Cayley-Moser Optimal Stopping

• The Cayley-Moser optimal stopping problem is a sequential decision-making model that uses irreversible accept/reject rules based on dynamic thresholds to maximize expected rewards.
• In the discrete model, a recursive threshold (A_m) is derived via dynamic programming and backward induction to efficiently decide when to stop.
• The continuous-time formulation with Poisson arrivals uses an ODE for threshold evaluation, yielding closed-form sale-price and stopping-time distributions under various offer distributions.

The Cayley-Moser Optimal Stopping Problem is a canonical sequential decision-making framework in which a decision-maker observes a finite or infinite sequence of random offers and faces the constraint of irreversible acceptance or rejection at each stage. The goal is to maximize the expected reward by selecting an optimal stopping time. This paradigm, in both discrete- and continuous-time versions, provides a model for settings such as hiring, asset sales, and online search, and stands out due to the full-information regime, with known value distributions, in contrast to the classical “best-choice” or Secretary problem.

1. Classical Cayley-Moser Problem: Model and Solution Structure

In its classical discrete-time formulation, the decision-maker is presented with a fixed number $N$ of candidates or offer values $\{X_i\}_{i=1}^N$, independently sampled from the $\mathrm{Uniform}[0,1]$ distribution. The decision-maker sequentially observes each $X_i$, and must decide to accept (stop) or reject irreversibly. The selected payoff is $X_\tau$, with $\tau$ the stopping time. The policy must be adapted, with no recall of rejected offers.

The objective is:

$$\max_{\tau\in\{1,\dots,N\}} \mathbb{E}\left[X_\tau\right].$$

The optimal policy is derived via dynamic programming and backward induction. Letting $V_m$ denote the maximal expected reward with $m$ applicants remaining:

$$V_0 = 0, \quad V_m = \mathbb{E}\left[\max\{X, V_{m-1}\}\right] = \int_0^1 \max\{x, V_{m-1}\} dx = \frac{V_{m-1}^2 + 1}{2}.$$

The “indifference” threshold or aspiration level $A_m$ at stage $m$ is set by $A_m = V_{m-1}$: accept $x$ if $x > A_m$, otherwise continue. The recursion, known as the Cayley–Moser recursion, is:

$A_{m+1} = \frac{A_m^2 + 1}{2}, \quad A_0 = 0.$

At step $k$ (with $N-k$ to go), the cutoff is $r_k = A_{N-k}$. This produces a deterministic threshold sequence and myopic “accept-if-above-cutoff” policy (Demers, 2018).

2. Statistical Properties and Duration Analysis

The distribution of the stopping time $T$ (the index of the chosen offer) is explicitly evaluated as:

$$\Pr(T = m) = (1 - A_{N-m}) \prod_{i=1}^{m-1} A_{N-i}, \quad 1 \leq m \leq N.$$

This formula enables computation of moments and other statistics of the search duration, capturing the process duration as a function of the threshold sequence. The expected stopping time is:

$$\mathbb{E}[T] = \sum_{m=1}^N m \cdot \Pr(T = m) = \sum_{m=1}^N m(1 - A_{N-m}) \prod_{i=1}^{m-1} A_{N-i}.$$

Exact closed forms for general $N$ are unavailable, but the recursion allows efficient $O(N)$ evaluation (Demers, 2018).

For large $N$, Gilbert–Mosteller’s asymptotic approximation applies:

$A_m \approx 1 - \frac{2}{m + \ln m + 1.76799},$

$$\Pr(T = m) \approx \frac{2(N + 1 - m)}{N(N + 1)}, \quad m = 1,\dots,N,$$

$$\mathbb{E}[T] \approx \frac{N + 2}{3} \sim \frac{N}{3}, \qquad \widetilde T / N \approx 1-\frac{1}{\sqrt{2}} \approx 0.2929.$$

The left-skewed, triangular distribution of $T$ contrasts markedly with the heavy right-tail profile found in the Secretary problem. Numerical values for small $N$ illustrate the trend (table below):

N	$\mathbb{E}[T]$	$\mathbb{E}[T]/N$	$\widetilde T$	$\widetilde T/N$
10	3.81	0.381	3	0.300
50	19.9	0.398	15	0.300
100	33.8	0.338	29	0.290

Key observations:

• Expected stopping time scales as $N/3$.
• Median is around $0.29\,N$, below the Secretary-problem threshold of $N/e \approx 0.37N$.
• Candidates are typically selected much earlier than the final period.

3. Comparison to the Classical Secretary and Sultan’s Dowry Problems

The Cayley–Moser problem differs crucially from the classic Secretary (best-choice) problem. In the latter, only ordinal information is available; the decision-maker observes relative ranks, not values. The optimal threshold is $c^* \sim N/e$, with expected interviews about $2N/e \approx 0.736 N$. In the Cayley–Moser regime, where full information $X_i \sim \mathrm{Uniform}[0,1]$ is exploited, the expected search time is dramatically shorter, as aspiration levels are continually updated to the evolving maximum-a-posteriori estimate of the attainable future reward.

Moreover, the distributional form of the stopping time $T$ is approximately triangular for Cayley–Moser, versus a heavy-tailed (geometric-like) distribution for the Secretary problem (Demers, 2018). This demonstrates the effect of full information: not only is the stopping policy more efficient on average, but also the process duration is more concentrated and predictable.

4. Continuous-Time Cayley-Moser with Poissonian Arrivals

A Poissonian arrival model generalizes the Cayley–Moser problem to continuous time, providing analytic tractability and novel insight. Here, offers arrive according to a Poisson process of rate $\lambda$ over a known horizon $[0,T]$. Each offer $X_i$ is i.i.d. from a known $F(x)$, and upon expiration the seller may settle for a “salvage” value $X_0$ with mean $\mu_0$ (Katriel, 4 Nov 2025).

The value function $V(t,x)$, representing the maximum expected sale price at time $t$ before deadline and observing offer $x$, satisfies:

$$V(t, x) = \max\{x, \mu(t)\}, \quad V(0, x) = \mu_0,$$

with $\mu(t) = \mathbb{E}[V((t-D)_+, X)]$, where $D \sim \mathrm{Exp}(\lambda)$ is the next arrival time. The optimal policy is accept $x$ if $x \geq \mu(t)$.

This induces the Volterra integral equation:

$$\mu(t) = e^{-\lambda t} \mu_0 + e^{-\lambda t} \int_0^t \lambda e^{\lambda s} \mathbb{E}[V(s, X)] ds,$$

which can be differentiated to the ODE:

$$\mu'(t) = \lambda \varphi(\mu(t)), \quad \mu(0) = \mu_0,$$

where

$\varphi(x) = \int_x^\infty [1 - F(u)] du.$

This ODE can be reduced by quadrature:

$$\int_{\mu_0}^{\mu(t)} \frac{du}{\varphi(u)} = \lambda t,$$

implying $\mu(t) = \Psi^{-1}(\lambda t)$ for

$\Psi(x) = \int_{\mu_0}^{x}\frac{du}{\varphi(u)}.$

Explicit solutions are attainable for specific distributions:

• Uniform $[a,b]$: $\varphi(x) = \frac{(b-x)^2}{2(b-a)}$, $$\mu(t) = b - \frac{2(b-a)}{\lambda t + 2(b-a)/(b-\mu_0)}$$.
• Exponential $(\eta)$: $\varphi(x) = \eta e^{-x/\eta}$, $\mu(t) = \eta \log[\lambda t + e^{\mu_0/\eta}]$.
• Pareto $(x_m,\alpha)$: $$\varphi(x) = \frac{x_m^\alpha}{(\alpha-1)x^{\alpha-1}}$$, $$\mu(t) = [\mu_0^\alpha + (\alpha\lambda/(\alpha-1))x_m^\alpha t]^{1/\alpha}$$.

A key distinction from the discrete case is the analytic tractability of the threshold function $\mu(t)$, as the continuous-time model leads to a solvable ODE, in contrast to the discrete nonlinear threshold recursion.

5. Distributional Forms in the Continuous-Time Problem

The explicit continuous-time formulation enables closed-form calculation of both the sale-price and stopping-time distributions:

• The sale-price distribution $G_t(x) = \Pr(S_t \leq x)$ satisfies the ODE:

$$\partial_t G_t(x) = \lambda [ (F(\mu(t))-1) G_t(x) + [F(x) - F(\mu(t))]_+ ], \quad G_0(x) = F_0(x),$$

with solution

$$G_t(x) = \varphi(\mu(t)) \left[ \frac{F_0(x)}{\varphi(\mu_0)} + \int_{\mu_0}^{\mu(t)} \frac{[F(x) - F(w)]_+}{\varphi(w)^2}dw \right].$$

• The stopping-time CDF $H_t(r) = \Pr(T_t \leq r)$ is

$$H_t(r) = 1 - \frac{\varphi(\mu(t))}{\varphi(\mu(t - r))}, \quad 0 \le r < t, \qquad H_t(r) = 1\ \text{if }r \ge t.$$

The conditional density $h_t(r)$ is given by:

$$h_t(r) = \lambda [1 - F(\mu(t - r))] \frac{\varphi(\mu(t))}{\varphi(\mu(t - r))}.$$

For exponential offers and $F_0 = F$, $H_t(r) = \lambda r/(\lambda t + e^{\mu_0/\eta})$ on $[0, t)$, indicating a uniform distribution of stopping times up to the horizon in this setting.

6. Interpretations, Economic Insights, and Regime Comparisons

In both discrete- and continuous-time models, the threshold function ($A_m$ or $\mu(t)$) is both the reservation price and the conditional expected reward under optimal stopping. For fixed horizon $t$, the threshold is monotonically increasing in available time: with greater time to sell, more selectivity is possible.

In the continuous model, increasing Poisson arrival rate $\lambda$ effectively increases the “number of opportunities” $\lambda t$, flattening the reservation curve and lowering selectivity for fixed remaining time. This continuous-time model allows for explicit analysis in situations with non-uniform offer distributions, salvage options, and time-dependent opportunity structure (Katriel, 4 Nov 2025).

As $\lambda t \to n$, the continuous-time threshold matches the asymptotic of the discrete regime, corroborating the limit behavior across formulations. In both settings, explicit forms for the full distribution of search duration and realized reward are available in the Poissonian case, contrasting with the exclusively asymptotic results obtainable in the original finite discrete process (Katriel, 4 Nov 2025).

A salient insight: compared to the Secretary problem, the Cayley–Moser regime exploits value information for more efficient and earlier selection, both in expectation and in the concentration of process duration. The informativeness of the offers—rather than solely their rank—permits substantially shorter search and higher efficiency in optimal stopping.