Dynamic Rental Games with Stagewise Individual Rationality

TL;DR

This paper investigates dynamic rental mechanisms with stagewise individual rationality, revealing that classical auction principles do not directly apply and developing new characterizations for optimal mechanisms under various economic objectives.

Contribution

It introduces a novel model of rental games with stagewise-IR agents, showing how optimal mechanisms differ from classical auctions and providing new characterizations for their design.

Findings

Optimal rental mechanisms can be modeled as dynamic auctions with seller costs.

Stagewise-IR behavior breaks classical auction properties like monotonicity and Myerson's payment rule.

New characterizations enable the design of optimal mechanisms for different economic objectives.

Abstract

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.