Information Cascades on Arbitrary Topologies

In this paper, we study information cascades on graphs. In this setting, each node in the graph represents a person. One after another, each person has to take a decision based on a private signal as well as the decisions made by earlier neighboring nodes. Such information cascades commonly occur in practice and have been studied in complete graphs where everyone can overhear the decisions of every other player. It is known that information cascades can be fragile and based on very little information, and that they have a high likelihood of being wrong. Generalizing the problem to arbitrary graphs reveals interesting insights. In particular, we show that in a random graph $G(n,q)$, for the right value of $q$, the number of nodes making a wrong decision is logarithmic in $n$. That is, in the limit for large $n$, the fraction of players that make a wrong decision tends to zero. This is intriguing because it contrasts to the two natural corner cases: empty graph (everyone decides independently based on his private signal) and complete graph (all decisions are heard by all nodes). In both of these cases a constant fraction of nodes make a wrong decision in expectation. Thus, our result shows that while both too little and too much information sharing causes nodes to take wrong decisions, for exactly the right amount of information sharing, asymptotically everyone can be right. We further show that this result in random graphs is asymptotically optimal for any topology, even if nodes follow a globally optimal algorithmic strategy. Based on the analysis of random graphs, we explore how topology impacts global performance and construct an optimal deterministic topology among layer graphs.