The Cayley-Moser problem with Poissonian arrival of offers

TL;DR

This paper analyzes a continuous-time Cayley-Moser stopping problem with offers arriving as a Poisson process at rate $\lambda$ until a deadline $t$. It derives an explicit threshold policy $\mu(t)$ via an ODE using $\phi(x)=\int_x^\infty [1-F(u)]du$ and yields a closed-form solution $\mu(t)=\Psi^{-1}(\lambda t)$, enabling exact expressions for the sale price $S_t$ and stopping time $T_t$ and their distributions. The work provides deep asymptotic insights across tail classes (bounded, exponential, and power-law) and illustrates results through Uniform, Exponential, and Pareto offer distributions, including precise rates for $\mathbb{E}[S_t]$, $\operatorname{Var}[S_t]$, and the limiting behavior of $\hat{T}_t= T_t/t$. The findings highlight the analytical tractability of the Poissonian model, offer exact distributions absent in discrete-time formulations, and yield practical guidance on pricing dynamics under time pressure.

Abstract

We study a version of the classical Cayley-Moser optimal stopping problem, in which a seller must sell an asset by a given deadline, with the offers, which are independent random variables with a known distribution, arriving at random times, as a Poisson process. This continuous-time formulation of the problem is much more analytically tractable than the analogous discrete-time problem which is usually considered, leading to a simple differential equation that can be explicitly solved to find the optimal policy. We study the performance of this optimal policy, and obtain explicit expressions for the distribution of the realized sale price, as well as for the distribution of the stopping time. The general results are used to explore characteristics of the optimal policy and of the resulting bidding process, and are illustrated by application to several specific instances of the offer distribution.

Figures (7)

• Figure 1: Matlab code for simulating a bidding process $N$ times, using the optimal policy, when the offer distribution and the residual distribution are uniform on $[a,b]$. The function outputs vectors $S,T$ of length $N$ (number of simulations), with $S$ containing the realized sale prices and $T$ containing the time durations until the sale was made.
• Figure 2: Histograms of sale prices attained in $N=10^5$ simulations of the bidding process, employing the optimal policy, where $\lambda=1$, and the offer distribution and residual distributions are uniform on $[1,3]$. The red line shows the analytical expression for the probability density, given by \ref{['uni_den']}. Left: t=2, Right: t=10.
• Figure 3: Histograms of sale prices attained in $N=10^5$ simulations of the bidding process, employing the optimal policy, where $\lambda=1$ and the offer and residual distributions are exponential with mean $\eta=2$. The red line shows the analytical expression for the probability density, given by \ref{['exp_den']}. Left: $t=2$, Right: $t=10$.
• Figure 4: Histograms of sale prices attained in $N=10^5$ simulations of the bidding process, employing the optimal policy, where $\lambda=1$ and the offer and residual distributions are Pareto with $x_m=1,\alpha=3$. The red line shows the analytical expression for the probability density, given by \ref{['pareto_den']}. Left: $t=2$, Right: $t=10$.
• Figure 5: Cumulative histogram of time to sale in $N=10^5$ simulations of the bidding process, employing the optimal policy, where $\lambda=1,t=10$, and the offer and residual distributions are uniform on $[1,3]$. The red line shows the analytical expression for the cumulative density, given by \ref{['H_uniform']}.

Theorems & Definitions (30)

• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Lemma 3.1
• Theorem 3.1
• proof