I would venture to guess that a pair of polarizing filters is probably used in concordance with some micro servos or some such. Two polarizing filters rotated in opposition to each other along a 90° arc can vary between minimal filtration (few percent opacity) to near total filtration (90% opacity or around there). Its a fairly common DIY project to create a "variable ND filter" by combining a circular polarizer lens-side with a linear polarizer scene-side when you don't want to drop the cash on a decent multi-filter/interchangeable-filter ND filtration system like Lee or Cokin.
I can't say if this is actually how its done. Such a setup would allow compact cameras to simulate aperture's effect on light volume without actually encountering severe diffraction blurring issues caused by literal pinhole-sized apertures at say f/8. It would require additional on-board logic to calculate the amount of rotation of one of the polarizing filters to achieve the amount of light reduction necessary to simulate a given aperture, and the results are probably not 100% accurate. That said, aperture stop-downs are not 100% perfect in DSLR's either, and there can be minor variation from one shot to the next of the same scene, with the same camera, and the same settings.
An important difference between using a real aperture and using some kind of ND filtration would be depth of field control. Not that you really have a whole lot of DOF control in a compact anyway given how small the apertures are and how tiny the sensors are, but with ND-filtration as a replacement for aperture would completely remove any control you have over DOF and background blur.