How to compute time-lagged correlation between two variables with many examples at each time t? I have a dictionary of following form:
datetimes = {year : {name : (score1, score2)}} #there are 50+ names/year

So, essentially, I'm trying to get an aggregate picture of how score1 at year_n is correlated with score2 at year_n.
An application of a specific correlation formula depends on the data-type (continuous or rank data etc). Given that your data is continuous, you can apply  Karl Pearson formula. However, if you're interested  interested in cause and effect relationship, you may prefer to use simple regression  model.