The classical solution to the problem is "don't ask don't tell". Or, as the Bavli puts it in Kiddushin 71a,
משפחה שנטמעה נטמעה
which means "a family (that includes mamzerim) that gets blended in (to the general population) stays blended in" - despite all the members of the general population who are now unknowingly marrying mamzerim. At first sight this looks intellectually dishonest, but not if you interpret the prohibition in Devarim 23:3,
No mamzer shall be admitted into the congregation of the Lordas meaning that no Jew who is known to be a mamzer shall be admitted into the congregation of the Lord. That works for Jews who are not known to be mamzerim, or for mamzerim who (properly IMHO) succeed in concealing their status, but it doesn't work for known mamzerim. In particular, it doesn't work for the mamzerim who are unfortunate enough to be on the Israel Rabbinate's blacklist. The mere existence of that blacklist subverts the traditional solution of "don't ask don't tell". As Daniel the tailor stated in Vayikra Rabbah 32:8 in interpreting Kohelet 4:1 as referring to mamzerim,
"their oppressors are empowered", this refers to Israel's Great Sanhedrin, who comes at them with the power of Torah, and pushes them away in the name of "no mamzer shall be admitted into the congregation of the Lord."
OTOH maybe the Sages were being intellectually dishonest. Shortly after
משפחה שנטמעה נטמעהthe Bavli has the following b'raita:
תאנא עוד אחרת היתה ולא רצו חכמים לגלותה אבל חכמים מוסרים אותו לבניהם ולתלמידיהן פעם אחת בשבועwhich means "There was another (family that was known to include mamzerim) that the Sages didn't want to identify, but they transferred the information to their children and to their students once every seven years". So "don't ask don't tell" was good enough for the masses, but not for the leaders.
It would be so much simpler, and more honest, if we would just admit that after 3000 years of Jewish history all of us are mamzerim. All it takes to make you a mamzer is one mamzer in your family tree. Statistically, the odds of you not having a mamzer in your family tree from that unidentified family are vanishingly small. Jack Lee used that kind of statistical argument to prove that he is a descendant of Charlemagne. Baruch Fischer and Moshe Zakai used more sophisticated mathematics to prove that by Second Temple times all Jews were descendants of King David. I used Jack Lee's method in part I to prove that all Jews are mamzerim.
Recently, I considered studying and actually understanding Fischer and Zakai in order to strengthen my argument. But then I changed my mind. How would my own increased understanding of the statistics help convince others? So I decided instead to code a mamzer simulator. Starting with an initial population with either no mamzerim or a miniscule proportion of mamzerim, and following the status of the population over several generations, depending on the input parameters, almost every run of the simulator winds up with the entire population being mamzerim and mamzerot within about 15 generations. The reason for that is that a child is a mamzer if either of his parents is a mamzer. As a result, the proportion of mamzerim in the population grows faster than the population, and eventually takes over the whole population.
This is what a typical output looks like (as of the date of this posting; I expect that the code will be improved). I started with 40 married couples. None of the 80 members of the founding generation were mamzerim or mamzerot. I introduced mamzerut by allowing one out of every 200 pregnancies to be the result of an adulterous union. The simulator runs for 15 generations and produces the following output that portrays the children of each generation:
Each colored row shows the children born in that generation. Boys who are mamzerim are represented by red rectangles. Boys who are not mamzerim are represented by blue rectangles. Girls who are mamzerot are represented by yellow rectangles. Girls who are not mamzerot are represented by green rectangles. The "ratio" in the table is the proportion of mamzerim+mamzerot in the whole population, in each generation. By the eighth generation, all the children are mamzerim and mamzerot.
I invite you to try the simulator for yourself, at http://avirosenthal.website2.me/
Update 5/6/18: That URL no longer exists. I will re-do. Meanwhile, this Web page suggests that a reasonable value for adulterous pregnancies is between 1 in 50 and 1 in 100.
(Later...) When I signed up with Wix to re-build the Web site, the old URL came back. Just ignore the "contact" page. I think Wix puts it there by default.
Update 10/6/18 In case you think that Jewish women are more virtuous than those of that Web page, I refer you to the end of Yevamot 118b:
תנא וכולן מזנות ותולות בבעליהן
