In the movie the Big Lebowski Walter explains that he is shomer shabbos and that he can't "roll" (meaning bowl) on shabbos. My question is, if he were able to get to the alley without violating shabbos and if he were allowed to bowl for free, other than possibly uv'din d'chol, is there any reason he can't bowl on shabbos?
The rolling ball is the "koach" of the original person, analogous to a thrown rock. When the ball then hits the pins/sensors/etc that action is "koach kocho" of the original person, meaning it is twice removed from the person himself (for the definition see the Rambam Hilchos Rotzeach 6:15).
Now see the Avnei Nezer OH 230:2 who argues that "koach kocho" is potur on Shabbos, meaning that the act of the pin is only d'rabanan anyway. And the response from the sensor is psik reisha, as noted in the other answers. And the overall action of the machine is just an issur d'rabanan, since it is just electrical/mechanical activity that is not in the form of lighting/burning (as is clear from the tshuvos of R. Henkin and discussed elsewhere on this forum), no different from an electric fan.
This could perhaps be coupled with the view of the Rashba quoted in Shulchan Aruch OH 316 (Magen Avraham there) that allows psik reisha if the intention is a permissible act as well as an impermissible one (i.e. one wants to close the door, which is totally permissible, and in the process a deer is trapped inside).
So adding all this together, it seems clear there is sufficient room to be lenient about bowling on Shabbos, aside from issues of "uvda d'chol" which were excluded by the original questioner.
(As an aside, we often rely on this koach kocho argument for opening refrigerators on Shabbos, when the action of opening the fridge will let warm air in and surely turn on the cooling unit, a psik reisha that is nicha leih, but people are meikal; also perhaps because of the Rashba.)
In every bowling alley I've been to, the operation of the lanes depends on electro-mechanical devices that reset the pins and return the ball. As these actions are triggered by the bowler's bowling and are helpful to the bowler, they'd constitute a "pesik reisha denicha leih" - an expected consequence that's desirable - and would therefore be forbidden as use of electricity on Shabbat.
Wikipedia addresses the question of permissability if only gutterballs are thrown thus voiding the questions of binyan and soser. I quote:
Of interest, this would include bowling, as the pins are knocked down, hopefully, during play. However, even if only gutterballs are thrown, the mere intention of knocking down pins is a violation of the prohibition against demolition. It is a matter of rabbinic debate as to whether intentionally throwing gutterballs during Shobbos is subject to the prohibition, as no demolition occurs. Commentary tends towards making this a violation as well, since the very act of bowling involves causing a machine to reconstruct the pin matrix upon each round, which action is initiated by the participating player. See further: Mishneh Torah Shabbos 10:15; Chayei Adam Shabbos 39, 43