Naive Generational Influence

"Good Warriors"? Who chooses these names?

Reading yet another article trying to assign generational responsibility for the current state of America I realized that I didn't have the best grasp of generation definitions or population distribution across the decades. That inspired me to whip up a quick spreadsheet to compute a simple measure of political influence across time, under the premise that ultimately responsibility lies with the electorate. Since generation definitions are debatable and birth rates can vary a lot I did it year by year and then turned it into a generational overview afterward. The goal isn't to directly make any responsibility assignment (an arguably pointless aim) but to examine what people's contribution over time would've been minus any idiosyncratic events (i.e. all other factors being equal) and therefore provide a better starting intuition for such factors.

Births in a given year: I neglected the issues of infant mortality, woman's suffrage, slavery, and voter suppression generally as this is intended as a rough analysis only and those can arguably be included in the historical idiosyncrasies I mentioned earlier. Here is the birth data by year, which required going back as far possible so that boundary issues are minimized.

Years of political activity: As an estimate of the years during which an individual can have political influence on society, I chose a starting age of 18 and an ending one of 75. All years are counted, not merely election years as political activity can (and should) extend beyond the mere act of casting a ballot. This chart shows the number of voters in each generation of interest that are in the age range of 18 to 75 over time, with preceding generations lumped together.

Political Inertia: To reflect the time it takes for the course of events to be altered, I employed an exponential decay model. Here is the generational influence with a couple choices of decay rate:

Since exactly what decay rate to use is difficult to determine, I plotted the share of generational responsibility for the state of affairs at the end of 2017 across the entire range of possible values:

The values approach a limit fairly quickly and quite a bit of the full range is rather extreme anyway, so here is the area with a decay rate less than ten percent:

Even given the simplistic nature of this analysis, it's pretty clear that one group shares little responsibility for the current state of affairs (for good or ill): New Boomers, or "Millennials" to use the more common label. Additionally, for all their numbers, Baby Boomers maximum value of impact for any choice of decay rate is 32.5%. While certainly not a small share, previous generations had a vast, ongoing influence.

More generally, I would argue that this basic model serves as a useful basis for considering more nuanced historical, societal, and psychological influences. I also would strongly advise against assigning more significance to it beyond that. Indeed, the major takeaway is that previous generations didn't simply evaporate once newer ones came of age. Patently obvious, but not properly reflected in the tone of many articles on the topic I come across.

The Real World Birthday Problem

Originally posted on my private blog September 19th, 2008.

The birthday problem is a well known mathematical illustration. Basically it asks how many people are needed in one room before two of them have a 50% chance of sharing a birthday. The answer is 23, which apparently is lower than what most people's intuition suggests. That result, however, is based on math that assumes that birthdays occur uniformly throughout the year and that also neglects leap days. I was curious what effect this might have, so I went looking for some birth statistics. The closest I found was a list of U.S. births by day for 1978. Series 1 (dark blue) is the raw information, which as you can see has a weekly cycle with much lower birth rates on weekends (especially Sunday). Unfortunately I needed to remove this effect as every date occurs on every day of the week in the long run. Not having other years to work with, I filtered the data, averaging every day with the births for three days before and after so that every weekday would be included. This is shown as series 2 (pink). Series 3 (yellow) is what the assumption of completely uniform birth rates would look like and is included for comparison. You can see that birth rates are significantly higher in the second half of the year. Unfortunately, 1978 wasn't a leap year, so I averaged the values of February 28th and March 1st and then weighted the leap day to be one quarter as likely (not shown).
Next I had to use this data to compute the probability distribution. I went with a simulation approach, which means I just performed the experiment a lot of times and kept track of how many people it took to get a matching pair of birthdays. Not sure of how many trials were needed to be accurate, I did successively larger runs, starting with ten thousand and ending with a billion, each run being an order of magnitude larger. You can see that the results converge, with the last two curves being right on top of each other.
Finally, here are the cumulative probability distributions of both the standard mathematical model (series 1) and my simulated one using real data (series 2). You can see that the two are basically identical and in fact the maximum error is less than two tenths of one percent. This means that all the effects of non-uniform birthday occurrence average out in practice. If the data had been more dramatically clustered this might not have been the case, but now I know for sure. In case you're wondering, a couple professors have written papers on this very topic, but I wanted to do it for myself as a mental exercise.