We consider human performance on an optimal stopping problem where people are presented with a list numbers independently chosen from a bounded uniform distribution. People are told how many numbers are in the list, and how they were chosen. People are then shown the numbers one at a time, and are instructed to choose the maximum, subject to the constraint that they must choose a number at the time it is presented, and any choice below the maximum is incorrect.
Problems like this occupy a useful niche in the study of human decision making. First, the optimal stopping problem is suited to controlled laboratory study, unlike studies of expertise in 'knowledge-rich' real-world domains. Secondly, the optimal stopping problem has features of real-world problem solving not evident in 'knowledge-lean' problems like the "Towers of Hanoi" or "Cannibals and Missionaries". In particular, it does not have a simple deterministic solution and requires people to reason under uncertainty. Thirdly, the optimal stopping problem neatly complements combinatorial optimisation problems (like the Traveling Salesperson Problem) for which human performance has recently begun to be studied. The optimal stopping problem is inherently cognitive rather than perceptual, induces a reliance on memory rather than presenting all relevant information, and has a known optimal solution rather than being NP-complete. This last feature allows people's ability to achieve a good solution to be distinguished from their ability to follow the known optimal solution process.
We present empirical evidence that suggests people use threshold-based models to make decisions, choosing the first number that exceeds a fixed threshold for that position in the list. We then develop a generative account of this model family, and use Bayesian and Minimum Description Length methods -- including model averaging, model selection and 'entropification' methods -- to learn about the parameters of the generative process, and make inferences about the threshold decision models people use. Finally, we consider a framework for modelling the clear individual differences in the data, and discuss the possibility of modelling group decision-making on a modified version of the optimal stopping problem.