I have to disagree a bit here. Although the gains are small, an increase in bore size does increase the compression ratio. Here goes:

Compression ratio (Cr) is defined as the ratio of the combustion volume plus the displacement over the combustion volume. (Cv + D)/Cv.

For a VW/Audi 1.8/2.2 liter motor, the displacement of one cylinder is 445.21 cc. (bore radius squared X pi X stroke) or ((81mm/2)**2)* pi*86.4mm. Assuming 8.5 compression ratio, it is possible to iterate the combustion volume, which equals 59.4 cc. Therefore (Cv + D)/Cv = (445.21 + 59.4)/59.4 = 8.49 (close enough to 8.5)

Boring the motor increases to bore size, and therefore the displacement. So we'll do +.020" (1/2 mm) and +.040" (1 mm) oversizes.

The bore size goes up to 81.5 mm and 82 mm respectively. The resultant displacements go up from 445.21 cc to 450.73 cc and 456.28 cc.

The combustion volume is in the piston top and the cylinder head, so for the most part, the combustion volume stays the same at 59.4 cc. So you can calculate the new compresion ratios:

For +.020" (1/2mm) overbore, Cr = Cv + D)/Cv = (450.73 + 59.4)/59.4 = 8.59

For +.040" (1mm) overbore, Cr = Cv + D)/Cv = (456.28 + 59.4)/59.4 = 8.68

So the answer is that compression ratio goes up a tenth of a point for .+.020" overbore and bout two tenths of a point for .+040" overbore. That's if I did the math right, anyway.