# Threshold Phenomena and Influence: Perspectives from Mathematics, Computer Science, and Economics

# Threshold Phenomena and Influence: Perspectives from Mathematics, Computer Science, and Economics

Threshold phenomena refer to settings in which the probability for an event to occur changes rapidly as some underlying parameter varies. Threshold phenomena play an important role in probability theory and statistics, physics, and computer science, and are related to issues studied in economics and political science. Quite a few questions that come up naturally in those fields translate to proving that some event indeed exhibits a threshold phenomenon, and then finding the location of the transition and how rapid the change is. The notions of sharp thresholds and phase transitions originated in physics, and many of the mathematical ideas for their study came from mathematical physics. In this chapter, however, we will mainly discuss connections to other fields. A simple yet illuminating example that demonstrates the sharp threshold phenomenon is Condorcet's jury theorem, which can be described as follows. Say one is running an election process, where the results are determined by simple majority, between two candidates, Alice and Bob. If every voter votes for Alice with probability p > 1/2 and for Bob with probability 1 — p, and if the probabilities for each voter to vote either way are independent of the other votes, then as the number of voters tends to infinity the probability of Alice getting elected tends to 1. The probability of Alice getting elected is a monotone function of p, and when there are many voters it rapidly changes from being very close to 0 when p < 1/2 to being very close to 1 when p > 1/2. The reason usually given for the interest of Condorcet's jury theorem to economics and political science [535] is that it can be interpreted as saying that even if agents receive very poor (yet independent) signals, indicating which of two choices is correct, majority voting nevertheless results in the correct decision being taken with high probability, as long as there are enough agents, and the agents vote according to their signal. This is referred to in economics as asymptotically complete aggregation of information.

*Keywords:*
central limit theorem, dictatorship function, edge-isoperimetric inequality, first passage percolation, game theory, harmonic analysis, influence, junta function, majority function

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