Quantifying the utility of imperfect reviews in stopping information cascades

In models of social learning where rational agents observe other agents' actions, information cascades are said to occur when agents ignore their private information and blindly follow the actions of other. It is well known that in some cases, incorrect cascades happen with positive probability leading to a loss in social welfare. Having agents provide reviews in addition to their actions provides one possible way to avoid such 'bad cascades.' In this paper, we study one such model where agents sequentially decide whether or not to purchase a good, whose true value is either 'good' or 'bad.' If they purchase, agents also leave a review, which is imperfect. We study the impact of such reviews on the asymptotic properties of cascades. For a good underlying state, we propose an algorithm that utilizes number theory principles and Markov chain analysis to solve for the probability of a wrong cascade. We discover that the probability of a wrong cascade is a non-monotonic function of the review strength. On the other hand, for a bad underlying state, the agents always eventually reach a correct cascade; we use a martingale analysis to bound the time until this happens.