Batya Berzack

Batya Berzack, Rotem Oshman, Inbal Talgam-Cohen

We study \emph{rental games} -- a single-parameter dynamic mechanism design problem, in which a designer rents out an indivisible asset over $n$ days. Each day, an agent arrives with a private valuation per day of rental, drawn from that day's (known) distribution. The designer can either rent out the asset to the current agent for any number of remaining days, charging them a (possibly different) payment per day, or turn the agent away. Agents who arrive when the asset is not available are turned away. A defining feature of our dynamic model is that agents are \emph{stagewise-IR} (individually rational), meaning they reject any rental agreement that results in temporary negative utility, even if their final utility is positive. We ask whether and under which economic objectives it is useful for the designer to exploit the stagewise-IR nature of the agents. We show that an optimal rental mechanism can be modeled as a sequence of dynamic auctions with seller costs. However, the stagewise-IR behavior of the agents makes these auctions quite different from classical single-parameter auctions: Myerson's Lemma does not apply, and indeed we show that truthful mechanisms are not necessarily monotone, and payments do not necessarily follow Myerson's unique payment rule. We develop alternative characterizations of optimal mechanisms under several classes of economic objectives, including generalizations of welfare, revenue and consumer surplus. These characterizations allow us to use Myerson's unique payment rule in several cases, and for the other cases we develop optimal mechanisms from scratch. Our work shows that rental games raise interesting questions even in the single-parameter regime.

Batya Berzack, Rotem Oshman, Inbal Talgam-Cohen

We study multi-unit multi-buyer auctions, where buyers are subject to constraints that affect their bidding strategy. These may take the form of \emph{bidding constraints} (e.g., no-overbidding, in cases where bids are partially verifiable), or of \emph{outcome constraints} (e.g., in the case of auctions that unfold over time, an inability to go into debt even temporarily). The constraints fundamentally redefine the design space: the revelation principle, the envelope theorem, and Myerson's lemma do not apply in this constrained setting. Consequently, the space of implementable mechanisms expands significantly, admitting auctions that are incentive compatible with respect to the constrained buyers' utility, but would not be incentive compatible in the classical sense. In this paper we focus on monotone constraints where it is not the buyers' total budget that is restricted, but rather the manner in which they can bid or spend their budget. Our results show a separation between \emph{revenue-aligned} objectives (e.g., revenue, welfare, or any linear combination of the two) and \emph{consumer-aligned} objectives (e.g., consumer surplus). For revenue-aligned objectives, we rely on measure-theoretic tools to establish a unified theory parallel to Myerson, showing that despite the expanded design space, Myerson-style auctions remain optimal. For consumer-aligned objectives, the picture is different: we show that the seller can leverage the buyers' strategic limitations to strictly outperform classically incentive compatible mechanisms. We design the optimal deterministic auction for a wide class of instances, focusing in particular on buyers who cannot tolerate temporary debt. Overall, our work provides theoretical underpinnings for this area and shows that a rigorous and systematic approach can reveal general insights regarding optimal auction design.