing strategy based on our algorithm for private valuations. This strategy preserves our approximation factor, despite an affine increase (due to externalities) in the optimum revenue.Comment: 18 pages, 1 figur

How to sell to buyers with crossholdings

This paper characterizes the optimal selling mechanism in the presence of horizontal crossholdings. We find that this mechanism imposes a discrimination policy against the stronger bidders so that the seller´s expected revenue is increasing in both the common crossholding and the degree of asymmetry in crossholdings. Furthermore, it can be implemented by a sequential procedure that includes a price-preferences scheme and the possibility of an exclusive deal with the weakest bidder. We also show that a simple sequential negotiation mechanism, although suboptimal, yields a larger seller´s expected revenue than both the first-price and the second-price auctions

Bidding among friends and enemies

We consider an auction setting in which bidders, even if they fail to obtain the good, care about the price paid by the winner. We study the impact of these price externalities on the first-price auction and the second-price auction in a symmetric information framework. We establish a distinction between price externalities that do not depend on the identity of the winner and price externalities that depend on the identity of the winner. We prove that the outcome of the first-price auction is not affected by the first type of price externalities while the outcome of the second-price auction is. In contrast, the second type of price externalities affects the outcome of both auction formats. In any case, in comparison with the first-price auction, the second-price auction exacerbates the effects of price externalities whatever their types are. The two auction formats are generically not equivalent

Optimal takeover contests with toeholds

This paper characterizes how a target firm should be sold when the possible buyers (bidders) have prior stakes in its ownership (toeholds). We find that the optimal mechanism needs to be implemented by a non-standard auction which imposes a bias against bidders with high toeholds. This discriminatory procedure is such that the target´s average sale price is increasing in both the size of the common toehold and the degree of asymmetry in these stakes. It is also shown that a simple mechanism of sequential negotiation replicates the main properties of the optimal procedure and yields a higher average selling price than the standard auctions commonly used in takeover battles.Optimal auctions, Takeovers, Toeholds, Asymmetric auctions