The thermal efficiency of an internal combustion engine is related to the compression ratio by the following equation:
Efficiency = 1-(1/compression ratio)^(1-gamma)
Gamma is the ratio of specific heats of the working fluid involved. For pure air it would be 1.4. However, this involves a lot of simplification, and I have found that setting gamma to 1.175 gives an accurate, real-world efficiency result for gasoline engines.
From this equation, it is obvious that raising the compression ratio of an engine will increase thermal efficiency, and power. However, this effect only takes place at compression ratios up to 17:1. Above 17:1, the efficiency and power actually drop (this is not represented in the formula).
Also, increasing the compression ratio requires an increase in fuel octane. The grades of fuel and their corresponding maximum compression ratios are shown below (This is only a rough guide. The actual octane requirement depends on a myriad of factors)
Regular (87 AKI): 8:1
Mid-Grade (89 AKI): 9:1
Premium (93 AKI): 10:1