Runs of gains and losses are particularly salient to decision makers because of their perceived departure from randomness, as well as their immediate impact on the financial status of the decision makers. Past research has focused on decision making biases that relate to faulty conceptions of chance and luck, such as the gambler’s fallacy and the hot hand effect. Participants in the current study bet on the outcomes of a long sequence of simulated coin tosses. Risk preferences were found to change as a function of run valence (i.e., losses vs. gains), run length, and financial status. Individuals were found to differ in the effect of all of these factors, in their responses to runs of gains and losses in sequential risky choice.

We consider a game-theoretical solution for an optimal stopping problem which we describe in terms of a job-search problem with an infinite population of candidates and an infinite population of posts of varying value. Each candidate finds posts from the post population at unit rate. If a post found is still vacant, a candidate can either accept it or reject it. The reward to a candidate is the value of the post if one is accepted and zero if he never accepts a post. There are no costs for searching, no discounting of future rewards and no recall of previously found posts. The only pressure on a candidate to accept a post comes from the changing rate at which he finds vacant posts (and the values associated with them) as a result of the actions of the other candidates. It is not possible to define optimality for a single candidate without reference to the policies followed by the other candidates. We say a policy π is an evolutionarily stable strategy (ESS) if it has the following property: if all candidates used π and an individual candidate was given the option of changing his policy, then it would not be to his advantage to do so. We first find the optimal value function and optimal policy for the case of a single candidate operating in an environment where the distribution of posts on offer and the chance of finding one both vary with time in a known way. We then show that for the infinite population there is a unique ESS given by a control-limit policy π c, where the control-limit function c is the solution of a given differential equation with a given initial condition. This function c also gives the expected future reward function for any single candidate when all candidates use π c.