Judgments and decisions under uncertainty are frequently linked to a prior sequential search for relevant information. In such cases, the subject has to decide when to stop the search for information. Evidence accumulation models from social and cognitive psychology assume an active and sequential information search until enough evidence has been accumulated to pass a decision threshold. In line with such theories, we conceptualize the evidence threshold as the “desired level of confidence” (DLC) of a person. This model is tested against a fixed stopping rule (one-reason decision making) and against the class of multi-attribute information integrating models. A series of experiments using an information board for horse race betting demonstrates an advantage of the proposed model by measuring the individual DLC of each subject and confirming its correctness in two separate stages. In addition to a better understanding of the stopping rule (within the narrow framework of simple heuristics), the results indicate that individual aspiration levels might be a relevant factor when modelling decision making by task analysis of statistical environments.

Let (S n ) n≥0 be a correlated random walk on the integers, let M 0 ≥ S 0 be an arbitrary integer, and let M n = max{M 0, S 1,…, S n }. An optimal stopping rule is derived for the sequence M n - nc, where c > 0 is a fixed cost. The optimal rule is shown to be of threshold type: stop at the first time that M n - S n ≥ Δ, where Δ is a certain nonnegative integer. An explicit expression for this optimal threshold is given.

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

n candidates, represented by n i.i.d. continuous random variables X1, …, Xn with known distribution arrive sequentially, and one of them must be chosen, using a non-anticipating stopping rule. The objective is to minimize the expected rank (among the ranks of X1, …, Xn) of the candidate chosen, where the best candidate, i.e. the one with smallest X-value, has rank one, etc. Let the value of the optimal rule be Vn, and lim Vn = V. We prove that V > 1.85. Limiting consideration to the class of threshold rules of the form tn = min {k: Xk ≦ ak for some constants ak, let Wn be the value of the expected rank for the optimal threshold rule, and lim Wn = W. We show 2.295 < W < 2.327.

This paper discusses general bounds for coverage probabilities and moments of stopping rules for sequential coverage problems in geometrical probability. An approach to the study of the asymptotic behaviour of these moments is also presented.

In a series of papers Whittle has proposed locating the optimal boundary between the continuation region and the stopping region, in the no-overshoot approximation, for a class of sequential analysis problems by using the criterion that the gradient of the objective function must be continuous at the boundary. It is shown here that the result originally proposed by Whittle is not true in the generality which he supposed, and an indication is given of some of the circumstances in which it does hold.