Very interesting article here regarding the permissibility of lotteries in general:
Opinions of interest brought:
Shiltei Giborim - says games of luck (mitzchakei kibuiyah) are not allowed on Shabbat.
Shimshon Pobino (Pachad Yitzchak; not Rav Hutner's of course) - all games should be prohibited on Shabbat (!)
Shabbat 149: - It's prohibited (even on a weekday) to make a lottery to determine who is to receive a large vs. small piece of meat, for instance
Rav Hadaya of the Beit Din in Jerusalem reports of a case where a Jew in Singapore lost all of his possessions while gambling and then proceeded to gamble what was left -- his wife! (He lost and she was forced to become the other man's wife!)
Rav Ovadya Yosef rules that the lottery is not permitted.
Ashkenazim in Israel are lenient in this matter and permit participation in the lottery.
While there's no problem due to a Gentile performing prohibited work (since the Gentile is doing it for the sake of all the other participants), there is obviously the issue of earning money on Shabbat. We could compare it to similar cases where Jews are allowed to earn money on Shabbat, such as working as a waiter, babysitter, madrich, etc. in which case we require that preparation work be done before Shabbat and we say that the money being earned is as a result of that preparation work. Similarly, we can say that choosing the lottery numbers and buying the ticket are the work required for earning money and the decision of whether you earn lots of money or little ($0) happens to simply be determined on Shabbat.