Risk When Some States Are Low-Probability Events

One usually assumes that the joint probability distribution is known or that agents will use Bayesian updating to estimate the true probabilities after a number of trials when the states of nature are finite in classical decision theory under uncertainty. If there are important states that have very low probabilities of occurrence, then each agent must make a subjective assessment of the probability distribution until a sufficient number of outcomes are observed in order to generate a precise estimate of the probability distribution. If one assumes that all agents know the states and their payoffs, the probability distribution is stationary, and they observe all outcomes that unfold over time, then it will take at least 10 times the mean time between occurrences of the lowest probability event in order to generate enough outcomes that all agents share the same objective knowledge of the distribution. The mean time of recurrence depends on both the probability distribution and the time unit used between recordings of the observations. If at least one state has a low probability of occurrence, then the time to convergence will exceed the period of stationarity of the process.