2009.01.10 pop_model_2.doc
An Investigation into the Ancestor/Descendant
Paradox
By
John D. Rogers P.Eng
1.3 Abbreviations
and Acronyms
5. Modifying the Number of Descendants
5.2 Parents
are Second Cousins
6. Impact of Multiple Cousin Marriages
6.1 Data
on the Frequency of Consanguineous Marriages
6.2 Modeling
the Consanguineous Marriage Frequency.
7.2 Looking
for a Solution from the Perspective of the Ancestors
7.3 Changing
the Consanguineous Marriage Frequency.
7.4 Changing
the Effect of Consanguineous Marriage Frequency
Doubling the number of ancestors for each previous generation eventually exceeds the number of descendants – this is the ancestor/descendant paradox.
This paper analyzes the growth of an individual’s ancestors and descendants over generations with an objective of investigating and resolving the paradox.
Results and charts used in this paper were obtained using Microsoft Excel spreadsheet pop_model_2.xls.
The following terminology is used in this paper:
Closed 
Term for a community (e.g., village or town) where people, in general, do not come in and out. For example, a community that is: owned by one landowner or geologically isolated. 
Consanguineous 
The union (marriage) of individuals having a common ancestor, e.g., cousins 
Cousinship 
The relationship of cousins 
Crossover 
The point in time when the number of descendants equals the number of ancestors 
Degree of cousinship 
The degree identifies whether the cousin is First, Second, Third, etc. 
Marriage 
Term used whether parents were married or not 
Open 
Term for a community (e.g., village or town) where people freely come in and out. 


The following abbreviations and acronyms are used in this paper:
CM 
Consanguineous Marriage 
CMF 
Consanguineous Marriage Frequency 
G 
Growth factor 
N 
Generation number 
Y 
Year 
The following table shows how the number of ancestors increases exponentially if it is assumed: each generation is twice the size of the previous; a generation is 25 years; and going back from the year 2000.
Generation 
Years 
Year 
Description 
Number 
Note 
0 
0 
2000 
Individual 
1 

1 
25 
1975 
Parents 
2 

2 
50 
1950 
Grandparents 
4 

3 
75 
1925 
Greatgrandparents 
8 

4 
100 
1900 
GreatGreatgrandparents 
16 

5 
125 
1875 
3 x Greatgrandparents 
32 

10 
250 
1750 
8 x Greatgrandparents 
1024 

15 
375 
1625 
13 x Greatgrandparents 
32,768 

20 
500 
1500 
18 x Greatgrandparents 
1,048,576 
@ 1 million 
30 
750 
1250 
28 x Greatgrandparents 
1,073,741,824 
@ 1 billion 
40 
1000 
1000 
38 x Greatgrandparents 
1.0995 x 10^{12} 
@ 1 trillion 
50 
1250 
750 
48 x Greatgrandparents 
1.1259 x 10^{15} 
@ 1 quadrillion 
N 
25N 
200025N 
(N2) x Greatgrandparents 
2^{N} 

The table shows that using the doubling approach for calculating the number of ancestors soon results in numbers that are greater than the current population of the British Isles (65 million) and the world (7 billion).
The approach for calculating the number of descendants assumes the numbers for the first three generations are 1, 2 and 4. From then on it is assumed that the number doubles every 100 years. Therefore there will be 8 descendants by the sixth generation. Again it is assumed that a generation is 25 years. The following table shows these assumptions.
Generation 
Years 
Descendants 
0 
0 
1 
1 
25 
2 
2 
50 
4 
3 
75 
See text 
4 
100 
See text 
5 
125 
See text 
6 
150 
8 
This approach results in a geometric series of numbers, i.e., each number is multiplied by a growth factor (G) to get the next number. The series (starting at 4) is therefore 4, 4G, 4G^{2}, 4G^{3}, 4G^{4} (8).
As 4G^{4} = 8, then _{}. So next number after 4 is 4 x 1.1892 = 4.7568.
Thus the series starts: 1, 2, 4, 4.7568, 5.6569, 6.7272, 8, 9.5137, 11.3137, 13.4543, 16, …..
The starting date for the series (0^{th} generation) is selected to give a number of descendants in the year that is close to the current population of the British Isles (65 million). Using a starting date of 425 BC gives 56,431,603 descendants after 98 generations.
The following table gives the full results for the numbers of ancestors and descendants calculated using the approaches described in the previous sections.
Descendent Generation 
Year

Number
of Descendants 
Ancestor Generation 
Number
of Ancestors 
0 
2000 
1 
97 
56,431,603 
1 
1975 
2 
96 
47,453,133 
2 
1950 
4 
95 
39,903,169 
3 
1925 
8 
94 
33,554,432 
4 
1900 
16 
93 
28,215,802 
5 
1875 
32 
92 
23,726,566 
6 
1850 
64 
91 
19,951,585 
7 
1825 
128 
90 
16,777,216 
8 
1800 
256 
89 
14,107,901 
9 
1775 
512 
88 
11,863,283 
10 
1750 
1,024 
87 
9,975,792 
11 
1725 
2,048 
86 
8,388,608 
12 
1700 
4,096 
85 
7,053,950 
13 
1675 
8,192 
84 
5,931,642 
14 
1650 
16,384 
83 
4,987,896 
15 
1625 
32,768 
82 
4,194,304 
16 
1600 
65,536 
81 
3,526,975 
17 
1575 
131,072 
80 
2,965,821 
18 
1550 
262,144 
79 
2,493,948 
19 
1525 
524,288 
78 
2,097,152 
20 
1500 
1,048,576 
77 
1,763,488 
21 
1475 
2,097,152 
76 
1,482,910 
22 
1450 
4,194,304 
75 
1,246,974 
23 
1425 
8,388,608 
74 
1,048,576 
24 
1400 
16,777,216 
73 
881,744 
25 
1375 
33,554,432 
72 
741,455 
26 
1350 
67,108,864 
71 
623,487 
27 
1325 
134,217,728 
70 
524,288 
28 
1300 
268,435,456 
69 
440,872 
29 
1275 
536,870,912 
68 
370,728 
30 
1250 
1,073,741,824 
67 
311,744 
31 
1225 
2,147,483,648 
66 
262,144 
32 
1200 
4,294,967,296 
65 
220,436 
33 
1175 
8,589,934,592 
64 
185,364 
34 
1150 
17,179,869,184 
63 
155,872 
35 
1125 
34,359,738,368 
62 
131,072 
36 
1100 
68,719,476,736 
61 
110,218 
37 
1075 
137,438,953,472 
60 
92,682 
38 
1050 
274,877,906,944 
59 
77,936 
39 
1025 
549,755,813,888 
58 
65,536 
40 
1000 
1.10E+12 
57 
55,109 
41 
975 
2.20E+12 
56 
46,341 
42 
950 
4.40E+12 
55 
38,968 
43 
925 
8.80E+12 
54 
32,768 
44 
900 
1.76E+13 
53 
27,554 
45 
875 
3.52E+13 
52 
23,170 
46 
850 
7.04E+13 
51 
19,484 
47 
825 
1.41E+14 
50 
16,384 
48 
800 
2.81E+14 
49 
13,777 
49 
775 
5.63E+14 
48 
11,585 
50 
750 
1.13E+15 
47 
9,742 
51 
725 
2.25E+15 
46 
8,192 
52 
700 
4.50E+15 
45 
6,889 
53 
675 
9.01E+15 
44 
5,793 
54 
650 
1.80E+16 
43 
4,871 
55 
625 
3.60E+16 
42 
4,096 
56 
600 
7.21E+16 
41 
3,444 
57 
575 
1.44E+17 
40 
2,896 
58 
550 
2.88E+17 
39 
2,435 
59 
525 
5.76E+17 
38 
2,048 
60 
500 
1.15E+18 
37 
1,722 
61 
475 
2.31E+18 
36 
1,448 
62 
450 
4.61E+18 
35 
1,218 
63 
425 
9.22E+18 
34 
1,024 
64 
400 
1.84E+19 
33 
861 
65 
375 
3.69E+19 
32 
724 
66 
350 
7.38E+19 
31 
609 
67 
325 
1.48E+20 
30 
512 
68 
300 
2.95E+20 
29 
431 
69 
275 
5.90E+20 
28 
362 
70 
250 
1.18E+21 
27 
304 
71 
225 
2.36E+21 
26 
256 
72 
200 
4.72E+21 
25 
215 
73 
175 
9.44E+21 
24 
181 
74 
150 
1.89E+22 
23 
152 
75 
125 
3.78E+22 
22 
128 
76 
100 
7.56E+22 
21 
108 
77 
75 
1.51E+23 
20 
91 
78 
50 
3.02E+23 
19 
76 
79 
25 
6.04E+23 
18 
64 
80 
0 
1.21E+24 
17 
54 
81 
25 
2.42E+24 
16 
45 
82 
50 
4.84E+24 
15 
38 
83 
75 
9.67E+24 
14 
32 
84 
100 
1.93E+25 
13 
27 
85 
125 
3.87E+25 
12 
23 
86 
150 
7.74E+25 
11 
19 
87 
175 
1.55E+26 
10 
16 
88 
200 
3.09E+26 
9 
13.4543 
89 
225 
6.19E+26 
8 
11.3137 
90 
250 
1.24E+27 
7 
9.5137 
91 
275 
2.48E+27 
6 
8 
92 
300 
4.95E+27 
5 
6.7272 
93 
325 
9.90E+27 
4 
5.6569 
94 
350 
1.98E+28 
3 
4.7568 
95 
375 
3.96E+28 
2 
4 
96 
400 
7.92E+28 
1 
2 
97 
425 
1.58E+29 
0 
1 
Note 1: negative years are BC.
Note 2: The notation 1.23E+12 represents 1,230,000,000,000 (12 places after what was the decimal point).
The table shows that the number of descendants is initially low compared with the number of ancestors, but then it rapidly increases. Between 1500 and 1475 AD the number of descendants exceeds the number of ancestors. Thereafter, the number of descendants grows to huge numbers while the number of ancestors steadily decreases.
For the 97^{th} generation in the year 425 BC, the number of descendants is 158,456,325,028,528,675,187,087,900,672 or 158 octillion.
The following chart shows the number of descendants and ancestors for the years 1300 to 2000 AD.
The following chart shows the years 1450 to 1550 AD amplifying the crossover point where the number of descendants exceeds the number of ancestors.
The crossover point occurs when the number of ancestors equals the number of descendants.
The number of ancestors = 2^{N}.
The number of descendants = 4.G^{(N2)} converting this to use the ancestor generation gives 4.G^{(97N2)}
At crossover 2^{N} = 4.G^{(95N)}
Dividing each side by 4 gives 2^{N2} = G^{(95N)}
Taking the logarithm of each side gives (N2).log 2 = (95N).log G
N.log 2 – 2.log 2 = 95.log G – N.log G
N (log 2 +log G) = 95.log G + 2 log 2
N = _{} dividing each side by log G gives _{}
Let_{}, as determined previously, _{} so _{}
Rearranging gives _{}
Taking the antilogarithms of each side gives _{} giving[1] k = 4
Substituting the value of k into the formula for N gives
_{}
20.6 generations = 20.6 x 25 = 515 years, so the crossover occurred 2000 – 515 years ago in 1485 AD.
Although doubling the number of descendants for each generation may be valid for recent history, the approach has to be modified so that the number of descendants never exceeds the number of ancestors.
The reason the number of descendants is less than expected is because of inbreeding. The following sections investigate the reduction in ancestors if parents (generation 1) are cousins.
First cousins have a common set of grandparents. If first cousins marry they will have 3 sets of grandparents (6 people) instead of 4 sets (8 people). This results in a reduction of ¼ or 25% in the number of descendants.
First cousin marriage is not illegal in the UK. The rate of white first cousin marriages is about 0.5%, however, for other cultures in the UK it is much higher, e.g., 55% for Pakistanis.
Second cousins have a common set of greatgrandparents. If second cousins marry they will have 7 sets of greatgrandparents (14 people) instead of 8 sets (16 people). This results in a reduction of 1/8 or 12.5% in the number of descendants.
Third cousins have a common set of greatgreatgrandparents. If third cousins marry they will have 15 sets of greatgreat grandparents (30 people) instead of 16 sets (32 people). This results in a reduction of 1/16 or 6.25% in the number of descendants.
The following table shows the numbers of descendants when parents are cousins.
Generation 
Description 
Number if Parents are
Unrelated 
Number if Parents are
Third Cousins 
Number if Parents are
Second Cousins 
Number if Parents are
first Cousins 
0 
Individual 
1 
1 
1 
1 
1 
Parents 
2 
2 
2 
2 
2 
Grandparents 
4 
4 
4 
4 
3 
Greatgrandparents 
8 
8 
8 
6 
4 

16 
16 
14 
12 
5 

32 
30 
28 
24 
6 

64 
60 
56 
48 
7 

128 
120 
112 
96 
20 

1,048,576 
1,015,808 
983,040 
786,432 
Reduction 

0^{} 
6.25% 
12.5% 
25% 
By the 20^{th} generation, if the parents are first cousins there has been a reduction of 262,144 in the number of descendants. This reduction is not all that significant compared with the number if the parents are unrelated. The reduction has only pushed the crossover back from 20.6 generations to 20.9 generations or 7.5 years.
As the degree of the cousinship increases, so the reduction becomes less.
In addition, if the cousins marrying belonged to earlier generations, the reduction is less because then only the previous generations are impacted.
Therefore, there must be multiple consanguineous marriages (CMs). To confirm this, consider the extreme case where all marriages are consanguineous. If both grandfathers are brothers and both grandmothers are sisters then there will 4 greatgrandparents instead of the usual 8. If both greatgrandfathers are brothers and both greatgrandmothers are sisters then there will 4 greatgreatgrandparents instead of the usual 16. In this extreme case, the number of ancestors stabilizes at 4, see the following figure.
Thus multiple CMs can result in a far reduced number of ancestors. The next section investigates the frequency of these marriages and the resulting number of ancestors.
The previous sections showed that if the parents (generation 1) are first cousins, the reduction in descendants was 25%. The following sections investigate the reduction in ancestors if there are multiple marriages between cousins.
The following information applies to the current consanguineous marriage frequency (CMF):
· the CMF for the UK white population is about 0.5%;
· the CMF for some immigrant (e.g., Pakistani and Bangladeshi) sections of the UK population is about 55%;
· the frequency of marriage between second or higher degree cousins is insignificant compared with the frequency of marriage between first cousins in the UK and elsewhere;
· the CMF[2] for some other countries is:
o 0.2% for the USA,
o 0.6% for Norway,
o 1.5% for Portugal,
o 4.1% for Spain,
o 4.8% for Brazil,
o 5.7% for Japan,
o 39.7% for Jordan, and
o 61.2% for Pakistan;
· for many countries, the CMF for rural areas is significantly higher than for urban areas; and
· it is estimated that 20 percent of all couples worldwide are first cousins.
Data on the historical CMF is very sparse, the following information has been obtained:
· in the UK, the rate of first cousin marriages was estimated to be:
o 0.32% during the 1920s,
o 1.12% during the 1890s,
o 1.1% from 1775 to 1924 in the ‘open’ parish of Kilmington, and
o 2.25% from 1775 to 1924 in the ‘closed’ parish of Stouton; and
· it is estimated that 80 percent of all marriages historically have been between first cousins.
The following assumptions are made in order to model the CMF:
· all CMs are between first cousins;
· prior to the industrial revolution in 1760, the CMF was steady at 80%;
· the CMF dropped from 1760 to the beginning of the 20^{th} centenary where it levelled off at 0.5%, this transition was centred around 1875 (Generation 5).
The change in CMF values follows a reverse Scurve. The function which best represents this type of behaviour is the generalized logistic function. This function is modified to fit the above assumptions as follows:
_{}
where:
CMF is the consanguineous marriage frequency (%)
L is the lower asymptote = 0.5%
U is the upper asymptote = 80%
G is the growth rate = 0.06 (value selected to fit the data)
Y is the year
The following chart shows the resulting curve of the CMF between 1700 and 1950.
Modifying the above formula to use ancestor generation numbers (gen) instead of years and to start at generation 0 gives:
_{}
The following chart shows the resulting curve of the CMF for generations 0  12 (2000 – 1700 AD).
The CMF is used in calculating the number of descendants for each generation as follows:
· calculate the number of marriages (half the number of descendants);
· calculate the CMF for the generation (see above);
· calculate the number of CMs (multiply the CMF by the number of marriages);
· calculate the number of descendants in the next generation (reduce the number of descendants in the current generation by the number of CMs and then double).
The following table shows the results around the generations where crossover occurs. It can be seen that crossover has been pushed back from 1485 (1.6 million ancestors/descendants) to around 960 AD (42,000 ancestors/descendants).
Descendent Generation 
Year

Number
of Descendants 
Ancestor Generation 
Number
of Ancestors 
36 
1100 
14,974 
61 
110,218 
37 
1075 
17,970 
60 
92,682 
38 
1050 
21,564 
59 
77,936 
39 
1025 
25,878 
58 
65,536 
40 
1000 
31,054 
57 
55,109 
41 
975 
37,266 
56 
46,341 
42 
950 
44,720 
55 
38,968 
43 
925 
53,664 
54 
32,768 
44 
900 
64,398 
53 
27,554 
45 
875 
77,278 
52 
23,170 
46 
850 
92,734 
51 
19,484 
Although this is an impressive reduction, after crossover, the number of descendants continues to increase by 20% per generation resulting in 1,012,749,142 descendants by the year 425 BC.
Increasing the CMF to 100% stops further growth in the number of descendants, but does not reduce it.
The results show that even with an 80% CMF, the reductions are still not enough to resolve the paradox.
This section investigates the changes required to resolve the paradox.
The genealogical chart shows a possible ancestry starting from a single female (Eve) in generation 0. The families are shown in the following chart along with their offspring (B = boy, G = girl). The numbers represent the family numbers of the 4 possible grandparents.
Family No 
1 
2 
3 
4 
5 
6 
7 
1x 
BBGG 






2x 
0, 0 0, 0 BBB 
0, 0 0, 0 GGG 





3x 
1, 1 1, 1 BBB 
1, 1 1, 1 GB 
1, 1 1, 1 GGG 




4x 
21, 22 21, 22 BBB 
21, 22 21, 22 GB 
21, 22 21, 22 GB 
21, 22 21, 22 GGG 



5x 
31, 32 31, 33 BBB 
31, 32 31, 33 GB 
31, 32 32, 33 GB 
31, 33 32, 33 GB 
31, 33 32, 33 GGG 


6x 
41, 42 41, 43 BBB 
41, 42 41, 44 GB 
41, 42 42, 44 GB 
41, 43 43, 44 GB 
41, 44 43, 44 GB 
42, 44 43, 44 GGG 

7x 
51, 52 51, 53 BBB 
51, 52 51, 54 GB 
51, 52 52, 53 GB 
51, 53 53, 55 GB 
51, 54 54, 55 GB 
52, 55 54, 55 GB 
53, 55 54, 55 GGG 
The following table provides a summary.
Generation 
Size 
Description 
0 
1 
Eve 
1 
2 
Eve has a boy (B) and a girl (G) 
2 
4 
They form 1 family with 2 boys (BB) and 2 girls (GG) 
3 
6 
They form 2 families with offspring BBB and GGG 
4 
8 
They form 3 families with offspring BBB, BG and GGG 
5 
10 
They form 4 families with offspring BBB, BG, BG, BG and GGG 
6 
12 
They form 5 families with offspring BBB, BG, BG, BG, BG and GGG 
7 
14 
They form 6 families with offspring BBB, BG, BG, BG, BG, BG and GGG 
Observations:
· it is possible to have a scenario where the previous generations reduce in size instead of increasing because of a high degree of inbreeding:
· if families have on average:
o less than 2 children, the population size will reduce,
o exactly 2 children, the population size will stabilize, and
o more than 2 children, the population size will increase;
As a result of the above the following changes will be made:
· the initial population sizes will be 1, 2, 4, 6, 8 and thereafter increase at a rate which doubles every 4 generations: 9, 11, 13, 16, 19, 23, 27, 32 etc.; and
· the impact of CMs will be increased (see next sections).
It is only in recent generations that the population has had the ability to, and has become, much more mobile in the pursuit of jobs, holidays and other entertainment and travel. This increases the probability of marriages outside of the family and the community. Previously, communities became increasingly closed for reasons such as:
· geographical isolation;
· lack of transport;
· religious (Catholics could only marry Catholics);
· wealthy families wishing to keep their wealth within the family; and
· mistrust, even hatred of strangers and other communities.
All of the above results in increased inbreeding causing reductions in the number of ancestors. Therefore the CMF will eventually increase to unity (100%). This increase will be achieved as shown in the following table.
Years 
CMF 
2000 – 1625 AD 
0 – 80% (logistic curve) 
1600 – 600 AD 
80  100% in increments of 0.5%/generation 
575 AD – 425 BC 
100% 
The effect of a first cousin marriage is to reduce the number of grandparents by 2. As the CMF reaches 100% the number of ancestors becomes constant. However as the amount of inbreeding increases, more complex cousinships occur. For example for a pair of grandparents:
· the two grandfathers my be first cousins as well as third cousins;
· the two grandmothers my be second cousins;
· one grandfather may be a second cousin of the other grandmother;
Instead of doubling the number of ancestors for each generation, the above all act to reduce the number of ancestors. Further reductions occur for situations when a man marries a second wife (or visa versa), when a family has multiple children all marrying within the family tree and when more distant cousins are taken into account.
To reflect the increased effect of CMs, a factor k is introduced so that instead of reducing the number of grandparents by 2 it will be reduced by 2(1 + k). Values of k are set as shown in the following table.
Years 
Value of k 
2000  1425 
0 
1400  1275 
0.1 
1250 – 425 BC 
0.2 
The following chart shows the new results for the years 0 to 2000.
The number of descendants peaks at 32,664 (about 6%) in 1275 AD when the number of ancestors was 524,288. The chart shows the number of descendants increases initially and then peaks following which there appears to be a relatively sharp reduction. However the rate is less than the rate for the descendants and appears less pronounced when the period between 1200 and 1325 AD is shown.
This paper has demonstrated the ancestor/descendant paradox and shown that when inbreeding is taken into account the number of descendants eventually reduces thereby refuting the paradox.
Further investigation is required to more precisely quantify the effect of inbreeding, to validate the assumptions made, refine the analysis and to obtain more historical data.
[1] Raising each side to the 4^{th} power gives k^{2} = 2^{4} = 16 so k = Ö16 = 4.
[2] See Empirical Estimates Of The Global Prevalence Of Consanguineous
Marriage In Contemporary Societies, Alan H. Bittles, Paper number 0074, Centre
for Human Genetics, Edith Cowan University, Perth, Australia