smooth randomized response guarantees learning at log(n) rate that is much faster than the sqrt(log n) rate in the non-private case & achieves finite expected times to the first correct action and last incorrect action, both of which are infinite in the non-private case + same eps can minimize both

a simple randomized response that flips actions with constant probability will prevents any cascade and asymptotic learning, but we can use a smooth version that hides signal locations within a range; this way, actions can be flipped with a probability that decays smoothly outside the range

for Gaussian signals, agent n switches its action from 0 to 1 when its private signal exceeds a threshold t(l_n) as a function of the log-likelihood ratio of public belief. In the non-private case, l_n grows as sqrt(log n), and agents eventually (but very slowly) take the correction almost surely

threshold k for a cascade (the number where having k signals of one type more than the other, makes following agents ignore their private signal and initiate a cascade) is fixed over intervals of eps. At interval boundaries, decreasing eps increases k by one and correct cascade probability jumps up

for binary signals, eps-DP is achieved using a randomized response mechanism that flips actions with probability 1/(1+e^eps) until a cascade occurs. The probability of correct cascades does not increase compared to the non-private baseline, but it does not vary monotonically with eps either

can we improve sequential learning by limiting information leakage to reduce the stickiness of information traps? randomizing actions limits what can be learned about each individual’s signal from their actions a la differential privacy + has potential to improve the collective learning outcome

tempting

a one-standard-deviation increase, equal to 11.70 more deaths per 100,000 population in the social proximity of ego counties in the contiguous United States, is associated with 13 more deaths per 100, 000 population in ego counties.

Measuring network dynamics of opioid overdose deaths in the United States - Our results show a statistically significant effect size for deaths in social proximity on OODs in counties in the United States, controlling for spatial proximity, as well as demographic and clinical covariates.