Who is a Jew(ish Chess Player)?

As the image above shows, not only are there many Jewish chess players--there are enough of them for a rather substantial book dealing only with those Jewish chess players who appeared on stamps.

But the mention of Yosef "Tommy" Lapid and Garry Kasparov in the previous post got me thinking: who decides who is a Jew? The reason is that, technically speaking, Kasparov isn't a Jew--his mother isn't Jewish (only his father is) so he isn't Jewish, either, according to the halacha (Jewish religious law). But Lapid told me that Kasparov, when visiting Israel, told him he considers himself a Jew--and, indeed, a right-wing Zionist Jew, who complained to Lapid Israel doesn't treat the Palestinians harshly enough.

The opposite of course is also true: Robert "Bobby" Fischer was, technically, a Jew since his mother was; and yet not only did he not consider himself Jewish, he was (at least in his later years) a viciously antisemitic holocaust denier. In his case, there is a general agreement that his antisemitism was more a symptom of mental illness than of a calculated world view: for instance, he was on friendly terms with many Jewish chess players, such as the Polgar sisters, Pal Benko, Andor Lillienthal, and others, who he somehow separated, in his mind, from "the Jews" who are out to get him. But the fact remains that he was a Jew according to Jewish religious law.

Of course, nobody says self-definition or Jewish religious law are the only possible criteria. Many ultra-orthodox Jews would deny that either Kasparov or Fischer are Jews--the former due to his gentile mother, the latter due to his behavior. For that matter, according to the Nazis' Nuremberg laws--or Israel's "Law of Return"--both Kasparov and Fischer would be considered Jews, having at least one Jewish parent. (The "Law of Return", which grants automatic Israeli citizenship to all Jews who want it, defines "Jew" deliberately in a way similar to the Nuremberg laws. The reason is that, if someone was Jewish enough for the Nazis to try and genocide, they should be Jewish enough for the Jewish state to give them citizenship.)

So how does one decide? Is Kasparov a Jew? Was Fischer? Are both Jews? Are neither? What does the reader think?