Dynamic Rental Games with Stagewise Individual Rationality

TL;DR

The paper studies dynamic rental games with stagewise-IR, where a designer rents an indivisible asset over $n$ days to sequential agents whose daily valuations follow known distributions. It shows that the problem can be reduced to a predetermined sequence of stagewise auctions with seller costs (SWACs), but stagewise-IR breaks classical Myerson connections: truthful SWACs need not be allocation- or reward-monotone, and payments need not follow Myerson’s rule. The authors provide structural results across four reward classes (welfare-like, revenue-like, positive tradeoffs, and negative tradeoffs such as consumer surplus), establishing when fixed-rate payments suffice and when horizon-specific threshold payments are required, along with algorithms to compute optimal rental mechanisms (DP-based fixed-rate mechanism and threshold mechanism). They also demonstrate a potential gap between variable pricing and fixed-rate pricing and provide a generalized virtual-value framework to guide optimal designs, with directions for extending the model to non-identical distributions, learning settings, and broader market structures. The work advances single-parameter dynamic mechanism design by clarifying how stagewise-IR interacts with allocation and payment rules and by delivering practical mechanism designs for several 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.

Figures (4)

• Figure 1: Flow of proofs for welfare-like, revenue-like, positive tradeoff and negative tradeoff reward. For each, we give monotonicity results (for either allocation or reward), establish the form of the optimal mechanism (w.l.o.g.), and finally give an optimal rental mechanism. Although some intermediate results appear to be the same across different types, each result is proven separately and leverages characteristics specific to the reward.
• Figure 2: An illustration of the transformation from $\mathop{\mathrm{\mathsf{A}}}\nolimits$ to $\mathop{\mathrm{\mathsf{A}}}\nolimits'$. We show for the points in $\mathop{\mathrm{\mathcal{V}}}\nolimits$, the reward the designer yields from them in $\mathop{\mathrm{\mathsf{A}}}\nolimits$ in Subfigure \ref{['subfig:1-remRtLft']}, and their allocation in $\mathop{\mathrm{\mathsf{A}}}\nolimits$, in Subfigure \ref{['subfig:2-remRtLft']}. On these figures we point out which valuations will be omitted from the bidding set of $\mathop{\mathrm{\mathsf{A}}}\nolimits'$.
• Figure 3: The reward-monotonicity violation, in the form of the set $\mathop{\mathrm{\mathcal{U}}}\nolimits$ and the sequence $v_0, v_1, \ldots \rightarrow v^*$ whose rewards approach $B$. Also depicted are the sets $\mathop{\mathrm{\textsc{RemRt}}}\nolimits$ (valuations above $v^*$ with reward below $B$) and $\mathop{\mathrm{\textsc{RemLft}}}\nolimits$ (valuations below $v^*$ with allocations above $x^*$; note that allocations are not shown in this figure).
• Figure 4: A comparison of $\mathop{\mathrm{\mathsf{M}}}\nolimits^1$ and $\mathop{\mathrm{\mathsf{M}}}\nolimits^2$

Theorems & Definitions (80)

• Example 1.1: Truthful but non-monotone SWAC
• Definition 1
• Theorem 3.1
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Definition 2