Forecasting Competitions with Correlated Events

Beginning with Witkowski et al. [2022] , recent work on forecasting competitions has addressed incentive problems with the common winner-take-all mechanism. Frongillo et al. [2021] propose a competition mechanism based on follow-the-regularized-leader (FTRL), an online learning framework. They show that their mechanism selects an ǫ-optimal forecaster with high probability using only O(log(n)/ǫ 2 ) events. These works, together with all prior work on this problem thus far, assume that events are independent. We initiate the study of forecasting competitions for correlated events. To quantify correlation, we introduce a notion of block correlation, which allows each event to be strongly correlated with up to b others. We show that under distributions with this correlation, the FTRL mechanism retains its ǫ-optimal guarantee using O(b 2 log(n)/ǫ 2 ) events. Our proof involves a novel concentration bound for correlated random variables which may be of broader interest. Perfectly Correlated Events Next, let us see why the problem is intractable when correlation is arbitrary. Suppose we have a set of m binary events that are perfectly correlated. Thus, their outcomes will either be all 0 or all 1. With just the single observation, we are reduced to the single event setting where there is not enough information to pick a good forecaster, regardless of the size of m.