Bayesian Learning with Random Arrivals

We add to a line of work considering the impact of observation imperfections in models of Bayesian observational learning. In particular, we study a discrete-time model in which in each time-slot, an agent may randomly arrive. Agents who arrive have the opportunity to buy a given item. If an agent chooses to buy, this action is recorded for subsequent agents. However, the decisions of agents that choose not to buy are not recorded. Hence, if no one buys in a given slot, agents are unaware if this was due to no agent arriving or an agent choosing not to buy. We study the impact of this uncertainty on the emergence of information cascades. Using a Markov chain based analysis, we show that the probability of incorrect cascades and the expected time until a cascade happens are not monotonic in the arrival probability of a user. We find that adding a small uncertainty in the arrival information from the perfect information setting will make a buy cascade happen with higher probability than a not-buy cascade. However, if the agents' private signals are weak, then a not-buy cascade is more likely to occur for most arrival rates, resulting in wrong cascades dominating when the item is good and vice-versa when the item is bad.