Sequential choice and self-reinforcing rankings

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Abstract: In many choice problems the probability that an individual chooses an option depends on its ranking as compared to other options.  The ranking itself is often determined by the option’s relative popularity.  In this contribution we explore the mathematical consequences of these two widespread properties of real-world choice systems.  We show that in the long-run such systems may converge to a stable ranking in which probabilities of choice and market shares stabilize.  However, there is unpredictability with regards to the ranking to which the system converges, potentially leading to suboptimal outcomes. For the case of two options, we provide an analytical solution for the probabilities of the possible rankings in the limit, while in the case of more than two options we provide a condition that indicates whether the system can converge to a specific ranking.  We then show that our results apply to a wide range of models studied in economics and beyond and can even help reinterpret classic models of sequential choice, such as the well-known herding model.