Historically, rational choice theory has focused on the utility maximization principle to describe how individuals make choices. In reality, there is a computational cost related to exploring the universe of available choices and it is often not clear whether we are truly maximizing an underlying utility function. In particular, memory effects and habit formation may dominate over utility maximization. We propose a stylized model with a history-dependent utility function, where the utility associated to each choice is increased when that choice has been made in the past, with a certain decaying memory kernel. We show that self-reinforcing effects can cause the agent to get stuck with a choice by sheer force of habit. We discuss the special nature of the transition between free exploration of the space of choice and self-trapping. We find, in particular, that the trapping time distribution is precisely a Zipf law at the transition, and that the self-trapped phase exhibits super-aging behavior.
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In the general case in which the agent started in a “site” different from and then got stuck in , one wants to replace the right-hand side by an average of the logit rule over the utility gaps . However, as , it is very unlikely that a site that was previously picked is chosen again. We can thus safely replace the average gap by the gap with the base level .
49.
Indeed, a direct integration of the kernel yields . As , this function can be approximated by up to a cutoff at . Direct integration then gives for and for .
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