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From a Jewish perspective, his wrongdoing falls into (at least) two categories g’neivat mammon, stealing money (prize money and endorsements, and perhaps contributions too by people who may not have given money if Armstrong was not who said he was) and g’neivat da’at, deceiving, by giving the impression that he was someone that he was not. The question is, to what degree are these sins forgivable or unimportant given the effect that they had.
In general, there are three different halachic concepts to be considered here for guidance. They are extremely complicated and well argued in rabbinic literature, such that a full treatment can’t be given in this space, but they offer a good frame for discussion:
What is the status of a mitvzah that comes about by someone doing a transgression — mitzvah ha-ba’ah ba’aveirah?
Might this be considered an aveira lishmah — a sin that is condoned because it is for the sake of a mitzvah?
To what degree does pikuach nefesh docheh hakol — saving a life justifies everything —apply here?
In order to see to what degree these ideas could be applied, I spoke to Rabbi Jeremy Wieder, a member of the rabbinical faculty at the Rabbi Isaac Elchanan Theological Seminary of Yeshiva University. What follows is a very condensed summary of our conversation:
With regard to the first halachic concept, mitzvah-haba’ah ba’aveirah, in general, the rabbis do not condone a transgression done for the sake of a mitzvah. For example, if you steal a lulav in order to fulfill the mitzvah of waving it, you have not fulfilled the mitzvah (See BT Sukkot 30a). There are some limited cases where committing transgressions do not invalidate the mitzvah itself, but the propelling actions are not excused, they are considered transgressions. If one steals money in order to save one’s own life (more on this later, some make this a strict case of pikuach nefesh as opposed to a mitzvah-haba’ah ba’aveirah), that may be permissible (or excusable, after the fact), but one still has committed a transgression, stealing, and needs to atone for that (including paying it back). It is permissible or at least an understandable sin.
From the other direction, instead of considering the mitzvah, we can consider the sin itself because there are sometimes that aveirot are condoned outright — an aveira lishma. Rabbi Wieder strongly warned that this is “an explosive subject,” (which makes sense because it might be used by people justifying their own bad behavior).
A paradigmatic example of this comes from BT Horayot 10b, which cites the story of Yael, who slept with Sisera in order to lull him to sleep and kill him (Judges 5:23-28). This sin was done for the sake of saving the Jewish people. Perhaps Armstrong’s sin, systematically cheating in order (remember this is the hypothetical) to help cure cancer and save the cancer patients is analogous.