A n important economic problem is that of finding optimal pricing mechanisms to sell a single item when there are a random number of buyers who arrive over time. In this paper, we combine ideas from auction theory and recent work on pricing with strategic consumers to derive the optimal continuous time pricing scheme in this situation. Under the assumption that buyers are split among those who have a high valuation and those who have a low valuation for the item, we obtain the price path that maximizes the seller's revenue. We conclude that, depending on the specific instance, it is optimal to either use a fixed price strategy or to use steep markdowns by the end of the selling season. As a complement to this optimality result, we prove that under a large family of price functions there is an equilibrium for the buyers. Finally, we derive an approach to tackle the case in which buyers' valuations follow a general distribution. The approach is based on optimal control theory and is well suited for numerical computations.
Auto-bidding is now widely adopted as an interface between advertisers and internet advertising as it allows advertisers to specify high-level goals, such as maximizing value subject to a value-per-spend constraint. Prior research has mostly focused on auctions which are truthful (such as SPA) since uniform bidding is optimal in such auctions, which makes it manageable to reason about equilibria. A tantalizing question is whether one can obtain more efficient outcomes by leaving the realm of truthful auctions. This is the first paper to study non-truthful auctions in the prior-free auto-bidding setting. Our first result is that non-truthfulness provides no benefit when one considers deterministic auctions. Any deterministic mechanism has a price of anarchy (PoA) of at least 2, even for 2 bidders; this matches what can be achieved by deterministic truthful mechanisms. In particular, we prove that the first price auction has PoA of exactly 2. For our second result, we construct a randomized non-truthful auction that achieves a PoA of 1.8 for 2 bidders. This is the best-known PoA for this problem. The previously best-known PoA for this problem was 1.9 and was achieved with a truthful mechanism. Moreover, we demonstrate the benefit of non-truthfulness in this setting by showing that the truthful version of this randomized auction also has a PoA of 1.9. Finally, we show that no auction (even randomized, non-truthful) can improve upon a PoA bound of 2 as the number of advertisers grow to infinity.
Over the past few years, more and more Internet advertisers have started using automated bidding for optimizing their advertising campaigns. Such advertisers have an optimization goal (e.g. to maximize conversions), and some constraints (e.g. a budget or an upper bound on average cost per conversion), and the automated bidding system optimizes their auction bids on their behalf. Often, these advertisers participate on multiple advertising channels and try to optimize across these channels. A central question that remains unexplored is how automated bidding affects optimal auction design in the multi-channel setting.In this paper, we study the problem of setting auction reserve prices in the multi-channel setting. In particular, we shed light on the revenue implications of whether each channel optimizes its reserve price locally, or whether the channels optimize them globally to maximize total revenue. Motivated by practice, we consider two models: one in which the channels have full freedom to set reserve prices, and another in which the channels have to respect floor prices set by the publisher. We show that in the first model, welfare and revenue loss from local optimization is bounded by a function of the advertisers' inputs, but is independent of the number of channels and bidders. In stark contrast, we show that the revenue from local optimization could be arbitrarily smaller than those from global optimization in the second model.
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