Friday, June 17, 2011

Dear Randy: Can you explain this Genealogical Paradox?

Reader Bud wrote an email asking a question:

"You seem like a person who would be familiar with what I call the Genealogical Paradox. Hopefully, you can tell what others are calling it these days and the solution others have come up with.

"Genealogical Paradox: We can trace our roots back to a time when everyone in the world is our ancestor. Equally, we are all related! Although this is a possibility, it seems unlikely. Let me use the attached worksheet to outline the argument.

"First, I researched our scientist’s best estimates of the world’s population. E.g.

"Where I did not have estimates, I filled in a reasonably consistent manner. Now to Sheet 1. In columns E-J, I assumed various intervals between a child’s birth and his/her parents. I begin in 1954. In column K, I assumed that each ancestor child had distinct parents from all other ancestor children…each child is independent from all others….only children. In columns M-R, I looked up the population in millions for each year in columns E-J from Sheet2. In column S, I took the 25 year spacing and blew it out for comparability in column K.

"All the estimates I found suggested that before the year (approx.) 1000, the world (not just Europe) had a population of 300 million or less.  What it shows is that at my 30 generation, everyone in the world is my 30th great-grandparents!

"Ok. Absurd, yes. My “independence” assumption is NOT true.  Because of the exponential affect, these exorbitant numbers are cut down a great deal.

"However, unless the world had more than 300 million people at some points in time, it doesn’t change the conclusion; it defers the conclusion: At some point in time, everyone in the world is our ancestor.

"My problem is looking at the magnitude of the numbers. It explodes well before we get into B.C era. Any direction or insights would be appreciated. And if I am the only nut who thinks of such things, then please pray for me. I must have too much time on my hands."

Bud did a lot of work trying to figure out his paradox.  Where are all of the missing ancestors?  The spreadsheet was well done, and accurate, too. 

I am not an expert at this, so my answer to Bud referred to more knowledgeable people than myself.  Here is my response:

"The simple answer to your question is: Pedigree Collapse. Before 1800, people often married their cousins because of the limited "breeding stock" within walking distance, and because of familiarity with their family.

"There are a number of excellent articles, with examples, at:

"* Pedigree Collapse --

"This article says:

'Since every person has two parents, the number of ancestors in each older generation can be calculated by multiplying by two again. So 2 parents in 2 generations = 2 x 2 = 4 ancestors (grandparents). 2 parents in 3 generations = 2 x 2 x 2 = 8 ancestors (great-grandparents). Going back 30 generations would mean 230, or a billion, ancestors. Few people have the same maternal and paternal grandfather, but many people have, for example, relatives who are both a great-grandfather and a great-great-grandfather — each through a different side of the family. This is because cousins — sometimes quite distant cousins — marry. So instead of a person having a billion distinct ancestors 30 generations ago, they are limited to a much smaller number, especially since historically many families lived in the same area for hundreds of years.'

"* Pedigree Collapse --

"This is an excellent article by John Becker, from the Ontario Genealogical Society Families periodical (Volume 38, Number 3, 1999) that explains it clearly, as in:

'We all are blessed with two parents, four grandparents, eight great grandparents and so on. If the average generation is twenty-five years, in 1200 years (back to 800 AD, the time of Charlemagne) each person has 281.5 trillion grandparents. That's the way geometric progressions work. The number of grandparents doubles every 25 years and in 1200 years or 48 generations, 281.5 trillion names would be on your pedigree.'

"* Pedigree Collapse --"

"This article is a bit more scientific, but the explanation is the same:

'Without pedigree collapse, a person's ancestor tree is a binary tree, formed by the person, the parents, grandparents, and so on. However, the number of individuals in such a tree grows exponentially and will eventually become impossibly high. For example, a single individual alive today would, over 30 generations going back to the High Middle Ages, have roughly a billion ancestors, more than the total world population at the time. This apparent paradox occurs because the individuals in the binary tree are not distinct: instead, a single individual may occupy multiple places in the binary tree. This typically happens when the parents of an ancestor are cousins (sometimes unbeknownst to themselves). For example, the offspring of two first cousins has at most only six great-grandparents instead of the normal eight. This reduction in the number of ancestors is pedigree collapse. It collapses the binary tree into a directed acyclic graph with two different, directed paths starting from the ancestor who in the binary tree would occupy two places.'

Does another reader have a better explanation, or better examples? 

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(c) 2011. Randall J. Seaver. All Rights Reserved. If you wish to re-publish my content, please contact me for permission, which I will usually grant. If you are reading this on any other genealogy website, then they have stolen my work.


Drew Smith said...

We cannot point to a specific time in ancient history when everyone alive is necessarily one of our ancestors, as some of those individuals would have no living descendants.

Otherwise, there certainly is a time where everyone alive (at that time) who *does* have living descendants is the ancestor of everyone alive today. I know there have been academic journal articles discussing this issue. (Maybe I can turn some of them up.)

Shane said...

The most recent common ancestors question you discuss is well covered here.