This article is worth looking at.
"The PTAB found that (i) the claimed method is directed to a set of rules for conducting a wagering game, which is a patent-ineligible abstract idea, and (ii) the manipulations of the deck of cards (e.g., shuffling, dealing) are conventional steps in playing a wagering game that do not add enough to the claims to transform the recited methods into patent-eligible subject matter."
Essentially, they were trying to patent a new method betting on Blackjack, which was deemed a "business method" and purely abstract. (i.e. It's simply a set of mental steps for managing risk.)
Ostensibly, truly novel game methods should still be patentable, regardless of whether they involve computers, although patent attorneys recommend using one of the claims to tie the game to computer systems for software-based implementations.
Another element to bear in mind is that patents are not a monolith. Typically you will have a few independent claims and a bunch of dependent claims (which are extensions of the independent claims.) Each claim is treated separately, so having any particular claim denied does not necessarily invalidate the entire patent. The review process generally takes a few years and involves some back-and-forth with the patent office.
Rules vary internationally. US patents can usually be defended in patent treaty regions, although if a game catches on, foreign filing may be worthwhile.
This answer is updated based on a this recent article from IP Watchdog.
The article mentions In re Smith, where a new method of betting on Blackjack was rejected and raised alarm. However, although such a method technically constitutes a "new game", all of the steps listed in the application were generic (such as shuffling cards.) In other words, the only novel aspect of the Smith patent application was the order of the steps that constitute the "new game". It's unclear that the result would have been the same had the claims included a step that was actually novel and inventive.
Deciding to pursue a patent can be risk. There is no guarantee of securing a grant and it can be expensive. A recent game that may be helpful to consider is Threes!, aka 2048. Would it have been possible to patent that game mechanic? Maybe. The more relevant question might be would it have been worthwhile to pursue a patent? (Based on the number of times the clones have been downloaded, I'd lean towards yes, but in this context, it is really an assessment of risk vs. reward.)
Copyright and Game Mechanics:
On a more positive note, this recent case, Spry Fox vs. Lolapps seems to extend protection of game mechanics into Copyright law. It could be overturned at any time by a higher court, but word on the street is it is a kind of "third rail" and no one wants to go near it. Further, there is some precedent for this extension of Copyright law in Tetris vs. Xio, which is bordering on Design Patent territory.
Copyright is a great, no-cost way to pursue some degree protection of your game.
Creative work of all types have very strong Intellectual Property protection, even something as simple as a photo snapped on a phone. Games are distinct in being algorithms, but still constitute creative work. Current climate aside regarding game patent eligibility, the spirit of the law" clearly indicates that rule sets for games should be granted the same intellectual property protection rights as other forms of art.